Diffraction Analysis of 2-D Pupil Remapping for High-Contrast Imaging

Transcription

1 Diffraction Analysis of -D Pupil Remapping for High-Contrast Imaging Robert J. Vanderbei SPIE San Diego Aug 4, 5 Page of Princeton University rvdb

2 Apodization 3 Pupil Mapping 5 3 Notations 6 4 Huygens Wavelets 7 5 Fresnel Approximation 8 6 Fresnel Examples 9 7 Better Than Fresnel (BTF) Page of

3 Apodization.5 Page 3 of

4 E(ξ, η) = λf E(ρ) = π λf e xξ+ỹη πi λf A( x + ỹ )dỹd x ( J π rρ ) A( r) rd r. λf Page 4 of Psf(ρ) = E(ρ)

5 Pupil Mapping Advantages: % throughput Implicit magnification... effectively iwa λ/d. Disadvantages: Diffraction effects limit achievable contrast to 5 for a pure pupil-mapping system..5.5 Page 5 of.5.5.5

6 Notations h r (r) = h r ( r) = r R( r) = ± A (s)sds. r R(r) Q + (n )(r R(r)) R( r) r Q + (n )(R( r) r) n is the refractive index and Q = (n )z R(r) r = S(R( r), r) + n( h( r) h(r( r))) h( r) Notes: h(r) r R( r) the second expression for Q is in fact independent of r. for mirrors, put n =. Z z Page 6 of

7 Huygens Wavelets Cartesian Coordinates E out ( x, ỹ) = eπiq( x,ỹ,x,y)/λ dydx, λq( x, ỹ, x, y) where Q denotes the optical path length: Q( x, ỹ, x, y) = (x x) + (y ỹ) + (h(r) h( r)) + n(z h(r) + h( r)) Polar Coordinates E out ( r) = where λq( r, r, θ) eπiq( r,r,θ)/λ rdθdr, Page 7 of Q( r, r, θ) = Double integrals are hard! r r r cos θ + r + (h(r) h( r)) + n(z h(r) + h( r)).

8 Fresnel Approximation Assuming a large separation between the lenses and that the lenses are themselves thin, we get r r r cos θ + r + (h(r) h( r)) (h(r) h( r)) + r r r cos θ + r. z Fresnel uses this approximation in the exponential (and the paraxial approximation in the leading factor) to get the standard Fresnel approximation: Page 8 of E out ( r) = π λz r (n ) h( r) eπi zλ +πi λ e r πi zλ πi (n )h(r) λ J (πr r/zλ)rdr.

9 Fresnel Examples Flat Glass A Phase in radians nd Pupil Amplitude Map nd Pupil Phase Map Intensity Relative to Peak Amplitude Map Zoomed x -3 Ideal and Fresnel PSFs Image-Plane Radius (mm) Page 9 of n =.5. D = 5mm. z = 5D. λ = 63.8nm.

10 Galilean Telescope: A a R( r) = a r R(r) = r/a h(r) = z + Q + (n )( /a) r Q (n )( /a) h( r) = Q + (n )(a ) r Q (n )(a ) Page of If n >, then both lenses are hyperbolic. If n <, then both lenses are elliptical.

11 Galilean Telescope A 3 nd Pupil Amplitude Map x -3 nd Pupil Phase Map Lens height (m).5 x -3 Lens profiles Ideal and Fresnel PSFs Phase in radians x -3 Intensity Relative to Peak Image-Plane Radius (mm) Page of Fresnel approximation is very bad.

12 Better Than Fresnel (BTF) We retain the paraxial approximation for the leading factor. We can subtract an arbitrary constant from the Q in the exponent. Let s choose to subtract Q( r, R( r), ). We compute: Q( r, r, θ) Q( r, R( r), ) = S( r, r, θ) S( r, R( r), ) + n(h(r( r)) h(r)) = S ( r, r, θ) S ( r, R( r), ) S( r, r, θ) + S( r, R( r), ) + n(h(r( r)) h(r)) Page of Then, we simplify the numerator: S ( r, r, θ) S ( r, R( r), ) = (r R( r))(r + R( r)) r ( r cos θ R( r) ) + ( h(r) h(r( r)) ) ( ) h(r) + h(r( r)) h( r).

13 Mo Better So far, everything is exact (except for the paraxial approximation). The only other approximation is to replace S( r, r, θ) in the denominator with S( r, R( r), ) so that the denominator becomes just S( r, R( r), ). Replacing the integral on θ with the appropriate Bessel function, we get a new approximation: E out ( r) π λz e πi r R( r) + rr( r)+(h(r) h(r( r)))(h(r)+h(r( r)) h( r)) + n (h(r( r)) h(r)) S( r,r( r),)λ λ ( ) π rr J rdr. λs( r, R( r), ) Page 3 of

14 Example: A 3 nd Pupil Amplitude Map x -3 nd Pupil Phase Map.4 Lens height (m).5 x -3 Lens profiles Ideal and Fresnel PSFs Phase in radians x -3 Intensity Relative to Peak Image-Plane Radius (mm) Page 4 of This approximation is much better than Fresnel.

15 Pupil Mapping for High Contrast (PIAA) nd Pupil Amplitude Map nd Pupil Phase Map Lens height (m) 3 x -4 Lens profiles.5..5 Ideal and Fresnel PSFs Phase in radians.6.4. Intensity Relative to Peak Page 5 of Image-Plane Radius (mm) Designed for. Delivers 5.

16 Brute Force Huygens Wavelet Integral Phase in radians nd Pupil Amplitude Map nd Pupil Phase Map Intensity Relative to Peak Lens height (m) 3 x -4 Lens profiles.5..5 Ideal and Fresnel PSFs Image-Plane Radius (mm) Page 6 of Full D integration: 5 r-values and 5 θ-values. Increased the wavelength by a factor. The degradation is fundamental physics, not numerical error.

17 How about larger lenses? 4 3 nd Pupil Amplitude Map.5.5 Lens height (m) Lens profiles nd Pupil Phase Map Ideal and Fresnel PSFs Phase in radians Intensity Relative to Peak Page 7 of Image-Plane Radius (mm) D =.5m ( times bigger). Discretization: 5 points. First sidelobe:.3 7.

18 Big Primary, Small Secondary 5 5 nd Pupil Amplitude Map...3 Lens height (m) Lens profiles nd Pupil Phase Map Ideal and Fresnel PSFs Phase in radians.6.4. Intensity Relative to Peak Page 8 of Image-Plane Radius (mm) A inch primary and a inch secondary. A DISASTER!

19 Design for Medium Contrast nd Pupil Amplitude Map.5..5 Lens height (m) 3 x -4 Lens profiles nd Pupil Phase Map Ideal and Fresnel PSFs Phase in radians.. -. Intensity Relative to Peak -5 - Page 9 of Image-Plane Radius (mm) Designed for 5. Delivers 5. max(a) is smaller...less off-axis distortion to correct.

20 Medium Contrast Pupil Mapping with Back-End Apodizer Phase in radians nd Pupil Amplitude Map nd Pupil Phase Map Intensity Relative to Peak Lens height (m) 3 x -4 Lens profiles.5..5 Ideal and Fresnel PSFs Image-Plane Radius (mm) Page of Hybrid system designed for. Delivers 7.5.

21 Front-End Apodizer nd Pupil Amplitude Map Lens height (m) 4 x -4 Lens profiles nd Pupil Phase Map Ideal and Fresnel PSFs Phase in radians Intensity Relative to Peak Image-Plane Radius (mm) Page of This hybrid system is about as good as the back-end apodizer. Easier to manufacture.