Chapter 5 –Interferometrylight.ece.illinois.edu/ECE460/PDF/Chap_17_ XVII - Interferometry.pdf ·...
Transcript of 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
5.1 Superposition of Fields
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
5.1 Superposition of Fields
ECE 460 – Optical Imaging
Consider 2 fields 1 2and E E2 2
5.1 Superposition of Fields
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.
3Chapter 5: Interferometry
5.2 Monochromatic Fields
ECE 460 – Optical Imaging
(5.5)1 2 1 2 02 cos[ ]I I I I I
5.2 Monochromatic Fields
( )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
4Chapter 5: Interferometry
2 1 0I I
5.2 Monochromatic Fields
ECE 460 – Optical Imaging
1 2I I I For :4II
5.2 Monochromatic Fields
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
5Chapter 5: Interferometry
5.2 Monochromatic Fields
ECE 460 – Optical Imaging
Practical Advantages of Interference:
b) if i di id d b 100 i fI
5.2 Monochromatic Fields
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
6Chapter 5: Interferometry
5.3 Wavefront‐Division Interferometry
ECE 460 – Optical Imaging
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 °
7Chapter 5: Interferometry
5.3 Wavefront‐Division Interferometry
ECE 460 – Optical Imaging
The field at the plane x=0d d
5.3 Wavefront Division Interferometry
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)
8Chapter 5: Interferometry
5.3 Wavefront‐Division Interferometry
ECE 460 – Optical Imaging
Similarity relationship again:
I
5.3 Wavefront Division Interferometry
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.
9Chapter 5: Interferometry
5.3 Wavefront‐Division Interferometry
ECE 460 – Optical Imaging
We can use convolution to express arbitrary shape:a
5.3 Wavefront Division Interferometry
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
5.3 Wavefront‐Division Interferometry
ECE 460 – Optical Imaging
b) Lloyd’s Mirror
5.3 Wavefront Division Interferometry
S
MS Young and S S
11Chapter 5: Interferometry
5.3 Wavefront‐Division Interferometry
ECE 460 – Optical Imaging
c) Thin films:
5.3 Wavefront Division Interferometry
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
12Chapter 5: Interferometry
5.4 Amplitude‐Division Interferometry
ECE 460 – Optical Imaging
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
13Chapter 5: Interferometry
k
5.4 Amplitude‐Division Interferometry
ECE 460 – Optical Imaging
BS‐ beam splitter
hi f !
5.4 Amplitude Division Interferometry
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
14Chapter 5: Interferometry
5.4 Amplitude‐Division Interferometry
ECE 460 – Optical Imaging
Other amplitude‐division interferometry
) l ll l l fl i
5.4 Amplitude Division Interferometry
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
15Chapter 5: Interferometry
5.4 Amplitude‐Division Interferometry
ECE 460 – Optical Imaging
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
16Chapter 5: Interferometry
5.5 Multiple Beam Interference
ECE 460 – Optical Imaging
Reflection
5.5 Multiple Beam Interference
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
17Chapter 5: Interferometry
5.5 Multiple Beam Interference
ECE 460 – Optical Imaging
R1 , R2 reflectivities Transmitted Intensity
( )I r
5.5 Multiple Beam Interference
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
18Chapter 5: Interferometry
5.6 Interference with Partially Coherent Light
ECE 460 – Optical Imaging
So far, we assumed monochromatic light = fully coherent
5.6 Interference with Partially Coherent Light
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
5.6 Interference with Partially Coherent Light
ECE 460 – Optical Imaging
b) Young Interferometer :
5.6 Interference with Partially Coherent Light
A
2
Ac
d OA
k1 2I I
k
i.e. Limited Spatial Coherence
dk
20Chapter 5: Interferometry
5.7 Temporal Coherence
ECE 460 – Optical Imaging
Coherence defines the degree of correlation between fields:
T i l l i i i ( i l h )
5.7 Temporal Coherence
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
21Chapter 5: Interferometry
5.7 Temporal Coherence
ECE 460 – Optical Imaging
Note: 2(0) ( )E t (5.14)
5.7 Temporal Coherence
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)
Chapter 5: Interferometry
5.7 Temporal Coherence
ECE 460 – Optical Imaging
! Compare 6.16 with 5.28
! I li h E( ) d h F i
5.7 Temporal Coherence
! 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)
23Chapter 5: Interferometry
5.7 Temporal Coherence
ECE 460 – Optical Imaging
Measuring the temporal coherence: Michelson interferometer1M 1 2t t lE E E
5.7 Temporal Coherence
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
24Chapter 5: Interferometry
5.7 Temporal Coherence
ECE 460 – Optical Imaging
The degree of coherence Spectral width( ) ( )S
5.7 Temporal Coherence
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
25Chapter 5: Interferometry
5.8 Optical Domain Refractometry
ECE 460 – Optical Imaging
Low‐Coherence interferometry ( interf. with broad band light).
S SLD LED hi li h f d l
5.8 Optical Domain Refractometry
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
26Chapter 5: Interferometry
5.8 Optical Domain Refractometry
ECE 460 – Optical Imaging
Scanning M, we retrieve
5.8 Optical Domain Refractometry
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.
27Chapter 5: Interferometry
5.9 Optical Coherence Tomography(OCT)
ECE 460 – Optical Imaging
Optical technique capable of rendering 3D images from think biological samples
5.9 Optical Coherence Tomography(OCT)
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).
28Chapter 5: Interferometry
5.9 Optical Coherence Tomography(OCT)
ECE 460 – Optical Imaging
If mirror is swept at constant speed v: z=vt
5.9 Optical Coherence Tomography(OCT)
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
5.9 Optical Coherence Tomography(OCT)
ECE 460 – Optical Imaging
Various Technological Improvements:
l d d f ll
5.9 Optical Coherence Tomography(OCT)
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
30Chapter 5: Interferometry
5.10 Dispersion Effects on Temporal Coherence
ECE 460 – Optical 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
31Chapter 5: Interferometry
5.10 Dispersion Effects on Temporal Coherence
ECE 460 – Optical Imaging
How does broad band fields propagate through dispersive materials (such as glass)? Think pulses!
5.10 Dispersion Effects on Temporal Coherence
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)
32Chapter 5: Interferometry
5.10 Dispersion Effects on Temporal Coherence
ECE 460 – Optical Imaging
221( ) ( ) ( ) ( )dn nn n
5.10 Dispersion Effects on Temporal Coherence
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
33Chapter 5: Interferometry
5.10 Dispersion Effects on Temporal Coherence
ECE 460 – Optical Imaging
E.g. Pulse: blue redblue redv v
5.10 Dispersion Effects on Temporal Coherence
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)
34Chapter 5: Interferometry
5.10 Dispersion Effects on Temporal Coherence
ECE 460 – Optical Imaging
is the generalization of (5.16) i.e
li d Wi Ki ili Th12
0
( ) ( ) iW e d
5.10 Dispersion Effects on Temporal Coherence
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
Chapter 5: Interferometry
5.10 Dispersion Effects on Temporal Coherence
ECE 460 – Optical Imaging
212( ) [ ]
i
h e
U f l F i T f l i hi f G f i
5.10 Dispersion Effects on Temporal Coherence
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
5.10 Dispersion Effects on Temporal Coherence
ECE 460 – Optical Imaging
( )ie
5.10 Dispersion Effects on Temporal Coherence
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
37Chapter 5: Interferometry