Scalar wave image formation

In this section we provide a brief review of image formation modeled using scalar wave theory as a basis to extend the work into the more accurate vector representation.

Object space irradiance distribution can be decomposed into an ensemble of delta functions. The intensity or height of each delta function maps out the structure of the object. The optical system operates on the complex amplitude and phase associated with that intensity to form an image at the detector. Most astronomical sources in the visible region of the spectrum radiate broadband, incoherent thermal light.

The theory of image formation is developed using the schematic shown in Figure 1 below. The coordinate system we will use in our analysis is shown in Fig 1 below. This system is in standard use by modern textbooks^{11, 12} on the physics of image formation. The object plane (#1 in the system) is represented by Cartesian coordinates from the Latin alphabet x₁ y₁ the pupil plane (#2 in the System) is represented by

Cartesian coordinates from the Greek alphabet ξ₂, η₂ and the image plane (#3 in the System) is represented by Cartesian coordinates from the Latin alphabet x₃, y₃.

Figure 1

schematic of an optical system in the meridional plane. The object space complex field U₁ (x₁,y₁) is shown as a delta function δ(x₁,y₁) to represent a star on-axis. This object space field is propagated by Fresnel diffraction to just to the left of the entrance pupil and is shown by . The pupil is shown to have a complex transmittance τ₂(ξ₂, η₂) at plane 2. The optical system, shown here schematically as a lens of focal length f, provides the optical power to create the field U₃ (x₃, y₃) at an image plane 3.

The scalar complex amplitude and phase across the image plane is found by standing at the image plane (#3) in Fig 1 and looking to the left, or back through the system toward the object. The Fresnel-Kirchoff diffraction integral is used to model the propagation of scalar electromagnetic waves through the optical system shown in Fig. 1. The image plane complex amplitude and phase field U₃(x₃, y₃) at the image plane is given by

Where K is a constant, the integral is taken over the complex field across the exit pupil, of the optical system whose focal length is f, is the quasimonochromatic wavelength of light. The amplitude and phase complex properties across the exit pupil are contained in the scalar term,

Where A₂ (ξ₂, η₂) varies between 0 and 1 and describes amplitude part of the complex wave as a function of position across the exit pupil. The phase properties at each point across the exit pupil are described by ϕ₂ (ξ₂, η₂). Eq 1 is written for the scalar wave solution to Maxwell’s equations and the not vector wave

solution to Maxwell’s equation.

To the left in Fig 1, we have a point source represented by a delta function. This point source is mapped onto the image plane. We record intensity at the image plane and define the image plane irradiance distribution for this point source to be:

Next if we let the object space irradiance be represented by I_Object(x₁,y₁) and the image space irradiance represented by I_Image (x₃, y₃) and use the theoretical development of Goodman, we can write,

Where the symbol ⊗ denotes the convolution operator.

To understand the need for vector-wave physical optics, we need to review the source of polarized light within an optical system and understand the complex (amplitude and phase) wavefront at the focal plane U₃(x₃, y₃)where coronagraphers place the occulting mask to control scattered light to one part in 10¹¹.

Role of vector waves in image formation

An experiment using linear orthogonal polarizers and a telescope shows the role of vector waves in image formation. Figure 2 shows the effects of adding polarizers to an optical system: Top left shows an open, unmasked exit pupil of a telescope with zero geometric wavefront error. Top right shows the shape of the PSF recorded with the pupil on the top left. Bottom left shows the same telescope pupil as that shown in the upper left, with two linear polarizers over the top, one aligned orthogonally to the other. Horizontally polarized light is admitted to the left-hand side of the pupil and vertically polarized light is admitted to the right-hand side of the pupil. The bottom right shows the PSF recorded using the pupil on the bottom left. Note that with no polarizer the angular resolution is not position angle dependent, however, with the polarizer the angular resolution is angle dependent.

Figure 2

PSF’s shown for a telescope with zero geometric wavefront aberration without (upper and with (lower) polarizers.

Orthogonally polarized white light does not interfere to create an image. In Fig 2, the lower left image of the exit pupil the polarized radiation from the left portion of the exit pupil does not interfere with the orthogonally polarized radiation from the right portion of the exit pupil. Therefore the PSF is elongated in the horizontal direction. In this case the PSF is the scalar sum (linear superposition) of two images of a “D” shaped aperture, not the vector sum across the circular aperture shown in the upper right panel in Fig 2.

Although this is a rather dramatic example and no one would intentionally place orthogonal linear polarizers over their telescope pupil, this shows that any source of polarization change across the exit pupil will result in distortion of the PSF at some level.

In the next section we identify sources within an optical system that polarize light. Current astronomical science measurement objectives requires high transmittance optical systems, which in turn require high reflectivity broad-band optical thin films. As the white-light electromagnetic wave propagates through the optical system it becomes partially polarized. The orthogonal components of the partially polarized beam do not interfere to contribute to the image, but rather flood the focal plane with unwanted background radiation. The Fresnel polarization equations give the magnitude and sign of this polarization and are described in the following section.

Fresnel polarization

Here we examine the source of phase and amplitude changes within astronomical telescopes and instruments. Systems require mirrors coated with metals (e.g. Al or Ag) to give high surface reflectance and thus maximize system transmittance. These mirrors are overcoated with a dielectric material that serves two purposes: 1. Transparent mechanical barrier coat to inhibit oxidation and surface abrasion. 2. Enhance the reflectivity at select wavelengths. In the next section we show that these metal surfaces partially polarize light.

In figure 2, above we saw that the change in polarization across the exit pupil affects image quality. Broadband unpolarized white-light is a characteristic of nearly all-astronomical sources and is divided equally into two orthogonally polarized beams for the derivation below. We represent unpolarized light by two orthogonal Eigenvector states and, for this example we select linearly polarized Eigenvector states. The Fresnel equations¹³¹⁴ are used to model the behavior of a vector electromagnetic complex wave interacting with a metal or dielectric surface (mirror).

A-J Fresnel in 1823 described the theory for interactions of electromagnetic radiation with dielectrics and metals. These relationships were developed further¹⁵ and are the basis of the commercial field of ellipsometry¹⁶.

Consider incoherent white-light incident at angle θ₀, onto a metal mirror with isotropic

properties. This metal mirror has a wavelength dependent complex index, N₁ (λ) = n₁ (λ) + ik₁ (λ). The Eigenstates of reflection are the s (perpendicular) and p (parallel) polarized components. A portion of the beam reflects at the incidence angle θ₀ (Snell’s Law) and another portion (a damped evanescent wave)penetrates a short distance into the metal at the complex refraction angle of θ₁ given by Snell’s law¹⁷ and is absorbed to heat the metal. This complex angle is given by

The complex reflectivities for light in the p and s polarizations are given by³³

Two polarization effects occur. 1. There is a phase shift between the waves associated with each of the two polarizations, Ψ, called retardance which is given by

Equation 6 gives us the retardance Ψ for a single ray propagating through the system. An image requires

an array of rays. When we trace multiple rays from a single point in object space, the tangent of the retardance becomes

Where (ξ, η)are coordinates across the pupil.

The reflectivity is polarization dependent, with the result that reflection acts as a partial polarizer and the diattenuation; D at each point (ξ, η) across the pupil is given by

Where r is the complex reflectivity given in Eq 5 above. Metallic reflection acts as a weak polarizer, and D varies from zero (nonpolarizing) to one for ideal polarizers. Astronomical optical systems require large Etendu (area times solid angle), which requires large optics. However, the volume for spacecraft bus is required to be compact to fit inside launch shrouds. These two requirements conflict and often lead to many fold mirrors in the instrument which, unless designed properly will, in turn, lead to large internal polarization with the concomitant loss in transmission and image quality.

Note there are two polarization aberrations: 1. Diattenuation is commonly used to model a polarization dependent reflectivity and 2. Retardance is used to model a polarization dependent change in the phase of the complex wave upon reflection.

Polarization ray trace (PRT)

Polarization ray tracing (PRT) is a technique for calculating the polarization matrices for ray paths through optical systems^{23,24,25,26,27,28}. Polaris-M²⁹ was built from the ground up to calculate polarization effects in optical systems. It is based on a 3x3 polarization ray tracing calculus⁴⁶. Diffraction image formation of PolAb beams is then handled by vector extensions to diffraction theory^{30, 31, 32,33}. A calculation of the polarization point spread matrix and optical transfer matrix can be seen in Section 4 of Ref. 42.

Polarization aberration theory (PolAbT)

Polarization aberration theory (PolAbT) describes the polarization effects of diattenuation, retardance, and apodization in a series expansion, where a cascade of terms separate mathematically the largest effects, from smaller effects and associate these polarization related image defects with constructional parameters and coating performance metrics.^34,35 For example one term, retardance tilt, is strongly associated with fold mirrors and causes the XX and YY image components to shift with respect to each other, making the PSF slightly elliptical¹. Another term retardance-defocus causes astigmatism from primary and secondary mirrors, which is polarization dependent; the orientation of the retardance rotates with the orientation of an incident linear polarization.³⁶.

PRT generates very large files of numbers, at least eight times more than a conventional ray trace, leaving the designer and management a substantial data interpretation task of the aberrations represented in a higher dimensional polarization space. PolAbT is more difficult analytically than PRT, but it simplifies the ray tracing results into a small number of “terms” which are understood and addressed in an uncoupled manner. This enables us to manage polarization aberrations in more complicated systems, such as WFIRST-CGI.

A distinction between the two is seen in the comparison between classical geometric aberration ray trace (analogue to PRT) and the structural aberration coefficients³⁷ (analogue to PolAbT) used by advanced designers to arrive quickly at near-optimized designs. Thus using PolAbT together with PRT is far more powerful than either method alone.

Polarization ray-trace

The output of a CAD ray-trace computer program is combined with Fourier optics to calculate point spread functions. Figure 3 shows a side view of a typical optical system with a fan of rays originating from a point on the object and passing through an optical system with k surfaces to the system exit pupil. Each ray strikes a real physical surface at a known angle of incidence (no paraxial approximation).

Figure 3

A fan of rays is shown passing from the object plane through an optical system with k surfaces before the exit pupil.

We know the physical properties of each surface that is a reflecting metal or dielectric. Using the Fresnel equations, discussed in section TBD we calculate values for each of the four complex entries in Eq. 10, for each ray intercept through the system. We compute the multiplicative amplitude and cumulative phases for both perpendicular and parallel light and map these into four arrays of complex numbers across the exit pupil. We then take a digital FFT to calculate the four PSF’s.

Three-mirror imaging telescope polarization ray trace (PRT)

The PolAb and diffraction image formation of a 3-mirror telescope system (fig 3 ref. 1) classical Cassegrain were analyzed with the Polaris-M software to evaluate the order of magnitude of polarization effects, which would be expected in a telescope/coronagraph. Polarization artifacts were discovered with “ghost PSFs” about twice the size of the Airy disk at 10⁵ of the peak intensity. This 3-mirror system is a much simpler system than the 18-mirror WFIRST-CGI. PRT for an 18-mirror system is very labor intensive and we developed our theory based on the much simpler 3-mirror system and derived the more general PolAbT and used that to predict performance of the 18 mirrors (before the image plane stop) WFIRST-CGI.

For the three-mirror bent Cassegrain telescope, we adjusted the surface figures and vertex intervals so that geometric aberrations were zero for the on-axis PSF. All residual aberrations were therefore caused by polarization, not geometric aberrations. The primary mirror is 2.4 m dia. and F/# 1.2, the curvature on the secondary was adjusted to give a system F#/8. The position of the secondary along the system axis was adjusted so the F/# 8 converging beam reflected from a flat 45-degree mirror to deviate the beam by 90-degrees to a focal plane underneath the primary. Our calculations assumed bare Al at λ = 800 nm, where the complex index of refraction, N₁ is N = 2.8 + 8.45i. Each surface contributes its own polarization for each ray that strikes it. The cumulative complex transmittance for each point in the exit pupil is found by multiplying the 3x3 polarization ray tracing matrices at each of the 3 surfaces³⁸. This provides in 3D the exiting polarization state for arbitrary incident states. The Jones field at the exit pupil is Fourier transformed to find the E&M field at the focal plane for all incident polarization states. The modulus squared of this field contains four PSF components³⁹.

This Cassegrain telescope is illuminated by a thermal white-light star on-axis. The image plane irradiance is found to be the linear superposition of the four amplitude PSF’s⁴⁰ seen in Fig 4. Not visible at this scale: the centroid of each of the four are shifted slightly, one with respect to the other.

Figure 4,

a single star on axis in object space produces a four-element amplitude response matrix (ARM). Here we plot the amplitude for these complex elements of the 2x2 ARM, as they appear nearly super-posed at the focal plane of the Cassegrain telescope. The scale or size of each of these ARM elements is wavelength dependent.

The spatial extent of each of the off-diagonal ghost PSFs, I_XY and I_YX, are twice as large as the spatial extent of the on-diagonal images. The detailed analysis of these images shows several issues of concern for the WFIRST-CGI, which contains 15 more additional mirrors. The two principal images I_XX and I_YY are shifted by 0.625 mas with respect to each other due to a linear variation of retardance at the mirror – additional mirrors (like we have in CGI) in the path will increase this separation. Each of the principal images, XX and YY, is slightly astigmatic, but with opposite astigmatism sign (rotated 90°). The PolAb cause polarization crosstalk between X and Y polarized light, the off-diagonal elements, which although weak, 0.0037 in amplitude, 10^-5 in flux, has a much larger extent than the Airy disks of the principal components - additional mirrors (like we have in CGI) will increase the flux in these “ghost PSF’s”

Primary and secondary mirror coatings induce astigmatism on-axis⁴¹ which couples light into the orthogonal polarization state in Maltese cross type patterns, yielding the ghost PSFs. Focusing through fold mirrors introduces a linear variation of retardance, putting a different linear phase shift on the two principal components, shifting them in opposite directions! We derived eighteen scaling relations or design rules for these system parameters using PolAb theory.

Fig 5 (top) below shows a plot of the log10 of the irradiance in the meridional plane at the image for I_XX (the solid line) and I_YX (the dotted line) while Fig 5 (lower) is a “face-on” map of I_YX with the classic Airy diffraction rings superposed.

Figure 5

(upper) plots (solid line) of the logio of the irradiance in the meridional plane at the image for the irradiance distribution. In the upper part of the figure, the lower dotted line shows the logio irradiance for the irradiance distribution. The lower image in this figure is the “face-on” appearance of Iyx or the PSF Image of the off diagonal terms shown in Figs 4 and Eq. 10.

Note that the positions of the zeros in the PSF associated with I_XX are not superposed on those associated with I_yx. Therefore ghost images could be misinterpreted as a “false alarm” candidate exoplanet.

Polarization changes across the PSF

To first order the optical system of a coronagraph is isoplanatic and the fore-optics PSF for the star is the same as that for the planet. The image plane mask suppresses the field from the star out to a distance of ∼3λ/D. In the case of the 2.4 m Cassegrain system at λ=800nm where 3λ/D ≈ 0.2 asec we find that for both the star and its displaced planet the radius r of the 90% encircled energy for r_XX = r_YY = 0.15 asec and r_YY = r_YX = 0.36 asec. Radiation from the cross-product terms extends well beyond the 0.2-asec coronagraph dark hole to create an irregular background pattern that will confuse exoplanet measurements.

Conclusions from the analysis of the 3 mirror bent Cassegrain

Breckinridge, Lam and Chipman (2015)⁴² and Chipman, Lam and Breckinridge (2015)⁴³ used the POLARIS-M software to provide a detailed complex vector polarization analysis of a typical astronomical bent Cassegrain telescope comprised of an F/# 1.2 primary and a secondary mirror whose optical power is sufficient to give an F/#8 beam converging past a 45-degree flat mirror to a focal plane. The curvatures on the primary and secondary were designed to give zero geometric aberrations on axis. Coatings were assumed to be bare Al and wavelength 800nm. Several important facts about image quality in astronomical telescopes based on this this simple configuration were found⁹’ ¹⁰ to affect high-contrast exoplanet coronagraphy.

Applications

At least two astronomical science measurement objectives are affected by internal polarization. These are: 1 Exoplanet characterization using coronagraphy and 2 Astrometry. The astrometric implications are discussed by Safonov⁴⁴ and will not be repeated here. Rather we will concentrate on applications to characterize exoplanets and discuss this topic in more detail in the next section.

Optical schematic

Figure 6 shows the optical schematic for a typical exoplanet Lyot coronagraph. Several physical optics phenomena associated with image formation contribute to modify the vector complex electromagnetic field incident onto the focal plane, and degrade image quality. These are caused by the interaction of light and matter: polarization induced by mirrors, windows & stops and diffraction produced by masks and stops and chromatic aberration which results from the wavelength-dependent indices of refraction of materials required to reflect (metal mirrors) and transmit (dielectrics, filters, prisms, etc.) light.

Figure 6.

Optical schematic for a typical Lyot coronagraph. The Star/planet complex electromagnetic field enters the Lyot coronagraph system from the left and is focused onto a complex occulting mask located at an image plane stop. This image plane stop is at the front focus of a collimator lens. The optical power on the collimator is such that an image of the entrance pupil is formed on the Lyot stop.

The three-dimensional electric field from exoplanets is thermal white-light broadband, either reflected from the planetary system’s parent star or emission from the planet or a mixture. The electric fields associated with the star and the planet are both spatially and spectrally incoherent^{45, 46, 47} at the source. This radiation travels through space, enters the telescope/coronagraph system and reflects from several mirrors to strike the stop at an image plane. The field is partially coherent at this stop. The stop must block by absorption or reflection almost all of the incident electric field from the star while passing as much as possible of the exoplanet field. Starlight diffracts around the entrance pupil and this electric field is scattered into the coronagraph to be blocked by the Lyot stop shown in Fig 6. The detector at the focal plane samples the modulus squared of the electromagnetic complex field, which appears as a speckle pattern cased by partial coherence of the wave-fields. This intensity speckle pattern is then digitized at an optimum dynamic range (number of bits) to obtain a high enough SNR for exoplanet characterization.

This intensity distribution contains information about the characteristics of the exoplanet as viewed through the “filter” of the telescope/coronagraph complex-vector transfer function (CVTF). This function, represented by the amplitude response matrix (ARM)⁴⁸ varies significantly with the physical optics and the opto-mechanical design implementation. It is currently unknown for space coronagraph optical systems.

Maximum extinction

To minimize scattered light, the complex vector field at the image plane where the occulting mask is located (plane 3 in Fig. 6) must match the complex vector filter of the occulting mask. We use the formalism developed for optical processing of images⁴⁹. From Eq. 12 we find the expression for the incoherent superposition of the four complex fields representing each of the 4 incoherent elements of the ARM to be:

Where is the Fourier transform operator, each of the four terms is complex and the superscript - sign on u refers to the complex electric field infinitesimally (ε) to the left (-) of the image plane. Note that it has been shown¹⁰ mathematically, confirmed observationally⁵⁰ and relationships verified⁵¹ that the centroids of the terms are displaced slightly one from the other.

Let the transmittance of the image-plane occulting-mask be represented by the complex transmittance, τ(x₃,y₃), then the electric field infinitesimally (ε) to the right (+) of the image plane is given by

To achieve the contrast levels needed for terrestrial exoplanet spectrometry we need to minimize u⁺ (x₃, y₃), over the wavelength bands of interest while maximizing the transmittance at the position angle and separation of the exoplanet in orbit around the star.

To further control the unwanted radiation, this process needs to be repeated at the exit pupil to create the optimum complex Lyot stop. We need to satisfy the joint condition. This is a form of E&M impedance matching both at the image plane occulting mask and at he exit pupil Lyot stop under the conditions to maximize the field of the exoplanet at the entrance aperture to the spectrometer, to maximize the intensity at the focal plane of the spectrometer.