1629.pdf

5. Wave

5. Wave - - Optics Analysis of Optics Analysis of

Coherent Optical Systems

Coherent Optical Systems

Lens Lens

R₁>0 (concave) R₂<0 (convex)

( )^{x, y = knΔ}( )^{x, y + k}[^Δ₀ ^{− Δ}( )^{x, y} ]

φ

( )^{x y} [^jk ] [^jk(ⁿ ) ( )^{x y} ]

t_l , = exp Δ₀ exp −1 Δ ,

( ) ( ) ( )^{x y t x y U x y}

U_l^' , = _l , _l ,

With Paraxial Approximation With Paraxial Approximation

( ) [ ] ( ) _⎥

⎦

⎢ ⎤

⎣

⎡ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ −

− +

− Δ

=

2 1

2 2

0

1 1

1 2 exp

exp

, R R

y n x

jk jkn

y x t_l

( ) _⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

−

≡

2 1

1 1 1

1

R n R

f

concave :

〉0 f

convex :

〈0 f

( ) ( )_⎥

⎦

⎢ ⎤

⎣

⎡− +

= ^{2 2}

exp 2

, x y

f j k y

x t_l

Types of Lenses Types of Lenses

Fourier Transform with Lenses Fourier Transform with Lenses

Input placed against lens

Input placed in front of lens

Input placed behind lens

Input Placed Against Lens Input Placed Against Lens

( )^{x y At} ( )^{x y}

U_l , = _A ,

Pupil function ( )

⎪⎩

⎪⎨

= ⎧

otherwise

0

aperture lens

the inside

1 , y

x P

( ) ( ) ( ) ( )_⎥

⎦

⎢ ⎤

⎣

⎡− +

= ^{2 2}

'

exp 2 ,

,

, x y

f j k y

x P y x U y

x

U_{l l}

( ) ( )

( ) ( ) (^{xu y} ) ^dxdy

j f y

f x j k y

x f U

j f u j k u

U_f ^⎥⎦ ∫ ∫ _l ^⎢_⎣^{⎡ + ⎥}_⎦^{⎤ ⎢}_⎣^{⎡− + ⎥}_⎦^⎤

⎢ ⎤

⎣

⎡ +

= ^∞

∞

−

λ υ π λ

υ

υ exp ²

exp 2 2 ,

exp

, ^{' 2 2}

2 2

( ) ( )

( ) ( ) (^{xu y} ) ^dxdy

j f y

x P y x f U

j f u j k u

U _f ^⎥⎦ ∫ ∫ _l ^⎢_⎣^{⎡− + ⎥}_⎦^⎤

⎢ ⎤

⎣

⎡ +

= ^∞

∞

−

λ υ π λ

υ

υ exp 2 , , exp ²

,

2 2

Fraunhofer diffraction pattern

Input Placed in Front of Lens (I) Input Placed in Front of Lens (I)

( _X , _Y ) 0( _X , _Y )exp[ ( _X² _Y²)]

l f f F f f j d f f

F = − πλ +

( ) ( )

⎟⎟⎠

⎜⎜ ⎞

⎝

⎥ ⎛

⎦

⎢ ⎤

⎣

⎡ +

= f f

F u f

j f u j k u

U _{f l}

λ υ λ

λ

υ

υ exp 2 ,

,

2 2

( ) ( )

( ) (^{ξ ηυ}) ^{ξ η}

λ η π

λ ξ

υ

υ u d d

j f f t

j f u d f

j k A

u

U _{f A} ⎥

⎦

⎢ ⎤

⎣

⎡− +

⎥⎦

⎢ ⎤

⎣

⎡ ⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛ −

= ∫ ∫^∞

∞

−

exp 2 ,

2 1 exp ,

2 2

Input Placed in Front of Lens (II) Input Placed in Front of Lens (II)

( ) ( ) (^{ξ ηυ}) ^{ξ η}

λ η π

λ ξ

υ u d d

j f f t

j u A

U _{f A} ⎥

⎦

⎢ ⎤

⎣

⎡− +

= ∫ ∫^∞

∞

−

exp 2 ,

, If d =f, then

Exact Fourier Transform!

Input Placed in Front of Lens (III) Input Placed in Front of Lens (III)

( ) ( )

f j

f u d f

j k A

u U_f

λ

υ υ

⎥⎦

⎢ ⎤

⎣

⎡ ⎟⎟⎠ +

⎜⎜ ⎞

⎝

⎛ −

=

2

1 2

exp 2 ,

( ) (^{ξ ηυ}) ^{ξ η}

λ υ π

η ξ

η

ξ u d d

j f f

u d f P d

t_A ⎥

⎦

⎢ ⎤

⎣

⎡− +

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + +

×∫ ∫^∞

∞

−

exp 2 ,

, Vignetting effect

Input Placed Behind Lens Input Placed Behind Lens

( )^{ξ η ξ η} (^{ξ η} ) ^{( )ξ,η}

exp 2 ,

, ^{2 2}

0 tA

d j k d

f d

P f d U Af

⎭⎬

⎫

⎩⎨

⎧ ⎥⎦⎤

⎢⎣⎡− +

⎟⎠

⎜ ⎞

⎝

= ⎛

( ) ( )

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 ,

,

Example of Optical Fourier Transform Example of Optical Fourier Transform

44--f Systemf System

Example of Spatial Filter Example of Spatial Filter

Image Formation Image Formation

( )^{u υ h}(^{u υ ξ η}) ( )^{U ξ η dξdη}

U_i ,

∫ ∫

^∞ , ; , ₀ ,

∞

−

=

(^{u υ ξ η}) ^{K δ} (^{u Mξ υ Mη})

h , ; , ≈ ± , ±

(hopefully)

Impulse Response of a Positive Lens Impulse Response of a Positive Lens

( )⁼ _{2 ⎢⎣}^⎡

(

^{2 + 2}

)

_⎥⎦^⎤ _⎢⎣^⎡

(

^{2 + 2}

)

_⎥⎦^⎤

1 2

2

1 exp 2

exp 2 , 1

;

, υ ξ η

η λ ξ

υ z

j k z u

j k z

u z h

( )

( )

_⎥

⎦

⎢ ⎤

⎣

⎡ ⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛ + −

×

∫ ∫

^∞

∞

−

2

1 2

1 1

exp 2 ,

2 1

y f x

z z

j k y

x P

dxdy z y

x z z

u jk z

⎭⎬

⎫

⎩⎨

⎧ ⎥

⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝⎛ +

⎟ +

⎠

⎜ ⎞

⎝⎛ +

−

×

2 1

2 1

exp ξ η υ

The Lens Law The Lens Law

1 0 1

1

2 1

=

− + z f

z Lens Law (Imaging Equation)

( ) ∫ ∫^∞ ( )

∞

−

≈ P x y

z u z

h 1 ,

,

; ,

2 1

λ2

η ξ υ

( ) ( )

[ ^{u M x M y}] ^dxdy

j z

⎭⎬

⎫

⎩⎨

⎧− − + −

× ξ υ η

λ π

2

exp 2

1 2

z

M = − z Magnification

* Impulse response is the Fraunhofer diffraction pattern of lens aperture.

Lens Law in Geometrical Optics Lens Law in Geometrical Optics

f z

z

1 1

1

2 1

=

+ ^{2 1 1}

1

2 y

z My z

y = = −

Relation between Object and Image (I) Relation between Object and Image (I)

( )

_⎟

⎠

⎜ ⎞

⎝

= ⎛

M M

U u u M

U_i υ 1 , υ

, ₀

( )

⎟

⎠

⎜ ⎞

⎝

⎛ − −

→ M M

u u M

h υ ξ η 1 δ ξ ,η υ

,

; ,

Relation between Object and Image (II) Relation between Object and Image (II)

η η

ξ

ξ~ = M ~ = M

M h z h

y z

x x ~ 1

~y

~

2 2

=

=

= λ λ

( ) ( )^{u,υ h~ u,υ U} ( )^u,υ

U_i = ⊗ _g

( ) ⎟

⎠

⎜ ⎞

⎝

= ⎛

M M

U u u M

U_g υ 1 , υ

, ₀

( )^{u P}( ^{z x z y}) [ ^j (^{ux y})]^dxdy

h~ , ~, ~ exp 2 ~ ~ ~ ~

2

2 λ π υ

λ

υ = ∫ ∫^∞ − +

∞

−

Point-Spread Function

Relation between Object and Image (III) Relation between Object and Image (III)

1. The ideal image produced by a diffraction-limited optical system(i.e. a system that is free from aberrations) is a scaled and inverted version of the object.

2. The effect of diffraction is to convolve that ideal image with the Fraunhofer diffraction pattern of the lens pupil.