Introduction to aberrations OPTI518 Lecture 5

Transcription

1 Introduction to aberrations OPTI518 Lecture 5 Second-order terms 1

2 Second-order terms W H W W H W H W, cos Piston Change of image location Change of magnification 2

3 Reference for OPD Exit pupil plane or reference sphere W H,,cos W W H W Hcos W What should the coefficients be? Reference sphere Reference sphere is centered at Gaussian image point Deformed wavefront Image plane 3

4 Exit pupil plane as reference y ' E Ж nuy nuy u ' y ' I W111 n' y' E u' y ' E u ' 1 W020 n ' y ' E u ' 2 4

5 Reference is a sphere Reference is a sphere centered at the Gaussian image point that passes by the on-axis pupil plane point W111 0 W020 0 Gaussian optics predicts the location and size of an image and therefore There is no magnification change or change of focus to second-order 5

6 Piston Reference for piston is chosen to be the entrance pupil plane on-axis point. Since the pupils are conjugated we have W 000 n nt i1 i i W

7 Question If the object is the reference then what is the piston term? 7

8 Second-order contribution from a spherical surface npa ' ' npb PB PQOQ Stop PB' OQ' PQ' O 8 npa ' ' npb W 0, n' PB' n' PA' n' PB' npb 8

9 Second-order contribution W 0, n' PB' n' PA' n' PB' npb PB 2 2 y y 2r 2s, 2 2 y y PB ' 2r 2 s', y y y y W n n 2r 2 s' 2r 2s 0, ' 2 y n' n n' n r s' s W W020 0 If s and s are Gaussian conjugates 9

10 Change of reference sphere and depth of focus 2 y n' n n' n W W ' W 2 r s' s' s 2 y n' n n' 1 n 2 r s' s ' 1 s s ' 2 y n' n n' n n' s' 2 2 r s' s s ' 2 y n' s' s' W n' u' 2 2 s '

11 Change of reference sphere and depth of focus W020 1 s ' 2 2 n' u' W020 2 s' 8 f /# n ' 2 2 s ' 2 f /# f /# in micrometers for W020 n ' 4 Note sign Alternate derivation 11

12 Microscope cover glass W s' 8 f /# 2 t n' 020 n 1 n f/#=1 W mm t=0.17 mm n=

13 Alternate derivation ye ye ye W020 n' n' n' s' 2 2 Rs' 2R 2R (we have neglected higher than second-order terms) 13

14 Change of magnification y ' E u ' y ' I y ' I W111 nu ' ' y ' I W nu ' ' 111 Ж nuy nuy 14

15 Review conceptual models F P P F Z' f' Z f 1 1 m m 1 Z \ f m 1 1 m f' Z' f Z. 1 1 Z' f' Provides a methodology to trace First-order rays 15

16 First-order chief and marginal rays Stop aperture Image plane 16

17 Meridional and Sagittal Rays

18 Path of a first-order ray and a real ray 18

19 Review of first-order quantities y r u s Assume you know how to trace first-order rays 19

20 Stop Shifting Optical axis Exit pupil Image plane Stop shift is a change in the location of the aperture stop along the axis Stop shift does not change the f/# or the Throughput Stop shift select a different portion of the wavefront for off-axis beams 20

21 Stop shifting 21

22 Stop shifting 22

23 Question How do we quantify stop shifting? 23

24 The stop sifting parameter S Can be calculated at any plane in the optical system S u u y y A A u y A new old new old new old is the stop shifting parameter If the old stop is at the center of curvature of a surface we have: S y new 0 Anew 0 y A Ж nuy nuy Ay Ay 24

25 Relation with the refraction invariants H s S y y y A A Spherical surface of radius R Auxiliary axis cc s Optical axis H H H R 1 H 1 y R Rs Rs R s s 1 1 y R s R s 1 1 Ani n y nu R s Hun Ж Ay Ay A A A A y y A y A ycc A0 ycc 0 S y A y A y cc Stop Shift from the CC cc 25

26 The stop shifting parameter can be calculated at any plane S y y y d y d cc d y y cte 26

27 Key question We need to know the aberration function as we select new stop positions We know the aberration function at the old exit pupil The key question is what is the aberration function at the new exit pupil? 27

28 Describing a ray from the old exit pupil requires an offset New chief ray SH SH Optical axis New exit pupil Old exit pupil 28

29 Math of stop shifting y ' E W H, W020 W020 y ' E y' E y' E W020 y' E y' E y' E y' E y' E H y' E y' E H y' E y' E H W H, W020 y' E y' E W 020 SH SH W W S H S W HH

30 Example: system with a focus error W H, W020 SH 2S H S H H 2 W H W H W W S H S W H H 2,,

31 Change of magnification W y' I nu ' ' W 020 nu ' ' S New pupil Old pupil s ' s ' s ' 31

32 Wavefront decomposition W SH W H W W S H S W H H 2,

33 Aberration function units and sign W represents and OPD and has linear dimensions of mm, m, micrometers, etc. Positive if it leads the reference sphere about the optical axis Product of a distance and the index of refraction (unit-less) W/λ in waves 33

34 Aberration metrics Wavefront deformation Angular Transverse Longitudinal Note that each aberration has a longitudinal and a lateral (or transverse) nature. Do not get confused. This is, the longitudinal nature of longitudinal chromatic aberration, or the transverse nature of longitudinal chromatic aberration, etc. 34

35 Airy pattern Object is a point source (not extended) Physical analysis Circular aperture No aberration Monochromatic Use physical Fourier optics theory (diffraction theory) No polarization considered (scalar theory) 1.22 f # First order Bessel function of The first kind (squared) 35

36 Plane wave focused by infinite lens xy, F 36

37 Airy pattern formation Circular aperture diffracts incoming light into an angular spectrum of plane waves (recall a diffraction grating) that focus at the rear focal plane. Ray representation Circular aperture Fourier transform of Complex amplitude at Front focal plane 37

38 Diffraction Encircled Energy in a PSF % % 0.8 ENCIRCLED ENERGY st zero 2nd zero DIAMETER OF CIRCLE (MM) 38

39 Estimate first-zero opd sin / f nd ndsin / 2 / 2 f D f opd ndsin / 2 39

40 40

41 Gouy phase shift effect Note the anomaly in the on-axis irradiance 41

42 Light field along the z-axis opd nz nz cos 2 nz 2 n Z 2 ; D /2 f f Z 8 f 2 n n D n 2 D 2 # f ΔZ 42

43 PSF F/10; f=100 mm; W= mm mm 0.4 mm 43

44 One wave of defocus 44

45 W020 Defocus Defocus does not depend on the field and so it happens uniformly across the entire field of view 45

46 3D imaging region 46

47 Camera obscura imaging psf 47

48 1D Camera Obscura OTF rect sin c Contrast reversal 48

49 Aberration free lens Geometrical optics psf Under defocus the psf is the cylinder function then the OTF is the sombrero function. We can use LSIS theory to understand the imaging and set image quality metrics. 49

50 Aberration free lens system OTF Physical optics 50

51 51

52 Spatial frequency and contrast 52

53 Eye contrast sensitivity Contrast 1 Treshold of detection 0 Spatial frequency 53

54 Summary Conceptual models Stop shifting References to measure W Second order terms: defocus Review of the Airy pattern Comments on OTF and aberration evaluation Some rules of thumb 54