Demo #3: 4F Spatial Filtering

Matt Klug 4/23/2012

2.71/2.710 - Optics Mechanical Engineering

MIT

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Demo #3: 4F Spatial Filtering Setup

635nm He-Ne Laser

Object

Objective Lens

Filter Collector

Lens

Magnifying Lens

Object is a

Wire Mesh

Beam

Expander

2

Demo #3: 4F Spatial Filtering Scheme

x”

y”

Aperture Plane

Input E-Field:

gin(x,y)

Pupil Plane

gFP(x’’,y’’)

Field Before Pupil Plane:

Plane Wave

Input

x’

y’

z

Output Plane

gout(x’,y’)

f1=12.5cm f1 f2

x

y

f2=15cm

Input Field: tillumin ggyxg ,

11

","","f

y

f

xGyxg inpp

Field After Pupil Plane: ","",""," yxgyxgyxg FPpppp

Field at Output Plane:

22

','','f

y

f

xGyxg ppout

Objective Lens Performs

Fourier Transform

Collector Lens

Performs Fourier

Transform

Spatial Filtering at

Pupil Plane. Remove

frequencies.

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Demo #3: Field Through Objective Lens

Input Field: 𝑔𝑖𝑛 𝑥, 𝑦 = rect 𝑥

Λ − 𝑑 rect

𝑦

Λ − 𝑑 ⨂ comb

𝑥

Λ comb

𝑦

Λ

Field Before Pupil Plane:

t(x,y)

x,y

Λ

d/2 d/2

x,y

Λ

⨂

𝑔𝑝𝑝− ∝ sinc Λ − 𝑑 𝑢 sinc Λ − 𝑑 𝑣 comb Λ𝑢 comb Λ𝑣

2D

𝑔𝑝𝑝− ∝ sinc Λ − 𝑑 𝑥"

𝜆𝑓1 sinc Λ − 𝑑

𝑦"

𝜆𝑓1

comb Λ𝑥"

𝜆𝑓1 comb Λ

𝑦"

𝜆𝑓1

Comb functions in the x and y-directions.

Sinc functions superimposed in x and y

directions produce a cross pattern.

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Demo #3: 4F Spatial Filtering and Output Field

Field After Pupil Plane:

gpp-(x”,y”)

x =

gFP(x”,y”) gpp+(x”,y”)

Field at Output Plane:

Since a is small,

sinc pattern has a

large central lobe

Recover near

original comb period

that has been

magnified by f2/f1

𝑔𝑜𝑢𝑡 ∝ sinc 𝑎𝜆𝑓1𝑣 comb 𝛬𝜆𝑓1𝑣 = sinc 𝑎𝑓1

𝑓2𝑦 ′ comb

1

𝛬

𝑓1

𝑓2𝑦′

gout (x’,y’)

𝑔𝑝𝑝 + ∝ rect 𝑣

𝑎 𝑐𝑜𝑚𝑏 Λ𝑣 = rect

𝑦"

𝑎𝜆𝑓1 𝑐𝑜𝑚𝑏 Λ

𝑦"

𝜆𝑓1

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2.71 / 2.710 OpticsSpring 2014

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