pupil engineering

The Point-Spread Function (PSF) of a low-pass filter

plane wave

illumination

pupil mask

ideal truncated

colle

ctor

objec

tive

PSFinput transparency ideal point source

ideal wave converging (infinitely wide) (infinitely wide) plane wave to form the PSF spherical wave plane wave at the output plane

Now consider the same 4F system but replace the input transparency with an ideal point source, implemented as an opaque sheet with an infinitesimally small transparent hole and illuminated with a plane wave on axis (actually, any illumination will result in a point source in this case, according to Huygens.)In Systems terminology, we are exciting this linear system with an impulse (delta-function); therefore, the response is known as Impulse Response. In Optics terminology, we use instead the term Point-Spread Function (PSF) and we denote it as h(x’,y’).The sequence to compute the PSF of a 4F system is:➡ observe that the Fourier transform of the input transparency δ(x) is simply 1 everywhere at the pupil plane➡ multiply 1 by the complex amplitude transmittance of the pupil mask

MIT 2.71/2.710 04/15/09 wk10-b-22

➡ Fourier transform the product and scale to the output plane coordinates x’=uλf2. ➡Therefore, the PSF is simply the Fourier transform of the pupil mask, scaled to the output coordinates x’=uλf2

Example: PSF of a low-pass filterPSF

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 30

0.25

0.5

0.75

1

x’’ [cm]

|gP(x

’’)|2 [a

.u.]

pupil mask

The output field, i.e. the PSF is

The pupil mask is . If the input transparency is δ(x), the field at the pupil plane to the

right of the pupil mask is

−20 −15 −10 −5 0 5 10 15 20−1

0

1

2

3

x’ [µm]

h(x’

) [a.

u.]

The scaling factor (3×) in the PSF ensures that the integral ∫|h(x’)|2dx equals the portion of the input energy MIT 2.71/2.710 transmitted through the system 04/15/09 wk10-b-23

Example: PSF of a phase pupil filter

The pupil mask is

The PSF is

MIT 2.71/2.710 04/15/09 wk10-b-24

−4 −3 −2 −1 0 1 2 3 40

0.25

0.5

0.75

1

x’’ [cm]

|gPM

| [a.

u.]

−4 −3 −2 −1 0 1 2 3 4 −pi

−pi/2

0

pi/2

pi

x’’ [cm]

phas

e(g PM

) [ra

d]

pupil mask

−20 −15 −10 −5 0 5 10 15 200

1

2

3

4

5

x’ [µm]

|h(x

’)|2 [a

.u.]

−20 −15 −10 −5 0 5 10 15 20 −pi

−pi/2

0

pi/2

pi

x’ [µm]

phas

e[h(

x’)]

[rad]

PSF

Comparison: low-pass filter vs phase pupil mask filter

−20 −15 −10 −5 0 5 10 15 200

1

2

3

4

5

x’ [µm]

|h(x

’)|2 [a

.u.]

−20 −15 −10 −5 0 5 10 15 20 −pi

−pi/2

0

pi/2

pi

x’ [µm]

phas

e[h(

x’)]

[rad]

PSF: phase pupil mask filter

−20 −15 −10 −5 0 5 10 15 200

2

4

6

8

10

x’ [µm]

|h(x

’)|2 [a

.u.]

−20 −15 −10 −5 0 5 10 15 20 −pi

−pi/2

0

pi/2

pi

x’ [µm]

phas

e[h(

x’)]

[rad]

PSF: low-pass filter

MIT 2.71/2.710 04/15/09 wk10-b-25

Shift invariance of the 4F system

plane wave

illumination

pupil mask

colle

ctor

objec

tive

PSFinput transparency ideal point source

plane wave

illumination

pupil mask

colle

ctor

objec

tive

PSF, shifted input transparency ideal point source

shifted

lateral magnification

MIT 2.71/2.710 04/15/09 wk10-b-26

The significance of the PSFpupil mask

diffracted field diffracted field

colle

ctor

objec

tive

output field

arbitrary complex input transparency

gt(x)

diffraction from the wave converging

input field

spatially coherent illumination gillum(x)

gPM(x”)

input field after objective after pupil mask to form the image at the output plane

Finally, consider the same 4F system with an input transparency whose transmittance is an arbitrary complex function gt(x) and the field gin(x) immediately to the right of the input transparency.

According to the sifting theorem for δ-functions, we may express gin(x) as

Physically, this expresses a superposition of point sources weighted by the input field, i.e. Huygens’ principle.

Using the shift invariance property, we find that the field produced at the output plane by each Huygens point source is the PSF, shifted by −x1×f2 ⁄f1, (note: −f2 ⁄f1 is the lateral magnification) and weighted by gin(x1); i.e.,

the output field is almost a convolution (within a sign & a scaling factor) of the input field with the PSF:

MIT 2.71/2.710 04/15/09 wk10-b-27

The Amplitude Transfer Function (ATF)pupil mask

diffracted field after objective

diffracted field after pupil mask

colle

ctor

objec

tive

output field


gt(x)

diffraction from the input field

wave converging to form the image

input field

spatially coherent illumination gillum(x)

0

gPM(x”)

at the output plane To avoid the mathematical complications of inversion and lateral magnification at the output plane, we define the “reduced” output coordinate . Inverted and re-sampled in the reduced coordinate, the output

field is expressed simply as a convolution .

Using the convolution theorem of Fourier transforms, we can re-express this input-output relationship in the spatial frequency domain as , where H(u) is the amplitude transfer function (ATF). Inverting the Fourier transform relationship between gPM(x”) and the PSF, we obtain That is, the ATF of the 4F system is obtained directly from the pupil mask, via a

coordinate scaling transformation. MIT 2.71/2.710 04/15/09 wk10-b-28

Example: ATFs of the low-pass filter and the phase pupil mask

×103

×103

MIT 2.71/2.710 04/15/09 wk10-b-29

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 30

0.25

0.5

0.75

1

x’’ [cm]

|gP(x

’’)|2 [a

.u.]

−4 −3 −2 −1 0 1 2 3 40

0.25

0.5

0.75

1

x’’ [cm]

|gPM

| [a.

u.]

−4 −3 −2 −1 0 1 2 3 4 −pi

−pi/2

0

pi/2

pi

x’’ [cm]

phas

e(g PM

) [ra

d]

low pass filter pupil mask

phase pupil filter pupil mask

−4 −3 −2 −1 0 1 2 3 40

0.25

0.5

0.75

1

x’’ [cm]|g

PM| [

a.u.

]

−4 −3 −2 −1 0 1 2 3 4 −pi

−pi/2

0

pi/2

pi

x’’ [cm]

phas

e(g PM

) [ra

d]

u[cm-1]

u[cm-1]

u[cm-1]

ATF

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 30

0.25

0.5

0.75

1

x’’ [cm]

|gP(x

’’)|2 [a

.u.]

×103

ATF

Today

• Lateral and angular magnification • The Numerical Aperture (NA) revisited • Sampling the space and frequency domains, and

the Space-Bandwidth Product (SBP) • Pupil engineering

next two weeks • Depth of focus and depth of field • The angular spectrum • “Non-diffracting” beams • Temporal and spatial coherence • Spatially incoherent imaging

MIT 2.71/2.710 04/22/09 wk11-b- 1

4F imaging as a linear shift invariant systempupil mask

colle

ctor

objec

tive

output field


gt(x,y)

input field

spatially coherent illumination gillum(x,y)

0

gPM(x”,y”)

reduced coordinates

diffraction from the diffracted field diffracted field wave converging input field after objective after pupil mask to form the image

MIT 2.71/2.710 04/22/09 wk11-b- 2

at the output planeIllumination: Input transparency: Field to the right of the input transparency (input field):

Pupil mask: Amplitude transfer function (ATF):

Point Spread Function (PSF): in actual coordinates

in reduced coordinates

Output field: in actual coordinates

in reduced coordinates

∝

∝

∝

∝

Lateral magnificationpupil mask

colle

ctor

objec

tive

output field


gt(x,y)

Huygens wavelet from input field, departing at x1


gPM(x”,y”)

spherical plane wave clear pupil mask Huygens wavelet wave converging to form

point image at MTx1=−x1f2/f1

An ideal imaging system with a point source as input field should form a point image at the output plane. Therefore, the ideal PSF is a δ-function. This is of course the limit of Geometrical Optics and in practice unachievable because of diffraction; alternatively, it implies that the spatial bandwidth is infinite, or that the lateral extent of the pupil mask is infinite. Both conditions are non-physical; however, for the purpose of calculating the geometrical magnification, let us indeed assume that

MIT 2.71/2.710 04/22/09 wk11-b- 3

So our calculation has reproduced the Geometrical Optics result for a telescope with finite conjugates:

∝

Angular magnificationpupil mask

colle

ctor

objec

tive

output field


gt(x,y)

plane wave input field (or angular spectrum

component of input field),


gPM(x”,y”)

spherical focusing plane wave exitingdeparting at θ1 wave at x”=θ1f1 at angle MAθ1=−θ1f1/f2

spatialNow let us consider a plane wave input field frequency

Still assuming the ideal geometrical PSF, the output field is

So the angular magnification is again in agreement with Geometrical Optics.

Based on the interpretation of propagation angle as spatial frequency, the magnification results are also in

∝

agreement with the scaling (similarity) theorem of Fourier transforms:

MIT 2.71/2.710 04/22/09 wk11-b- 4

∝ ∝

11-b- 5MIT 2.71/2.710 04/22/09 wk

Example: PM, ATF and PSF for clear apertures

a

b

Rectangular

PM

a/λf1

b/λf1

ATF PSF

2R Circular

PM

2R/λf1

ATF PSF

Numerical aperture (NA) and PSF sizepupil mask

colle

ctor

objec

tive

output field


gt(x,y)

on-axis point source truncated by

pupil-plane mask


gPM(x”,y”)

(NA)in (NA)out

plane wave clear pupil mask converging spherical wave

The pupil mask is the system aperture (assuming that the lenses are sufficiently large); therefore,

MIT 2.71/2.710 04/22/09 wk11-b- 6 PSF

Let Δx’ denote the half-size of the PSF main lobe; then,

PSF

Let Δr’ denote the radius of the PSF main lobe; then,

fx,max

A note on sampling input field

y v y”

x u x”

spatial frequency domain

pupil mask

δx :pixel size

Δx :field size

Nyquist sampling

relationships

MIT 2.71/2.710 04/22/09 wk11-b- 7

δu :frequency resolution

2Δu :frequency bandwidth

δx” :PM pixel size

2Δx” :pupil mask size

Space-Bandwidth

Product (SBP)

Pupil engineeringpupil mask

diffracted field diffracted field

colle

ctor

objec

tive

output field


gt(x,y)

diffraction from the wave converging

input field


0

gPM(x”,y”)

reduced coordinates

Amplitude transfer function (ATF):

Point Spread Function (PSF): in actual coordinates

Output field: in actual coordinates

∝

∝

input field after objective after pupil mask to form the image at the output plane

Pupil engineering is the design of a pupil mask gPM(x”,y”) such that the ATF, the PSF or the output field meet requirement(s) specified by the user of the imaging system

MIT 2.71/2.710 04/22/09 wk11-b- 8

Example: spatial frequency clipping

f1=20cmλ=0.5µm

monochromatic coherent on-axis

illumination

object plane pupil mask Image plane transparency circ-aperture observed intensity

MIT 2.71/2.710 04/22/09 wk11-b- 9

Spatial filtering examples: the amplitude MIT pattern

MIT 2.71/2.710 04/22/09 wk11-b- 10

Weak low-pass filtering

f1=20cm pupil mask Intensity @ image plane λ=0.5µm

MIT 2.71/2.710 04/22/09 wk11-b- 11

Moderate low-pass filtering

(moderate blurring)


MIT 2.71/2.710 04/22/09 wk11-b- 12

Strong low-pass filtering

(strong blurring)


MIT 2.71/2.710 04/22/09 wk11-b- 13

Moderate high-pass filtering


MIT 2.71/2.710 04/22/09 wk11-b- 14

Strong high-pass filtering

(edge enhancement)


MIT 2.71/2.710 04/22/09 wk11-b- 15

One-dimensional (1D) blur: vertical


MIT 2.71/2.710 04/22/09 wk11-b- 16

One-dimensional (1D) blur: horizontal


MIT 2.71/2.710 04/22/09 wk11-b- 17

Phase objectsthickness

protrusion (transparent)

t(x)TOP CROSS VIEW SECTION

protrusion

glass plate (transparent)

This model is often useful for imaging biological objects (cells, etc.)

If we illuminate this object

uniformly (e.g., with a plane wave) the

resulting intensity|gin(x)|2 is also

uniform, i.e. the object is invisible.

MIT 2.71/2.710 04/22/09 wk11-b- 18

phase-shiftscoherent illumination

by amount φ(x) =2π(n-1)t(x)/λ

To visualize phase objects in

subsequent slides, we use the

color representation of phase as shown here. Physically, phase may be measured with interferometry.

The Zernike phase pupil mask

5mm The Zernike mask is a phase pupil mask π/2 1mm used often to visualize input transparencies that are themselves phase objects. The Zernike mask imparts phase delay of π/2 near the center of the pupil plane, i.e. at the lower spatial frequencies. The result at Use of the Zernike phase mask is alsothe output plane is intensity contrast at the called phase contrast imaging.edges of the phase object.

MIT 2.71/2.710 04/22/09 wk11-b- 19

Phase contrast (Zernike mask) imaging

MIT 2.71/2.710 04/22/09 wk11-b- 20

plane wave

illumination

pupil mask

colle

ctor

objec

tive

π/2 phase shift

arbitrary

pupi

l pla

ne b

andw

idth

DC

term

φ=π/2

phase object gt(x,y)=exp{iφ(x,y)}

phase contrast image Iout(x’,y’)

Phase contrast imaging the MIT phase object


MIT 2.71/2.710 04/22/09 wk11-b- 21

Phase contrast imaging the MIT phase object


MIT 2.71/2.710 04/22/09 wk11-b- 22

The transfer function of Fresnel propagation

Fresnel (free space) propagation may be expressed as a

convolution integral

MIT 2.71/2.710 04/08/09 wk9-b-26

The Talbot effect

y z y z

x x

MIT 2.71/2.710 04/08/09 wk9-b-27

MIT 2.71/2.71004/08/09 wk9-b-

Talbot effect: still shots

28

intensity atz = 5, 000!


x(!)

y(!)

−1000 −500 0 500 1000

1000

500

0

−500

−1000

x(!)

y(!)

−1000 −500 0 500 1000

1000

500

0

−500

−1000

intensity atz = 0

x(!)

y(!)

−1000 −500 0 500 1000

1000

500

0

−500

−1000


x(!)

y(!)

−1000 −500 0 500 1000

1000

500

0

−500

−1000

Talbotsubplane

Talbotplane

(in phase)

Talbotplane(phase

reversal)

Physical explanation of the Talbot effectone three

plane plane xwave waves

0th order

+1st order

–1 st order

z

refe

renc

e sph

ere

(radi

usz)

!r ! !

z2 + x2 " z # 2z

2

" x = z tan ! # ! #

"

phase delay x !r "z

!# = 2$ $ . "

# "2

!! = 0, ±2", ±4", . . . replicates field at grating

!! = ±", ±3", . . . 1 period shift wrt grating 2

!! = ± ! 2 , ± 32

! , . . . +1st , ! 1st orders interfere

destructively

0th, +1st, -1st orders are in phaseMIT 2.71/2.710 04/08/09 wk9-b-29

Mathematical derivation of the Talbot effect /1

• Field immediately past grating

g(x; 0) = 1 !

1 + m cos "2!

x #$

2 !

– also expressed as g(x; 0) = 1 !

1 + m

exp "i2!

x # +

m exp

"!i2!

x #$

2 2 ! 2 !

• Fourier transform (spatial spectrum) of field past grating 1

! m

" 1

# m

" 1

#$

G(u; 0) = 2

!(0) + 2

! u ! !

+2

! u + !

• Fourier transform (spatial spectrum) of Fresnel propagation kernel

H(u, v; z) = exp ! ! i!"z

"u 2 + v 2

#$

1 !

"z "

– value at grating’s spatial frequencies H(± !

, 0) = exp !i! !2

MIT 2.71/2.710 04/08/09 wk9-b-30

Mathematical derivation of the Talbot effect /2 • Fourier transform (spatial spectrum) of field propagated to distance z,

G(u, v; z) = G(u, v; 0) ! H(u, v; z)1

! m

" 1

# m

" 1

#$

=2

!(0) + 2

! u " !

+2

! u + !

! H(u, v; z)

1 !

m "

1 # "

1 #

=2

H(0, 0; z)!(0) + 2

H !

, 0; z ! u " !

m "

1 # "

1 #$

+2

H " !

, 0; z ! u + !

1 !

m "

#z # "

1 #

=2

!(0) + 2

exp "i" !2

! u " !

m "

#z # "

1 #$

+ 2

exp "i" !2

! u + !

• Inverse Fourier transforming,

g(x; z) = 21

!1 +

m 2

exp

"

!i! !"z

2

#

exp $i2!

! x %

+ m 2

exp

"

!i! !"z

2

#

exp $!i2!

! x %

&

MIT 2.71/2.710 04/08/09 wk9-b-31

Mathematical derivation of the Talbot effect /3 • Our previous result

g(x; z) = 12

!1 +

m 2

exp

"

!i! !"z

2

#

exp $i2!

! x %

+ m 2

exp

"

!i! !"z

2

#

exp $!i2!

! x %

&

– also expressed as

g(x; z) = 12

!1 + m exp

"

!i! !"z

2

#

cos $2!

! x %

&

– intensity is I(x; z) = |g(x; z)| 2

=1

!1 + m 2 cos2

"2!

x # + 2m cos

$

!"z

%

cos "2!

x #&

4 ! !2 ! – note intensity immediately after grating

I(x; 0) =

! 1 "

1 + m cos #2!

x $%&2

2 !

=1 '

1 + m 2 cos2 #2!

x $ + 2m cos

#2!

x $(

4 ! !

MIT 2.71/2.710 04/08/09 wk9-b-32

Mathematical derivation of the Talbot effect /4

• Special cases of propagation distance

Talbot plane

(shifted) Talbot plane

MIT 2.71/2.710

Talbot subplane

04/08/09 wk9-b-33

The Talbot planes

MIT 2.71/2.710 04/08/09 wk9-b-34

T alb

ot p

lane

T alb

ot p

lane

T alb

ot su

bpla

ne

T alb

ot su

bpla

ne

frequencydoubling

1/2 periodshift

frequencydoubling

replica ofgrating

etc.

MIT OpenCourseWarehttp://ocw.mit.edu

2.71 / 2.710 Optics Spring 2009

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

http://ocw.mit.edu/terms

http://ocw.mit.edu

pupil engineering

Documents