Deriving the Point Spread Function (PSF) of a Thin Lens

ECE 637 Supplementary Material

 Venkatesh Sridhar

Diffraction-limited Imaging

2  

-4-3

-2-1

01

23

4-0.5 0

0.5 1

z

•  Point-spread function (PSF) caused due to imperfect interference pattern

•  Decreasing pixel-size on detector array does not enhance resolution beyond a certain point

Phases NOT equal on plane Phases equal on

plane Planar wavefront

Interference Pattern

Detector plane

…  

Wavefront bent due to lens

Wave propagation …  

Lens Geometry – Parabolic Approximation

y

δ2(y) δ1(y)

Thin Lens

L

R1

R1

R2

Axis through center of lens

•  R1 , R2 = radius of curvature for lens surfaces

•  n = refractive index of lens

(n > 1)      •  L = Lens thickness

•  Parabolic approximation: Approximate Lens surface as parabolic

y z

x

2 2

1 21 2

( ) and ( )2 2y yy yR R

δ δ= =1-D case:

2-D lens surface: 2 2

( , ) , 1, 2 2i

i

x yx y iR

δ += =

(into  plane  of  paper)  

3  

Phase-shift imparted by lens

y

δ2(x,y) δ1(x,y)

L

R1

R1

P P’

( ) ( )1 2 1 22 2( , ) ( , ) ( , ) + ( , ) ( , )

( / )x y x y x y L x y x y

nπ πφ δ δ δ δλ λ

= + − −

y z

x (into  plane  of  paper)  

•  Phase change from P → P’:

2 2

1 2

2 ( ) 1 1( , ) ( 1)2

x yx y nL nR R

πφλ

⎧ ⎫⎛ ⎞+⎪ ⎪= − − +⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

•  Applying the parabolic approximation to δ1 and δ2

Path through Air Path through Lens

refractive index

R2

4  

Lens Focal length : Interpretation

•  Can re-write phase-shift ϕ(x,y) as

             where df is defined as            •  What is so special about df ?

!  Optics perspective ‒  According to the famous Lensmaker’s equation, df is the focal length of the lens

!  Signal processing perspective ‒  Typically a narrow PSF is preferable (ideal PSF is an impulse function) ‒  PSF dependent on system parameters such as lens-diameter, wavelength, detector position, etc. ‒  We can show that for a thin lens with sufficiently large diameter, the PSF has minimal width

when detector plane is at distance df from lens.

2 22 ( )( , ) ,2 f

x yx y nLd

πφλ

⎧ ⎫+⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭

1 2

1 1 1( 1)f

nd R R

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

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Interference Pattern on Detector Plane

P1

P2

PN

P1’

P2’

PN’

.  .  .  .  .  .  

Phase=ϕ(x1,y1)

Phase=ϕ(xN,yN)

y z

x (into  plane  of  paper)  

Phase=0

-4-3

-2-1

01

23

4-0.5 0

0.5 1

z

Pd = (u,v,z)

{ }( )02( , , ) exp ( , ) exp ( , )dP

lens

E u v z A j x y j D x y dxdyzα πφ

λ⎧ ⎫= ⎨ ⎬⎩ ⎭∫∫

•  Huygens-Fresnel principle : each point on a wavefront is a point source •  Field E at a point Pd = (u,v,z) on the detector plane

Phase shift due to Lens

Phase shift due to path difference

Distance between (x,y,0) and Pd

Energy conservation for point source

(x,y,0)

Phase=0

Planar wavefront

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Binomial Approximation to Path Length

DPd (x, y) = x − u( )2

+ ( y − v)2 + (0− z)2

= z 1+x − u( )2

z2 + ( y − v)2

z2

DPd (x, y) ≈ z 1+

x − u( )2

2z2 + ( y − v)2

2z2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= z + u2 + v2

2z⎡

⎣⎢

⎤

⎦⎥ −

xu + yvz

⎛⎝⎜

⎞⎠⎟+ x2 + y2

2z⎛⎝⎜

⎞⎠⎟

•  Distance between points (x,y,0) on pupil plane and Pd = (u,v,z) on detector plane

•  Recall binomial expansion

1+ a = 1+ a

2− a2

8 ! ≈1+ a

2 for sufficiently small a

•  For sufficiently large value of z

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Calculation: Interference Pattern on Detector Plane

•  Field E at a point Pd = (u,v,z) on the detector plane

       •  Lens mask    •  Re-arrange terms in above equation and change integral limits to full real space  

0

2 2 2 2 2 20

2( , , ) exp ( , ) ( , )

2 ( ) exp 2 2 2

dPlens

flens

AE u v z j x y D x y dxdyz

A x y u v xu yv x yj nL z dxdyz d z z z

α πφλ

α πλ

⎧ ⎫⎛ ⎞= +⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭⎧ ⎫⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + +⎪ ⎪⎛ ⎞= − + + + +⎜ ⎟⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎪ ⎪⎝ ⎠⎝ ⎠⎩ ⎭

∫∫

∫∫

Lens Detector distance

E(u,v, z) =A0α

z exp j

2πλ

nL+ z + u2 + v2

2z⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪f (x, y) exp j

2πλ

xu + yvz

⎛⎝⎜

⎞⎠⎟+ x2 + y2( ) 1

z− 1

d f

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪−∞

∞

∫−∞

∞

∫ dxdy

Lens Detector distance

1 if ( , ) within lens region( , )

0 e.w. x y

f x y ⎧= ⎨⎩

Amplitude vanishes if z = df

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Special Case: Interference Pattern on Detector Plane

•  Let F denote the continuous-space Fourier transform (CSFT) of f , the lens mask

•  Consider special case of detector-plane position : z = df

2 20 2( , , ) exp ( , ) exp 2

2ff f f f

A u v u vE u v d j nL z f x y j x y dxdyd d d dα π π

λ λ λ

∞ ∞

−∞ −∞

⎧ ⎫ ⎧ ⎫⎛ ⎞ ⎛ ⎞+⎪ ⎪ ⎪ ⎪= + + +⎜ ⎟ ⎜ ⎟⎨ ⎬ ⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪ ⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭ ⎩ ⎭∫ ∫

2 20 2( , , ) exp ,

2ff f f f

A u v u vE u v d j nL z Fd d d dα π

λ λ λ⎧ ⎫⎛ ⎞ ⎛ ⎞+ − −⎪ ⎪= + +⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭

This is a 2-D Fourier Transform of f !!

( , ) ( , )CSFT

f x y F µ ν→

Can drop the – sign in front of u and v since f is real-valued

9  

Point Spread Function (PSF) Model

222 0( , , ) ,f

f f f

A u vE u v d Fd d dα

λ λ⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

•  To compute PSF we need only field intensity

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•  The PSF h(u,v) is then given by

( )( )

22

2 2

/ , /( , , )( , )

0,0(0,0, )

f ff

f

F u d v dE u v dh u v

FE d

λ λ= =

•  Note that F is dependent on shape of lens, and so, PSF h is dependent on the same

PSF of a Circular lens

•  Refer to ECE 637 lecture on CSFT (2nd lecture of this course)

•  For a circular lens of diameter W

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( )( )2 2

1

2 2( , ) circ , ( , ) jinc ,

2

CSFT J Wx yf x y F W WW W W

π µ νµ ν µ ν

µ ν

+⎛ ⎞= = =⎜ ⎟⎝ ⎠ +→

where J1 is Bessel function of first kind order 1

•  PSF for circular lens ( , ) jinc ,

f f

W Wh u v u vd dλ λ

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

•  Intuition : Large (W/λ) ratio corresponds to narrow PSF !!

•  Exercise: Why do parabolic RADAR antennas have large diameters ?

References

•   Joesph W. Goodman, “Introduction to Fourier Optics”, Roberts & Company, third edition, 2005.

•  Applied Optics Group, “Modern Optics: Properties of a Lens”, Dept. of Physics, University of Edinburgh, chapter 3.

•  Leno S. Pedrotti, “Fundamentals of Photonics: Basic Geometrical Optics”, SPIE Publications, 2008.

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