A. Correspondence Between the Spatial Harmonic Function and the plane wave

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4. Fourier Optics

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4.1 PROPAGATION OF LIGHT IN FREE SPACE

A. Correspondence Between the Spatial Harmonic Function and the plane wave

Consider a two-dimensional plane wave.

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Spatial frequency

2 2

cos sin

2

_x

2

_y

k I J

k f I f J

π θ π θ

λ λ

π π

= +

$

θ

e

(4)

Plane waves : 3D

x

y z

e

cos

1

a =

⁻

α

cos

1

b =

⁻

β cos

1

c =

⁻

γ

(α, β, γ)… directional cosine

x y z

f f f

α λ = β λ = γ λ =

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Physical meaning of spatial frequency

cos sin

= sin

y y y

f f θ φ f

β λ φ λ

λ λ

= → = → =

φ θ

spherical parabolic planar

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Spatial frequency and propagation angle

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Principle of Fourier Optics

Î An arbitrary wave in free space can be analyzed as a superposition of plane waves.

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Propagation of light in free space

Consider a plane wave of complex amplitude:

The complex amplitude in the z = 0 plane, U(x, y, 0), is a spatial harmonic function

U(x, y, 0) =

U(x, y, z) = f(x, y) exp(- jk

_z

z)

At z = d, g(x, y) = U(x, y, d) = f(x, y) exp(-jk

_z

d)

The plane wave at z, U(x, y, z) is constructed by using the relation,

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The complex amplitude in the z = 0 plane,

U(X, y, 0) =

Î This is a harmonic function of spatial frequencies, v

_x

and v

_y

,

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U(x, y, z) = f(x, y) exp(- jk

_z

z)

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Amplitude modulation

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Frequency modulation

f(x,y)

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B. Transfer Function of Free Space

At z = d, U(x, y, d) = g(x, y) = f(x, y) exp(-jk

_z

d) At z = 0, U(x, y, 0) =

g(x, y) / f(x, y) = exp(-jk

_z

d) =

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Transfer Function of Free Space

: evanescent wave

We may therefore regard 1/λ as the cutoff spatial frequency (the spatial bandwidth) of the system

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Fresnel approximation

Fresnel approximation for transfer function of free space

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Input - Output Relation

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Impulse-Response Function of Free Space

= Inverse Fourier transform of the transfer function

Free-Space Propagation as a Convolution

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Huygens-Fresnel Principle and the impulse-response function

The Huygens-Fresnel principle states that each point on a wavefront generates a spherical wave.

The envelope of these secondary waves constitutes a new wavefront.

Their superposition constitutes the wave in another plane.

The system’s impulse-response function for propagation between the planes z = 0 and z = d is

In the paraxial approximation, the spherical wave is approximated by the paraboloidal wave.

Î Our derivation of the impulse response function is therefore consistent with the H.-F. principle.

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In summary:

Within the Fresnel approximation, there are two approaches to determining the complex amplitude g(x, y) in the output plane, given the complex amplitude f(x, y) in the input plane:

Space-domain approach

in which the input wave is expanded as a sum of plane waves.

in which the input wave is expanded in terms of paraboloidal elementary waves

Frequency-domain approach

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4.2 Optical Fourier transform

A. Fourier Transform in the Far Field

If the propagation distance d is sufficiently long,

Î the only plane wave that contributes to the complex amplitude at a point (x, y) in the output plane is the wave with direction making angles

Proof!

d

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Proof :

If f(x, y) is confined to a small area of radius b, and if the distance d is sufficiently large so that the Fresnel number is small,

Condition of Validity of Fraunhofer Approximation for Fraunhofer approximation

when the Fresnel number

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B. Fourier transform using a lens

How can a convex lens perform the FT ?

f

o

f

o

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Input placed against lens

Input placed in front of lens

Input placed behind lens

back focal plane

Three configurations

( ) ( ) _⎥

⎦

⎢ ⎤

⎣

⎡ − +

= ² ²

exp 2

, x y

f j k

y x t _l

Æ Phase representation of a thin lens in paraxial approximation

convex :

>0 f

concave :

<0 f

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(a) The input placed directly against the lens

Pupil function ; ( ) 1 inside the lens aperture , 0 otherw ise

P x y ⎧

= ⎨ ⎩

( ) ( ) ( ) ( )

' 2 2

, , , exp

l l

2 U x y U x y P x y j k x y f

⎡ ⎤

= ⎢ − + ⎥

⎣ ⎦

( ) ( )

( ) ( ) ⁽ ⁾

2 2

' 2 2

exp 2 2

, , exp exp

f l 2

j k u

f k

U u U x y j x y j xu y dxdy

j f f f

υ π

υ υ

λ λ

∞

−∞

⎡ ⎤

⎢ + ⎥ ⎡ ⎤ ⎡ ⎤

⎣ ⎦

= ⎢ + ⎥ ⎢− + ⎥

⎣ ⎦ ⎣ ⎦

∫ ∫

( ) ( )

( ) ( ) ( )

2 2

exp 2 2

, , , exp

f l

j k u

U u f U x y P x y j xu y dxdy

j f f

υ π

υ υ

λ λ

∞

−∞

⎡ + ⎤

⎢ ⎥ ⎡ ⎤

⎣ ⎦

= ⎢− + ⎥

⎣ ⎦

∫ ∫

Quadratic phase factor

From the Fresnel approximation when d = f ,

Fourier transform

U

_l

U

_l’

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(b) The input placed in front of the lens

( ) ( )

2 2

exp 1

2 2

, , exp

f l

k d

A j u

f f

U u U x y j xu y dxdy

j f f

υ π

υ υ

λ λ

∞

−∞

⎡ ⎛ ⎞ ⎤

− +

⎢ ⎜ ⎝ ⎟ ⎠ ⎥ ⎡ ⎤

⎣ ⎦

= ⎢ − + ⎥

⎣ ⎦

∫ ∫

If d = f

( ) ^, ( ) ^, ^exp ² ( )

f l

U u A U x y j xu y dxdy

j f f

υ π υ

λ λ

∞

−∞

⎡ ⎤

= ⎢ − + ⎥

⎣ ⎦

∫ ∫

Exact Fourier transform !

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( ) ( )

d f d

j d u j k A

u U

_f

λ

υ

υ ^⎥⎦

⎢⎣ ⎤

⎡ +

=

2 2

exp 2

, ( ) ( ξ υη ) ξ η

λ η π

ξ η

ξ u d d

j d d

f d

P f t

_A

⎥⎦ ⎤

⎢⎣ ⎡ − +

⎟ ⎠

⎜ ⎞

⎝

× ∫ ∫

^∞

⎛

∞

−

exp 2 ,

, (c) The input placed behind the lens

Scaleable Fourier transform !

As d reduces, the scale of the transform is made smaller.

( ) ^ξ ^η ^ξ ^η ( ^ξ ^η ) ^{( )} ^ξ ^, ^η

exp 2 ,

,

² ²

0

t

A

d j k d

f d

P f d U Af

⎭ ⎬

⎫

⎩ ⎨

⎧ ⎥⎦ ⎤

⎢⎣ ⎡ − +

⎟ ⎠

⎜ ⎞

⎝

= ⎛

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In summary, convex lens can perform Fourier transformation

The intensity at the back focal plane of the lens is therefore proportional to the squared absolute value of the Fourier

transform of the complex amplitude of the wave at the input plane, regardless of the distance d.

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Note : Invariance of the input location to FT

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4.3 Diffraction of Light

Regimes of Diffraction

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A. Fraunhofer diffraction

Aperture function :

b

d

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' 0

2 f 0.64 f

W

λ

D

λ

D

≈

π

≈

Note,

for focusing Gaussian beam with an infinitely large lens,

Radius :

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B. Fresnel diffraction

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Spatial filtering in 4-f system

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Transfer Function of the 4-f Spatial Filter With Mask Transmittance p(x, y) :

ÎThe transfer function has the same shape as the pupil function.

Impulse-response function is

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High-pass filter

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C. Single-lens imaging system

Impulse response function

At the aperture plane :

Beyond the lens : Assume d₁= f

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Single-lens imaging system

Transfer function

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Imaging property of a convex lens

magnification

From an input point S to the output point P ;

Fig. 1.22, Iizuka

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Diffraction-limited imaging of a convex lens

From a finite-sized square aperture of dimension a x a

to near the output point P ;

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4.5 Holography

If the reference wave is a uniform plane wave,

Original object wave!!

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Off-axis holography

Assume that the object wave has a complex amplitude

Ambiguity term Æ 2θ_s ( θ_min Æθ_s/2 )

Æ Spreading-angle width : θ_s

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Fourier-transform holography

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Holographic spatial filters

IFT

Called “ Vander Lugt filter” or “Vander Lugt correlator”

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Volume holography

THICK

Recording medium

Transmission hologram :

Reflection hologram :

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Volume holographic grating

k _r _k

_g

_{= k}

₀

_{- k}

_r

Grating vector

k _g

Λ = 2 π / |k

_g

| Grating period

Proof !!

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Volume holographic grating = Bragg grating

Bragg condition :

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“Holographic data storage prepares for the real world”

Laser Focus World – October 2003

“Holographic storage drives such as this prototype from Aprilis

are expected to become commercially available for write-once-read-many (WORM) applications in 2005.”

200-Gbyte capacity in disk form factor 100 Mbyte/s data-transfer rate