Sta$s$cal Op$cs Speckle phenomena in Op$cs. Groupe d op$que atomique Bureau R2.21

Transcription

1 Sta$s$cal Op$cs Speckle phenomena in Op$cs Groupe d op$que atomique Bureau R2.21

2 Diffrac$on from a rough surface (Fourier speckle) Eample Star imaging through turbulence When does laser speckle occurs? Coherent imaging of a rough object (Subjec$ve speckle) Eample Earth observa$on by radar Imaging

3 Diffrac$on from a rough surface (Fourier speckle) Two imaging configura$ons Speckle destroys the image : informa$on is lost (at least it seems so) Coherent imaging of a rough object (Subjec$ve speckle) Speckle is superimposed on the image : informa$on is s$ll here (incoherent imaging)

4 Recall usual epression on the Point Spread func$on (PSF) Fraunhofer s diffrac$on (at infinity ) Pupille Plan d observation 1D configura$on : rectangular aperture p( ) = rect D ) E() / p (u ) / sinc( u D) avec u = f p (u )

5 Recall usual epression on the Point Spread func$on (PSF) Fraunhofer s diffrac$on (at infinity ) Plan d observation Pupille 2D configura$on :circular aperture p(, ) = disc D J1 ( u D) ) E(, y) / p (u ) / u D avec u = p 2 + y 2 f p (u )

6 Onde plane incidente Diffrac$on by a slot : From Fresnel to Fraunhofer Régime de diffraction de Fresnel Régime de diffraction de Fraunhofer Projection au foyer d une lentille a Lentille Ecran ~ Diffraction à l infini : Diffraction «vraiment» à l infini :

7 Coherent versus incoherent imaging (short overview) Coherent Transfer Func$on (CTF) Modula$on Transfer Func$on (FTM in french) 1 CTF = H vs 1 FTM = FTO(ν) 0 ν 0 ν ν ma = λ ν ma = 2 λ Coherent imaging : Func6on transfer in amplitude E im (ν) t obj (ν) CTF(ν) Incoherent imaging : Transfer func6on in intensity I im (ν) t obj (ν) 2 FTO(ν)

8 Op$cal imaging system p() Converging illumina6on in the plane of the lens E 0 () F Object characterized by the transmission t obj () Lens f How to get the epression of the image? Image plane E im () E im (), I im ()??

9 Coherent imaging : Fresnel s propaga$on p() Converging illumina6on in the plane of the lens E 0 () F Object characterized by the transmission t obj () Method 1 : Fresnel s propaga$on Lens f Image plane E im () Coherent propaga$on of the transverse field over the en$re op$cal system (see lecture 1) E (+) () =t obj () E 0 ()

10 Coherent imaging : Fresnel s propaga$on FT p() Converging illumina6on in the plane of the lens E 0 () F Object characterized by the transmission t obj () Method 1 : Fresnel s propaga$on Lens f Image plane E im () Coherent propaga$on of the transverse field over the en$re op$cal system [ E() FT t obj () ] λ p()

11 Coherent imaging : Fresnel s propaga$on Converging illumina6on in the plane of the lens E 0 () FT p() F FT Object characterized by the transmission t obj () Method 1 : Fresnel s propaga$on Lens f Image plane Coherent propaga$on of the transverse field over the en$re op$cal system E im () t obj ( ) FT E im () [ p()) ] λ

12 Coherent imaging : linear response theory p()?? Point source t (0) obj = δ() Method 2 : linear response theory 1. Look at the response (image) for an point source object (pinhole) PSF c () =?? The coherent Point Spread Func$on also called Réponse Percusionnelle Cohérente (RPC)

13 Coherent imaging : linear response theory p() FT [ p() ] λ Point source t (0) obj = δ() Size λ 2 1. Look at the response (image) for an point source object (pinhole) Method 2 : linear response theory PSF c () =FT [ p() ] λ The coherent Point Spread Func$on also called Réponse Percusionnelle Cohérente (RPC)

14 Coherent imaging : linear response theory t (1) obj = δ( 1) p() 1 FT [ p() ] λ Point source t (0) obj = δ() 1 = 1 PSF c ( 1 ) 1. Look at the response (image) for an point source object (pinhole) Method 2 : linear response theory 2. Response (image) of a translated point source give the same PSFc, but translated with the magnifica$on factor γ = /

15 Coherent imaging : linear response theory p() FT [ p() ] λ 1. Look at the response (image) for an point source object (pinhole) Method 2 : linear response theory 2. Response (image) of a translated point source give the same PSF c, but translated 3. Sum the responses in amplitude or in intensity for all source points

16 Coherent imaging : summary p() FT [ p() ] λ Coherent imaging : convolu$on in amplitude E im () t obj ( ) PSF c () Image = object convoluted by the resolu$on Geometrical op$cs with γ = / Op$cal resolu$on ( RPC () )

17 Coherent imaging : frequency transfer p() FT [ p() ] λ In frequency space : a Coherent Transfer Func$on E im (ν) t obj ( γν ) CTF(ν) Image = object convoluted by the resolu$on [ ] Coherent transfer func$on or H(ν) =FT PSF c () ν

18 Coherent imaging : frequency transfer p() FT [ p() ] λ Coherent Transfer Func$on (CTF or H ) [ ] CTF(ν) =FT PSF c () [ ) FT FT [ p() ] ν λ ]ν p ( λ ν ) : rescaled pupil ν (c) ma = λ 1 0 CTF = H ( ν λ ν (c) ma = D/2 )

19 Coherent imaging : frequency transfer p() FT [ p() ] λ For instance : the radial gauge 1 CTF = H ν (c) ma = λ 0 ( ν λ ν (c) ma = D/2 )

20 Coherent imaging : frequency transfer p() FT [ p() ] λ For instance : the radial gauge Coherent imaging = -OFF response in frequency ν (c) ma = λ What about Incoherent imaging?

21 Incoherent imaging system p() FT [ p() ] λ Incoherent imaging : convolu$on in intensity I im () t obj ( ) 2 PSF c () 2 Image = object convoluted by the resolu$on Geometrical op$cs with γ = / Op$cal resolu$on ( RPI () )

22 Incoherent imaging system p() FT [ p() ] λ Incoherent imaging : convolu$on in intensity I im () T obj ( ) PSF I () PSF I = T obj = 2 PSF c 2 t obj Geometrical op$cs with γ = / Op$cal resolu$on ( RPI () )

23 Incoherent imaging system p() FT [ p() ] λ In frequency space : an Incoherent Transfer Func$on 2 PSF 2 I im (ν) T ( ) I = PSF c obj γν FTO(ν) T obj = t obj [ ] Op$cal Transfer Func$on (Fonc$on de transfert op$que) =FT PSF I () ν

24 Incoherent imaging system p() FT [ p() ] λ Incoherent Op$cal Transfer func$on [ ] FTO(ν) =FT PSF I () FT[ FT [ p() ] 2 λ ]ν ( ) p p ( λz2 ν ) ν PSF I = : rescaled autocorrela$on of the pupil T obj = 2 PSF c 2 t obj

25 Incoherent imaging system p() FT [ p() ] λ Incoherent Op$cal Transfer func$on [ ] FTO(ν) =FT PSF I () FT[ FT [ p() ] 2 λ ]ν ( ) p p ( λz2 ν ) ν ν (Inc) ma 1 = 2 λ FTM = FTO(ν) 0 ( λ ν (Inc) ma ν = D )

26 Incoherent imaging system p() FT [ p() ] λ For instance : the radial gauge 1 FTM = FTO(ν) ν (Inc) ma = 2 λ 0 ( λ ν (Inc) ma ν = D )