8 March 2016 Design of off-axis mirrors for the phase-induced amplitude apodization complex mask coronagraph
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The phase-induced amplitude apodization complex mask coronagraph (PIAACMC) provides an efficient way to control diffraction propagation effects caused by the central obstruction/segmented mirrors of the telescope. PIAACMC can be optimized in a way that takes into account both chromatic diffraction effects caused by the telescope obstructed aperture and the tip-tilt sensitivity of the coronagraph. As a result, unlike classic phase-induced amplitude apodization (PIAA), the PIAACMC mirror shapes are often slightly asymmetric even for an on-axis configuration and require more care in calculating off-axis shapes when an off-axis configuration is preferred. A method to design off-axis PIAA mirror shapes given an on-axis mirror design is presented. The algorithm is based on geometrical ray tracing and is able to calculate off-axis PIAA mirror shapes for an arbitrary geometry of the input and output beams. The method is demonstrated using the third generation PIAACMC design for WFIRST-AFTA telescope. Geometrical optics design issues related to the off-axis diffraction propagation effects are also discussed.



The Wide-Field Infrared Survey Telescope-Astrophysics Focused Telescope Assets (WFIRST-AFTA) coronagraph is designed to have a powerful high contrast performance for direct imaging of exoplanets by their reflected light. Any coronagraph system considered for the WFIRST-AFTA project should solve the main design problem of removing the starlight diffracted from the telescope central obstruction. The phase-induced amplitude apodization (PIAA) complex mask coronagraph (CMC)1 based on lossless pupil apodization by the beam shaping concept2 combined with a complex amplitude focal plane mask gives a solution to this problem.

Such a combination can be designed to concentrate the diffracted starlight in the pupil area that matches the telescope obstruction and block it with the Lyot stop that replicates the obstruction shape. It also allows the inner working angle as small as 0.8λ/D and much milder aspheric mirror shapes in comparison with the original PIAA design. However, it requires a focal plane mask that affects both phase and amplitude, as opposed to the simpler opaque focal plane mask used for PIAA.

The phase-induced amplitude apodization complex mask coronagraph (PIAACMC) (both the mirror shapes and the phase mask) can be optimized in a way that takes into account both chromatic diffraction effects caused by the telescope obstructed aperture, and the tip-tilt sensitivity of the coronagraph,3,4 and still attain the contrast of 109 between 2.0 and 4.0 λ/D with 10% broadband visible light (550 nm)4 that satisfies the WFIRST-AFTA requirements.5

The remapping optics of the PIAACMC system can be realized with lenses or mirrors. While either option is suitable at the 107 raw contrast level of ground-based systems, mirrors are the preferred option for reaching 108 contrast and beyond, as they are free from ghosts and wavefront errors induced by substrate inhomogeneities. Lens-based PIAACMC systems offer the convenience of being on-axis and can be inserted in an existing beam.6 Mirror-based systems can only be built on-axis for beams with large central obstruction, in a Cassegrain-like optical configuration. For all other configurations (unobstructed pupil, small central obstruction), mirror-based PIAACMC have to be built off-axis.

The typical design process for PIAA mirrors is to first optimize their shapes for an abstract on-axis collimated space configuration and then convert this design to a real unobstructed off-axis physical configuration like WFIRST-AFTA PIAACMC configuration (see Fig. 1, Ref. 3). The pupil asymmetry introduced by the telescope obstruction produces the system asymmetry that could be shared between the PIAA mirror shapes and the occulter (see the difference between Gen 2 and 3 designs in Ref. 3). Due to this asymmetry and Fresnel propagation effects, the procedure used to design the previous generation of off-axis PIAA optics is not suitable to design the off-axis system considered in this paper.

Fig. 1

Optical layout.


The goal of this paper is to present a method to calculate PIAA optics for an arbitrary off-axis geometry of the optical system based on the given on-axis optics shapes. By “arbitrary,” we mean different off-axis geometries for the system entrance and exit beams as well as a possible beam compression that the system should perform (one of the requirements for the proposed WFIRST-AFTA PIAACMC configuration3). Though diffraction critically affects the final system performance, our consideration will be based on geometrical optics only, assuming that the diffraction propagation effects are taken into account during the on-axis system optimization and are unaffected by conversion to off-axis. On the contrary to the full diffraction-based approach, the considered procedure is much less calculation intensive and provides the global convergence to the solution. The obtained solution can be used as a starting point for the full diffraction-based approach.

The phase mask design/characterization is outside of the scope of this paper and is considered in a paper by Kern et al.3


Optical Layout

The proposed optical layout of the PIAA system for the WFIRST-AFTA PIAACMC (Fig. 1) includes two PIAA mirrors M1 and M2. In this collimated-beam-to-focus system, the collimated input beam is reshaped by the M1 mirror and is focused by the M2 mirror. The angle of the input beam α is equal to 20 deg (approx.) and the output beam angle β is equal to 25 deg (approx.). The input beam diameter is about 40 mm, the distance d between mirrors is about 750 mm, and the focal length f of M2 (the distance between the M2 center and the system focus) is about 1200 mm. The M1 mirror also performs the compression of the input beam needed to match the focal plane point spread function (PSF) size with the diameter of the focal plane occulter. As a result the beam diameter on M2 is approximately equal to 15 mm.

We define a coordinate reference frame shown in Fig. 1 with the origin at the center of PIAA M1 and the z-axis parallel to the central ray between M1 and M2 mirrors. All mirror shapes will be expressed in this reference frame as functions of (x,y).

The base shape of M1 and M2 mirrors (i.e., without any PIAA remapping) is a plane and an off-axis parabola (OAP), respectively. All the following mirror shapes are calculated as corrections to these shapes that are given by


The base shapes are computed such that they focus the collimated input beam to the desired system focus whose coordinates are (x,y,z)=(0,fsinβ,dfcosβ) as shown in Fig. 1. Such a choice of base shapes turns the calculation of these shapes into a trivial procedure that is independent on a particular focus position and the beam compression/extending rate.

Two additional mirrors are used to perform remapping calculations. They are the flat mirror FLAT and the off-axis parabola OAPf


Both mirrors form collimated input and output beams that are parallel to the z-axis.


Off-Axis Mirror Shapes

In this section the procedure to calculate off-axis PIAA shapes is given. The shape of both mirrors can be presented as a sum of three terms


In Eq. (3), PLANE(x,y) and OAP(x,y) are base shapes of M1 and M2 mirrors [Eq. (1)], f1(r) and f2(r) are circularly symmetric mirror terms that are mainly responsible for the beam remapping in the on-axis collimated-beam-to-focus system, and OAT1(x,y) and OAT2(x,y) are off-axis PIAA corrective terms that preserve the correct PIAA apodization when mirrors are converted to off-axis.


Remapping Functions

The ray-tracing calculations of the PIAA optics are based on the remapping function concept. For a source on the optical axis the remapping function establishes a correspondence between a ray position r1=(x1,y1) in the entrance pupil and the position r2=(x2,y2) of the same ray in the exit pupil


Note, that the origin of the reference frames in Eq. (4) matches with the position of the central ray in the entrance and exit pupils.

Since we take the on-axis mirror shapes as given, the ray position in the entrance and exit pupils, and thus two-dimensional (2-D) remapping functions x2(x1,y1) and y2(x1,y1), can be readily calculated with a ray-tracing procedure.

The starting point to design mirror shapes for the WFIRST-AFTA coronagraph was the diffraction-based optimized solution (Fig. 2, Ref. 3; “revised” PIAACMC design, Ref. 4) obtained for an on-axis collimated-beam-to-collimated-beam PIAA system without beam compression. Although this solution is not circularly symmetric because of the central obstruction asymmetry, the deviation from the symmetry for both mirrors is small. That makes it possible to define the azimuthally averaged remapping function


that determines the radial mirror terms f1(r) and f2(r). In Eq. (5) ϕ means azimuthal averaging.

Fig. 2

Extended remapping function R2(r1) and inverse remapping function R1(r2). The working beam radii rb1 and rb2 are shown.


The remapping function r2(r1) constructed in according with Eq. (5) provides a good initial approximation to calculate the off-axis mirror shapes for all the rays within the working beam. However, the discontinuity of derivatives of the remapping function at the beam edge and the absence of the remapped values outside of the working beam creates numerous difficulties for the ray-tracing calculations that are suppose to work with real square arrays of points.

To avoid these problems we created an extended remapping function R2(r1) (Fig. 2) by transposing the remapping function r2(r1) onto the intervals [rb1,R] and [rb2,R] backward, beyond the working beam radii rb1 (on PIAA M1) and rb2 (on PIAA M2), and making sure that the derivatives match on the beam edge. The R2(r1) is given as follows:


R2(r1)=r2(r1)for  r1<rb1,{r1=rb1+(1+α1r1)r1r2=rb2+(1+α2r2)r2rb2r2=R2(rb1r1)for  rb1r1R,
and R is the radius of an area big enough to cover all arrays involved in the mirror shapes calculation.

It should be noted that for a system with beam compression rb2 is not equal to rb1. The desired beam compression can also be included in the final expressions for 2-D remapping functions X2(x1,y1) and Y2(x1,y1) through the rescaling factor rb2/rb1


where x2(x1,y1) and y2(x1,y1) are remapping functions for the system without beam compression.


Optics Shapes Calculation

The following iteration procedure is proposed to calculate off-axis PIAA mirror shapes given on-axis ones:

  • 1. Calculate the 2-D remapping functions x2(x1,y1) and y2(x1,y1) by ray-tracing on-axis mirror shapes and azimuthally average them to get the radial remapping function R2(r1).

  • 2. Calculate the 2-D remapping functions X2(x1,y1) and Y2(x1,y1) for the compressed beam.

  • 3. Determine base shapes PLANE(x,y) and OAP(x,y) of mirrors per Eq. (1).

  • 4. Calculate radial mirror profiles f1(r) and f2(r) for the on-axis collimated-beam-to-focus system with the radial remapping R2(r1) (Sec. 3.3).

  • 5. Start a loop. For i’th iteration step construct an estimate OAT1,i(x,y) as a sum of small corrective terms dM1,j(x,y) calculated during previous iteration steps (Sec. 3.4)



  • 6. Determine the off-axis PIAA term OAT2,i(x,y) that provides equal optical path length (OPL) for all the rays propagated from the source to the main PIAA focus. This step is performed assuming that the off-axis PIAA term OAT1(x,y) is equal to OAT1,i(x,y).

  • 7. Close the loop by returning to step 5.


Radial Mirror Terms

For the case of a radially symmetric collimated-beam-to-collimated-beam PIAA system, the mirror shapes can be directly computed from R2(r1) by numerically integrating a pair of differential equations.2

In this section, we consider the case of the on-axis collimated-beam-to-focus radially symmetric system. The proposed solution is similar to the procedure we use to calculate the 2-D off-axis correction to the M1 shape (Sec. 3.4).

The radial mirror profiles f1(r) can be derived from


if the derivative df1(r)/dr is known. The derivative df1(r)/dr can be easily estimated for a collimated-beam-to-focus system by using the “desired” normal [i.e., one that generates the desired remapping in according with Eq. (5)] to the M1 surface.

In according with the reflection law, the “desired” normal ni=(nri,nzi) to the surface of the M1 mirror can be written as


where r1s is the vector from a point r1 on the M1 surface to the source, and the vector r12 connects the point r1 with the remapped point r2. The position of the remapped point r2 on the M2 mirror is determined by the remapping function R2(r1). The derivative df1(r)/dr is given by


To derive both f1(r) and f2(r) radial profiles an iterative approach is used. In this approach, the “desired” normal ni is estimated from the previous approximation for the M1 and M2 shapes and the M1 shape is obtained by solving Eq. (9). The next approximation of the M2 shape is calculated assuming that for all the rays propagated from the source to the main PIAA focus OPL should be equal to the OPL of the principal ray (Fermat’s principle). The OPL of the principal ray is known from the initial conditions. Radial mirror terms for the WFIRST-AFTA coronagraph are shown in Fig. 3.

Fig. 3

Mirror shapes for WFIRST-AFTA coronagraph: radial mirror terms. The focal distance f is equal to the distance between mirrors d.



Calculation of dM1(x,y)

To calculate the corrective term dM1,j(x,y) (Sec. 3.2) we should

  • 1. For given 2-D remapping functions [Eq. (7)] determine the “desired” normal ni=(nxi,nyi,nzi) to the M1 surface assuming that M1 and M2 mirror shapes match the current shapes approximation. Similarly Eq. (10), the “desired” normal


    where vectors r1s and r12 are determined in Sec. 3.3. To determine a particular ray position on the M2 surface, the geometrical propagation in direction from the exit pupil to the off-axis parabola OAPf and then to the M2 mirror should be used. The ray position in the exit pupil can be determined as the remapped [in accordance with Eq. (7)] ray position in the entrance pupil. By the entrance pupil, we mean a plane upstream of the flat mirror FLAT(x,y) [Eq. (2)]. The exit pupil is the plane downstream of the off-axis parabola OAPf(x,y) [Eq. (2)] optically conjugated with the M2 mirror. Both the entrance and exit pupils are orthogonal to the collimated input/output beams and all the rays are considered to be collimated between the entrance pupil and the flat mirror, and between the exit pupil and OAPf.

  • 2. Obtain the “current” normal (i.e., gradient vector of the surface) nc=(nxc,nyc,nzc) to the current M1 mirror shape.

  • 3. Calculate derivatives dM1,i(x,y)/x and dM1,i(x,y)/y





  • 4. Calculate dM1,i(x,y) through a line integral



    Since, for any physically possible remapping, the integration in Eq. (15) along any path that connects the mirror center with the point (x,y) should give the same result, dM1,i(x,y) values can also be averaged for different integration paths.

  • 5. To reduce the M1 errors caused by the numerical integration/interpolation, an appropriate smoothing algorithm should be applied for each dM1,i(x,y) estimation.


Calculation of OAT2 (x, y)

To determine OAT2,i(x,y), we use an iterative procedure II that corrects the OPL of the output rays for every step of the main iteration. Note that iteration procedure II is different from the main iteration discussed earlier in Sec. 3.2.

During k'th step of the iteration II, the function dM2,k(x2,y2) is calculated


where OPLk(x,y) is the OPL of the ray that starts at the source position, reflects from the current approximation of the M1 shape at the point (x1,y1), reflects from the current approximation of the M2 shape at the point (x2,y2) and then hits the system focus. Similarly to OAT1,i(x,y), the off-axis term OAT2,i(x,y) can be determined as


The iteration stops as soon as the optical path difference (OPD) for all the rays that cross the M2 surface becomes negligible. Because of the remapping, the sampling of OAT2(x2,y2) on the M2 surface appears to be inhomogeneous, even if the original sampling of arrays on the M1 surface is homogeneous. To get homogeneous OAT2(x2,y2), sampling on the M2 surface a triangulation procedure should be used.

One question, related to the iterations described above needs to be mentioned specially. During each iteration step numerous derivative calculations are performed to estimate the normal to the mirror surfaces at the location of each ray. The integration, differentiation, and interpolation executed during each iteration step produce and propagate numerical errors. Those errors result in so-called numerical catastrophe when the calculated derivatives show infinite growth starting from some iteration number. To avoid the numerical catastrophe, the data arrays related to calculation of the dM1(x,y) and dM2(x,y) terms should be smoothed with any appropriate algorithm. It is clear though that the convergence of iterations should slow down (or even be broken), if the spatial scale of the smoothing becomes comparable with the expected size of features on the mirror surfaces. For the system considered here, the minimal mirror features size is determined by the diffraction-like structure described in Sec. 4. As a result, the residual output beam OPD is constrained to the 1010  m level, which is better than the physical limit of optics manufacturing. Also, note that the 1010  m RMS errors in mirror shapes produce 40 to 50  μm ray scattering in the focal plane of the system, i.e., comparable with the size of the diffractive PSF of the system (50 to 60  μm depending on the wavelength) and a 50  μm beam walking in the M2 mirror plane.


Mirror Shapes

The calculated mirror shapes for the WFIRST-AFTA coronagraph are shown in Fig. 4. To present the deviation of shapes from common optical shapes we presubtracted following terms


from the M1 and M2 surfaces respectively. Note that these parabolas are not the base optics shapes considered earlier and are not chosen based on their optical properties, they were chosen simply to highlight higher order departures visually.

Fig. 4

Calculated off-axis mirror shapes for the WFIRST-AFTA coronagraph: (a) PIAA M1 and (b) M2 mirrors. Deviation from parabolic surfaces P1(x,y) and P2(x,y) are only presented. Working beam radii rb1 and rb2 are shown on crosscuts on the right.


An additional point to emphasize is that the diffraction propagation introduces radial oscillations both in amplitude and phase of the output beam (Figs. 5 and 6) which the initially optimized on-axis mirror shapes take into account and correct for.

Fig. 5

The diffraction caused output beam phase oscillations: (a) the geometrical OPD (projection on the PIAA M1 mirror) of the output beam for the optimized on-axis mirror shapes and (b) the PHASE(x,y) mirror term. Note that OPD(x,y) and PHASE(x,y) have different scales.


Fig. 6

Three-dimensional ray tracing check of the PIAA mirror shapes with the term PHASE(x,y)=0: (a) the amplitude and phase in the exit pupil calculated in the geometrical optics approximation and (b) the amplitude (normalized, dots) and phase crosscuts through the exit pupil center. The PIAA optics is designed to deliver the flat output wavefront with the “smooth” amplitude profile (solid line in the amplitude crosscut) in the presence of the Fresnel diffraction.


As a result, the geometrical OPD of the output beam is not equal to zero as expected in according to the pure geometrical optics model (Fig. 5), although it appears to be equal to zero if we account for diffraction propagation. To take into account the above effect, an additional term PHASE(x,y) for the M2 mirror shape should be introduced


This term describes the OPD change caused by the diffraction propagation in the system. To calculate the M2 shape in this case, Eq. (16) should be modified in the following way:


where OPDd(x,y) is the geometrical OPD in the output beam for the baseline on-axis system. The PHASE(x,y) term is equal to the difference between the M2 shape calculated in according with Eq. (20) and the M2 shape obtained under the assumption of the pure geometrical propagation [Eq. (16)].


Ray-Tracing Check

The derived optical shapes with the “phase” term PHASE(x,y)=0 have been checked with the ray-tracing code that has been developed to simulate the previous generations of PIAA optics manufactured by AXSYS Inc.7 and Tinsley Inc.8 Results of the ray-tracing simulation are shown in Fig. 6 and include the wavefront amplitude and phase in the exit pupil of the system calculated in the geometrical optics approximation.

Though the wavefront amplitude looks noisy with a 4.7% RMS white noise in amplitude caused by the limited number of rays used for the simulation (about 115  rays  per  pixel2), the calculated values show close correspondence with the designed exit pupil amplitudes except for the area that is blocked by the telescope central obstruction. At the same time, the expected profile seems to be deformed by the clearly observable diffraction-like structure. The presence of this structure [as well as the presence of the “phase” term PHASE(x,y)] is easy to understand if one takes into account the diffraction-based optimization of the on-axis mirror shapes mentioned above. This optimization produces the shapes that deliver the designed “smooth” (without diffraction-like structure) amplitude and phase profiles in the presence of the Fresnel diffraction. As a result, the “smooth” shape of the mirrors as well as the exit pupil amplitude and phase profiles (in the geometrical optics approximation) appears to be modulated with the “inverse” (in respect to the Fresnel diffraction) diffraction-like structure that is responsible for the correction of diffraction propagation effects.

In an absence of the PHASE(x,y) term, the exit pupil phase appears to be flat with the wavefront errors of about λ/3000. The exit pupil phase errors are only limited by the residual output beam OPD with a 1010  m RMS discussed in Sec. 3.5. The value of 1010  m seems to be a reasonable estimate for the accuracy of the PIAA mirror shapes calculation with the described algorithm.



In this paper, we presented a method to calculate off-axis mirror shapes for the PIAACMC assuming that the on-axis mirror shapes are known. The method is based on geometrical ray tracing and is able to calculate off-axis PIAA mirror shapes for an arbitrary geometry of the input and output beams. The accuracy of off-axis PIAA shapes calculation is limited to the value of 1010  m, which is better than the physical limit of optics manufacturing.


This work was supported by the National Aeronautics and Space Administration’s WFIRST-AFTA program. It was carried out at the NASA Ames Research Center in collaboration with NASA Jet Propulsion Laboratory. The authors would like to thank Brian Kern for precursor work and helpful discussions. The preliminary version of this paper has been published in Proceedings of SPIE 9605, 2015.9


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Biographies of the authors are not available.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.
Eugene A. Pluzhnik, Eugene A. Pluzhnik, Olivier Guyon, Olivier Guyon, Ruslan Belikov, Ruslan Belikov, Eduardo A. Bendek, Eduardo A. Bendek, } "Design of off-axis mirrors for the phase-induced amplitude apodization complex mask coronagraph," Journal of Astronomical Telescopes, Instruments, and Systems 2(1), 011018 (8 March 2016). https://doi.org/10.1117/1.JATIS.2.1.011018 . Submission:


Design of off-axis PIAACMC mirrors
Proceedings of SPIE (September 15 2015)
4-m off-axis telescope
Proceedings of SPIE (August 24 1998)
Reflecting collimator systems with no central obscuration
Proceedings of SPIE (April 14 1993)

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