Control Of Op+cal Aberra+ons With Coded Apertures

Transcription

1 Control Of Op+cal Aberra+ons With Coded Apertures Andy Harvey, Gonzalo Muyo and Tom VeAenburg Heriot- WaA University Edinburgh EH14 4AS, UK SPIE Coded Aperture

2 Outline Introduc+on and mo+va+on Design of binary- amplitude masks Contour masks (analy+cal) Mul+- mask (numerical) Examples Fundamental aberra+ons Ocular aberra+ons Aberra+ons in conformal op+cs Conclusions SPIE Coded Aperture

3 Control of Aberra+ons High- cost op+cs Hybrid/computa+onal imaging Wavefront coding Coded apertures With lens! Agile/adap+ve Cf. adap+ve op+cs Foveal Low cost SPIE Coded Aperture

4 Aberra+ons: decomposi+on of OTF Defocus: At spatial frequency v, OTF components form a circle OTF ν ( ) = P( u + v) OTFHnL w = = 1ê2 1ê3 5ê6 0 1 P * ( u v)du Cubic phase modulation: OTF components curl into generalised Cornu spiral Defocus unwinds spiral no zeros in MTF OTFHnL w 20 = 1ê6 W 20 =1/6 =1/3 =1/2 =2 =0 v n OTF G. Muyo and A. R. Harvey, Opt. Lett. 30, (2005) M Demenikov and A R Harvey Opt. Exp. 18, 17, pp (2010) 4 M SPIE Demenikov Coded Aperture and 2011 A R Harvey Opt. Exp. 18, 8, pp 8207 (2010)

5 G Muyo, A Singh, M Andersson D Huckridge,A Wood and A R Harvey, Opt. Exp., 23, SPIE Coded Aperture 2011 (2009) 5

6 Decomposi+on of the OTF OTF( ν )= P ( u+ v)p( u v)du, with P( u)=expil( u), u u 1 ( ) = exp iδ u,v Ωv du, with Δ( u,v)= l( u+ v) l u v ( ) Op+cal path length varia+on Difference in op+cal path length between two points on the pupil 2ν apart. OTF with defocus OTFHnL w 20 = 1ê2 OTF with defocus and mask OTFHnL Hw w 20 = 1ê3 1ê2 W 20 =1/2 W 20 =1/2 n n SPIE Coded Aperture

7 Mi+ga+ng aberra+ons with amplitude masks Binary- amplitude masks aaenuate 2 nd order image- plane interference enable sufficient informa+on to be recorded for good image restora+on. Advantages Wide variety of fixed or agile amplitude modula+on techniques Wideband implementa+on: UV to far- infrared Lower complexity and cost Binary- amplitude mask Example: Coded mask for defocus MTF mask MTF no- mask Detector SPIE Coded Aperture 2011 Aberrated op+cal system 7

8 Design of the binary mask The ideal mask permits only construc+ve 2 nd - order interference (of phasors within OTF integral): Thus, for an aberra+on characterized by an op+cal path length varia+on l(u), the mask M(u) blocks areas of the pupil along contour lines: Parameters to op+mise Δ u, ν + π 2mod2π π Tip- +lt t and reference phase M 1, l ϕ ϕ u= u u t u+ Δ mod2π Δϕ 0 2 ϕ 0 π l π 2 ( u)mod2π 2 SPIE Coded Aperture

9 OTF of a contour mask Contour mask can be designed with no nulls in OTF in the limit for large aberra+ons 1 2 MTF OTF π ν dl sin2 ϕ 2 c ν Δ MTF ϕ Δ π2 2 π2 dl ν For Δϕ=π, MTF~0.2MTF dl MTF is independent of the magnitude of aberra+ons SPIE Coded Aperture T.VeAenburg, A. Harvey, J. Opt. Soc. Am. A 28, (2011). 9

10 Op+misa+on M 1, ϕ u = u u ϕ t u+ Δ mod 2π Δϕ 2 ( ) l( ) 0 Low dimensional op+miza+on Tip- +lt, t Phase reference, ϕ o minimizes expected imaging error ε. Cost func+on for as+gma+sm as a func+on of +p- +lt (t x,t y ) and minimized for phase- reference. SPIE Coded Aperture

11 Examples of contour masks Pupil phase (hue) Mask (black) MTF (no mask) MTF (mask) RMS OPD of 2λ RMS OPD of 2λ RMS OPD of 3.5λ SPIE Coded Aperture

12 Example: doublet and broadband illumina+on F/5 cemented doublet, 5 off- axis at λ 0 =550nm As+gma+sm and field curvature with a peak- to- valley OPD of 10λ 0 Broadband MTF (495nm λ 605nm) strongly suppressed Higher MTF for extended bandwidth Contrast lower than for monochroma+c opera+on SPIE Coded Aperture

13 Benefit from mul+ple MTFs? Häusler (1972) Scan image plane through volume Reduced MTF No zeros Approximately constant Inverse filter to recover high- quality image MTF Freq. 13

14 Mask design: numerical op+misa+on Op+miza+on of pixelated mask High dimensional Maximize MTF Op+misa+on of a set of masks with complementary MTFs 1 Some mask MTFs may contain nulls { Mi : i 1,..., N} = = M = argmin 1 σ + T MTF = argmin ( M ) σ 2 N + i T i 1 MTF ( Mi) 2 1 SPIE Coded Aperture J. W. Stayman, N. Subo+c, and W. Buller, Proc. SPIE 7468, 74680D (2009).

15 Image restora+on algorithm for a sequence of N coded images The sequence of coded images are subsequently processed via a mul+- frame restora+on algorithm to produce a sharp image. Forward model, g = g 1 g 2 g N = H f + n = Tikhonov- type regulariza+on scheme leads us to a least- squares restora+on algorithm: H 1 H 2 H N f + f = H * H+σ I 1 H * g SPIE Coded Aperture n 1 n 2 n N

16 Applica+on: Conformal op+cs Imaging through ellipsoidal dome in MWIR (λ 0 =4 microns) Imaging system with ±45 o FoV Aberra+ons: PV OPD of 4λ 0 at ±45 o Two binary- amplitude masks for each FoV Reconfigurable shuaer array at the aperture stop that op+mises mask with FoV Foveal imaging Op+misa+on up to Nyquist frequency uses a differen+al evolu+on algorithm with itera+ons 100mm 100mm SPIE Coded Aperture

17 Op+mised coded masks (0 o, 45 o ) (- 45 o, 45 o ) Average mask transmission of 60 % SPIE Coded Aperture

18 PSFs across FoV Coded aperture system (ayer spa+ally- variant restora+on) Conven+onal system 90 o 90 o 90 o SPIE Coded Aperture

19 MTF at (0 o, 45 o ) Solid black line: Diffrac+on limited MTF Solid lines (red and blue): mask 1 and mask 2 MTFs Dashed blue line: aberrated MTF (no masks) SPIE Coded Aperture

20 Restored images for (0 o, 45 o ) deg With masks Without masks RMSE=15% RMSE=35% SNR of one recorded conven+onal image~37db SNR of one coded image~30db (signal scaled down by transmission of coded masks) SPIE Coded Aperture

21 Restored images for (0 o, 45 o ) With masks Without masks RMSE=13% RMSE=27% SPIE Coded Aperture

22 Conclusions Where aberra+ons are unavoidable binary coded masks can be op+mised to yield improved overall MTF. MTF typically 20% of diffrac+on limited MTF Absence of nulls and modest reduc+on in MTF allows recovery of high quality images. Agile - adaptable programmable masks are a low cost alterna+ve to deformable mirrors Good performance with broadband illumina+on is possible. SPIE Coded Aperture