Holography - MITweb.mit.edu/2.710/Fall06/2.710-wk14-a-sl.pdf · Holography: recording incident illumination (laser beam♣)) imaging system film records the interference pattern (interferogram)

MIT 2.71/2.710 Optics12/05/05 wk14-a-1

Holography

• Preamble: modulation and demodulation• The principle of wavefront reconstruction• The Leith-Upatnieks hologram• The Gabor hologram• Image locations and magnification• Holography of three-dimensional scenes• Transmission and reflection holograms• Rainbow hologram

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Modulation & Demodulation

• Principle borrowed from radio telecommunications• Idea is to take baseband signal (e.g. speech, music, with

maximum frequencies up to ~20kHz) and modulate it onto a carrier signal which is a simple tone at the frequency where the radio station emits, e.g. 104.3 MHz (that’s Boston’s WBCN station)

• One of the benefits of modulation is that radio stations can be multiplexed by using a different emission frequency each

• After selecting the desired station, the receiver follows a process of demodulation which recovers the basebandsignal and sends it to the speakers.

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Types of modulation

• Amplitude modulation (AM)

• Frequency modulation (FM)

• Phase modulation (PM)

• Digital methods (Amplitude Shift Keying – ASK, Frequency Shift Keying – FSK, Phase Shift Keying – PSK, etc.)

used in radio at low frequencies only (“AM band” = 535kHz to 1.7MHz) ; as we will see, it is an almost-exact analog of holography

dominant in commercial radio (“FM band” = 88MHz to 108MHz) ; there is an analog in optics, called “spectral holography,” but it is beyond the scope of the class

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Amplitude modulation

( )xf(baseband)

( )( ) ( )xuxf

xf

c2cos modulated

π×==

uc: carrier frequency

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AM in the frequency domain

( )xf of spectrum( )xf modulated

of spectrum

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( )xf of spectrum( )xf modulated

of spectrum

(zoom-in) (zoom-in)

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( ) ( )

( ) ( ) ( ) [ ]

( ) ( ) ( )[ ]

( ) ( )[ ]cc

cc

22c

21

21

ee212cos cc

uuFuuF

uuuuuF

xfxuxf

uFxf

xuixui

−+−=

=−+−∗→

+×=

→

−

δδ

π ππmodulation in the space domain

modulation in the frequency domain:two replicas of the basebandspectrum, centered on the carrier frequency

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Modulation

simple carrier tone

multiplication

× ( )xf modulated

( )xuc2cos π

( )xf

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Demodulation

simple carrier tone

multiplication

× low-passfilter( )xf modulated

( )xuc2cos π

( )xf

must accommodatebaseband spectrum

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Demodulation

( ) ( )xuxf c2 2cos π× ( ) ( )xuxf c

2 2cos of spectrum π×

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Demodulation

( ) ( )xuxf c2 2cos π× ( ) ( )xuxf c

2 2cos of spectrum π×

LP filter pass-band

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The wavefront reconstruction problem

• Wavefront is the amplitude (i.e. magnitude and phase) of the electric field as function of position

• Traditional coherent imaging results in intensity images (because detectors do not respond fast enough at optical frequencies) → magnitude information is recovered but phase information is lost

• Can we imprint intensity information on an optical wave? YES → photography (known since the 1840’s)

• Can we imprint wavefront information on an optical wave? YES → holography (Gabor, late 1940s)

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Photography: recordingincident illumination

(laser beam or white light)

imaging system

film records intensity information

S

2S

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Photography: reconstructing the intensity

incident illumination

(laser beam or white light)

imaging system

at the image plane, an intensity pattern is formedthat replicates the originally recorded intensity

2S2S

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Holography: recordingincident illumination

(laser beam(laser beam♣♣))

imaging systemfilm records the interference patternthe interference pattern

(interferogram) of the object wavefrontand the reference wavefront

S

2SR +

reference beam(split from the

same laser)

R

♣in general, the illumination must be quasi-monochromatic, and spatially mutually coherent

with the reference beam throughout the wavefront

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Holography: reconstructing the wavefront

illumination:replicates the

reference beam

imaging system

R ?

what is the field at the image plane?

2SR +

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R ?

2SR +

( )=+++×=+× **222 RSSRSRRSRR

( ) *2222 SRSRSRR +++×=

The field being imaged is:

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xuiR 02e π=

( ) ( ) *2222 00 e 1e SSS xuixui ππ +++×=

take the simplest possible reference wave, a plane wave:

then the reconstructed field is:

R ?

2SR +

, spatial frequencyλθ0

0sin

=u

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fields departing from thehologram

R

2SR +

( ) *22 0e Sxui π

S

( )22 1e 0 Sxui +×π

propagatesat angle:

0θ

02θ

on-axis

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fields departing from thehologram

R

2SR +

( ) *22 0e Sxui π

S

( )22 1e 0 Sxui +×π0θ

02θ

on-axis

wantedwanted

not wantednot wanted

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Filtering the wavefront: bandlimited signal

{ } wSwS

radius of circle within 0i.e. ,bandwidth has

≠ℑ

u

v

w

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Filtering the wavefront: bandlimited signal

{ } { } { } { } { }SSSSS

wS

ℑ⊗ℑ=ℑ∗ℑ=ℑ

+*2

2

because ,2bandwidth has 1

u

v

Term

2w

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Filtering the wavefront: Fourier transform description

u

v

2ww w

u0 2u0

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u

v

2ww w

u0 2u0

originalspectrum

autocorrelation of theoriginal spectrum

original spectrum but phase--conjugated:

inside-out, or “pseudo-scopic”

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u

v

2ww w

u0 2u0

wantedwantednotnot wantedwanted

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uw

a low-pass filter ofpassband w or slightlygreater permits thedesired term to pass, and eliminatesthe undesirable terms

and .

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illumination:replicates the

reference beam

4F system with Fourier plane filter

R S

the field at the image planereplicates the original S stored in the hologram

2SR +hologram:

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w

2w


u

v

w

u0 2u0

Potential problem: spectra overlap!

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Spectra should not overlap, i.e. wuwwu23

2 00 >⇔>−

u

v

2ww w

u0 2u0

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Leith-Upatnieks vs Gabor hologram

S

2SR +

S

2SR +

Leith-Upatnieks

Gabor

xuiR 02e π=

1=R

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Analogy between the Leith-Upatniekshologram and amplitude modulation (AM)

AM Radio HolographyModulation Recording

ReconstructionDemodulation

× low-passfilter( )xf modulated

( )xuc2 cos π

( )xf

× ( )xf modulated

( )xuc2 cos π

( )xf S

2SR +

xuiR 02e π=

xuiR 02e π=

2SR +

S

low-passfilter

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Image locations and magnification

Referencesource

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Image locations and magnification

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Holography of Three-Dimensional Scenes

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Orthoscopic and Pesudoscopic

Real Image(Pseudoscopic)

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Holography of Three-Dimensional Scenes

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Transmission and Reflection Holograms

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Transmission and Reflection Holograms

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Rainbow hologram (Record)

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Rainbow hologram (Reconstruct)

Documents

Holography - MITweb.mit.edu/2.710/Fall06/2.710-wk14-a-sl.pdf · Holography: recording incident illumination (laser beam♣)) imaging system film records the interference pattern (interferogram)