July 2003 Chuck DiMarzio, Northeastern University 11140b-06-1

ECEG105Optics for Engineers

Course NotesPart 6: Interference

Prof. Charles A. DiMarzio

Northeastern University

Fall 2007

August 2007

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-2

Basic Equations of Interference

Re E

Im E

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-3

Mach Zehnder (1)

Gas Cell n>1

Example

Straight-Line Layouts:

Source, s

M1

M2BS1

BS2

s BS1 BS2

s BS1 M2 BS2

s BS1 M1 BS2

s BS1 M2 BS2

Geometric

M1 Transit Time

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-4

Mach Zehnder (2)

Straight-Line Layout: Transit Time

s BS1 BS2

s BS1 M2 BS2

M1

Diffuser

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-5

Mach Zehnder (3)

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-6

Michaelson Interferometer

Front =50% R?

Rear=AR

BS

M1

M2

Straight-Line Layout: Transit Time

s BS

s BS M2

M1 BS

BS

Geometric Optics

Via M2 with Three Passes Through Beamsplitter and M1 with only one

s BS

s BS M2

M1 BS

BS

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-7

Interferometry: Conflicting Goals

• Match Optical Path Lengths (Transit Time)– Pulses must arrive simultaneously for interference, or

– Paths must be within coherence length of the source (more on this later in the course)

• Match Wavefront Curvature (geom. opt.)– Fringes must be few enough to be within resolution of

instrumentation

• Thus we can’t compensate for extra glass with extra path length.

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-8

Compensator Plate

BS

s BS

s BS M2

M1 BS

BS

s BS

s BS M2

M1 BS

BS

Goal: Match Paths in Transit Time and Wavefront Curvature

t

sourcePower received

PowerC

C C

CC

M1

M2

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-9-50 0 50

-50

0

50 0

0.5

1

1.5

2

Optical Testing

BS

M1, Known Reference

-50 0 50

-50

0

50 0

0.5

1

1.5

2

-50 0 50

-50

0

50 0

0.5

1

1.5

-50 0 50

-50

0

50 0

0.5

1

1.5

2

M2,Mirror Under Test

Spheres 4-wave difference Same with 2-wave bump

Flats with 8-wave tilt Same with 2-wave bump

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-10

Fabry-Perot Interferometer (1)I1 I4

Straight Line Model

Multiple Sources, with decreasing amplitudes (usually) and increasing distances (and

thus phases).

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-11

Fabry-Perot: Transmission Peaks

460 470 480 490 5000

0.05

0.1

0.15

0.2

0.25

f, Freq., THz

T, T

ran

smis

sion

600nm650nm

10m

FSR

August 2007

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-12

Fabry-Perot: Resolution

460 470 480 490 5000

0.05

0.1

0.15

0.2

0.25

f, Freq., THz

T, T

ran

smis

sion

600nm650nm

10m

f

FSR

August 2007

Max

1

1

2

0.7 Max

GraphicalAnalysisof I

4/I

1

Denominator

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-13

Fabry-Perot: Extinction

460 470 480 490 5000

0.05

0.1

0.15

0.2

0.25

f, Freq., THz

T, T

ran

smis

sion

August 2007

600nm650nm

10m

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-14

Interference Magnitudes

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-15

Coherent Detection

0 20 40 60 80 1000

2000

4000

6000

8000

10000

Psig

Signal4kTB

QuantumNoise

0 20 40 60 80 1000

1

2

3

4

5

6x 10

4

Psig

Dotted Lines:Higher LO

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-16

Amplitude Modulator

Fiber Splitters

E/O Modulator

Input Output

Unused Alt.Output

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-17

Thin Films: Three Approaches

Sum the multiple Reflections

EAEB EC ED

Use Wave Equationand BoundaryConditions

E1

Network Approachwith Fresnel Coefficients

E2 E3 E4

E’1 E’2 E’3 E’4

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-18

Thin Films: Fresnel Coefficients

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-19

Coherence Issues

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-20

Etalon Transmission

Only Coherent Sourcescontribute to the fringes

Re (1/ )

Im (1/)

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-21

Thin Film Matrices (1)

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-22

Thin Film Matrices (2)

M=M1M2M3M4

M1M4

November 2007

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-23

High Reflection Stack (1)

November 2007

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-24

High Reflection Stack (2)

Phase=0

Sign of Reflectivity is invertedon going from High to Low Index.

LH H L

November 2007

July 2003 Chuck DiMarzio, Northeastern University 11140b-06-25

Anti-Reflection Coatings

nnQuarter-Wave Coating

Magnesium Flouride n=1.35is a close match, low cost and durable.

Phase=0

R

4%

Examples

Uncoated Glass

Violet Red

OneLayer

TwoLayers

ThreeLayers