chapter 5 –interferometrylight.ece.illinois.edu/ece460/pdf/chap_17_ xvii - interferometry.pdf ·...

Chapter 5 – Interferometry

Gabriel Popescu

University of Illinois at Urbana‐Champaigny p gBeckman Institute

Quantitative Light Imaging Laboratory

Electrical and Computer Engineering, UIUCPrinciples of Optical Imaging

Quantitative Light Imaging Laboratoryhttp://light.ece.uiuc.edu

5.1 Superposition of Fields

ECE 460 – Optical Imaging

Interference is the phenomenon by which electromagnetic fields interact with one another


fields interact with one another.

Interferometry has various applications from motion sensors to spectroscopy and material characterization.

Interference is the result of the superposition principle

[ , ] [ , ]jE r t E r t (5 1)

Intensity is the measurable quantity

[ , ] [ , ]jj (5.1)

Where stands for time (or ensemble) average

2[ , ] [ , ]I r t E r t (5.2)

Where stands for time (or ensemble) average

2Chapter 5: Interferometry



Consider 2 fields 1 2and E E2 2


Note:

2 2 * *1 2 1 2 1 2[ , ]I r t E E E E E E (5.3)

nI 11

iA e 22

iA e the dot product

Polarization is critical

Parallel polarization offers the most interference

*1 2E E

1 2

Parallel polarization offers the most interference

Assume parallel polarization1 2 1 2 cosE E E E

2E

1E

2 [ [ ]]I I I E E is generally a random variable

For uncorrelated (incoherent) fields => simplest case

(5.4)

cos[ ] 0

1 2 1 22 cos[ [ , ]]I I I E E r 2 1

is a monochromatic field because it is fully coherent.


5.2 Monochromatic Fields


(5.5)1 2 1 2 02 cos[ ]I I I I I


( )1 2 1 2 0[ ]

Time difference In nI E

0

The phase shiftFrequency

I I Fringe contrast (visibility): max

max

min

min

I II I

(5.6)

2 1 2 ( 1) 2I I I I I I I I I I

1 2 1 2 1 2 1 2 1 2

1 2 1 2

2 1 2 ( 1) 2(0 :1)

2( )I I I I I I I I I I

I I I I

1I I 0I I 1 2 1I I


2 1 0I I



1 2I I I For :4II


2I

4I02 (1 cos[ ])tI I

(5.7)

2 3 0

Practical Advantages of Interference:

a) Interference term is so if is too small to 1 22 cos[ ]I I 1Ibe measured directly then can act as an amplifier. gives a higher sensitivity

2I1 2I I




Practical Advantages of Interference:

b) if i di id d b 100 i fI


b) => if is divided by 100, interference is divided by 10 => high dynamic range

1 2Interference I I 1I

c) Imagine we frequency shift by =>1E 1 2 cos[ ] cos[ ]I I t t t 1 2 1 2cos[ ] cos[ ]I I t t t

=> can tune to high frequency ( > 1 kHz) => Low Noise


5.3 Wavefront‐Division Interferometry


Interference is obtained between different portions of the same wavefront (next: amplitude division)

5.3 Wavefront Division Interferometry

same wavefront (next: amplitude‐division)

limk d d

Young Interferometer0 d

The oldest interferometer

K yx

2 700 microns

S ll Sli 1 functions: [ ]; [y+ ];d dy °2d

2d

0y

K

z

1000

Small Slits 1 -functions: [ ]; [y+ ];2 2

y °




The field at the plane x=0d d


Assume the observation plane is in the far zone =>

1 2 0 ( [ ] [y+ ])2 2d dE E E E y (5.8)

Assume the observation plane is in the far zone => Fraunhoffer diffraction (Fourier)

2 2[ ] [ [ ]] [ ] 2cos[ ]y yd dig ig dE q E y E e e E q

( )

Remember (chapter 3)

2 20 0[ ] [ [ ]] [ ] 2cos[ ]

2y yE q E y E e e E q (5.9)

yyqz

Fringes cos[ ]2d y

z (5.10)




Similarity relationship again:

I


I1d

yI

2and d

Note: using Fourier it is easy to generalize to arbitrary y

slit/particle shape; similar to scattering from ensemble of particles => use convolution to express the arbitrary shape.




We can use convolution to express arbitrary shape:a


ydd

a

=dd

y

V

a

Other Wavefront‐Division Interferometers

22

22

a) Fresnel Mirrors1S

1M S

Image S thru M1 and M2

virtual sources S1 and S2

2S2M

1 SS1 and S2 act as Young’s pinholessame equations S1 and S2 are derived from the

10

same sources and therefore coherent

Chapter 5: Interferometry



b) Lloyd’s Mirror


S

MS Young and S S




c) Thin films:


Films of oil break the white light into colors due to interference and phase ( )f

S P

interference and phase ( )f

d) Newton’s rings:

n

d) Newton s rings: Circular fringes = ringsk

Localized on the surface of lens

h


5.4 Amplitude‐Division Interferometry


Interference is obtained by replicating the wavefront=>less amplitude in each beam

5.4 Amplitude Division Interferometry

amplitude in each beam.

Michelson Interferometer:

2M1L

L

21L

2LS

BS

Detector ( )t k L

Very sensitive to path length differences between ‘arms’

Eg. It has been used to measure pressure in rarefied ( l ll d k gases(place cell on one arm‐> produce


k



BS‐ beam splitter

hi f !


assume thin for now!

Let , be the lengths of the 2 areas

=> The intensity at the detector:1L 2L

=> The intensity at the detector:

1 2 1 2I( L)=I +I +2 cos(2 )LI I

(5.11)

2 1 2 1Note: L ( )time delay cos( )

L L C t t C

We’ll come back to Michelson with low‐coherence light, temporal coherence, OCT, etc




Other amplitude‐division interferometry

) l ll l l fl i


a) Plane parallel plate reflection.S S’

point source‐ inter fringe ( ) t lf d

dn

inter‐fringe = ( , ) metrologyf n d

b) Plane parallel plate‐ transmission.

Note: With plane wave

n

S

incident, fringes are localized at infinity => Need lens

L

n

P

L




c) Fizeau Interferometer:R M

5.4 Amplitude Division Interferometry( )Widrelson Cos t

k R M

Fringes

( )H Z Cos k x

k

Reference surface

P

Used to test mirrors and other surfaces.

P

d) Mach‐Zender Interferometer Very Common Very Common Shear interferometry Tilt one mirror

( )Cos k x

Fringes analysis of surfacesSample


5.5 Multiple Beam Interference


Reflection


Lens

Fabry‐Penot interferometry2

( ) sinI 2

r Fnd

( )2

( ) ( ) ( )

I 2 ;1 sin

2

i

i

I F

2

4 Finess coeff.(1 )

RFR

( ) ( ) ( ) t i rI I I

Transmission

(5.12)

0

( )2 cos one pass phase shiftk d n

Transmission




R1 , R2 reflectivities Transmitted Intensity

( )I r


1 , 2

R increases narrower lines i.e. More reflection orders participate in interference

2R

( )

( )

I r

iI1R

participate in interference

In practice, Fabny‐Perot gave accurate information about spectral lines(also called etalon)

2k 2 2k

about spectral lines(also called etalon)

D Rayleigh Criterion:

1 2 li

S

1

12

2 lines are separated if:

FWHM

is fixed etalon


5.6 Interference with Partially Coherent Light


So far, we assumed monochromatic light = fully coherent


coherent.

What happens when an arbitrary, broad‐band, extended source is used for interference?

a) Michelson interferometry

1M ( )I L

BSS

L

DL L

The contrast of the fringe is decreasingThe contrast of the fringe is decreasing

i.e. limited temporal coherence.19Chapter 5: Interferometry



b) Young Interferometer :


A

2

Ac

d OA

k1 2I I

k

i.e. Limited Spatial Coherence

dk


5.7 Temporal Coherence


Coherence defines the degree of correlation between fields:

T i l l i i i ( i l h )


Typical correlation at one point in space (typical coherence)

Temporal correlation between fields at two points (spatial coherence)coherence)

Given the field at one point , the mutual coherence function is:

( , )E r t

t l ti f ti

( ) ( ) *( ) tE t E t

( ) *( )E t E t dt

(5.13)

autocorrelation function( ) *( )E t E t dt




Note: 2(0) ( )E t (5.14)


So

I irradiance

E E So

Appl y again the correlation theorem (Eq 1.30)E E

22[ ] ( ) *( ) ( )E E E (5.15)

( ) Spectrum FTIRS

So, the autocorrelation function relates to the optical spectrum of the field:

( )

Wi Ki t hi th

iw

22

Wiener‐Kintchin theorem.0

( ) ( ) iwS e d (5.16)




! Compare 6.16 with 5.28

! I li h E( ) d h F i


! It applies even when E(t) does not have a Fourier Transform

Typically, spectrum is centered on 0Typically, spectrum is centered on

Assume spectrum is ; apply shift theorem (Eq 1.31)0

0( )S

0( ) ( ) ie

0

( ) ( ) ,

where ( ) ( ) ;

iu

e

S u e du u (5.17b)

(5.17a)

Complex degree of coherence

0

( ) ( ) (0 1)( )( )(0)

( ) (0;1) (5.18) (5.19)




Measuring the temporal coherence: Michelson interferometer1M 1 2t t lE E E


2M1E

2ES

Irradiance:

A

1 2totalE E E

21 2( ) ( )I E t E t

E E(5.20)

Assume:1 2E E

1 '2' 1 2 1 2( ) *( ) * ( ) ( )2 2 Re[ ( )]

I I I E t E t E t E tI

(5 21)12 2 Re[ ( )]I

can be measured separately access to directly, by moving

1,2I Re( )M R [ ( )]

(5.21)

by moving 2M Re[ ( )]

S

0




The degree of coherence Spectral width( ) ( )S


c

0 coherence time

width of ( )

c Coherence Length:

c cl C (5.23)( )( ) [ ( )]

constant

S Rule of thumb:c c

20

cl

(6.22)

(5.24) constant c

0 20

2 2; ( ) 2

cc T c c

c

Interference occurs only if L l

(6.22)

Interference occurs only if

Broad spectrum => short ( 2 3 )cl m ( 100 )nm

cL l


5.8 Optical Domain Refractometry


Low‐Coherence interferometry ( interf. with broad band light).

S SLD LED hi li h f d l


Sources: SLD, LED, white light, femtosecond laser, etc.

consider a transparent, layered structure under investigation.

1 2 3 4

M

S

d

a a a

D cd l




Scanning M, we retrieve


R The interface are resolved Reflectivity give info about refractive index

L1L 2L 3L 4L

refractive index determine position of interfaces.

1,2,3,4L

ODR:

Successful for quantifying lasers in waveguides, fiber q y g g ,optics, etc.

1987‐HP‐ fiber optic reflectometer.


5.9 Optical Coherence Tomography(OCT)


Optical technique capable of rendering 3D images from think biological samples


think biological samples.

Penetrates 1‐2 mm deep in tissue.

=> Typically implemented in optical fiber configuration. Typically implemented in optical fiber configuration.

M2x2S

1991, Fujimoto’s group, MIT

CouplerD

Scan beam x‐yScan M ‐> z

Tissue = continuous superposition of interfaces.

Scanning M, a depth‐resolved reflectivity signal is retrieved

> can resolve regions inside tissue(e g Tumors)=> can resolve regions inside tissue(e.g. Tumors).




If mirror is swept at constant speed v: z=vt


z=vt Phase delay: (2 means back and forth) Frequency shift: Dopler Shift

2 2kz kvt 2 kvt (5.25)

The detector is recording a high‐frequency signal Low‐noise Dynamic range can easily reach 10 orders of magnitude! i.e.

d fl f / b ll ( d )can record reflectivities from 1 to 1/10 billion!(100 dB)

Various Technological Improvements:

Spectral domain OCT: instead of scanning M Spectral domain OCT: instead of scanning M,

measure

Galvo‐scanning group delay‐ fast

S( ) ( )

Galvo scanning group delay fast

Swept source OCT29Chapter 5: Interferometry



Various Technological Improvements:

l d d f ll


Spectral encoding‐ instead of scanning on x, illuminate with

Spectroscopic OCT‐ trade z‐resolution for ( )S

( )x

Spectroscopic OCT trade z resolution for information

Since 1991, ~ 5,000 OCT papers published.

( )S

Clinically applied in: Oftalmology

Research stage: Dermathology, cardio, breast cancer, etc

R l bi d i h SHG l l i i Recently combined with SHG, molecular imaging


5.10 Dispersion Effects on Temporal Coherence



What happens if on one area of the Michelson interferometer, there is extra material (eg Glass)?there is extra material (eg. Glass)?

1M

d2M

S

d

1 pass

1 pass




How does broad band fields propagate through dispersive materials (such as glass)? Think pulses!


materials (such as glass)? Think pulses!

k

( )n 0( ) ( )k k n

d

( )n

Above resonance

( )S Incident

0

The phase delay through a transparent

l Taylor expansion of ( )n

Above resonance

Below resonancespectrum

material:( ) ( )

( )k d

k d n

around the central frequency: ( )S

0 ( )k d n 0 1

(5.26)




221( ) ( ) ( ) ( )dn nn n


0

0 0 02( ) ( ) ( ) ( ) .....2

n nd

Note:

constant not important0 0 0 0( ) ( )n n k d 2

21( ) ( ) ( )dk d kd d 0 0 02( ) ( ) ( ) ....2

d dd d

Definitions:

l itdk (5 27) = v = group velocity

= = group velocity dispersion (GVD)

d

2

2

d kd 2

(5.27)

= = group velocity dispersion (GVD)

Different colors have different group velocities

2d 2




E.g. Pulse: blue redblue redv v


Riding with pulse (v=0) parabolic phase: So

( ) ( )S

21( ) (5 28) So

Cross‐Spectral density: 02( )

2

1 2( ) ( ) *( )W E E

(5.28)

(5.29)

Then, the cross‐correlation function is 1 2( ) ( ) ( )

( ) ( ) iW e d

(5.30)12

0

( ) ( )W e d (5.30)




is the generalization of (5.16) i.e

li d Wi Ki ili Th12

0

( ) ( ) iW e d


generalized Wiener‐Kinntilin Theorem.

For the “unbalanced” Michealson the cross spectral density is

0

For the unbalanced Michealson, the cross‐spectral density is2 2

2 21 1

* * 2 21 2 1 2( ) ( ) ( ) ( )

i iW E E E E e S e (5.32)

The cross‐correlation function:2

212( ) [ ( )] [ ( ) ]

i

W S e (5 33)Spaul

Remember convolution theorem:

12 ( ) [ ( )] [ ( ) ] W S e

21i

(5.33)2ie

SpaulFrensel

spatial frequency 2

20( ) [ ( )] [ ] ( ) ( )

iS e hv (5.34)

35

v




212( ) [ ]

i

h e

U f l F i T f l i hi f G f i


Useful Fourier Transform relationship for Gauss functions:2

241

2

tb be e

b

(5.35)

2

22( ) ( ) a lso p a rab o lic

ieh

i

i

V

( )ie

cl 36Chapter 5: Interferometry



( )ie


V

The coherence time is increased

Frequency is “chirped”

So, in OCT, is important to balance the interferometer=> minimum coherence length

gives the depth resolutionl gives the depth resolutioncl


chapter 5 –interferometrylight.ece.illinois.edu/ece460/pdf/chap_17_ xvii - interferometry.pdf ·...

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