第七章最佳化（Optimization）

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1 7.1 Optimization 7.2 Damped least squares; DLS (derivative increment) (variable damping) 7.4 (operands) (component classes) (RMS spot size) (RMS OPD) (MTF) 7.6 (automatic error function generation) 7.7 (multiconfiguration optimization) 7.8 (Global optimization) merit function global optimization OSLO Six adaptive simulated Annealing 7-1

2 iterative 7.1 A C x=a or x=c f (X ) Region A Region B Region C a b c X B x=b dimensionality downhill least-square OSLO damped least squares 7.2 Damped least squares; DLS OSLO DLS DLS OSLO DLS ( ) DLS φ x v n v φ 7.1 ( x) = w f ( x) i= 1 i 2 v i x v x i 7-2

3 v x = x x, x,, 7.2 1, 2 3 L f operands f i ( c c ) f i 1i 2i x n = 7.3 c 1i, c 2i components 7.3 w 1 φ x v v T φ x = f 7.4 ( ) f f m ( ) f = f f, f,, 7.5 1, 2 3 L j i f i f i ( x j x j ) = f i ( x j ) + x j x f i 0 x j j f i OSLO RMS; root mean square f x v v v A x = f 7.7 A 7-3

4 f1 f1 f1 f1 L x 1 x2 x3 xn f 2 f 2 L A = x1 x2 7.8 M f m f m LLLL x x 1 n (1) x v 0 x v 0 (2) x v v, x, 1 2 L v v v v v v,, v v v x1 = x0 + x1 x2 = x1 + x2 L x k +1 = xk + xk x v k A T k A x v k k = A T k f k 7.9 k x v x v I T v T ( Ak Ak + k I ) xk = Ak f k µ x v k 2 n x n m>n T 7.8 A nxn k A k 3 x v k x v

5 k I D d D = 11 d 22 O d m m, fi d = jj i= 1 x j OSLO OSLO A T A DLS 7.2 OSLO DLS ite std Damping x v =0 7-5

6 A T A singular? A T A x v x v OSLO ite ful Opds 10-8 Opdm 7-6

7 opst Constraints ( f ) 1 Solves -0.1 f 5 2 Penalty terms in the error function TH(7)> Constraint Operands OSLO Lagrange active 7-7

8 7.3 variable independent variable dependent variable OSLO solves pickups variable set of variable OSLO programming Reference Manual Boundary Conditions variable spreadsheet min 2 (penalty)= opbw min 7.12 max (penalty)= opbw max (7.12) 7.13 opbw 7-8

9 Edge-thickness OSLO 0.5 TH(3)>0.5 0 TH(3) cross-reference component TH(3)>0.5 (one-sided) (one-sided) OSLO one-sided tolerance for one-sided constraint operands opct DLS optimization ASA global optimization ASA Derivative increments 7-9

10 f i x j v x f i ( x x, L, x + fx, L, x ) f ( x, x,, x, x ) 1, 2 j j n i 1 2 L δx j j n 7.14 opdi step size x j x j x j solution vector OSLO aspheric surface sag n power n th 1 phase function variable damping 7.11 A T A OSLO piece-wise linear (approximate solution point) OSLO A T A (column vector) 7.4 Operands 7-10

11 OSLO operands OSLO f [ 1] c1 c2 = c 1 c * ** < c 1 c 2 f=c 1 -c 2 f=0 > c 1 c 2 f=c 1 -c 2 f= Component Classes system GRIN surface sag Aberration and Paraxial data 7 axial secondary ray 7-11

12 field points set ray set Spot diagram MTF RMS wavefront error field points field point set aperture divisions grid size CCL SCP CCL SCP external dynamic-link library c CCL SCP cross-reference statistical constant Operand Component Syntax argument comma OSLO configuration 7-12

13 OSLO Argument default none none none none conventions 0 cv(-2) 2 7 cv(-2) 5 Y(3,1) 3 1 y 0 TH(5,2) TH(5,2) 2 5 System Operand Components ETH PWR LN <component>(surface#,configuration#) 7-13

14 ETH PWR LN <component>(first surface#, last surface#, configuration#) TH(5,1) OSLO TH(5) 1 TH(5) TH(5,2) OSLO LN(1,0,1) LN(1,0) LN(1) OSLO LN(1) LN(1,0,2) Aberration and paraxial data components primary axial color, PAC primary lateral color, PLC secondary axial color, SAC secondary lateral color, SLC PAC PLC SAC SLC 1 1 SA3(0,1,1) 1 1 OSLO SA3 2 2 PLC(0,2) ray operand components OSLO 7-14

15 10 FBY FBX FBZ x,y,z fractional value YRF XRF FY1 FY2 FX1 FX2 (Sagittal vignetting) (vignetting) FY FX [ FY= ( ) ][ ] + FY1 FY + 1 FY 2 FY1 2 [ FX= ( ) ][ ] + FX1 FX + 1 FX 2 FX (7.17 FY1 FY2 FX1 FX2-1.0,1.0, FY FX FY FX (vignetting) FY 1+ FY 2 (vignetted) YRF= 2 (vignetted) FX 1+ FX 2 XRF= 2 WGT 1 2 fractional aperture coordinates 3 OSLO ordinary reference FY FX 7-15

16 X Y Z RVK RVL RVM NVK NVL NVM XA YA PL OPL ( update > operation conditions > lens drawing Optimize>> Error Function Tables >> Ray Sets ; OSLO LT 5.4 ) FBY=0.7 FBX=0.0 5 FY=-0.5 FX= (FBY = 0.7 FBX =0.0) 2(FY = FX = 0.5) OPL 2,2,3,5,1 OSLO 1 OPL 2,2,3,5 OPL 7-16

17 Optimize > Operand PL 0.5 PL 1,1,1,4,1 >0.5 OSLO PL 1,1,1,4 > OPL(1,1,1,1) (mode min) 7.7 Spot diagram operand components OSLO SIX (user-defined tolerancing error function) MTF differential optimizations MTF 1 (Sagittal) MTF (MTX) 2 MTF tangential MTF, MTY 3 WVF MTX MTY WVF : WVF spot diagram set spreadsheet sde OSLO sds 7-17

18 spot diagram set 5 FPT APDIV FIRST WVL NBR WVLS (FIRST WVL, FIRST WVL+1,,FIRST WVL+NBR WVLS-1) MTF MTX MTY MTF MTF 10 cycles/mm FBY=1.0 FBX=0.0 MTF MTF polychromatic tangential MTF 1 polychromatic MTY(1,10) CCL SCP CCL SCP CCL SCP spreadsheet Buffer OSLO CCL SCP : OCM< > 7-18

19 < > array element# CCL/SCP OCM CCL/SCP 1 CCL SCP procedure OCM CCL/SCP 7.9 CCL Callbaks.ccl ( op_callbaks.ccl ) (RMS spot size) OCM 0 OCM 1 OCM CCL 2 1 CCL/SCP CCL SCP SCP * 7.9 oprds-spot-size 3 CCL SCP OCM OCM< > 7.9 ocm1,ocm2 ocm3 CCL SCP CCL SCP cfg OCM CCL/SCP OCM CCL 1 2 OCM(0) OCM(1) CCL

20 7.10 CCL CCL/SCP output_text OSLO OFF ON 7.9 set-preference output_text, off set_preference output_text_on CCL/SCP External components CCL/SCP CCL/SCP C dynamic-link library, DLL; UNIX DLL shared libraries OSLO DLL OSLO cross-reference components O SA OSLO SA3+SA5+SA7 weight SA o statistical operands AVE RMS AVE 7-20

21 RMS RMS AVE RMS AVE a RMS r r>a+1 AVE a f a = r 1 w c i 1i i= a+ 1 r 1 w i i= a RMS r f r 1 r 1 2 r = wi fi wi i= a+ 1 i= a r 1 r 1 2 f r = wi ( c1 i f a ) wi 7.20 i= a+ 1 i= a+ 1 FBY=1.0 FBX=0.0 y AVE RMS

22 2 3 4 demotrip.len demotrip.len ope AVE AVE RMS RMS AVE RMS 7.14 operands all RMS RMS 7-22

23 constant components OSLO SA3> SA3< <SA3 DY(1.1)* *DY(1.1) CMA3**-0.5 CMA**O1 O Error function construction 1 2 (image quality) RMS Spot size X ( h,λ, v ρ ) ( Y h,λ, v ρ) x y 7-23

24 fractional object coordinate h v h v h x,h y ρ x,ρ y s 2 i i= 1 j= 1 k = 1 w ijk v v 2 v v {[ X ( h,, ) ( )] [ (,, ) ( )] 2 i λ j ρ k X hi + Y hi λ j ρ k Y hi } 7.21 w ijk X Y v ( ) i i hi wijk X ( hi, λ j, ρ k ) v j= 1 k = 1 v j= 1 k = 1 ( ) i i hi wijky ( hi, λ j, ρ k ) j= 1 k = 1 v j= 1 k = 1 w ijk w ijk OSLO Automatic error function generation RMS Optical Path Difference, OPD (point image) ( d h,λ, v ρ ) h v fractional D 2 i i= 1 j= 1 k = 1 w ijk v [ d( h,, ρ ) d ( h )] i v 2 λ 7.24 j k i w ijk 7-24

25 v v d w 7.25 ( ) i i hi wijk d( hi, λ j, ρ k ) j= 1 k = 1 j= 1 k = 1 ijk MTF MTF MTF MTF MTF MTF MTF differential optimization MTF 7.6 Automatic error function generation h, h (fractional coordinates) x y,,,h x=x(,,,h) y=y(,,,h) y φ = 2π [ ( ρ θ, h) y( h) ] 2 y, ρ dρ dθ 7.26 φ Forbes 7-25

26 Gaussian qnadrature methods exact solution n 1 ( a ρ + a ρ + + a ρ ) dρ 1 3 L 7.27 n on-axis 9 2n-1<9 n 4 4 2n θ k =(k-½) N θ N θ 0 φ N ρ j= 1 Nθ [ y( ρ j, θ k ) j k = 1 2 w y] N ρ N θ ring spoke skew rays y N ρ, N θ j,k ρ j w j θ k N ρ N θ

27 7.16 ring ( ) spoke 7.17 Cartesian, square grid 7.17 reference marginal ray Radu Lobatto Radu Lobatto OSLO (automatic error function generation) 1 Lobatto

28 ( 7.16) rms spot size RMS wavefront error y Conrady D-d distortion (vignetted) (vignetting) field point set (vignetted) OSLO OSLO 7.7 multiconfiguration optimization ensemble OSLO

29 configuration data spreadsheet solves Global optimization merit function grid search 5 1µs N 5 N µs N=10 10 N=15 8 N= µs 7-29

30 stimulated annealing controlled random search Monte Carlo T 7.18 φ φ x x+ s?

31 T T T T random walk T T T step T 0 1 T T step invariance OSLO adaptive simulated Annealing, ASA (merit function) adaptive ASA T step T T 7-31

32 T OSLO n ASA central limit theorem variance ASA m T distance 2 ASA 2 ASA T T ASA ASA iterate full iterate standard 7-32

33 ASA Jones Forbes Journal of global optimization 6, 1-37 (1995) 7.9 A design Project ASA efl 35mm f/ (vignetting) ASA ± Lobatto 9 50 D - d 7.20 lens data Paraxial setup of lens wavelength variables operands ASA 20 sparc2 ASA glass variable ASA

34 front surface PU constraint mode operand a (improved glasses) b (Vignetting) David Shafer a b 7-34

35 oblique ASA ASA ASA 7-35

36 [1] OSLO Optical Reference version 5 (Sinclair Optics, 1996); chapter 7. [2] OSLO Optical Reference version 6.1 (Lambda Research Corp, 2001). [3] OSLO Program Reference version 5 (Sinclair Optics, 1996) 7-36

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