6.1 Image Evaluation 6. 6.3 6.3.1 6.3. Spot size Analysis 6.3.3 wavefront Analysis 6.4 Point spread functions 6.4.1 6.4. 6.4.3 6.5 fiber coupling 6.6 Energy Distribution 6.7 Transfer functions 6.7.1 6.7. 6.1 object 6. Geometrical V.S. diffraction evaluation 6-1
Schrodinger Helmholtz ( x, y, z) + k u( x, y, z) = v u 6.1 u(x,y,z) k= / ray Fermat stationary value geometric wavefront 6.3 Spot diagrams and wavefront 6.3.1 Spot diagrams Spot diagram 6.1 6-
6.1 Petzval lens OSLO pupil ray intersection density OSLO entrance pupil spot diagram operating conditions calculate menu setup spd\wavefront spreadsheet aperture division APDIV 6. APDIV=16. 6. APDIV=16. OSLO direction cosine 6-3
exit pupil Abbe Abbe exit pupil entrance pupil pupil aberrations OSLO Six image space spot diagrams Zernike 6. OSLO x y reference sphere through-focus MTF(modulation transfer function) OSLO Gaussian apodization ration x y 6-4
1 1/e entrance beam radius x y 6.3. Spot size Analysis Gaussian apodization function moment average values x p(x) centroid ( x) x = xp dx 6. x ( x) = x p dx 6.3 σ = = x ( x x ) p( x) x dx 6.4 standard deviation x n w i DX i,dy i x y 1 = w 1 = w w = n i= 1 n i= 1 w DX i w DY i n w i i= 1 i i 6.5 6.6 6-5
1 σ x = w 1 σ y = w n i= 1 n i= 1 w w i i ( DX x ) ( DY y ) i i 6.7 σ x σ y root-mean square, RMS radial RMS spot size σ = σ + σ 6.8 r x y OSLO Calculate>>spot size Analysis r OSLO RMS 6.1 6.1 x, y r GEO RMS X GEO RMS Y GEO RMS R centroid x y CENTX CENTY y 6.3 CENTX= DIFFR LIMIT Airy Airy Disk 6.3 CENTX CENTY 6-6
6.3.3 wavefront Analysis optical path difference; OPD wavefront aberration OPD OSLO real exit pupil General Operating Condition OSLO RMS 6.4 6.4 diffraction focus OSLO fidelity distortion Calculate>>Wavefront Analysis 6. 6-7
6. RSY, RSX, RSZ y RSX= nominal location RSZ= interferometric testing Zernike 1 order Zernike Zernike Zernike Zernike n i Zernike Z i (, ) ( θ ) = ( n w ρ, c i z i ρ, θ ) 6.9 i= c i OSLO Six Zernike Zernike Zernike 6.4 Point spread functions point spread function impulse response 6.4.1 ( ) exit pupil A(x,y) w y pupil function (x,y) 6-8
( x, y) = A( x, y) exp[ ikw( x, y) ] ρ 6.1 k=π/λ (x,y)= Kirchhoff u(x',y') u i λ ( x', y' ) = ρ( x, y) A exp R' ( ikk' ) da 6.11 A R' (x,y) (x',y') 6.5 6.5 6.11 u ( x', y' ) { ik[ R+ ( x', y' )]} i exp π = ρ( x, y) exp i ( xx' + yy' ) dxdy λm R λr R 6.1 R ( x', y' ) MR z 6.1 Hopkins Yzuel Optical Acta 17, 157-18 (197) A (x,y)= 6.1 x' y' ν x =, ν y λr λr u(x',y') squared modulus ( x', y' ) u( x', y' ) PSF = 6.13 6-9
transfer function 6.1 FFT FFT point-by-point Fourier kernel integral N NxN Strehl OSLO 6.1 ν y y = ' N N λ R 1 ν y = 6.14 N y y λr y' = 6.15 N y D M y=d/m λ λrm M y' = = 6.16 ND NA N ( ) D λ NA NA = n M/N k fill factor 6.16 M N 6-1
= N y' = M 6.17 NA 6.17 M λ M N aliasing Whittaker-Shannon ν max 1 y ' 6.18 ν max transfer function OTF NA ν = OTF PSF y' λ λ 6.18 y' M 4ΝΑ ( λm ) λ ( ) = ΝΑ Ν 4 ΝΑ ( ) M =N/ M M N M sampling intervals squared modules a r'=(x' +y' ) / ( b) J 1 Bessel J1 PSF ( r') = 6.19 b sπar' π b = = NAr' 6. λr λ NA a NA = nsinu = N 6.19 Airy J 1 (πx) x=1. k Airy.61λ r' Airy = 6.1 NA b=1. r' Airy DIFFR LIMIT 6-11
6.4. OSLO 6.19 infinite object conjugate File>>New 1mm image numerical aperture. 3 6.3 6.3 OSLO 6.19 Calculate>>Spot Size Analysis Calculate>>Spread Function 6.4 DIFFR LIMZT=.179 6.1 r' Airy =.61*.58756mm/.=.179mm 6.6 6-1
6.4 6.6 figure of merit Strehl Strehl Strehl variance of the wavefront OSLO PSF PSF 1 6.4.3 line spread functions and knife edge distributions line spread function, LSF x y knife edge - x' knife edge distribution, KED KED x' ( x' ) = LSF( x ) ~ dx ~ 6.3 irradiance point spread function 6-13
6.5 fiber coupling fiber coupling efficiency u(x',y') (x',y') η normalized overlap integral ( x', y' ) Ψ *( x', y' ) u dx' dy' η = 6.4 u dx' dy' ( x', y' ) u * ( x', y' ) dx' dy' Ψ( x', y' ) Ψ * ( x', y' ) T T = ηη * = η 6.5 T power 6.4 x',y' OSLO u(x',y') Schwarz T 1 T=1 (x',y')=ku(x',y') k (x',y') gradient index step Ψ ( x', y' ) x' + y' = exp r r' = exp r Gaussian 6.6 r r 1/e cladding 1 n core a 3 cladding n cladding n core n cladding n cladding <<1 6-14
Ψ step index ( r' ) ur' J a, J ( u) = wr' K a, K ( w) r' a r' > a 6.7 r'=(x' +y' ) J Bessel K Hankel Modified Hankel function u w π ν = a 6.8 λ n core n cladding u w ( 1 ) v u = 6.9 1+ 1 4 ( 4 +ν ) 4 w u = ν 6.3 Gloge Appl. Opt. 1, 5-59 (1971) OSLO Six CCL ball lenses y 3 x 1 w 1 λ θ = tan 6.31 πw.83 m y w y =.458µm x w x =1.498µm w =(w x w y ) 1/ =.88µm ( ) 6 nominal paraxial magnification m = 5.88 Melles-Groit 1mm 6.5 6.7 6-15
6.5 Melles-Groit 6.7 Melles-Groit aperture stop Gaussian apodization 3 [ ] ( ) mm [] tan( 1 ) = 1.85.176 =. mm 1 th tan 3 = 1.85.577 =.74 th 7 6.6 6-16
6.6-6 (.75*8.99) 1/ =5µm 6.7 6.7 diffraction pattern Gaussian shape 6.8 6.8 5µm 18 6.8 6-17
6.8 6µm 4 6.9 6.9 6.6 Energy Distribution spread function 8 spot size square pixels energy distribution x',y' r' ' π ( x', y' ) dx' dy' = PSF( r', θ ') PSF r' dr' dθ ' = 1 6.3 6-18
6.1 6.19 r' r' r' OSLO 6.3 discrete Fourier transform Parseval 6.3 image patch size OSLO radial energy distribution, RED a encircled energy RED a a π ( ) = PSF( r', ') θ r' dr' dθ ' 6.33 s ensquared energy, SQE s / s / () s = PSF( x', y' ) SQE dx' dy' 6.34 s / s / OSLO 6.19 Rayleigh RED a π π ( ) = 1 J NAa J NAa 1 λ λ 6.35 84 Airy 91 Airy 6.35 J 1 (πx) x=3.38 r'=1.619λ NA=1.619*.58756mm.48mm 1µm.1 calculate>>energy Distribution>>Diffraction Spreadsheet 6-19
6.9.18mm.33mm.48mm 1 e. x y 6.1 6.1 1 EBR e 4mm 6.1 6-
6.1 6.9 6.11 6.11 Airy lobes ( ) ( ) ( ) PSF r' = exp r' π w w w RED a = 1 exp a w ( ) ( ) 6.7 Transfer functions 6-1
6.7.1 optical transfer function, OTF ν x ν y OTF (, ν ) PSF ( x', y' ) exp[ iπ ( ν x' + ν y' )] x y ν x y = 6.36 PSF ( x', y' ) dx' dy' dx' dy' OTF(,)=1 6.1 autocorrelation OSLO 6.19 cutoff frequency ν NA ν = 6.37 λ modulus modulation transfer function, MTF phase transfer function, PTF MTF modulation PTF 6.1 modulation M M E max min = 6.38 E max E + E MTF PTF object min M image MTF = 6.39 M 6-
g PTF = π = πgν 6.4 p 6.1 OTF o(x,y) O(ν x,ν y ) I(ν x,ν y ) ( ν, ν ) = OTF( ν, ν ) O( ν, ν ) Ι 6.41 x y x y x y OTF 6.36 6.7. singlet File>>New 1mm f 1 1mm (*1)=5mm paraxial axial ray angle solve -1 (*FNB)=-.5 axial ray height solve 6-3
.5mm Schott F 3 1 85.89181 1.58756µm.5 6.11 6.11 6.13 3 6-4
MTF 5cycle mm 5cycles mm peak-to-valley OPD default switchover 3λ 6.1 6.1.1 6.13 6.13 ( ) 1-5.36147 6 6.14 6-5
6.14 MTF 5 MTF 3 MTF 6.14 6.14 MTF 6-6
6-7
[1] OSLO Optical Reference version 5 (Sinclair Optics, 1996) Chapter 6. [] OSLO Optical Reference version 6.1 (Lambda Research Corp., 1) [3] Wagner Tomlinson Appl. Opt. 1, 671-688 (198) 6-8
1. catalog lens triplet lens spot diagram point spread function. ball lens 15 µm( ) 6 µm( ) 3. MatLab MTF 6-9