Today

• Spatially coherent and incoherent imaging with a single lens – re-derivation of the single-lens imaging condition – ATF/OTF/PSF and the Numerical Aperture – resolution in optical systems – pupil engineering revisited

next week • Two more applications of the MTF

– defocus – diffractive optics and holography

• Multi-pass interferometers: Fabry-Perot – optical resonators and Lasers

• Beyond scalar optics: polarization and resolution

MIT 2.71/2.710 05/04/09 wk13-a- 1

Imaging with a single lens: imaging conditionpupil mask

diffracted field after lens + pupil mask

output field

arbitrary complex input transparency

gt(x)

diffraction from the input field

wave converging to form the image

input field

spatially coherent illumination

gillum(x)

gPM(x”)

z1 z2

at the output plane The field at the output plane of this imaging system is (derivation in the supplement to this lecture)

We also need to eliminate this quadratic term

To eliminate the quadratic term we satisfy the Lens Law of geometrical optics

After eliminating the quadratic, the remaining integral is the Fourier transform of the pupil mask gPM(x”, y”).

MIT 2.71/2.710 05/04/09 wk13-a- 2

MIT 2.71/2.710 05/04/09 wk13-a-

Eliminating the quadratic term

3

pupil mask

colle

ctor

output field

arbitrary complex input transparency

gt(x) gPM(x”)

z1 z2

gillum(x)

Solution 2: attach a lens of focal length z1 to gt(x) Solution 3: limit gt(x) to ¼ the lateral size of gPM(x”) [see Goodman 5.3.2 and Ref. 303]

pupil mask

output field

gillum(x)

gPM(x”)

z1 z2

Solution 1: shape gt(x) as a sphere of radius z1

gt(x)

Imaging with a single lens: PSF and ATFpupil mask

diffracted field after lens + pupil mask

output field

arbitrary complex input transparency

gt(x)

diffraction from the input field

wave converging to form the image

input field

spatially coherent illumination

gillum(x)

gPM(x”)

z1 z2

at the output plane Assuming one of the three conditions is satisfied and the last remaining quadratic term can be eliminated, the output field is

MIT 2.71/2.710 05/04/09 wk13-a- 4

where is the PSF,

i.e. the Fourier transform of the pupil mask scaled so that (x”,y”)=(λz1u,λz1v). As in the 4F system,

the scaled complex transmissivity of the pupil mask is the ATF

Imaging with a single lens: lateral magnificationpupil mask

diffracted field after lens + pupil mask

output field

arbitrary complex input transparency

gt(x)

diffraction from the input field

wave converging to form the image

input field

spatially coherent illumination

gillum(x)

gPM(x”)

z1 z2

at the output plane If the pupil mask gPM(x”, y”) is infinitely large and clear, its Fourier transform is approximated as a δ-function. Therefore, the optical field at the output plane is

replica of the input fieldwithin the factor of

lateral magnification

So by ignoring diffraction due to the finitelateral size and, possibly, phase-delay

elements inside the clear aperture of the pupil mask gPM(x”, y”), we have

essentially found that the imaging condition and lateral magnification

relationships from geometrical optics remain valid in wave optics as well for the intensity of the optical field.

phase factor, does not affect the

image intensity

MIT 2.71/2.710 05/04/09 wk13-a- 5

Block diagrams for coherent and incoherentlinear shift invariant imaging systems

thin transparency

cPSF

ATF

Fourier transform

output field

convolution

multiplication

Fourier transform

spatially coherent

illumination

input field

✳

4F imaging system:

Single lens imaging system:

spatially incoherent

thin input intensityoutput

transparencyillumination

iPSF intensity

✳ convolution

Fourier Fourier transform transform

OTF

multiplication

MIT 2.71/2.710 05/04/09 wk13-a- 6

4F imaging system:

Single lens imaging system:

Example: Zernike phase mask4F

pupil mask gPM(x”,y”) phase contrastphase object co

llecto

r

objec

tive

π/2 phase shift λ=1µm

f1=10cm, f2=1cm z1=11cm, z2=1.1cm

f1 f2

quasi-monochromatic illumination

(sp. coherent or incoherent)

gt(x,y)=exp{iφ(x,y)} image Iout(x’,y’)

MIT 2.71/2.710 05/04/09 wk13-a- 7

objec

tive

π/2 phase shift

phase object gt(x,y)=exp{iφ(x,y)}

phase contrast image Iout(x’,y’)

pupil mask gPM(x”,y”)

f1

quasi-monochromatic illumination

(sp. coherent or incoherent)

SL

MIT 2.71/2.71005/04/09 wk13-a- 8

ATF and OTF of the Zernike phase mask

a=2cm

b=4cm

0.5µm

glass,

n =1.5

x”

air

z

T2

opaque opaque

0.2cm

1cm

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.25

0.5

0.75

1

x’’ [cm]

Re(

g PM) [

a.u.

]

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.25

0.5

0.75

1

x’’ [cm]

Im(g

PM) [

a.u.

]

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.25

0.5

0.75

1

x’’ [cm]

|gPM

| [a.

u.]

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −pi

−pi/2

0

pi/2

pi

x’’ [cm]

phas

e(g PM

) [ra

d]

−200 −150 −100 −50 0 50 100 150 2000

0.25

0.5

0.75

1

u [mm−1]R

e(AT

F) [a

.u.]

−200 −150 −100 −50 0 50 100 150 2000

0.25

0.5

0.75

1

u [mm−1

Im(A

TF) [

a.u.

]−200 −150 −100 −50 0 50 100 150 200

0

0.25

0.5

0.75

1

u [mm−1]

Re(

ATF)

[a.u

.]

−200 −150 −100 −50 0 50 100 150 2000

0.25

0.5

0.75

1

u [mm−1

Im(A

TF) [

a.u.

]

−200 −150 −100 −50 0 50 100 150 2000

0.25

0.5

0.75

1

u [mm−1]

OTF

[a.u

.]

−200 −150 −100 −50 0 50 100 150 2000

0.25

0.5

0.75

1

u [mm−1]

OTF

[a.u

.]

ATF

OTF

4F SL

4F SL

See also PP03, pp. 16-17

Clear circular aperture

pupil mask gPM(x”,y”)= circ(r”/R)

colle

ctor

objec

tive

radius R (NA)in=R/f1

object gt(x,y)

phase contrast image Iout(x’,y’)

λ=1µm f1=10cm, f2=1cm

z1=cm, z2=cm

f1 f2

4F

quasi-monochromatic illumination

(sp. coherent or incoherent)

(NA)out=R/f2

objec

tive

object gt(x,y)

phase contrast image Iout(x’,y’)

f1 SL

quasi-monochromatic illumination

(sp. coherent or incoherent)

radius R (NA)in=R/z1 (NA)out=R/z2

pupil mask gPM(x”,y”)= circ(r”/R)

MIT 2.71/2.710 05/04/09 wk13-a- 9

ATF, cPSF of clear circular aperture

-a-10MIT 2.71/2.710 05/04/09 wk13

Common expression for the cPSF

4F

SL

Resolution

[from the New Merriam-Webster Dictionary, 1989 ed.]:

resolve v : 1 to break up into constituent parts: ANALYZE; 2 to find an answer to : SOLVE; 3 DETERMINE, DECIDE; 4 to make or pass a formal resolution

resolution n : 1 the act or process of resolving 2 the action of solving, also : SOLUTION; 3 the quality of being resolute: FIRMNESS, DETERMINATION; 4 a formal statement expressing the opinion, will or, intent of a body of persons

MIT 2.71/2.710 05/04/09 wk13-a- 11

Rayleigh resolution limit

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.25

0.5

0.75

1

x [µm]

I [a.

u.]

Total intensitySource 1Source 2

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.25

0.5

0.75

1

x [µm]

I [a.

u.]

Total intensitySource 1Source 2

Two point sources are well resolved if they are spaced such that: (i) the PSF diameter (i) the PSF radius equals the point source spacing equals the point source spacing

MIT 2.71/2.710 05/04/09 wk13-a-12

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