This paper describes an idea of creating a liquid analyte sensor based on a one-dimensional (1D) photonic crystal - dielectric mirror - integrated on the optical fiber end face and creating a resonant cavity with the reversed structure. This geometry produces the evident resonances within the photonic band gap spectral region which are sensitive to refractive index changes inside the resonant cavity. The main idea is to use this arrangement as a sensor in Lab-on-a-chip (LOC) device with two single mode optical fibers (SMFs). An inlet and outlet is proposed for different liquid analyte delivery and in-situ monitoring of the refractive index change of the analyte.
One of many possible applications of a one-dimensional photonic crystal (1DPhC) is sensing utilizing the Bloch surface wave (BSW). In this paper, we present a method of relative humidity (RH) sensing based on the phase shift of the BSW supported by a truncated 1DPhC represented by a multilayered structure. The structure is composed of six TiO2/SiO2 bilayers with a termination layer of TiO2. The BSW is excited by a total internal reflection of light in the Kretschmann configuration. A spectral interferometric technique is used to obtain a channeled spectrum due to projections of both reflected s and p polarized light waves. The phase is retrieved using a Windowed Fourier transform and spectral derivatives forming peaks are calculated. The sensing concept is based on tracking the derivative peak as a function of RH. A sensitivity to humidity of 0.028 nm/%RH and figure of merit of 0.0042 %RH−1 were determined.
A theoretical study of a new sensing concept based on Bloch surface waves (BSWs), which utilizes a onedimensional photonic crystal (1DPhC) represented by a multilayer structure, is presented. 1DPhCs have a number of useful features, including physical and chemical robustness. We model the response of a BK7 prism/multilayer/analyte system in the Kretschmann configuration for s- and p-polarized waves. We show that the 1DPhC supports surface electromagnetic waves at frequencies within the forbidden transmission band of the multilayer structure. A dip in the reflectance spectrum for s-polarized wave represents the excitation of the BSW and this is demonstrated for air and gaseous analytes.
A phase detection method is proposed and demonstrated to measure the response of surface plasmon resonance (SPR) in the spatial domain. In an experimental setup comprising a laser source, a launching optics, a gold coated SF10 glass plate, a coupling prism, a Wollaston prism and a CCD camera, a spatial interference fringes are recorded. Two such spatial interferograms, one including interference of p- and s-polarized waves from the SPR structure for air when the SPR phenomenon does not occur, and the other one for an analyte when the SPR phenomenon occurs, are used to detect the SPR phase shift at the source wavelength. The interferometry setup is used to measure the SPR phase shift for aqueous solutions of ethanol. The measurements are accompanied by theoretical modeling of the SPR responses using the material characteristics, that is, the refractive index of the SF10 glass, the complex refractive index of gold, and the refractive index of the analyte.
In sensing applications utilizing the effect of surface plasmon resonance (SPR), a thin film of gold is widely used as the plasmonic layer. Despite advantages of gold over other metals, characterization of its optical properties is not sufficient as is evident from the measured responses at different angles of incidence. In this paper, a new method of determining the optical constants of a thin golden layer is presented. The method is based on measuring the phase shift between p- and s- polarized optical waves induced by SPR for air in the Kretschmann configuration with an SF10 glass prism and an SPR structure. The SPR structure comprises a gold coated SF10 slide with a chromium adhesion layer. In addition, a birefringent crystal is included in the setup to attain the spectral interference. Information about the phase shift induced by SPR is inscribed in the spectral interferogram recorded by a spectrometer and can be retrieved using the Fourier transform analysis. The measurements are performed for different angles of incidence to obtain the spectral dependence of the optical constants of the golden layer. Measured data are fitted to a model and in addition, the feasibility of the method is demonstrated in measuring the phase response for distilled water.
A theoretical study of a new type of surface electromagnetic wave sensor, similar to a surface plasmon resonance (SPR) sensor, which utilizes a one-dimensional photonic crystal (1DPhC) instead of a metal film, is presented. Replacing the metal film in the SPR sensor by the 1DPhC has number of advantages including physical and chemical robustness, enhanced sensitivity, etc. 1DPhCs can be engineered to exhibit metal-like optical properties over given frequency intervals. Equivalently, the optical response of the 1DPhC can be described by an effective dielectric constant with a negative real value that permits the 1DPhC to support surface electromagnetic waves at frequencies within the forbidden transmission band. In our theoretical study, the 1DPhC is represented by a multilayer interference filter and we model the response of the system comprising a BK7 prism/multilayer/analyte in a Kretschmann configuration. For the system under study we express the reflectances of p- and s-polarized waves as a function of angle of incidence on the prism. Dips in the reflectance spectra represent the coupling of light waves to surface modes and this can be confirmed by sensitivity to refractive index changes at the multilayer-analyte surface. The theoretical study is accompanied by experiment with some results for the BK7 prism/multilayer/analyte system in the Kretschmann configuration.
Surface plasmon resonance has emerged as a powerful optical detection technique. Among others, one can find its applications in sensorics, material characterization or detection of molecules and their interactions. The existence of surface plasmons can be identified in a setup consisting of a glass prism, thin metallic layer and analyte. This effect manifests itself as a plasmonic wave propagating on a boundary between two layers, which are metal and analyte, and its detection can be performed via measurement of reflectivity of the incident light. Indeed, if we illuminate the prism face under certain angle, spectrum of the reflected light contains a minimum, i.e. a dip. This dip occurs for certain wavelength for which the plasmonic wave is excited. In the other words, it appears for a wavelength, fulfilling a certain resonance condition. The resonance condition provides, after some approximations, widely used expression describing dependence of the resonant wavelength on the angle of incidence. Nevertheless, even this expression has some limits and works well only if certain constraints are imposed. We study possible corrections to this expression providing better match even in the cases where we go behind the constraints.
A new method of sensing small refractive index changes of a liquid analyte using the effect of surface plasmon resonance (SPR) at different angles of incidence is presented. The method is based on detection of the phase shift induced by SPR in the Kretschmann configuration with an SPR structure comprising an SF10 glass prism, a gold coated SF10 slide with chromium adhesion layer, and an analyte (aqueous solutions of ethanol). First, the theoretical modeling of the phase shift at different angles of incidence induced by SPR is performed using the material dispersion characteristics. The phase shift at different angles of incidence is evaluated at a specific wavelength as a function of the analyte parameter and sensitivity is specified. Second, the theoretical modeling is accompanied by an experiment utilizing a polarimetry setup to detect the spectral phase shift induced by SPR. In addition, the phase shift is measured at a specific wavelength as a function of the analyte parameter, and the sensitivity is determined for different angles of incidence.
The surface plasmon resonance (SPR) phenomenon in the Kretschmann configuration comprising an SF10 glass prism, a gold coated SF10 slide and an analyte is analyzed theoretically and experimentally in the spectral domain utilizing the ratio of the re ectances of p- and s-polarized waves. Using the dispersion characteristics of a metallic layer according to the known model, the analysis for one angle of incidence can give the results that agree well with the experimental data. However, if the different angles of incidence are considered, the agreement fails because the SPR response is very sensitive to the dispersion characteristics of a metallic layer. A simple technique to obtain the dispersion of the complex permittivity of a metallic layer is proposed. To minimize the effect of an analyte, the SPR phenomenon is considered for air when a desirable angle of incidence is adjusted. Using this technique, we measure parameters of the ratio of the reflectance's of p- and s-polarized waves at different angles of incidence, that is, the minimum of the reflectance ratio and the resonance wavelength, to obtain the real and imaginary part of the complex permittivity. In the processing of every dip, we used the linear approximation of the real part and the quadratic approximation of the imaginary part of the complex permittivity of gold. The dispersion of the gold layer thus retrieved is compared with the model dispersion.
A theoretical study of a spectral method based on surface plasmon resonance (SPR) to measure the dispersion of a liquid analyte is presented. A setup with an SF10 glass prism and a gold coated SF10 slide is proposed and the SPR phenomenon in the Kretschmann configuration is analyzed in the spectral domain. Using the material dispersion of the SPR structure and the analyte, the resonance wavelength for the ratio of the reflectances of p- and s-polarized waves is determined for different angles of incidence. Using two new procedures, these theoretical values are utilized in obtaining the refractive index of the analyte as a function of the resonance wavelength. The dispersion of the analyte thus retrieved is compared with the one used in the model and it is concluded that one of the procedures is more accurate than the other. The applicability of the new method is demonstrated for two different analytes, water and ethanol, and the measurement range is specified.
A spectral interferometric technique to detect the phase shift induced by surface plasmon resonance (SPR) in the Kretschmann configuration is used in sensing small refractive index changes in a liquid analyte. The technique employs a polarimetry setup with an SPR structure comprising an SF10 glass prism, an immersion oil and a gold coated SF10 slide with a adhesion layer of chromium. In this setup two channeled spectra are recorded to detect the spectral phase shift induced by SPR. One spectrum includes reflection of p- and s- polarized waves from the SPR structure for air when the SPR phenomenon does not occur in the source spectral range, and the other one is for an analyte when the SPR phenomenon occurs. The polarimetry setup is employed to measure the spectral phase shift for aqueous solutions of ethanol. In addition, the phase shift is measured at a specific wavelength as a function of the analyte parameter, and the sensitivity is determined. The measurements are accompanied by theoretical modeling of the phase shift induced by SPR using the material dispersion characteristics, i.e., the refractive index dispersions of the SF10 glass, gold, and the analyte.
A two-step spectral interferometric technique to detect the spectral phase shift of surface plasmon resonance (SPR) in the Kretschmann configuration is proposed and demonstrated. The technique utilizes a polarimetry setup to record two channeled spectra, one including reflection of p- and s-polarized waves from an SPR structure for air when the SPR phenomenon does not occur, and the other one for an analyte when the SPR phenomenon occurs. The channeled spectra are used to detect the SPR spectral phase shift and first, an SF10 glass prism and a gold coated SF10 slide with a chromium adhesion layer is used to measure the SPR phase shift for aqueous solutions of ethanol. In addition, the position of a sharp maximum in the spectral derivative of the SPR phase shift is measured as a function of the analyte parameter. Second, the setup with a gold coated SF10 glass prism is used to measure the SPR phase shift for the same analyte. It is revealed that the detection accuracy of the measurement of the spectral derivative of the SPR phase shift in the second setup is lower than that in the first setup. For the first case, the measurements are accompanied by theoretical modeling of the SPR responses using the material dispersion characteristics.
Spectral interferometric techniques utilizing the interference of polarization modes in a highly birefringent (HB) elliptical-core fiber to measure temperature are analyzed experimentally. First, an experimental setup comprising a white-light source, a polarizer, a sensing birefringent fiber, an analyzer and a spectrometer is considered. Temperature sensing by this method is based on the wavelength interrogation. Second, the above setup is extended by a birefringent quartz crystal to increase the sensitivity of the temperature sensing. Third, the above setup is extended by an analyzer, and the combination of a polarizer, a birefringent quartz crystal and an analyzer represents another interferometer, which is used to increase the sensitivity of the temperature sensing. In this case the Vernier effect is present and the resultant spectrum is with an envelope, which is utilized in temperature sensing. We reached a sensitivity of 0.56 nm/K in the third setup, compared to -0.12 nm/K and -0.19 nm/K in the first and the second setup, respectively.
Spectral interferometric methods utilizing the interference of two beams in a Michelson interferometer to measure the displacement are analyzed theoretically and experimentally. First we consider an experimental setup comprising a white-light source, a dispersion balanced Michelson interferometer and a spectrometer. The position of one of the interferometer mirrors is controlled via a piezo positioning system and the displacement measurement is based on the wavelength interrogation, i.e., the position of a selected interference fringe in the resultant channeled spectrum is measured as a function of the mirror displacement. Second we consider a setup with another interferometer, included in the Michelson interferometer, to increase the sensitivity of the displacement measurement. In this setup, the resultant channeled spectrum is with envelope which shifts with the displacement of the interferometer mirror. We analyze the new measurement method theoretically and show that the displacement measurement is once again possible by using the wavelength interrogation and the sensitivity is substantially increased. We also realized the new measurement setup in which the position of the interferometer mirror is controlled via a closed-loop piezo positioning system and confirmed the theoretical results.
Spectral interferomeric methods utilizing the interference of polarization modes in a highly birefringent fiber to measure temperature are analyzed experimentally and theoretically. First, we consider an experimental setup comprising a white-light source, a polarizer, a sensing birefringent fiber, an analyzer and a spectrometer. Temperature sensing by this method is based on the wavelength interrogation, that is the position of a chosen spectral interference fringe in a channeled spectrum is measured as a function of temperature. Employing the setup, we carried out temperature sensing in the range from 300 to 370 K when a part of the sensing fiber is exposed to temperature changes. A wavelength shift of a selected spectral interference fringe is measured and the temperature sensitivity reaches −0.11 nm/K. Second, we consider a setup with another interferometer (represented by a polarizer, a birefringent quartz crystal and an analyzer) to increase the sensitivity of the temperature sensing. In this setup, the resultant channeled spectrum is with envelope which shifts with temperature. We analyze the new sensor theoretically and show that temperature sensing is once again possible by using the wavelength interrogation and the temperature sensitivity to be reached is 0.68 nm/K.
A fiber-optic refractive index sensor based on surface plasmon resonance (SPR) in a thin metal film deposited
on an unclad core of a multimode fiber is presented. The sensing element of the SPR fiber-optic sensor is a
bare core of a step-index optical fiber made of fused silica with a deposited gold film. First, a model of the
SPR fiber-optic sensor based on the theory of attenuated total internal reflection is presented. The analysis is
carried out in the frame of optics of multilayered media. The sensing scheme uses a wavelength interrogation
method and the calculations are performed over a broad spectral range. Second, in a practical realization of
the sensor with a double-sided sputtered gold film, a reflection-based sensing scheme to measure the refractive
indices of liquids is considered. The refractive index of a liquid is sensed by measuring the position of the dip
in the reflected spectral intensity distribution. As an example, the aqueous solutions of ethanol with refractive
indices in a range from 1.333 to 1.364 are measured.
A spectral-domain interferometric technique using the interference of polarization modes of a highly birefringent (HB) elliptical-core fiber to measure the temperature is presented. The method is based on the wavelength interrogation, i.e., the position of a chosen spectral interference maximum as a function of temperature is measured. Temperature sensing is carried out in a range from 300 to 370 K in an experimental setup comprising a white-light source, a polarizer, a delay line, a sensing HB fiber, an analyzer and a spectrometer. As the delay line, a birefringent quartz crystal of a suitable thickness is utilized to resolve a channeled spectrum in a range as wide as possible. A part of the sensing HB fiber, which is placed in a chamber, is exposed to temperature changes, and first, the polarimetric sensitivity to temperature is measured. It is revealed that the HB fiber is suitable for temperature sensing at a wavelength of 600 nm. Second, the shift of the wavelength position of the chosen spectral interference maximum with temperature is measured. It is revealed that the temperature sensitivity is higher at shorter wavelengths.
A model of a surface plasmon resonance fiber-optic sensor with a double-sided metallic layer is presented. Most of such fiber optic sensing configurations are based on a symmetric circular metal layer deposited on a bare fiber core used for excitation of surface plasmon waves. To deposit a homogeneous layer, the fiber sample has to be continually rotated during deposition process, so the deposition chamber has to be equipped with an appropriate positioning device. This difficulty can be avoided when the layer is deposited in two steps without the rotation during the deposition (double-sided deposition). The technique is simpler, but in this case, the layer is not at and a radial thickness gradient is imposed. Consequently, the sensor starts to be sensitive to polarization of excitation light beam. A theoretical model is used to explain the polarization properties of such a sensing configuration. The analysis is carried out in the frame of optics of layered media. Because the multimode optical fiber with large core diameter is assumed, the eccentricity of the outer metal layer boundary imposed by the thickness gradient is low and the contribution of skew rays in the layer is neglected. The effect of the layer thickness gradient on the performance of the sensor is studied using numerical simulations.
Refractive index sensors based on surface plasmon resonance (SPR) in a thin metal film deposited on an unclad core of a multimode fiber are presented. The sensing element of the fiber optic SPR sensors is a bare core of a step-index optical fiber made of fused silica with a double-sided sputtered gold film. First, an in-line transmissionbased sensing scheme with the fiber optic SPR probe is used. Second, a reflection-based sensing scheme with a terminated fiber optic SPR probe is employed. The fiber optic SPR probes have different lengths and the thickness of the sputtered gold film is about 50 nm. Both sensing schemes utilize a wavelength interrogation method so that the refractive index of a liquid is sensed by measuring the position of the dip in the transmitted or reflected spectral intensity distribution. As an example, the aqueous solutions of ethanol with refractive indices in a range from 1.333 to 1.364 are measured. For the transmission-based sensing scheme a polarization-dependent response is revealed.
Polarimetric sensitivity of a birefringent holey fiber to strain, temperature and hydrostatic pressure is measured in the spectral domain. In an experimental setup comprising a broadband source, a polarizer, a birefringent holey fiber under a variable physical parameter, a birefringent delay line, an analyzer and a compact spectrometer, the spectral interferograms are resolved. These are characterized by the equalization wavelength at which spectral interference fringes have the largest period due to the zero overall group modal birefringence. The spectral interferograms are processed to retrieve the phase as a function of the wavelength. From the retrieved phase functions corresponding to different values of the physical parameter, the polarimetric sensitivity as a function of wavelength is obtained. Using this approach, the polarimetric sensitivity to strain, temperature and hydrostatic pressure is measured.
The intermodal sensitivity of a two-mode birefringent holey fiber to strain, temperature and hydrostatic pressure is measured in the spectral domain. In an experimental setup comprising a broadband source, a polarizer, a twomode birefringent holey fiber under a variable physical parameter, an analyzer and a compact spectrometer, the spectral interferograms are resolved. These are characterized by the equalization wavelength at which spectral interference fringes have the largest period due to the zero group optical path difference between the fundamental, the LP01 mode and the higher-order, the LP11 mode. The spectral interferograms are processed to retrieve the phase as a function of the wavelength. From the retrieved phase functions corresponding to different values of the physical parameter, the intermodal sensitivity as a function of wavelength is obtained. Using this approach, the intermodal sensitivity to strain, temperature and hydrostatic pressure is measured for two orthogonal (x and y) polarizations.
The spectral dependence of the polarimetric sensitivity of a birefringent side-hole fiber to temperature and hydrostatic pressure was measured using a simple experimental setup comprising a broadband source, a polarizer, a birefringent fiber under test, a birefringent delay line, an analyzer and a compact spectrometer. The spectral interferograms, characterized by the equalization wavelength at which spectral interference fringes have the highest visibility (the largest period) due to the zero overall group birefringence, were processed to retrieve the phase as a function of wavelength. First, from the retrieved phase functions corresponding to different temperatures of the fiber under test, the spectral polarimetric sensitivity to temperature was obtained. Second, from the retrieved phase functions corresponding to different hydrostatic pressures in a chamber with the fiber under test, the spectral polarimetric sensitivity to hydrostatic pressure was obtained.
A spectral interferomeric technique to measure the chromatic dispersion, the zero-dispersion wavelength and the dispersion slope of a highly nonlinear suspended-core fiber is presented. This method utilizes an experimental configuration with a supercontinuum source in combination with a dispersion balanced Mach-Zehnder interferometer. A low-resolution spectrometer is employed at the output of the setup to record spectral interferograms for the path lengths adjusted in the interferometer. The spectral interference fringes of the highest visibility are resolved in the vicinity of the stationary-phase point corresponding to the equalization wavelength. First, from a series of spectral interferograms the dependence of the equalization wavelength on the path length adjusted in the Mach-Zehnder interferometer is measured. Then, the dependence of the path length difference on the equalization wavelength is obtained, which enables to determine dispersion of the differential group index of the fiber. Next, the chromatic dispersion including the zero-dispersion wavelength is determined exploiting a least-square fitting. Finally, the dispersion slope is obtained.
Theoretical study of a polarimetric setup intended to measure the refractive index change of a liquid analyte is presented. The detection scheme is based on the excitation of surface plasmon resonance in Kretschmann configuration combined with spectral interferometry. The principle of the method is to observe the spectral interference fringes as a result of mixing of two orthogonal linearly polarized waves with an analyzer. The waves are reflected from the base of a coupling prism covered by a thin metal layer used for generation of surface plasmon waves. The polarimetric setup consists of a linear polarizer, a birefringent crystal, a SF10 coupling prism covered by a gold layer and a linear analyzer. The attenuated total reflection at the prism base serves for the excitation of surface plasmon waves. The output optical field is then analyzed by a spectrometer. The phase change of resulting interference spectrum contains the information about the refractive index change of investigated analyte. The shift of phase curve is related to the analyte refractive index change. The model computation is performed in the frame of thin-film optics and the dispersion properties of all included materials are taken into account.
In this paper a simple spectral interferometric method for precise determination of the zero-dispersion wavelength
of fiber polarization modes using a supercontinuum source is presented. This technique is based on processing
a single spectral interferogram recorded in an experimental setup utilizing a dispersion balanced Mach-Zehnder
interferometer with a birefringent fiber under test, a supercontinuum source and a low-resolution spectrometer.
The zero-dispersion wavelengths retrieved from the sigle interferograms are compared with those obtained by a
wide spectral range measurement technique applied in the same setup, and good agreement between the results
is confirmed.
In this paper, the strain sensitivity of a two-mode birefringent holey fiber is measured in the spectral domain. In a
simple experimental setup comprising a broadband source, a polarizer, a two-mode birefringent holey fiber under
varied elongations, an analyzer and a compact spectrometer, the spectral interferograms are resolved. These are
characterized by a specific wavelength, the equalization wavelength, at which spectral interference fringes have the
highest visibility (the largest period) due to the zero group optical path difference between the fundamental, the
LP01 mode and the higher-order, the LP11 mode. The spectral interferograms with the equalization wavelength
are processed by a new method to retrieve the phase as a function of the wavelength. From the retrieved phase
functions corresponding to different elongations of a two-mode birefringent holey fiber under test, the spectral
strain sensitivity is obtained. Using this approach, the intermodal spectral strain sensitivity was measured for
two orthogonal (x and y) polarizations.
A model of surface plasmon resonance fiber optic sensor operated in spectral domain is presented. The sensing scheme is based on a multimode optical fiber whose core is covered by thin silver film. Theoretical description of the proposed fiber optic structure is carried out in the frame of multilayer optic approximation. For the sake of clarity, the excitation by collimated centro-symmetric beam focused on the sensing fiber axis is supposed and only the propagation of meridional rays is considered. The contribution of s and p-polarization is included in the computation, however only p-polarization is affected by surface plasmon resonance phenomenon. The optical dispersion of all involved materials is taken into account. It is known that the usage of silver surface plasmon resonance detection layer leads to narrower dip (comparing to gold layer) and higher sensitivity, but the chemical stability of silver is the potential problem. The solution to this problem can be the creation of oxide overlayer on the top of the surface plasmon resonance detection layer. The effect of the overlayer thickness on the performance of the sensor is analyzed in detail using numerical simulation and discussed in terms of sensitivity and detection accuracy.
A theoretical model of surface plasmon resonance (SPR) fiber-optic sensor based on the theory of attenuated
total internal reflection is presented. The analysis of the sensor response is carried out in frame of optics of
multilayered media. Some of the studied SPR sensors use the oxide or semiconductor overlayer for the protection
of the metallic layer generating the surface plasmon wave. In the same time, the overlayer can help to improve
the sensitivity of the sensor. The semiconductor overlayer protects the metal against the oxidation, but a native
oxide layer can be formed on its top surface, when exposed to the atmosphere. This effect has been scarcely
addressed, even if it can have an influence on the functionality of the sensor. In our case, the structure contained
a metallic layer covered by silicon with oxide layer. The influence of the oxide layer thickness on the sensor
performance in wavelength domain is studied in detail. The calculation of optical power transmitted through the
multimode sensing fiber is carried out in order to evaluate the response of the sensor. The effect of the dispersion
of all involved media is taken into account. The thickness of the metal layer in the sensing structure has been
optimized to achieve the most pronounced dip at the resonance condition. The performance of the sensor is
described in terms of sensitivity and detection accuracy.
Two spectral interferometric techniques employing a supercontinuum source are used for dispersion characterization
of birefringent microstructured and specialty optical fibers over a broad spectral range (e.g. 500-1600 nm).
First, a technique employing an unbalanced Mach-Zehnder interferometer is used for measuring the chromatic
dispersion and zero-dispersion wavelength of one polarization mode supported by a microstructured optical fiber.
Second, a technique employing a tandem configuration of a Michelson interferometer and a fiber under test is
used for measuring the group modal birefringence dispersion of the fiber and the chromatic-dispersion difference
as a function of wavelength. From these measurements, the chromatic dispersion and the zero-dispersion
wavelength of the other polarization mode supported by the microstructured optical fiber are retrieved. We
revealed from four measurements the dependence of the zero-dispersion wavelength on the geometry of air-silica
microstructured optical fiber. We also measured by the second technique the zero-chromatic-dispersion difference
wavelength for elliptical-core optical fibers. We revealed from four measurements that the dispersion parameter
can be tuned by the fiber geometry.
A model of surface plasmon resonance fiber-optic sensor based on the theory of attenuated total internal reflection is presented. Because of the mathematical complications related to rigorous theory of cylindrical optical waveguides, the analysis is carried out in frame of optics of multilayered media. The sensing scheme uses a wavelength interrogation method and the calculations are performed over a broad spectral range. The influence of the dispersion of the analyte on the sensor performance is considered and discussed. Model computations are performed for two mostly used types of excitation conditions. In the first case, the usage of a focused beam from the collimated light source is considered. The second case is related to the excitation of the fiber by a diffusive source. According to the excitation conditions, the contribution of meridional rays, or skew and meridional rays has to be taken into account. The calculation of optical power transmitted through the multimode sensing fiber is carried out. The effects of the layered structure thicknesses on surface plasmon resonance as well as the influence of the parameters describing the considered fiber are discussed. The influence of the analyte dispersion on the sensitivity, detection accuracy and signal to noise ratio is discussed.
A simple polarimetry configuration is used for measuring the thickness of a nonabsorbing thin film on an absorbing
substrate from the ratio between the spectral reflectances of p- and s-polarized components reflected from the
thin-film structure. The spectral reflectance ratio measured at a fixed angle of incidence is fitted to the theoretical
one to obtain the thin-film thickness provided that the optical constants of the thin-film structure are known.
This procedure is used for measuring different thicknesses of a SiO2 thin film on a Si substrate. Moreover, an
approximate linear relation between the thin-film thickness and a wavelength of the maximum of the reflectance
ratio for a specific angle of incidence is revealed when the substrate is weakly absorbing. The application of this
method is once again demonstrated in determining the thicknesses of the SiO2 thin films. The results of the
techniques are compared with those obtained by a technique of spectral reflectometry, and a very good agreement
is confirmed.
Chromatic dispersion of polarization modes in holey fibers is measured over a broad spectral range (e.g. 500-1600 nm) using two white-light spectral interferometric techniques. First, a technique employing an unbalanced
Mach-Zehnder interferometer with a fiber in the test arm is used to measure the wavelength dependence of the
differential group effective index, or equivalently the chromatic dispersion of one polarization mode supported
by the fiber. Second, a technique employing a tandem configuration of a Michelson interferometer and the
optical fiber under test is used to measure the group modal birefringence in the fiber. From these measurements,
the chromatic dispersion of the other polarization mode supported by the fiber is retrieved. We measured by
these techniques the chromatic dispersion of polarization modes in four air-silica holey fibers and revealed the
dependence of zero-dispersion wavelength on the geometry of the holey fiber.
We present two white-light spectral interferometric techniques for measurement of the chromatic dispersion of
polarization modes in holey fibers over a broad spectral range (e.g. 500-1600 nm). First, a technique employing
an unbalanced Mach-Zehnder interferometer with a fiber in the test arm is used to measure the wavelength
dependence of the differential group effective index, or equivalently the chromatic dispersion of one polarization
mode supported by the fiber. Second, a technique employing a tandem configuration of a Michelson interferometer
and the optical fiber under test is used to measure the group modal birefringence in the fiber. From these
measurements, the chromatic dispersion of the other polarization mode supported by the fiber is retrieved. We
measured by these techniques the chromatic dispersion of polarization modes in three air-silica holey fibers and
revealed the dependence of zero-dispersion wavelength on the geometry of the holey fiber.
We describe a new concept of spectral ellipsometry based on a channeled spectrum detection in a polarimetric
configuration with a birefringent crystal to measure the thickness of a thin film. We use a two-step technique to
retrieve the ellipsometric phase of a thin-film structure from the recorded channeled spectra. In the first step,
the phase difference between p- and s-polarized waves propagating in the birefringent crystal alone is retrieved.
In the second step, the additional phase change that the polarized waves undergo on reflection from the thin-film
structure is retrieved. Moreover, in the same polarimetric configuration, the ratio between the reflectances of
both polarization states is determined. The new concept of ellipsometry is used in measuring the data for a
SiO2 thin film on a Si substrate in a range from 550 to 900 nm and in determining the thin-film thickness
provided that the optical parameters of the structure are known. The thicknesses of different samples obtained
are compared with those resulting from previous interferometric and reflectometric measurements, and good
agreement is confirmed.
We present the results of measurement of dispersion characteristics of two-mode highly birefringent (HB) fibers
by three spectral interferometric techniques. First, a technique employing a tandem configuration of a Michelson
interferometer and HB fiber under test is used for a broad spectral range measurement (e.g. 500-1300 nm) of
the group modal birefringence for two spatial modes supported by the fiber. Second, a method of a lateral
point-like force acting on the fiber and based on spectral interferometry is used for measuring the phase modal
birefringence at one wavelength for the fundamental mode only. The measured value is combined with the
dispersion of the group modal birefringence to obtain the phase modal birefringence over a broad wavelength
range. Third, a spectral interferometric technique employing an unbalanced Mach-Zehnder interferometer with
HB fiber in the test arm is used for measuring the wavelength dependence of the chromatic dispersion of the one
of the polarization modes supported by the HB fiber over a broad wavelength range (e.g. 500-1400 nm). From
this dependence and from the chromatic-dispersion difference, which is obtained from the measured group modal
birefringence, the chromatic dispersion of the other polarization mode supported by the HB fiber is retrieved.
We measured by these techniques dispersion characteristics of two HB fibers, including elliptical-core HB fiber
and microstructured HB fiber.
We present the results of interferometric measurements of chromatic dispersion of polarization modes in holey
fibers over a broad spectral range (e.g. 500-1600 nm). First, a spectral interferometric technique employing an
unbalanced Mach-Zehnder interferometer with a birefringent holey fiber in the test arm is used for measuring the
wavelength dependence of the differential group effective index of the one of the polarization modes supported
by the fiber. We apply a five-term power series fit to the measured data and by its differentiation we obtain
the chromatic dispersion. Second, a spectral interferometric technique employing a tandem configuration of a
Michelson interferometer and the holey fiber under test is used for measuring the group modal birefringence in
the fiber. From the measurements, the differential group effective index and the chromatic dispersion of the
other polarization mode supported by the fiber are retrieved. We confirmed that the measurement results agree
well with that specified by the manufacturer. We also measured by these techniques the chromatic dispersion in
other birefringent holey fiber.
We present a white-light spectral interferometric method for measuring the chromatic dispersion of microstructured
fibers made of polymethyl methacrylate (PMMA). The method uses an unbalanced Mach-Zehnder interferometer
with the fiber of known length placed in one of the interferometer arms and the other arm with adjustable
path length. We record the spectral interferograms to measure the equalization wavelength as a function of the
path length difference, or equivalently the differential group refractive index dispersion over a wide wavelength
range. First, we verify the applicability of the method by measuring the wavelength dependence of the differential
group refractive index of a pure silica fiber. We apply a five-term power series fit to the measured data and
confirm by its differentiation that the chromatic dispersion of pure silica glass agrees well with theory. Second, we
measure the chromatic dispersion for the fundamental mode supported by two different PMMA microstructured
fibers, the multimode fiber and the large-mode area one.
A white-light spectral interferometric technique is used for measuring the thickness of a SiO2 thin film grown by
thermal oxidation on a Si substrate. The technique is based on recording of the spectral interferograms at the
output of a Michelson interferometer with one of its mirrors replaced by a thin-film structure. From the spectral
interferograms, the nonlinear-like phase function related to the phase change on reflection from the thin-film
structure is retrieved. The function is fitted to the theoretical one to obtain the thin-film thickness provided that
the optical constants of the thin-film structure are known. This procedure is used for measuring four different
thicknesses of the SiO2 thin film on the Si substrate. The results of the technique are compared with those
obtained in the same setup by spectral reflectometry and good agreement is confirmed. To minimize the errors
introduced by optical elements of the interferometer, the measurements are performed with the reference sample
of the known phase change on reflection and reflectance.
We present the results of measurement of birefringence dispersion in elliptical-core fibers. The measurement is
performed over a broad wavelength range by two different spectral interferometric techniques. First, a technique
employing a tandem configuration of a Michelson interferometer and an optical fiber under test is used for
measuring the group modal birefringence for two linearly-polarized modes supported by the fiber. Second,
a method of a lateral point-like force acting on the fiber and based on spectral interferometry is used for
measuring the phase modal birefringence at one wavelength for the fundamental mode only. The measured value
is combined with the dispersion of the group modal birefringence to obtain the phase modal birefringence over
a broad wavelength range. We measured dispersion of the birefringences for two different elliptical-core fibers
and resolved for example that the maximum of the group modal birefringence for the fundamental mode is at a
certain wavelength, which depends on the parameters of the fiber.
We present the results of broadband dispersion measurement of a
two-mode birefringent holey fiber (BHF). First,
a spectral interferometric technique employing an unbalanced
Mach-Zehnder interferometer with the fiber in the
test arm is used for measuring the wavelength dependence of the group effective index of the fundamental mode
supported by the fiber. Second, a spectral interferometric technique employing a tandem configuration of a
Michelson interferometer and the BHF under test is used for measuring the group modal birefringence dispersion
for two lowest-order linearly polarized (LP) modes supported by the BHF. The data measured over a broad
spectral range are fitted to polynomials to obtain the dispersion of the phase modal birefringence for both LP
modes. We reveal that the results are in agreement with a general model of birefringence in air-silica BHFs.
A white-light spectral interferometric technique is used for measuring small thickness changes of a SiO2 thin film
grown by thermal oxidation on a Si substrate. The technique is based on recording of the spectral interferograms
in a Michelson interferometer with one of its mirrors replaced by a thin-film structure. From the spectral
interferograms, the nonlinear-like phase function related to the phase change on reflection from the thin-film
structure is retrieved. The phase function is fitted to the theoretical one to obtain the thin-film thickness
precisely provided that the optical constants of the thin-film structure are known. This procedure is used for
measuring small thickness changes of a SiO2 thin film attributed to different dopant concentrations of a Si
substrate. The results of the technique are compared with those obtained by spectral reflectometry and very
good agreement is confirmed.
We present a simple white-light spectral-domain technique based on tandem interferometry that employs a
low-resolution spectrometer for a direct measurement of the group dispersion of optical components over a
wide wavelength range. The technique utilizes a tandem configuration of a Michelson interferometer and an
unbalanced Mach-Zehnder interferometer with a component under test inserted into one arm and the other arm
with adjustable path length. We resolve the spectral interference fringes when suitable path length diff;erence is
adjusted in both interferometers. From the recorded spectral interference signals, the equalization wavelength is
determined for which the overall group optical path difference between the beams is zero. By measurement of the
equalization wavelength as a function of the path length difference adjusted in the Mach-Zehnder interferometer,
the group dispersion of a glass sample is determined and agreement is confirmed with measurement using a
Michelson interferometer only.
We present two different white-light spectral interferometric techniques to measure birefringence dispersion in
two different elliptical-core polarization-maintaining fibers (PMFs). The first method is based on application
of a lateral point-like force on a fiber under test and resolution of the spectral interference fringes and is used
for measuring the wavelength dependence of the phase modal birefringence. The second method employs a
Michelson interferometer that compensates the differential group optical path difference in the tandem PMF and
is used for measuring the wavelength dependence of the group modal birefringence. The second method is also
utilized for determining the sign of the group modal birefringence.
A white-light spectral interferometric technique employing a Michelson interferometer with same metallic mirrors
is used to measure the effective thickness of a beamsplitter cube. The thickness is measured for four different
beamsplitters being in two different orientations. Moreover, it is revealed that the phase function of a thin-film
structure measured by a similar interferometric technique depends on the path length difference adjusted in the
Michelson interferometer. This phenomenon is due to a dispersion error of a beamsplitter cube, the effective
thickness of which varies with the adjusted path length difference. A technique for eliminating the effect in
measurement of the phase function is described. First, the effective thickness of the beamsplitter cube as a
function of the path length difference is measured. Second, the phase function of the thin-film structure is
measured for the same path length differences as those adjusted in the first case. In both cases, the phase
is retrieved from the recorded spectral interferograms by using a windowed Fourier transform applied in the
wavelength domain.
A new spectral-domain interferometric technique used for measuring the group index of holey fibers over a wide
wavelength range is presented. The technique utilizes an unbalanced Mach-Zehnder interferometer with a fiber under
test of known length placed in one of the interferometer arms and the other arm with adjustable path length. First, the
differential group index of the fiber is measured. Second, the fiber excitation is changed to guide light in the fiber
cladding or the fiber is replaced by the reference sample of known thickness and known group dispersion to determine
precisely the group index of the fiber at one specific wavelength. The group index as a function of wavelength is
measured for two different holey fibers, one of them made of pure silica glass.
We present a new method of phase retrieval from the spectral interferograms, which is based on the use of a
windowed Fourier transform applied in the wavelength domain. First, the numerical simulations are performed to
demonstrate high precision of the phase retrieval from the spectral data related to a slightly dispersive Michelson
interferometer comprising a thin-film structure. Second, the feasibility of the method is confirmed in processing
experimental data from the Michelson interferometer with SiO2 thin film on a silicon wafer to determine precisely
the thin-film thickness. We confirm very good agreement with the previous results obtained by the fitting of the
recorded spectral interferograms to the theoretical ones. Finally, the method is used in processing experimental
data from the Michelson interferometer with two metallic mirrors. From the retrieved phase function, the effective
thickness of the beamsplitter made of BK7 glass is determined precisely.
We present a white-light interferometric method for measuring the wavelength dependence of the group index
of a pure silica holey fiber. The method is based on the recording of a series of the spectral interferograms in
a Mach-Zehnder interferometer with the fiber of known length placed in one of the interferometer arms and
the other arm with adjustable path length. We measure the equalization wavelength as a function of the path
length difference, or equivalently the group index dispersion. Subtracting the group dispersion of the optical
components present in the interferometer along with the fiber, first we measure the wavelength dependence of
the differential group index of the pure silica glass provided that that the light is guided by the outer cladding of
the fiber. Second, we measure the wavelength dependence of the group effective index of the fundamental mode
supported by the fiber provided that some of the recorded interferograms are also due to the mode.
We present a new method for measuring the wavelength dependence of phase modal birefringence in highly
birefringent fibers. The method is based on application of a lateral pointlike force on the fiber and resolution of
the spectral interference fringes. The displacement of the lateral force needed for one-period phase change of the
spectral fringes at given wavelength is used to determine the beat length or the phase modal birefringence. In this
paper, the new method is analyzed theoretically and experimentally. First, we model the spectral interferograms
for two different locations of the lateral force with respect to the fiber end and show how the wavelength
dependence of the phase modal birefringence can be retrieved from the interferograms. Second, we measure the
wavelength dependence of phase modal birefringence in two different highly birefringent fibers and compare it
with that resulting from the measurement of the wavelength dependence of group modal birefringence.
We present two di.erent white-light spectral interferometric techniques employing a low-resolution spectrometer
for a direct measurement of the group dispersion of isotropic and anisotropic optical elements. First, the dispersion
of the group refractive index for glass plate is measured in a Michelson interferometer with the plate of
known thickness inserted in one of the interferometer arms. The technique utilizes the spectrometer to record
a series of spectral interferograms for measuring the equalization wavelength as a function of the displacement
of the interferometer mirror from the reference position, which corresponds to a balanced Michelson interferometer.
The use of the technique is extended for measuring the dispersion of the group refractive indices for the
ordinary and extraordinary polarizations in a quartz crystal. We con.rm that the measured group dispersions
agree well with those resulting from the semiempirical dispersion equations. We also show that the measured
mirror displacement depends, in accordance with the theory, linearly on the theoretical group refractive index
and that the slope of the corresponding straight line gives precisely the thickness of the quartz crystal. Second,
the group dispersion of the quartz crystal is measured in an unbalanced Mach-Zehnder interferometer with the
adjustable path length when the crystal is inserted in the test arm. The use of the second technique is extended
for measuring the di.erential group dispersion of a glass of a holey optical fiber.
We present a white-light spectral interferometric technique for measuring the thickness of a thin film on a
substrate. First, the channeled spectrum is expressed analytically for a setup of a slightly dispersive Michelson
interferometer with a cube beam splitter of given effective thickness and a fiber-optic spectrometer of a Gaussian
response function when one of the interferometer mirrors is replaced by the thin film on the substrate. Then we
model the wavelength dependences of the reflectance, the visibility of the spectral interference fringes, the phase
change on reflection and the so-called nonlinear phase function, respectively, for a SiO2 thin film on a silicon
wafer. In the modeling, the optical constants are known and multiple reflection within the thin-film structure
is taken into account. Second, we perform interferometric experiments with a SiO thin film on aluminium and
the SiO2 thin film on the silicon wafer. Channeled spectra are recorded for determining the thin-film thickness,
provided that the optical constants of the thin-film structure are known. We confirm very good agreement
between theoretical and experimental channeled spectra and determine precisely the thicknesses for two cases
including the SiO thin film on the aluminium and the SiO2 thin film on the silicon wafer.
In this paper, different interferometric techniques were used for measuring dispersion characteristics of specialty
optical fibers, including Corning PMF-38 highly birefringent fiber. We measured the wavelength dependence of
both the phase and group modal birefringences for two lowest-order linearly polarized (LP) modes. The phase
modal birefringence was measured by a lateral force method. The group modal birefringence was measured by a
method of spectral-domain tandem interferometry. The latter method was also used to measure the intermodal
group dispersion for two orthogonal polarizations of the LP modes. The experiment revealed a distinct dispersion
splitting between X-polarized and Y-polarized LP modes.
We present a white-light spectral interferometric technique which is used for measuring the absolute spectral
optical path difference (OPD) between the beams in a slightly dispersive Michelson interferometer with a thin-
film structure as a mirror. Two spectral interferograms are recorded to obtain the spectral interference signal
from which the spectral phase is retrieved that includes the effect of both a cube beam splitter and the phase
change on reflection from the thin-film structure. Knowing the effective thickness and dispersion of the beam
splitter made of BK7 optical glass, a simple procedure is used to determine both the absolute spectral phase
difference and OPD. The spectral OPD is measured for a SiO2 thin film on a silicon substrate and is fitted to the
theoretical spectral OPD to obtain the thin-film thickness. The theoretical spectral OPD is determined provided
that the optical constants of the thin-film structure are known. We measured also the nonlinear-like spectral
phase and fitted it to the theoretical values in order to obtain the thin-film thickness.
We present the results of theoretical and experimental analysis of waveguiding in a two-mode birefringent holey
fiber in which the birefringence is induced by two large holes adjacent to the fiber core. First, using a full-vector
finite-element method we modeled the wavelength dependence of the phase and group effective indices for the
fundamental and higher-order linearly polarized (LP) modes in two orthogonal polarizations. Then we evaluated
the wavelength dependence of the phase and group modal birefringences for both LP modes and the intermodal
dispersion in two orthogonal polarizations as well. Second, we used different interferometric techniques, including
time-domain and spectral-domain ones and a lateral force method, to measure in a broad spectral range the
wavelength dependence of the phase and group modal birefringences for the fundamental and higher-order LP
modes. Employing a white-light spectral interferometric method, we also measured the wavelength dependence
of the intermodal dispersion for two orthogonal polarizations of the two LP modes. Furthermore, using an
unbalanced Mach-Zehnder interferometer we measured the wavelength dependence of the relative group effective
index for the fundamental mode.
We present a white-light spectral interferometric technique for measuring the thickness of a thin film on a
substrate. First, the spectral interferogram is expressed analytically for a setup of a slightly dispersive Michelson
interferometer with a cube beam splitter and a fiber-optic spectrometer of a Gaussian response function when
one of the interferometer mirrors is replaced by the thin film on the substrate. We reveal that the visibility of
the spectral interference fringes is dependent on the reflectance of the thin-film structure and that the phase
change on reflection from the structure is inscribed in the phase of the spectral interference fringes. We model
the wavelength dependences of the reflectance and of the so-called nonlinear phase function for SiO2 thin film on
a silicon wafer of known optical constants taking into account multiple reflection within the thin-film structure.
Second, we perform experiments with the SiO2 thin film on the silicon wafer and record the spectral interferograms
for determining the thin-film thickness. We confirm very good agreement between theory and experiment and
determine precisely various thicknesses of the SiO2 thin film.
We present white-light spectral interferometric techniques for measuring dispersion characteristics of a birefringent
holey fiber supporting two lowest-order linearly polarized (LP) spatial modes. The first technique utilizes
a low-resolution spectrometer at the output of a tandem configuration of a Michelson interferometer and the
holey fiber to measure the equalization wavelength as a function of the optical path di.erence adjusted in the
interferometer. Thus we measured the dispersion of the group modal birefringence for the LP01 and even LP11
spatial modes and the intermodal group dispersion for two orthogonal polarizations of the LP modes. We present
also a new technique for measuring the differential group refractive index dispersion of the holey fiber which is
inserted in one of the arms of a Mach-Zehnder interferometer. The technique is based on measuring the equalization
wavelength as a function of the displacement of the interferometer mirror from the reference position,
which corresponds to a balanced Mach-Zehnder interferometer.
We present two different white-light spectral interferometric techniques employing a low-resolution spectrometer
for a direct measurement of the group dispersion of anisotropic optical materials. First, the dispersion of the
group birefringence of a calcite crystal is measured in a tandem configuration of a Michelson interferometer and
the calcite crystal. Second, the dispersion of the ordinary and extraordinary group refractive indices of a quartz
crystal is measured in a Michelson interferometer with the crystal inserted in one of the interferometer arms. Both
techniques utilize the spectrometer to record a series of spectral interferograms for measuring the equalization
wavelength as a function of the displacement of the interferometer mirror from the reference position, which
corresponds to a balanced Michelson interferometer. We confirm that the measured group dispersions agree well
with those resulting from the semiempirical dispersion equations. Furthermore, we propose a new white-light
spectral-domain fiber-optic sensor configuration utilizing a tandem configuration of a uniaxial (calcite or quartz)
crystal and a sensing birefringent fiber.
A phase-locked loop method is applied in processing the spectral interferograms resolved in a narrow spectral
range in a dispersive Michelson interferometer including optical elements. The unwrapped spectral fringe phases
with 2π ambiguity are determined and a simple procedure in removing the ambiguity is applied to obtain the
effective thicknesses of the optical elements knowing their dispersion. The effective thicknesses determined in two
cases are compared with those obtained by measuring the equalization wavelengths. A Fourier transform method
is applied in processing the spectral interferogram resolved over a wide spectral range in a slightly dispersive
Michelson interferometer including a cube beamsplitter. The ambiguous spectral fringe phase is determined and
a procedure in removing its ambiguity is applied to obtain the effective thickness of the beamsplitter.
In this contribution, an overview of applications of white-light spectral interferometry in dispersion characteriza-
tion of optical fibers is presented. First, spectral-domain intermodal interference is analyzed theoretically at the
output of a few-mode optical fiber alone and at the output of the optical fiber in a tandem configuration with a
Michelson interferometer. The theoretical analysis is performed under general measurement conditions when a
broadband source and a spectrometer of a Gaussian response function are considered and when the first-order
and second-order intermodal dispersion effects in the optical fiber are taken into account. Second, the theoretical
analysis is performed for three different examples of dispersion curves of two-mode optical fibers and the effect
of the limiting factors is specified. Finally, the theory is accompanied by the corresponding experiments for mea-
suring either the intermodal dispersion in circular-core and elliptical-core fibers or the dispersion of birefringence
in the elliptical-core fiber.
We analyze theoretically and experimentally spectral interference of a white light in a tandem configuration
of a birefringent crystal and a sensing birefringent fiber. We express analytically the spectral interference law
taking into account the dispersion of birefringence in the crystal and in the fiber. We reveal that the visibility of
spectral interference fringes is highest for the group optical path difference (OPD) in the crystal compensating
the group OPD in the fiber. We model two types of spectral interferograms knowing dispersion characteristics
of the sensing fiber and using a quartz crystal of the positive or a calcite crystal of the negative birefringence.
Sensing capabilities of the configuration are demonstrated by the change of the phase of spectral interference
fringes due to the change of fiber length. We perform two experiments with a highly birefringent fiber of a
suitable length and a birefringent quartz crystal of two suitable thicknesses. We confirm, in accordance with the
theory, that the wavelength-dependent phases of spectral interference fringes vary with the fiber length.
Various original applications of dispersive white-light spectral interferometry employing a low-resolution spectrometer and based on the equalization wavelength determinations are reviewed. First, the differential group
refractive index dispersion or the chromatic dispersion of a dispersive sample of known thickness is measured
determining the equalization wavelength as a function of the displacement of the interferometer mirror. Similarly,
the effective thickness of a dispersive sample is determined measuring the mirror displacement as a function of
the group refractive index of material. Second, the technique, which needs no phase retrieving procedure to be
applied, is used to measure mirror distances and displacements when dispersion in a two-beam interferometer is
known. Third, a technique of dispersive spectral interferometry is used for a slightly dispersive interferometer
including the effect of thin films. The technique, which needs phase retrieving procedure to be applied in processing spectral interferograms resolved over a wide spectral range, is used to determine the spectral phase of
thin films or the positions of the interferometer mirror.
A spectral-domain white-light interferometric technique employing a low-resolution spectrometer is used for
measuring dispersion characteristics of a two-mode elliptical-core optical fiber. The technique utilizes the fact that
the spectral interference fringes are resolved at the output of a tandem configuration of a Michelson interferometer
and a two-mode optical fiber under test only in the vicinities of so-called equalization wavelengths at which
the optical path difference (OPD) in the interferometer is the same as the intermodal or differential group
OPD. Measuring the OPD adjusted in the interferometer as a function of the equalization wavelength serves for
dispersion characterization of the optical fiber guiding two spatial modes in two polarizations. We measured the
dispersion of the intermodal group OPD between the LP0l and LP11 spatial modes, which were X-polarized or
Y-polarized, and the dispersion of the group modal birefringence for the LP0l and LP11 spatial modes. The
measured data were fitted to polynomials to obtain the dispersions of both the differences between propagation
constants of the LP modes and the phase modal birefringences.
Different interferometric methods were used to measure the dispersion characteristics of a holey fiber, in which birefringence is induced by two large holes adjacent to the fiber core. Employing a lateral force method, we measured the spectral dependence of the phase modal birefringence for two lowest-order spatial modes. Using a wavelength
scanning method and a technique of spectral-domain tandem interferometry, we measured the spectral dependence of the group modal birefringence for both spatial modes over two different wavelength ranges. The technique of spectral-domain tandem interferometry, which employs a white-light source, a polarizer, a tandem configuration of a Michelson interferometer and the optical fiber, an analyzer and a low-resolution spectrometer, was also used to measure the intermodal group dispersion. Two different group optical path differences between both spatial modes, which were X- or
Y-polarized, were measured as a function of wavelength.
We present the results of experimental and theoretical analysis of dispersion characteristics of a two-mode birefringent holey fiber, in which the birefringence is induced by two large holes adjacent to the fiber core. Different interferometric techniques were used to measure in a broad spectral range the wavelength dependences of the
phase and the group modal birefringence for the fundamental and the higher-order linearly polarized (LP) spatial modes. We also measured the wavelength dependence of the intermodal dispersion for two orthogonal polarizations of the fundamental and the higher-order LP spatial modes employing a white-light spectral interferometric
method. Furthermore, we calculated all the dispersion characteristics using a full-vector finite-element method and confirmed good agreement between experimental and theoretical results.
An overview of some important results is presented in measurement of optical fibers by interferometric methods in two domains. Both time-domain and spectral-domain measurements are analyzed theoretically including the effect of the first-order and second-order intermodal dispersion. In the time-domain measurements, a tandem configuration of a Michelson interferometer and a few-mode optical fiber is used and the spatial interference fringes are resolved to measure the intermodal group optical path difference. In the spectral-domain measurements, the intermodal interference at the output of an optical fiber alone shows up as a periodic modulation of the source spectrum when a high-resolution spectrometer is used. Utilizing a white-light spectral interferometric method employing a low-resolution spectrometer and a tandem configuration of a Michelson interferometer and an optical fiber under test, the equalization wavelengths are resolved and the intermodal dispersion in the optical fiber can be measured over a wide spectral range. We have used the method for measuring either intermodal dispersion in circular-core, elliptical-core and bow-tie fibers or dispersion of birefringence in elliptical-core fibers.
We analyze theoretically fiber-optic sensor configurations working in the time domain or in the spectral domain and utilizing interference between two spatial or polarization modes. We show for example that a two-mode fiber interferometer in which the phase difference between the two modes is large and the intermodal group optical path difference (OPD) is zero at a specific wavelength of a light source of a short coherence length can be used without losing both time-domain and spectral-domain interference. Similarly, we analyze theoretically a tandem configuration of two interferometers with one of the interferometers as a receiving interferometer and the other one as a sensing two-mode fiber interferometer. We show that time-domain or spectral-domain interference lost due to the OPD between the two modes exceeding the coherence length can be restored in the configuration. We analyze both theoretically and experimentally a tandem configuration of the receiving interferometer and the sensing two-mode fiber interferometer in which spectral interference between two polarization modes is detected by a low-resolution spectrometer. The configuration is characterized by suitable adjusting of the measuring sensitivity over a broad wavelength range. We present some preliminary experimental results regarding sensing of strain using an elliptical-core optical fiber.
A new null ellipsometer has been recently proposed that uses photoelastic modulator (PEM). The phase modulation adds a good signal-to-noise ratio, high sensitivity, and linearity near null positions to the traditional high-precision nulling system. The ellipsometric angles Delta and psi are obtained by azimuth measurement of the analyzer and the polarizer--PEM system, for which the first and second harmonics of modulator frequency cross the zeros. In this paper we discuss influence of component imperfection on precision of null measurement. Particular interest is devoted to azimuth angle error of compensator and modulator. Effect of residual birefringence of PEM is discussed. We show that the null system is insensitive to ellipsometer misadjustment and component imperfections and modulator calibration is not needed.
Spectral modulated interferograms (channeled spectra) characterize dispersion of optical samples and allow measuring distances and displacements. Useful information is contained in spectral fringe frequency and phase that can be evaluated dynamically by the extended Kalman filtering method based on recurrence prediction-correction processing procedure. The method provides direct dynamic estimates of unwrapped fringe phase within single fringe sample series. The accuracy of the extended Kalman filtering method has been investigated experimentally by processing recorded spectral interferograms.
A new spectral-domain interferometric technique employing a simple experimental setup is used to measure the group birefringence of a uniaxial crystal of known thickness over the wavelength range of the visible spectrum. The experimental setup comprising a white-light source, a Michelson interferometer, a polarizer, a uniaxial crystal, an analyzer, and a low-resolution spectrometer is utilized to record a series of spectral interferograms for different optical path differences (OPDs) adjusted in the Michelson interferometer. The spectral interferograms include interference fringes resolved only in a narrow spectral range around the so-called equalization wavelength at which the overall group OPD between interfering beams is zero. We measure the equalization wavelength as a function of the OPD in the Michelson interferometer to obtain directly the wavelength dependence of the group birefringence of a calcite crystal. Using the calcite crystal of two different thicknesses, we confirm that the measured dispersion of the group birefringence agrees well with the theoretical one. The thicknesses of the calcite crystal are also determined precisely from the slopes of linear dependences of the measured OPDs on the theoretical group birefringences.
Two different spectral-domain techniques based on reflectometry and white-light interferometry are used to measure spectral characteristics of thin-film systems. A technique of spectral reflectometry uses a standard configuration with a halogen lamp, a reflection probe and a thin-film system under test to record the reflection spectrum over a wide range of wavelengths. A new white-light spectral interferometric technique uses a slightly dispersive Michelson interferometer with a cube beamsplitter to measure the phase spectra of reflective or transparent thin-film systems over a wide range of wavelengths. This technique is based on a Fourier transform method in processing the recorded spectral interferograms to obtain the ambiguous spectral fringe phase function. Then, using a simple procedure based on the linear dependence of the optical path difference between beams of the interferometer on the refractive index of the beamsplitter material, the ambiguity of the spectral fringe phase function is removed and the beamsplitter effective thickness and the phase spectrum of the thin-film system are determined.
Two different measurement techniques based on tandem interferometry are used to measure the dispersion of group and phase modal birefringence in elliptical-core optical fibers. The first technique is based on time-domain tandem interferometry and uses processing of a series of interferograms at different wavelengths recorded in a tandem interferometer placed at the output of the optical fiber under test. The second technique is based on spectral-domain tandem interferometry and employs a low-resolution spectrometer. It uses a series of the spectral interferograms recorded at the output of a tandem configuration of a Michelson interferometer and the optical fiber under test to resolve the so-called equalization wavelengths, at which the overall group optical path difference is equal to zero. This technique enables the direct dispersion measurements of the group modal birefringence over a wide spectral range. The results obtained by both measurement methods are compared each other and good agreement is confirmed. We also modeled the dispersion of phase and group modal birefringence in the optical fibers using the modified perturbation approach first proposed by Kumar.
Two-beam spectral interference at the output of a slightly dispersive Michelson interferometer is used to measure the phase spectra of transparent thin films over a wide range of wavelengths. First, using a Fourier transform method in processing of the recorded spectral interferograms the ambiguous spectral fringe phase function is obtained. Then, using a simple procedure based on the linear dependence of the optical path difference between beams of the interferometer on the refractive index of material of the interferometer optical element, the ambiguity of the spectral fringe phase function is removed and the thickness of the interferometer optical element and the phase spectrum of the transparent thin films are determined. Knowing both the thickness of the interferometer optical element and the phase spectrum of the transparent thin films for a given configuration, the theoretical spectral interferogram is compared with the recorded one and good agreement between these interferograms is confirmed.
A spectral-domain white-light interferometric technique employing a low-resolution spectrometer at the output of a tandem configuration of the compensated (non-dispersive) Michelson interferometer and a two-mode highly birefringent optical fiber is used to measure intermodal dispersion characteristics of the optical fiber. The technique utilizes the fact that the spectral interference fringes are resolved in this configuration only in the vicinity of the so-called equalization wavelength at which the optical path difference (OPD) in the interferometer is the same as the intermodal group (OPD). The white-light spectral interferometric technique is used to measure the wavelength dependences of both the difference between propagation constants of X-polarized and Y-polarized LP modes and the intermodal group OPDs for an elliptical-core (highly birefringent) optical fiber. The measured intremodal dispersion characteristics of the optical fiber are compared with those corresponding to the results of an adequate theoretical analysis using the known parameters of the optical fiber.
A new spectral-domain interferometric method of measuring absolute distances is utilized when the effect of low dispersion in an interferometer is known and the spectral interference fringes are resolved over a wide spectral range. First, processing the recorded spectral interferograms by a phase-locked loop (PLL) method, which is a special simplified version of the general recurrence non-linear data processing method, the unmodulated spectrum, the spectral fringe visibility function and the unwrapped spectral fringe phase function are obtained. Then, knowing the dispersion relation for the material present in the interferometer, the material effective thickness is determined. Finally, the positions of the interferometer mirror are determined precisely by fitting the recorded spectral interferograms to the theoretical ones knowing all the mentioned spectral functions.
A phase-locked loop (PLL) method application for processing the recorded spectral interferograms obtained under a partial dispersion compensation in a white-light interferometer is considered. The PLL method, which is a special simplified version of the general dynamic recurrence non-linear data processing method, has been realized in iterative m ode which provides the unwrapped phase recovery in temporal Fourier holography. The accuracy of the iterative PLL method has been investigated via the comparison of the unwrapped spectral fringe phase obtained by processing the recorded spectral interferograms with the absolute phase resulting from knowledge of both dispersion in the interferometer and the absolute distance between mirrors of the interferometer.
Spectral-domain interference of two beams from a white-light source is analyzed theoretically and experimentally when the effects of both dispersion in an interferometer and the response function of a, spectrometer are taken into account. The spectral interference law is expressed analytically under the condition of a Gaussian response
function of a spectrometer. It is revealed that the spectral interference fringes can be resolved, visibility of which depends on the bandpass of the spectrometer and on the group optical path difference (OPD) between interfering beams. The theoretical analysis is accompanied by experiments employing a dispersive Michelson
interferometer and a low-resolution Spectrometer. Two experiments with different amounts of dispersion in the Michelson interferometer are realized giving the spectral interference fringes resolved only in the vicinity of the so-called equalization wavelength, at which the group OPD between interfering beams is zero. The spectral interferograms recorded within both experiments are compared with the theoretical ones, which are modeled knowing dispersion in the interferometer and the bandpass of the spectrometer. Dispersion in the
interferometer is affected by the thickness of the optical sample made of fused silica and the refractive index dispersion governed by the Sellmeier dispersion relation. It is revealed good agreement between the theory and experiment indicating a possibility of proposing a new method of measuring distances and displacements when
dispersion in the interferometer is known.
Spectral domain characteristics and parameters of optical fibers such as losses and their wavelength dependence, cutoff wavelengths for different fiber modes, wavelength dependencies of both the beat length and intermodal dispersion, etc., are important from the point of view of the development of new types of optical fibers. More
recently, a new spectral domain white light interferometric technique employing a low resolution spectrometer has been used in measurement of the intermodal dispersion in optical fibers. The technique utilizes the fact that the spectral interference fringes are resolved at the output of a tandem configuration of the compensated
(non-dispersive) Michelson interferometer and a few-mode optical fiber only in the vicinities of the so called equalization wavelengths at which the optical path difference (OPD) in the interferometer is the same as the intermodal group OPDs. Thus in the case of a two mode optical fiber the OPD adjusted in the interferometer and measured as a function of the equalization wavelength gives directly the spectral dependence of the intermodal group OPD in the optical fiber. The aim of this work is to use the new spectral domain white light interferometric technique to measure the intermodal dispersion in three different optical fibers. The aim of the work is also to perform an adequate theoretical analysis using a model of a weakly guiding, step-index optical fiber and to compare the corresponding results with the measured intermodal dispersion.
Spectral-domain white-light interferometric technique with channeled spectrum detection is used to measure characteristics and parameters of a fiber optic spectrometer. In the experimental setup comprising a white-light source, a Michelson interferometer and a spectrometer to be characterized, the spectral interferograms are recorded for different optical path differences (OPDs) between interfering beams. By processing the recorded spectral interferograms using discrete filtering and fringe amplitude demodulation method, the spectral fringe visibilities, first as a function of wavelength for given OPDs between interfering beams, and second as a function of the OPD between interfering beams for given wavelengths, are obtained. It is confirmed, in accordance with previous experimental and theoretical results, that the spectral fringe visibility functions, which are dependent on the OPD between interfering beams, are Gaussian functions. From the widths of the Gaussian visibility functions the spectrometer bandpasses at different wavelengths are evaluated.
A new white-light spectral-domain interferometric technique is used to measure the group velocity dispersion (GVD) in optical samples of known thicknesses. In the experimental setup comprising a halogen lamp, a Michelson interferometer with an optical sample, and a low-resolution spectrometer, the equalization wavelength as a function of the displacement of the interferometer mirror is measured by the spectrometer. From the measured equalization wavelengths either the differential group refractive index of the optical sample as a function of the equalization wavelength or the difference of the mirror displacements at two different equalization wavelengths is obtained to determine the GVD in the optical sample. The new white-light spectral-domain interferometric technique is applied to measure the GVD in two fused-silica optical samples of known thicknesses. In a broad spectral range it is confirmed that the GVD in both samples agrees well with the GVD resulting from the Sellmeier dispersion equation.
A new spectral-domain white-light interferometric technique employing a low-resolution spectrometer is used for dispersion characterization of optical fibers. The technique utilizes the fact that the spectral interference fringes are resolved at the output of a tandem configuration of the compensated (non-dispersive) Michelson interferometer and a two-mode optical fiber only in the vicinities of so-called equalization wavelengths at which the optical path difference (OPD) in the interferometer is the same as the intermodal group OPDs. Processing of these interference fringes in case of a two-mode optical fiber and measuring the OPD adjusted in the interferometer as a function of the equalization wavelength serve for dispersion characterization including the spectral dependence of both the difference between propagation constants of modes and the intermodal group OPD in the optical fiber. The white-light spectral interferometric technique is used for both circular-core and elliptical-core (highly birefringent) optical fibers. The measured dispersion characteristics of the optical fibers are compared with those corresponding to the results of an adequate theoretical analysis using the known parameters of the optical fibers.
A new white-light interferometric technique characterized by the range of measurable distances dependent on the amount of dispersion in an interferometer is proposed. We show that processing of the spectral interferograms recorded by a low-resolution spectrometer and including so-called equalization wavelengths can be extended to precise the determination of positions or displacements of the interferometer mirror to reach sub-micron or even nanometer resolution. In contrary to standard spatial-domain white-light interferometry employing interferometers balanced for dispersion we use a dispersive Michelson interferometer with fused-silica optical element (optical sample or beamsplitter) of know thickness. Knowing both dispersion and the thickness of the interferometer optical element and using a least-squares fit of the theoretical spectral interferograms to the recorded ones we obtain the positions of the interferometer mirror. Within two different configurations of a dispersive Michelson interferometer, one with an optical sample and the other with one a beamsplitter, we show that the range of measurable displacements depends on the thickness of the optical element.
The application of white-light spectral interferometry with the equalization wavelength determination is extended to propose new techniques for dispersion characterization of optical samples and fibers. First, a configuration of a non-dispersive Michelson interferometer and an optical sample of known thickness is proposed and realized to measure the differential group refractive index dispersion and the group velocity dispersion in the optical sample. Second, a tandem configuration of a non-dispersive Michelson interferometer and a birefringent optical sample of known thickness is proposed to measure the differential group refractive index dispersion in the birefringent optical sample. The latter experimental setup can be effectively modified to measure dispersion in both birefringent and polarization maintaining optical fibers.
Modeling of the light interaction in magnetic structures represents a problem of optics of anisotropic media based on solution of the Maxwell's equations. Magneto-optic medium is described using permittivity and permeability tensors, however, the permeability tensor is mostly set to its vacuum value for optical frequencies. The permittivity tensor can be expanded into a power series as a function of magnetization components. Linear and quadratic effects of the magnetization are characterized by linear and quadratic magneto-optic tensors, respectively. A complexity of the tensors and a number of the independent tensor components are reduced by including the crystal symmetry and the Onsager's principle. We describe, how the magnetic ordering affects eigenmode polarizations in magneto-optic media. Magneto-optic angles dependence on the magnetization components is discussed in details. Quadratic or second-order terms also affect significantly the magnetization measurement. Particular attention is devoted to magneto-optic effects in cubic crystals. While the optical and linear magneto-optic properties of cubic crystals are independent on the direction of crystal axes, the quadratic effects exhibit strong anisotropy. The theory was completed by an experimental observation of the quadratic effect anisotropy in an epitaxial Fe layer prepared on a MgO substrate. The influence of the magnetization components on the magneto-optic vector magnetometry is discussed for a general magnetization direction.
The processing of the recorded spectral interferograms including the equalization wavelengths is extended by using autoconvolution method to determine the equalization wavelengths precisely. The equalization wavelength determination is used to estimate the spectral fringe phase function in a two-beam interference experiment with the compensated Michelson interferometer and an optical sample inserted in it under the knowledge of both dispersion in the interferometer and the displacement of the interferometer mirror. The interferometer dispersion is given by the thickness of fused-silica optical sample and its refractive index, which can be evaluated from the Sellmeier dispersion relation. Good agreement between the recorded and theoretical spectral interferograms is confirmed and a new method of determining the optical path differences between beams of the interferometer is proposed.
The feasibility of the technique utilizing the spectral-domain two-beam interference resolved by a low-resolution miniature fiber optic spectrometer is demonstrated in measuring distances and displacements when two different light sources are used. First, when a low-coherence source, namely a multimode laser diode TOLD 9 140 is used in the configuration of a dispersive Michelson interferometer, a cross-correlation procedure between the measured and the theoretical spectral interferograms is applied to estimate the group optical path differences (OPDs) between beams corresponding to different positions of one of the mirrors of the interferometer and needed in the displacement evaluations. Second, when a white-light source, namely a halogen lamp is used in the configuration of a non-dispersive Michelson interferometer, the measured spectral interferograms are processed by a phase locked loop (PLL) method, which is a special simplified version of the general recurrence non-linear data processing method previously presented, to obtain the corresponding group OPDs between beams in the interferometer or the displacements. The PLL method is realized in iterative mode which provides the unwrapped spectral fringe phase recovery. The accuracy of the iterative PLL method is investigated via processing the measured spectral interferograms.
Response functions and bandpasses of a miniature fiber optic spectrometer with three read fibers of different core diameters are measured both interferometrically and directly. In interferometrically measured response functions and bandpasses, timedomain and spectral-domain interference oftwo light beams from a laser diode TOLD 9140 operated below the threshold is utilized. It is revealed, in accordance with the theory, that the visibility function for the spatial interference fringes measured in the Michelson interferometer configuration with a broadband detector consists of a Gaussian central peak whose width gives the source spectral width. Similarly, it is revealed, in accordance with the theory, that the visibility functions for the spectral interference fringes measured in the Michelson interferometer configuration by the fiber optic spectrometer are the functions related to the spectrometer response functions whose widths give overall bandpasses. Subtracting the effect of the source spectral width, the spectral bandpasses of the fiber optic spectrometer are determined. In directly measured response functions and bandpasses, the delta-function spectrum of the same laser diode operated far above the threshold is utilized. It is revealed, in accordance with the theoretical assumptions, that the response functions are Gaussian functions. These are compared with the interferometrically measured bandpasses confirming good agreement.
Spectral modulated interference fringes are observed in the form of the periodical modulation of broadband spectrum at the output of interferometer provided with a subsequent spectrometer. Group optical path difference of interfering light waves corresponding to the distance from the surface to be measured is characterized by the phase function or fringe frequency of the spectral modulated interferogram. The phase locked loop (PLL) method was used to demodulate spectral fringes. It is supposed a priori unknown carrier fringe frequency, and PLL technique is realized in iterative mode to calculate full fringe phase. At the first iteration step the fringe phase equal to zero is supposed. The second iteration takes the demodulated phase found form the first iteration, etc. As a result, the phase function of the spectral fringes is found. Method was verified experimentally with application to the analysis of the spectral modulated interferograms inherent in Michelson interferometer excited by a low-coherent source.
The spectral interference of two beams form a white-light source, a tungsten halogen lamp, is measured at the output of a dispersive Michelson interferometer by a low-resolution miniature fiber optic spectrometer when the interferometer beam splitter is made of fused silica. It is confirmed that the spectral interference fringes are resolved only in the vicinity of the equalization wavelength whose value is dependent on the group optical path difference in the interferometer. Thus, the equalization wavelength is measured as a function of the displacement in the interferometer in the spectral range approximately from 490 to 870 nm. Moreover, when the group refractive index of fused silica is evaluated at the equalization wavelengths by using the Sellmeier dispersion equation, it is confirmed, in accordance with the theory, that the measured displacement as a function of the group refractive index of fused silica is a straight line whose slope gives the effective thickness of the beam splitter. It is also revealed that when the effective thickness of the beam splitter made of fused silica is know with a sufficiently high accuracy, the differential group refractive index of fused silica in a spectral range from 490 to 870 nm can be obtained from the measurement of the equalization wavelength as a function of the displacement in the interferometer.
The aim of this paper is to review some important experimental results we have obtained in the application of time-domain and spectral-domain low-coherence interferometry for dispersion characterizing optical fibers whose intermodal optical path differences (OPDs) exceed the source coherence length. In time-domain measurements, a tandem configuration of a Michelson interferometer and a few-mode optical fiber is used and the spatial interference fringes are obtained at its output when the OPDs between beams and modes are matched within the source coherence length. In spectral-domain measurements, the intermodal interference at the output of an optical fiber alone shows up as a periodic modulation of the source spectrum when a high-resolution spectrometer is used. By processing the measured spectral modulation, the wavelength dependence of the overall phase can be obtained and then, e.g., the intermodal group OPD can be evaluated. When the spectral modulation is not resolved by the spectrometer, a tandem configuration of a Michelson interferometer and the optical fiber is used to obtain the low-frequency spectral modulation and to evaluate the intermodal dispersion in the optical fiber. In this paper, two different low-coherence sources with different temporal and spectral characteristics are used to obtain the intermodal group OPD and its wavelength dependence for two different optical fibers.
Time-domain and spectral-domain two-beam interference experiment with a Michelson interferometer and a detecting system consisting of a monochromator of a variable spectral bandpass and a broadband detector is analyzed theoretically and experimentally and the effect of the monochromator is included in the evaluation the visibility functions of both the spatial and spectral interference fringes. The time-domain theoretical analysis of the two-beam interference shows that the width of the visibility function, that is, the visibility of the spatial interference fringes as a function of the delay in the interferometer varies with the bandpass of the monochromator. The spectral-domain theoretical analysis of the two-beam interference gives the visibility function and shows that the visibility of the spectral interference fringes resolved for a given delay in the interferometer by a spectrometer also varies with the bandpass of the spectrometer. The theoretical conclusions are confirmed experimentally in the Michelson interferometer configuration using a TOLD 9140 laser diode operated below the threshold and a detecting system of a variable spectral bandpass consisting of a prism or grating monochromator and a PIN photodetector. It is shown from the time-domain measurements how the width of the visibility function decreases with the increasing width of the entrance slit of the monochromator. Similarly, it is shown what are the visibility functions for the spectral interference fringes resolved by spectrometers and how they vary with the bandpass of the spectrometer.
The effect of a detecting system consisting of a monochromator of a variable spectral bandpass and a broadband detector on time-domain and spectral-domain two- beam interference in a Michelson interferometer is analyzed theoretically and experimentally. The time-domain theoretical analysis of the two-beam interference shows that the visibility function for the spatial interference fringes, that is, the visibility of the spatial interference fringes as a function of the delay in the interferometer, is affected by the bandpass of the monochromator. Similarly, the spectral-domain theoretical analysis of the two-beam interference gives the visibility function for the spectral interference fringes that varies with the bandpass of the spectrometer. The time-domain theoretical conclusions are confirmed experimentally in the Michelson interferometer configuration using a TOLD 9140 laser diode operated below the threshold and the detecting system of the variable spectral bandpass consisting of a prism or grating monochromator and a PIN photodetector. The spectral-domain theoretical conclusions are confirmed experimentally using the TOLD 9140 laser diode and the prism or grating spectrometer, and using the TOLD 9211 laser diode and the grating spectrometer. The measured time-domain and the spectral-domain visibility functions are also used to obtain the reciprocal linear dispersion of the monochromator or spectrometer.
Spectral-domain optical interference phenomena for broadband spectra take in the case of a two-beam interference the forms of the spectral interferograms. The group optical path differences between interfering beams that include information about the distance to be measured and about the properties of propagating wavefronts are inscribed in phase functions of the corresponding spectra interferograms. To recover the phase functions, a standard Fourier-transform method, or a new, recurrence non-linear data processing method can be used.
Interferometric radar technique is an important tool for a surface topography reconstruction of optically rough surfaces. Spectral interferometric radar is based on optical interference phenomenon that is observed in the form of the periodical modulation of broadband spectrum at the output of interferometer. Group optical path difference of interfering light waves corresponding to the distance from the surface to be measured is characterized by the phase function or fringe frequency of the spectral modulated interferogram. To measure fringe frequency a noise-immune recurrence non-linear filtering method was used where interferometric data are assumed to be a series of samples of the spectral interferometric signal defined by a known model. It is predicted the signal value from previous step of discretization to the next step. The prediction error is used for signal parameters correction. Recurrence non-linear filtering generally gives the optimal dynamic estimate of the phase function, frequency and visibility maximum position of the interferogram. Method was verified experimentally with application to the analysis of the spectral modulated interferograms inherent to Michelson interferometer excited by low-coherent source with a subsequent spectrometer.
Intermodal dispersion in a two-mode optical fiber can be measured in the spectral domain when the spectral interference between modes at the output of the optical fiber shows up as a periodic modulation of the source spectrum that can be processed. However, this technique cannot be used to measure intermodal dispersion in the two- mode optical fiber when the period of modulation is too small to be resolved by a spectrometer. Consequently, we proposed a new measuring technique utilizing a tandem configuration of a dispersive Michelson interferometer and the two-mode optical fiber in which the spectral interference can be resolved even if a low-resolution spectrometer is used. In the tandem configuration of the Michelson interferometer and the two-mode optical fiber, the optical path difference (OPD) in the Michelson interferometer is adjusted close to the group OPD between modes of the optical fiber so that the low-frequency spectral modulation that can be processed is produced. Using the Fourier transform method in processing the measured spectral modulations and subtracting the effect of the dispersive Michelson interferometer, the feasibility of this technique has successfully been demonstrated in obtaining the intermodal dispersion in the two-model optical fiber.
Low-coherence spectral interferometry can be used for dispersion characterizing two-beam interferometers or two- mode optical fibers when the spectral interference between beams or modes at the output of a dispersive interferometer or an optical fiber shows up as a periodic modulation of the source spectrum that can be processed. Thus, using the Fourier transform method in processing the measured spectral modulations, the wavelength dependences of the overall phases and then the first-order and the second-order dispersion characteristics, including the wavelength dependence of the group optical path differences (OPDs) between two beams in the dispersive interferometers or two modes in the optical fibers, can be obtained. The applications of low-coherence spectral interferometry can be extended to resolve the equalization wavelength at which the group OPD between two beams in a dispersive interferometer or two modes in an optical fiber is zero. The measurement of the equalization wavelength for the dispersive interferometer or the two-mode optical fiber, together with the measurement of the spectral dependence of the group OPD between two beams in the interferometer or two modes in the optical fiber, can be used for a compete dispersion characterizing the interferometer or the optical fiber.
Many sensors of magnetic field and the electric current sensors are based on the magneto-optical (MO) effects. MO Kerr effects in reflection are widely used for thin film magnetization study. A magneto-optical hysteresis loop measurement and an observation of the magnetic domain (MO microscopy) have an important advantage for the study of the magnetism of thin and ultrathin films. The observation and careful separation of three magnetization components (sometimes called vectorial magnetometry) are the tasks for a precise MO measurement. The paper is devoted to the influences of various magnetization components on the ellipsometric observables. The component separation is discussed including both linear and quadratic terms in magnetization. Presented theory is based on a solution of the wave equation in MO medium described by the permittivity tensor. The eigenmode (characteristic) equation is solved for a general magnetization direction in the linear and quadratic approximations using both numeric and symbolic methods. The boundary conditions for the electric and magnetic fields at interfaces are written in compact form based on 4 X 4 Yeh's matrix formalism. Three basic magnetization configurations are usually distinguished -- polar, longitudinal and transversal. The component perpendicular to the plane of incidence (transversal) affects only the rpp reflection coefficient in the linear approximation. The components lying in the plane of incidence (polar component is normal and longitudinal one is parallel to the interface) affect the conversion reflection coefficients rsp, rps and the ellipsometric angles (Kerr rotation and ellipticity). The quadratic effects are observed when both the transversal magnetization component (perpendicular to the plane of incidence) and the longitudinal one (parallel to the plane of incidence) are present simultaneously. These quadratic effects are observed as a product of the magnetization components in the conversion reflection coefficients.
In this paper, time-domain analysis of a tandem configuration of a nondispersive Michelson interferometer and a two-mode optical fiber, and spectral-domain analysis of intermodal interference at the output of the optical fiber alone are presented when a low-coherence exciting source is used and effects of first-order and second-order intermodal dispersion are included. Time-domain analysis shows that the visibility dependence of the output interferogram consists of two side peaks corresponding to positive and negative imbalance of the interferometer that compensate for the group optical path difference (OPD) of two modes in the optical fiber. It is shown that the effect of second-order dispersion gives rise to a flattening and a symmetrical broadening of these two side peaks. Spectral-domain analysis shows that the wavelength- dependent modulation of the source spectrum is obtained at the output of the two-mode optical fiber alone. By processing the spectral modulation, the intermodal dispersion parameters can be obtained. Theoretical results are verified experimentally: the visibility dependences measured in the setup comprising a bulk-optic Michelson interferometer and a two-mode optical fiber give the intermodal group OPDs, and the wavelength- dependent modulations of the source spectrum measured at the outputs of the two-mode optical fibers alone give the wavelength dependences of the group OPDs between two modes of the optical fibers. Moreover, good agreements between the experimental results of both measuring techniques are confirmed.
Low-coherence spectral interferometry with channelled spectrum detection, extensively used for dispersion characterizing optical fibers, utilizes the fact that the spectral interference between two modes of an optical fiber shows up at its output as a periodic modulation of the source spectrum with the period dependent on the group optical path difference (OPD) between modes. However, this measurement technique cannot be used to measure intermodal dispersion in the optical fiber for which the period of modulation is too small to be resolved by a spectrometer. We proposed and realized a new measurement technique utilizing a tandem configuration of a dispersive Michelson interferometer and the two-mode optical fiber in which the intermodal spectral interference can be resolved even if a low-resolution spectrometer is used. In the tandem configuration of the dispersive Michelson interferometer and the two-mode optical fiber, the OPD in the Michelson interferometer is adjusted close to the group OPD between modes of the optical fiber so that the low-frequency spectral modulation that can be processed is produced. Using the Fourier transform method in processing the measured spectral modulations and subtracting the effect of the dispersive Michelson interferometer, the intermodal dispersion of the two-mode optical fiber over a limited spectral region has been obtained.
The aim of this paper is to extend spectral-domain measurements of fundamental characteristics and parameters of optical fibers such as the wavelength dependences of losses, the cutoff wavelengths of a single-mode regime, dispersions, etc., to include new measurement techniques of low-coherence spectral interferometry that can be used for comprehensive characterizing optical fibers.
The spectral interference between two modes of an optical fiber, which shows up as a periodic modulation of the source spectrum at its output, cannot be used to measure intermodal dispersion in the optical fiber when the period of modulation is too small to be resolved by a spectrometer. We proposed a novel measuring technique utilizing a tandem configuration of a dispersive Michelson interferometer and a two-mode optical fiber in which the intermodal interference can be restored, and consequently spectral interference fringes can be resolved, even if a low-resolution spectrometer is used. In the tandem configuration of the Michelson interferometer and the two-mode optical fiber, the optical path difference (OPD) in the Michelson interferometer is adjusted close to the group OPD between modes of the optical fiber so that the low-frequency spectral modulation that can be processed is produced. The feasibility of this technique has successfully been demonstrated in obtaining the wavelength dependence of the group OPD between two modes of the optical fiber. Using the Fourier transform method in processing the measured spectral modulations and subtracting the effect of the dispersive Michelson interferometer, the intermodal dispersion of the two-mode optical fiber has been obtained.
The aim of this paper is to review our recent experimental studies on the spectral interference between two beams at the output of a Michelson interferometer and between two modes at the output of an optical fiber. The paper starts with important theoretical part in which fundamentals of the time-domain and the spectral-domain optical interferometry are applied to the two-beam and the two-mode interference experiments and then some important results of our recent experimental studies are presented. It is successfully demonstrated that the interference effects of optical beams or optical modes occur no matter how long or short the optical path difference (OPD) between beams or modes may be, so that the effect is reproduced as a periodic spatial variation when the OPD between beams or modes is shorter than the source coherence length, and as a spectral variation when the OPD between beams or modes is greater than the source coherence length. Moreover, it is successfully demonstrated that the spectral interference is resolved for both multimode and singlemode low-coherence sources. The aim of the paper is also to address some important applications of the two-beam and the two-mode spectral interferometry, including for example dispersion measurements.
Successful applications of low-coherence interferometry both in the time and spectral domains are demonstrated in the evaluation of the group optical path differences (OPDs) between two linearly polarized (LP) modes of weakly guiding optical fibers excited by a multimode laser diode TOLD 9201. Using the first measuring technique, the visibility of spatial interference fringes is measured as a function of the OPD in the experimental setup comprising a bulk-optic Michelson interferometer and a two-mode optical fiber under test. From the visibility dependences, the group OPDs between two LP modes of the optical fibers are obtained. Using the second measuring technique with the same exciting source TOLD 9201, the wavelength-dependent modulations of the source spectrum are obtained at the outputs of the two- mode optical fibers. By processing three spectral modulations using the Fourier-transform method, the wavelength dependences of the group OPDs between two LP modes are obtained and good agreements with the time-domain measurements are confirmed. When a halogen lamp is used as an exciting source, the wavelength-dependent modulation is resolved only in the vicinity of the equalization wavelength, where the period of modulation is sufficiently large to be resolved by the spectrometer. From the corresponding dependence, a value of the equalization wavelength of two LP modes of a single optical fiber is obtained.
The spectrum of light changes on its propagation through few-mode optical fibers due to the coherence-induced changes. They lead to the presence of modulated spectrum affected by intermodal dispersion when two completely spatially coherent modes of an optical fiber are included. This spectral modulation can be resolved even if the group optical path difference (OPD) between modes is greater than the source coherence length. In this contribution, the spectral changes of light occurring on its propagation through a two-mode optical fiber are analyzed both theoretically and experimentally. It has been confirmed experimentally, in accordance with theory, that the spectral modulations characterized by the wavelength-dependent periods of modulations can be resolved at the output of a two-mode optical fiber excited by low-coherence sources having different spectral widths. It has also been confirmed that the characteristics such as the unmodulated spectra and the wavelength dependences of both the visibilities of spectral fringes and the group OPDs between modes can be obtained. Moreover, it has been revealed theoretically that the configuration of a two-mode fiber, for which no spectral fringes are resoled, and a Michelson interferometer is suitable for the spectral analysis with a low-resolution spectrometer.
The study of guided modes in planar systems allows a precise characterization of refractive index, thickness and optical anisotropy. On the basis of the matrix elements, the conditions for waveguiding in magneto-optical simple layers, double layers and in sandwiches at transversal geometry have been specified. In this case there are two independent dispersion relations for TE and TM modes for discussed thin film configurations. Usually we assume that the relative permeability is equal to 1 in the frame of optical frequencies. This approximation is problematic in far infrared region or for ultrathin films in the optical one. Through the extension of the Yeh's formalism to magneto optical media the formulae of dispersion relations for guided modes in planar structures with both permittivity and permeability tensors have been derived. The permeability tensor effect at transversal magnetization on waveguiding in mono- and double layer systems and in sandwiches is analyzed in detail. The experimental possibilities of dark mode spectroscopy in the analysis of the planar magneto-optical systems inclusive of permeability tensor element specification are studied.
Successful applications of low-coherence interferometry both in the time and spectral domains are demonstrated in the evaluation of the group optical path difference (OPD) between two linearly polarized (LP) modes of an optical fiber excited by a multimode laser diode TOLD 9201. Firstly, the visibility of spatial interference fringes is measured in a tandem configuration of a Michelson interferometer and a two-mode optical fiber under test as a function of the OPD in the interferometer. From the visibility dependence, the group OPD between two LP modes of the optical fiber is obtained. Secondly, using the same exciting source TOLD 9201, the wavelength-dependent modulation of the source spectrum is obtained at the output of the two-mode optical fiber. By processing ten measured spectral modulations using the Fourier-transform method, the wavelength dependence of the group OPD between two LP modes is obtained and good agreement with time-domain measurement is confirmed. When a halogen lamp is used as an exciting source, the wavelength- dependent modulation is resolved only in the vicinity of the equalization wavelength, where the period of modulation is sufficiently large to be resolved by the spectrometer. From the corresponding dependence, a value of the equalization wavelength of two LP modes of the optical fiber is obtained.
The mutual interference of two beams in the spectral domain has been demonstrated experimentally at the output of a Michelson interferometer excited by two low-coherence sources which differ substantially from each other by the width of the spectrum, and whose coherence lengths are shorter than the optical path difference (OPD) between both beams. The spectral interference between two beams, which shows up for the first source the wavelength-independent and for the second source the wavelength-dependent periodic modulation of the source spectrum, serves as an illustration of the feasibility of an experimental method utilizing a high-resolution spectrometer in the evaluation of the group OPDs and displacements in a range dependent on the source spectral width. A cross- correlation procedure between the experimental and theoretical spectral modulations has been applied in the estimate of the group OPDs as well as in the evaluation of their wavelength dependence affected by dispersion of the beamsplitter. Moreover, good agreement between theoretical description of two-beam spectral inference experiment and the measured spectral modulation has been confirmed and the effect of both the spatial and the spectral integration has been resolved.
The mutual interference of two linearly polarized (LP) modes in the spectral domain has been demonstrated experimentally at the output of a two-mode optical fiber excited by two low- coherence sources which differ substantially from each other by the width of the spectrum, and whose coherence lengths are shorter than the optical path difference between both LP modes. The spectral interference between two LP modes, which shows up for the first source the wavelength-independent and for the second source the wavelength-dependent periodic modulation of the source spectrum, serves as an illustration of the feasibility of a novel experimental method utilizing a high-resolution spectrometer in the evaluation of the spectral dependence of the group optical path difference between both modes. Moreover, it has been shown that the spectral analysis performed across the output modal pattern of the two-mode optical fiber leads to spectral modulations in which the effects of both polarization properties and overlaps of both LP modes are inscribed.
The mutual interference of tow linearly polarized modes in the spectral domain has been demonstrated experimentally at the output of a two-mode, step-index optical fiber excited by low-coherence sources having different spectral widths. The corresponding spectral interferograms, which are characterized by the wavelength-dependent periods of modulation of the source spectrum affected by intermodal dispersion, serve as an illustration of the feasibility of a novel experimental method utilizing a high-resolution spectrometer in the evaluation of the spectral dependence of the group optical path difference (OPD) between two modes of an optical fiber. Using the Fourier transform method for the spectral fringe pattern analysis, good agreement between the experimental and the theoretical spectral interferograms has been achieved and characteristics such as the unmodulated spectra and the wavelength dependences of both the visibilities of spectral fringes and the group OPDs between modes exceeding the source coherence length have been obtained. These last two characteristics have also been compared with those obtained using the cross-correlation technique.
The mutual interference of two beams in the spectral domain has been demonstrated experimentally at the output of the uncompensated Michelson interferometer excited by two low- coherence sources having different spectral widths. For the first, a wide-spectrum source, three spectral interferograms with the wavelength-dependent periods of modulation of the source spectrum affected by dispersion of the beamsplitter have been resolved. Using the Fourier transform method for the spectral fringe pattern analysis, good agreement between the experimental and the theoretical spectral interferograms has been achieved and characteristics such as the unmodulated spectra, which agree well with those obtained by blocking one of the arms of the interferometer, and the wavelength dependences of the group optical path differences (OPDs) have been obtained. For the second, a narrow-spectrum source, the spectral interferograms with the wavelength- independent periods of modulation of the source spectrum have been resolved and a cross-correlation procedure between the experimental and theoretical spectral interferograms has been applied in the estimate of the group OPDs between both beams corresponding to different positions of one of the mirrors of the interferometer.
The temporal coherence of the scattered light is analyzed in the case of a point aperture, as well as in the Fraunhofer region of an aperture which is placed in front of a uniformly moving nonabsorbing random phase screen illuminated by a quasimonochromatic plane wave. It is shown that the temporal coherence of the scattered light is affected by not only the motion of the phase screen but also the statistical properties of the random phase screen. Moreover, the coherence of the scattered light in the frequency domain is considered in the case of a polychromatic plane wave. Theoretical considerations are made with a particular emphasis on the evaluation of the spectral changes of the scattered light occurring on its free space propagation from the phase screen to its far zone. It is shown that the spectral changes affected by both the motion of the random phase screen and its statistical properties are present. Both strongly and weakly scattering phase screens are considered. Analytic or closed-form expressions for both the temporal coherence and the spectrum of the scattered light are obtained, and numerical examples are presented.
Two detection schemes utilizing a two-mode fiber waveguide as an intrinsic sensor are presented in the case when a low-coherence source is used and no observable interference pattern exists at the exit face of the fiber waveguide. In the first detection scheme the sensor configuration is used which consists of a two-mode fiber waveguide and a high-resolution spectrometer placed at its output. In the second detection scheme the output optical field of the two-mode fiber waveguide is analyzed in the Michelson interferometer configuration by a low- resolution spectrometer. It has theoretically been shown that in both detection schemes the spectral modulation is present which is closely connected with both the measured and intermodal dispersion. However, the sensor operation of the second detection scheme is constrained by the optical path difference (OPD) in the Michelson interfermeter which is not only greater than the source coherence length, but is also adjusted to match the group OPD between guided modes.
The role played by optical interference in determining the mutual coherence function of an optical field propagating in few-mode fiber waveguides is examined under different conditions of the source coherence. By using the classical coherence formalism and guided-mode field representation, the mutual coherence function of the optical field guided in weakly-guiding fiber waveguides excited by stationary, cross- spectrally pure source is given in terms of the exciting coefficients, the source degree of coherence and guided-mode parameters that take also into account the first-order as well as the second-order modal dispersion. In particular, the optical interference is examined by evaluating the intensity distribution at the exit face and in the far- field of few-mode fiber waveguide as well as by evaluating the modulus of spatial and temporal coherence of an optical field at its output.115
The classical coherence formalism and guided-mode field representation is used to discuss the operation of few-mode fiber waveguide excited by a low-coherence, cross-spectrally pure, spatially coherent source in Michelson interferometer configuration as a sensor, even if a suppressed interference pattern at its exit face exists. In the case of a low- coherence excitation of few-mode fiber waveguide the principle of coherence modulation can be used, that is, the optical path difference between guided modes that exceeds the source coherence length can be compensated in Michelson interferometer configuration. The analysis of temporal coherence in a particular case of two-mode, weakly-guiding, step-index fiber waveguide takes also into consideration the effect of second-order modal dispersion; the potential applications to low- coherence source based interferometric sensors are discussed.
The role played by optical interference in determining the complex degree of coherence of an electromagnetic field propagating in a fiber waveguide is examined under different conditions of the source spatial and temporal coherence. In particular, the optical interference is examined by evaluating the modulus and the phase of the complex degree of both spatial and temporal coherence at the exit face of two-mode step-index fiber waveguide excited by spatially coherent and quasi-monochromatic source. Moreover, by introducing the averaging operation over a statistical ensemble of fiber waveguides, the statistical quantities of the optical field at the output of the fiber waveguide can be determined.
For interferometric distance measurements on rough surfaces multiple wavelength interferometry is a powerful tool. It allows to reduce the sensitivity and to extend the range of unambiguity for interferometric measurement. A new kind of a double wavelength heterodyne interferometer is presented which uses only one laser source which generates two wavelengths X, and X, simultaneously. The two different heterodyne frequency shifts of the double heterodyne interferometer (DHI) are achieved by combining an acousto-optical modulator (driven in suppressed carrier mode) with a fixed diffraction grating. In a first prototype a multi-wavelength Helve laser was used as light source. The synthetic wavelength A in this system is 55.5 gm and the resolution about 0.15 gm. Measurement results obtained with the setup described are presented .
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