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Blue Road Research
Applications of the Fiber Optic Sagnac Interferometer
Blue Road Research
Session6
Session8
Session7
Session1
Session5
Session4
Session3
Session2
Blue Road Research
Sagnac Interferometer
Blue Road Research Session 3, Page 1
Part I
Rotation Sensing
Part II
Quasi-Static and Time Varying Sensing
Blue Road Research
Rotation Sensor Characteristics
Blue Road Research Session 3, Page 2
Rate Gyro
= KV
= Rotation rate
K = Scale factor
V = Output signal
Blue Road Research
Definition of Terms
Blue Road Research Session 3, Page 3
• Rate integration gyro - Integrates angular rate
to get angular output
• Fixed bias - Output rotation rate with zero
input rotation rate
• Bias drift - Change in output rate over time
(temperature, wear, etc.)
• Scale factor - Linearity and hysteresis
Blue Road Research
Rotation Sensor Performance Factors
Blue Road Research Session 3, Page 4
• Sensitivity» Lowest measurable rotation rate
• Spectral noise characteristics
• Dynamic range
• Turn on time
Blue Road Research
The Sagnac Effect
Blue Road Research Session 3, Page 5
R
cwccwcw path: 2R+ RL/c
ccw path: 2R - RL/c
Net path difference: 2RL/c
ZR = 2RL/(c)
Blue Road Research
The Sagnac Effect
Blue Road Research Session 3, Page 6
cwccw
ZFcw = (Fo+F)(Ln/c)
ZFccw = FoLn/c
ZF = FLn/c
Setting ZR = ZFRenders F = 2R/n
F
Blue Road Research
Fiber Optic Gyro Competition
Blue Road Research Session 3, Page 7
• Mechanical Gyros» Advantages
• Established industrial base
» Disadvantages• Bearing wear
• Start-up time
• Reliability
Blue Road Research
Fiber Optic Gyro Competition
Blue Road Research Session 3, Page 8
• Ring Laser Gyros» Advantages
• Established industrial base
• Replaced mechanical gyros for navigation
» Disadvantages• Mechanical dither
• Ultraclean vacuum tube technology
Blue Road Research
Fiber Optic Gyro Tradeoffs
Blue Road Research Session 3, Page 9
• All solid state
• Packing flexibility
• Potentially very long lifetimes
• Small size
• Low cost
Blue Road Research
Ring Laser Gyro Assembly
Blue Road Research Session 3, Page 10
Anode
Cathode
Partially transparentmirror
Blue Road Research
Ring Laser Gyro Readout Optics
Blue Road Research Session 3, Page 11
cw beam ccw beam
Rolling fringe pattern
Splitdetector
Blue Road Research
Ring Laser Lock In Zone
Blue Road Research Session 3, Page 12
Blue Road Research
Open Loop Fiber Optic Gyro
Blue Road Research Session 3, Page 13
Detector
Light sourcePolarizer
Fiber optic loop
Modulator
Blue Road Research
Detection Signals
Blue Road Research Session 3, Page 14
Blue Road Research
Open Loop Fiber Optic Gyro Output
Blue Road Research Session 3, Page 15
Blue Road Research
Closed Loop Fiber Optic Gyro
Blue Road Research Session 3, Page 16
Detector
Light sourcePolarizer
Fiber optic coil
Modulator
Frequency shifter
Oscillator
IntegratorVCO
Blue Road Research
Scale Factor
Blue Road Research Session 3, Page 17
Open loop fiber gyro
= ZR[c/2RL]
Closed loop fiber gyro
= F(n/2R)
Dependence on wavelength
Blue Road Research
Correction of Scale Factor
Blue Road Research Session 3, Page 18
-F F
Blue Road Research
First Closed Loop Fiber Optic Gyro
Blue Road Research Session 3, Page 18A
Blue Road Research
First Solid State Fiber Optic Gyro
Blue Road Research Session 3, Page 18B
Blue Road Research
2.5” 1980 Fiber Optic Gyro
Blue Road Research Session 3, Page 18C
Blue Road Research
1982 Oil Drilling FOG
Blue Road Research Session 3, Page 18C
Blue Road Research
1983 Closed Loop FOGs
Blue Road Research Session 3, Page 18C
Blue Road Research
First Honeywell Production FOG
Blue Road Research Session 3, Page 18C
Blue Road Research
Litton (NG) FOG IMU
Blue Road Research Session 3, Page 18C
Blue Road Research
The Open Loop Fiber Optic Gyro Marketplace
Blue Road Research Session 3, Page 19
• Automobiles and trucks
• Pointing and tracking
• Robot navigation
• Aircraft attitude control
• Short range air navigation
Blue Road Research
The Closed Loop Fiber Optic Gyro Marketplace
Blue Road Research Session 3, Page 20
• Medium to long range aircraft
• Spacecraft
• Missiles
• Launch vehicles
• Platforms making rapid turns
Blue Road Research
FOG Manufacturers
Blue Road Research Session 3, Page 21
• Hitachi» Closed loop automotive and low cost FOGs
» Delivered over supports high end automobiles like Lexus navigators, thousands of units per year
• Japan Aviation Electronics
» Intermediate grade FOGs for Japan self-defense force,
variety of commercial applications, soccer field grass
cutters, cleaning robots, mini-crop spraying helicopters…
Blue Road Research
FOG Manufacturers (continued)
Blue Road Research Session 3, Page 22
• Honeywell
» Supplies 3 axis FOG navigator for German Dornier commuter
aircraft, 777 back up navigator
» Leader in commercial aircraft navigation and space based FOG
• Northrup
» 3 axis closed loop AHRS units with 0.1-1.0 deg/hr
performance to full military specifications
» Working on full navigation grade 0.01 deg/hr FOGs targets
competing with Honeywell on commercial aircraft
Blue Road Research
FOG Manufacturers (continued)
Blue Road Research Session 3, Page 24
• Mitsubishi Precision Company
» Flight tested first space based FOG on Feb. 22, 1990
aboard S-520-11 rocket
» Makes both open and closed loop FOGs
• Photonetics
» Closed loop 0.1 deg/hr FOGs to support ship
navigation
Blue Road Research
Estimated FOG Market Size
Blue Road Research Session 3, Page 26
• 1995 - $50,000,000
• 2000 - $100,000,000
• 2005 - $150,000,000
• Combination of commercial
and military/government
funded markets
Blue Road Research
Fiber Optic Gyro References
Blue Road Research Session 3, Page 27
• S. Ezekiel and H.J. Arditty, Editors, “Fiber Optic Rotation
Sensors”, Springer-Verlag, New York, 1982.
• E. Udd, Editor, “Fiber Optic Gyros: 10th Anniversary
Conference”, SPIE Proc., Vol. 719, 1986.
• R.B. Smith, Editor, “Selected Papers on Fiber Optic
Gyros”, SPIE Milestone Series, Vol. MS 8, 1989.
Blue Road Research
Fiber Optic Gyro References (continued)
Blue Road Research Session 3, Page 28
• S. Ezekiel and E. Udd, “Fiber Optic Gyros: 15th
Anniversary Conference”, SPIE Proc., Vol. 1585, 1991.
• H. Lefevre, “The Fiber Optic Gyroscope”, Artech House,
1993.
• W.K. Burns, Editor, “Optical Rotation Sensing”,
Academic Press, 1994.
Blue Road Research
Part II
Blue Road Research Session 3, Page 29
Quasi-Static and Time Varying Sensing Using the
Fiber Optic Sagnac Interferometer
Blue Road Research
Time Varying Environmental Effects-Acoustics
Blue Road Research Session 3, Page 30
Detector
Light source
Center
AcousticWave
Blue Road Research
Optimized Fiber Coil Configurations
Blue Road Research Session 3, Page 31
Variablecoating
ShieldedFiber
Blue Road Research
Effects of Shielding/Position
Blue Road Research Session 3, Page 32
Induced phase shift unshielded coil
Induced phase shiftshielded coil
Blue Road Research
Time Varying Effects
Blue Road Research Session 3, Page 33
dyyFiber Coil Length L
Response of Fiber G(y,P)
..y n
c
.( )L y n
cd
d tP
R( )P( )t d
0
L
y.G( ),y Pd
d t.P
.( ).2 y L n
c
Blue Road Research
Example Cases
Blue Road Research Session 3, Page 34
I. G(y,P) = A = Constant, R[P(t)] = 0
II. G(y,P) = 0, 0<y<L/2
G(y,P) = A = Constant, L/2<y<L,R[P(t)] = AnL2/4c]dP/dt
for P = Bsin(t)R[P(t)] = [ABnL2/4c] sin(t)
Blue Road Research
Quasi-Static Sensing-Strain
Blue Road Research Session 3, Page 35
Detector
Light source
Frequency shifter
Fo+F
FoPolarizer
Blue Road Research
Quasi-Static Sensing
Blue Road Research Session 3, Page 36
ZF = F(Ln/c)
Suppose ZF = Constant
0 = dF(Ln/c)+FdL(n/c)
dF/F = -dL/L
Blue Road Research
Sagnac Strain Sensor Cabling
Blue Road Research Session 3, Page 37
Detector
Light source
Frequency shifter
Fiber cable
Blue Road Research
Earth Movement Detection System
Blue Road Research Session 3, Page 38
Fault line
Fiber cables
Earthmovement
Blue Road Research
Monitoring Oil Platform Motion
Blue Road Research Session 3, Page 39
Fiber optic strain sensor
Blue Road Research
Stress on Power Lines
Blue Road Research Session 3, Page 40
Fiber strain sensor surroundedby conductive cable elements
Blue Road Research
Distributed Sagnac Sensors
Blue Road Research Session 3, Page 41
• Changing modes from time varying to quasi-static
• Interlaced Sagnac loops
• Combination of the Sagnac and Mach-Zehnder
interferometers
Blue Road Research
Changing Mode Distributed Sensor
Blue Road Research Session 3, Page 42
Detector
Light source
Frequency shifter
Polarizer
Fo+F
Fo
Switches
Blue Road Research
Interlaced Sagnac Loops
Blue Road Research Session 3, Page 43
Detector, 1
Light source, 1
Detector, 2
Light source, 2
Position
I
WDMs
Blue Road Research
Sagnac/Mach-Zehnder
Blue Road Research Session 3, Page 44
Detector, 1
Light source, 1
Sagnac
Mach-Zehnder
Detector, 2Light source, 2
Blue Road Research
Detection of Leaks in Pressurized tanks
Blue Road Research Session 3, Page 45
Sagnac distributed sensor
Fiber coil
Blue Road Research
Coherence Length
Blue Road Research Session 3, Page 46
L1
L2
Light source Detector
Blue Road Research
Basic Sagnac Interferometer Secure Communication System
Blue Road Research Session 3, Page 47
Detector
Light source
Data out
Receiver Transmitter
Phasemodulator
Data in
Blue Road Research
Basic Intrusion Scenario
Blue Road Research Session 3, Page 48
Detector
Light source
Data out
Receiver Transmitter
Phasemodulator
Data in
LA LB
Blue Road Research
References for Part II
Blue Road Research Session 3, Page 49
Acoustic Sensors
• E. Udd, “Fiber Optic Acoustic Sensors Based on the
Sagnac Interferometer”, SPIE Proc., Vol. 425, p. 90,
1983.
• K. Krakenes and K. Blotekjaer, “Sagnac Interferometer
for Underwater Sound Detection: Noise Properties”,
Optics Letters, Vol. 14, p. 1152, 1989.
Blue Road Research
References for Part II (continued)
Blue Road Research Session 3, Page 50
Strain Sensors
• R.J. Michal, E. Udd, and J.P. Theriault, “Derivative Fiber
Optic Sensors Based on the Phase Nulling Optical Gyro”,
SPIE Proc., Vol. 719, p. 150, 1986.
• E. Udd, R. Blom, D. Tralli, E. Saaski and R. Dokka,
“Application of the Sagnac Interferometer Based Strain
Sensor to an Earth Movement Detection System”, SPIE
Proc., Vol. 2191, 1994.
Blue Road Research
References for Part II (continued)
Blue Road Research Session 3, Page 51
Spectrometers and Scale Factor
• E. Udd, “Usage of Dispersive Effects for Scale Factor
Correction in the Fiber Optic Gyro, SPIE Proc., Vol.
1585, p. 255, 1991.
Distributed Sensing
• E. Udd, “Sagnac Distributed Sensor Concepts”, SPIE
Proc. 1586, p. 46, 1991.
Blue Road Research
References for Part II (continued)
Blue Road Research Session 3, Page 52
Secure Communication
• E. Udd, “Secure Communication System”, U.S. Patent
5,223,967, June29, 1993.
• E. Udd, “Secure Communication System, U.S. Patent
5,274,488, December 28, 1993.
Blue Road Research
The Mach-Zehnder and Michelson Interferometers and Multiplexing
Blue Road Research
Session6
Session8
Session7
Session1
Session5
Session4
Session3
Session2
Blue Road Research
Interference requirements:
Polarization state of two beams identical
Path length difference < Coherence length:
Interferometer Basics
nL nL2
cLL
Detector
2
sin2 1010 IIIIID
Light Source
Detector
Light Source
2
cL
Mach-Zehnder Michelson
Blue Road Research
Flexible Geometries, High Sensitivity
Blue Road Research Session 4, Page 1
Blue Road Research
Basic Elements of the Mach-Zehnder Interferometer
Blue Road Research Session 4, Page 2
Light source / coupler module
Homodyne demodulator
Transducer
Blue Road Research
Grating Based Homodyne Demodulator
Blue Road Research Session 4, Page 3
Dual input fibers
Grin lens
Interferencepattern
Split photomasked detector,sine and cosine outputs
Blue Road Research
Quadrature Demodulation Electronics
Blue Road Research Session 4, Page 4
Blue Road Research
The Signal Fading Problem
Blue Road Research Session 4, Page 5
Blue Road Research
Active Feedback
Blue Road Research Session 4, Page 6
Light source / coupler module
Transducer
Coupler
Amplifier/integrator
Detectors
Modulator
Blue Road Research
3 by 3 Coupler
Blue Road Research Session 4, Page 7
1
2
3
Light source
120 degree offsets between outputs
Blue Road Research
Phase Generated Carrier
Blue Road Research Session 4, Page 8
Light source
L1
L2
Current driver
, 2
OutputF(L1-L2)n/c
Blue Road Research
The Michelson Interferometer
Blue Road Research Session 4, Page 9
Light sourceCoupler
L1
L2
Mirrors
Detector
Blue Road Research
SMARTEC Strain Sensors
Blue Road Research Session 4, Page 10
Blue Road Research
SMARTEC Strain Sensors (continued)
Blue Road Research Session 4, Page 11
Coupleur
Photo- LED1300nm
A/D
Ampli
PC
LED1300nm
A/D
Ampli
PCInternal
LED1300nm
Mirr
ors
Structure under test Coupleur
Portable PC
diode
A/D
filtre Ampli
Mirror
Internal PC
Blue Road Research
Coatings
Blue Road Research Session 4, Page 12
Glass fiber
Secondary compliantcoating
Primarycoating
Pressure
Blue Road Research
Compliant Mandrels
Blue Road Research Session 4, Page 13
Pressure
Optical fiber
Blue Road Research
Transducer Materials
Blue Road Research Session 4, Page 14
• Acoustics» Nylon
• Magnetic fields» MetGlass, Nickel
• Electric fields» PVF
• Seismic/Vibration» Soft rubber
Blue Road Research
Transducer Geometries
Blue Road Research Session 4, Page 15
Hollow mandrel
Strip
Blue Road Research
Seismic/Vibration Sensor
Blue Road Research Session 4, Page 16
Fiber coil
Soft rubbermandrel
Seismicmass
Blue Road Research
Serial Layout
Blue Road Research Session 4, Page 17
Blue Road Research
Seismograph Layout
Blue Road Research Session 4, Page 18
Line arrays
Seismograph recorders
Transmitter
Blue Road Research
Multiplexing Techniques
Blue Road Research Session 4, Page 19
• Time division
• Frequency division
• Wavelength
• Coherence
• Polarization
• Spatial
Blue Road Research
Time Division Multiplexing
Blue Road Research Session 4, Page 20
Light source
Signal processingelectronics
Detector
Microbendfiberattachment
Pipe joints
Blue Road Research
Frequency Division Multiplexing
Blue Road Research Session 4, Page 21
Light source
F1
F2
F3
Detector
L1 L2 L3
L L L
Frequency chirpedlight source
Blue Road Research
Wavelength Division Multiplexing
Blue Road Research Session 4, Page 22
Light source
1
2
34
2 3 4
1
Wavelength divisionmultiplexer/detectors
Blue Road Research
Coherence Length
Blue Road Research Session 4, Page 23
L1
L2
Light source Detector
Signal falls exponentially to zero when
|L1-L2|>Lc
Blue Road Research
Coherence Multiplexing
Blue Road Research Session 4, Page 24
L1 L2Light source
L L
L
L
L1
L2
Detector 2
Detector 1
Blue Road Research
Polarization Multiplexing
Blue Road Research Session 4, Page 25
Light source
Detector 1
Detector 2
Polarizing beamsplitter
Evanescent sensors
Polarization states
Blue Road Research
Spatial Multiplexing
Blue Road Research Session 4, Page 26
Light sources
S1
S4
S3
S2
1
2
S1(1),S3(2)
S2(1),S4(2)
DetectorsUnbalanced interferometers
Blue Road Research
Extensions of Spatial Multiplexing
Blue Road Research Session 4, Page 27
1
2
3
J
Sou
rces
11
J1
1K
JK
1 32 KDetectors
Blue Road Research
Distributed Fiber Sensors
Blue Road Research Session 4, Page 28
• Optical time domain reflectometers» Rayleigh
» Raman
» Brillouin
» Fluorescence
Blue Road Research
Raman Scattering
Blue Road Research Session 4, Page 29
Blue Road Research
Brillouin Scattering
Blue Road Research Session 4, Page 30
Wavelength
Brillouinlines
Incident laser wavelength
Blue Road Research
Summary
Blue Road Research Session 4, Page 31
• Mach-Zehnder and Michelson
interferometers
» High sensitivity
» Excellent multiplexing potential
• Distributed sensors may be used to
cover wide areas with less sensitivity
Blue Road Research
Fiber Optic Smart Structures for Natural, Civil, and Aerospace Applications
Blue Road Research
Session6
Session8
Session7
Session1
Session5
Session4
Session3
Session2
Blue Road Research
Fiber Optic Smart Structures
Blue Road Research Session 5, Page 1
Part I
Fundamental concepts and technology
Part II
Applications
Blue Road Research
Access to Space/Rockets
Blue Road Research Session 5, Page 2A
Blue Road Research
Space Platforms
Blue Road Research Session 5, Page 2B
Blue Road Research
Transport Aircraft SHM
Blue Road Research Session 5, Page 2C
Blue Road Research
Transport Aircraft SHM
Blue Road Research Session 5, Page 2D
Blue Road Research
Military Aircraft SHM
Blue Road Research Session 5, Page 2E
Blue Road Research
Aspects of Fiber Optic Smart Structures
Blue Road Research Session 5, Page 2
• Smart manufacturing
• Nondestructive testing
• Health and damage
assessment
• Control systems
Blue Road Research
Advantages of Fiber Optic Sensors for Smart Structures
Blue Road Research Session 5, Page 3
• Lightweight/nonobtrusive
• All passive
• EMI resistant
• Environmental ruggedness
• Multiplexing potential
Blue Road Research
Fiber Optic Smart Structure System
Blue Road Research Session 5, Page 4
Composite panelwith multiplexedfiber sensors
Fiber optic link toactuator system
Control system-Performance-Health
Optical/ElectronicProcessor
Environmental effect
Blue Road Research
Fiber Optic Smart Structure System Technology
Blue Road Research Session 5, Page 5
• Fiber/material issues
• Fiber optic sensors
• Multiplexing
• Signal processing
• System architecture
Blue Road Research
Fiber/Material Issues
Blue Road Research Session 5, Page 6
• Fiber coatings
• Ingress/egress
• Connectors
• Wide range of materials
» Carbon epoxy, polyimides
» Aluminum, titanium
» Ceramics, carbon-carbon
Blue Road Research
Fibers Embedded in Carbon Epoxy
Blue Road Research Session 5, Page 6A
Blue Road Research
Polyimide Coating in Thermoplastic
Blue Road Research Session 5, Page 6B
Blue Road Research
Fibers in Titanium Metal Matrix
Blue Road Research Session 5, Page 6C
Blue Road Research
Fiber Optic Sensors-Issues
Blue Road Research Session 5, Page 7
• Parameters to be sensed
» Strain, temperature, viscosity, etc.
• Gauge length
• Number of sensors per string
• Fiber/sensor diameter
• Dynamic range/sensitivity
Blue Road Research
References for Part I
Blue Road Research Session 5, Page 28
• Fiber Optic Sensors: An Introduction for Engineers and Scientists, Edited by Eric Udd, Wiley, 1991 (Chapter 14).
• SPIE - The International Optical Engineering Society Proceedings, Volumes on Fiber Optic Sensors and Smart Structures (call for current catalogue 206-676-3290).
Blue Road Research
Part II
Blue Road Research Session 5, Page 29
Applications of Fiber Optic Smart Structures
Blue Road Research
Smart Manufacturing
Blue Road Research Session 5, Page 30
Temperaturesensor demodulator
Composite partAutoclave
Degree ofcure monitor(Fluorescence)
Autoclave controller
Blue Road Research
Nondestructive Evaluation
Blue Road Research Session 5, Page 31
Fiber optic sensor acoustic demodulator
Pulsed laser
External acoustic source
Composite part
Blue Road Research
Damage Assessment in Composite Panel
Blue Road Research Session 5, Page 32
Light sources
Detectors
Damage site
Beamsplitters
Microbend sensitive fiber
Blue Road Research
Monitoring Bond Line
Blue Road Research Session 5, Page 33
Light source
Tension
Fixed structure
Detector
Bond area
Blue Road Research
Embedded Fiber Sensors for a Large Structure
Blue Road Research Session 5, Page 34
• Large numbers» Thousands to tens of thousands
• Cost» System must be a small fraction of
platform cost
• Must add substantial value» Safety, reliability, maintainability
Blue Road Research
Overall Architecture
Blue Road Research Session 5, Page 35
• First layer» Distributed sensors
• Localize damage• Measure ambient conditions
• Second layer» Multiplexed discrete sensors
• Reconfigurable• Detailed assessment
Blue Road Research
First Layer Coverage
Blue Road Research Session 5, Page 36
• Low cost distributed sensors or long
gauge length sensor networks
• Low to medium accuracy
• Temperature distribution
• Acoustics
• Wide area strain changes
Blue Road Research
First Layer Coverage Candidates
Blue Road Research Session 5, Page 37
• Blackbody sensor - networks• Microbend sensor - networks• Distributed sensors
» Rayleigh» Raman» Brillouin» Fluorescence» Interleaved interferometric
Blue Road Research
Second Layer Coverage
Blue Road Research Session 5, Page 38
• Discrete, high performance sensor
arrays
• Detailed damage/health assessment of
designated area
• Reconfigurable to minimize
processing requirements
• Redundant
Blue Road Research
Second Layer Sensor Criteria
Blue Road Research Session 5, Page 39
• Single Point of Ingress/Egress
• Amplitude independence
• Compatible with in-line multiplexing
• Low cost and manufacturable
• No larger than fiber diameter
Blue Road Research
Second Layer Sensor Candidates
Blue Road Research Session 5, Page 40
• Fabry-Perot etalons
• Fiber gratings
Blue Road Research
Modular Architecture
Blue Road Research Session 5, Page 41
Sensor string
Vehicle healthmanagement bus
Subsystem signalprocessor
Demodulator
Data formatterand transmitter
Optical switch
Fiberopticlink
Blue Road Research
Avionics Example
Blue Road Research Session 5, Page 42
Pilot
Display
Processor
Avionics bus
Vehicle health management bus
Distributionsystem
Blue Road Research
Civil Structures
Blue Road Research Session 5, Page 43
Buildings
Police, fire,maintenance
Bridges
Blue Road Research
Natural Structures
Blue Road Research Session 5, Page 44
• Geophysical fault line monitors
• Earth movement around oil platforms
• Strain induced by earth movement on
high voltage lines
Blue Road Research
Manufacturing
Blue Road Research Session 5, Page 45
• Environmental Control
» Water and air chemistry
• Process control
» Oven temperature, pressure
» Valve position, liquid levels
» Flow rate
• Health monitoring
» Vibration
Blue Road Research
Medicine
Blue Road Research Session 5, Page 46
• Chemistry of the blood» Oxygen content» Dosage levels
• Internal inspection» Blood vessels» Intestines» Stomach
• Power delivery• Potential for artificial limbs,
nerves
Blue Road Research
References for Part II
Blue Road Research Session 5, Page 47
• Fiber Optic Sensors: An Introduction for Engineers and Scientists, Edited by Eric Udd, Wiley, 1991.
• Eric Udd, “Fiber Optic Smart Structures”, Chapter 14.