Grigory Adamovsky and John Lekki

Glenn Research Center, Cleveland, Ohio

James A. Lock

Cleveland State University, Cleveland, Ohio

Detection, Evaluation, and Optimization of

Optical Signals Generated by Fiber Optic

Bragg Gratings Under Dynamic Excitations

NASA/TM—2002-211565

August 2002
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Grigory Adamovsky and John Lekki

Glenn Research Center, Cleveland, Ohio

James A. Lock

Cleveland State University, Cleveland, Ohio

Detection, Evaluation, and Optimization of

Optical Signals Generated by Fiber Optic

Bragg Gratings Under Dynamic Excitations

NASA/TM—2002-211565

August 2002

National Aeronautics and

Space Administration

Glenn Research Center
Acknowledgments

The authors express their gratitude to Mr. Bertram Floyd of Akima Corporation for his excellent technical support.

James A. Lock also acknowledges the partial support of NASA Grant NCC3–909.

Available from

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7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov/GLTRS
NASA/TM2002-211565 1 

Detection, Evaluation, and Optimization of Optical Signals Generated by 

Fiber Optic Bragg Gratings Under Dynamic Excitations

Grigory Adamovsky and John Lekki 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

James A. Lock 

Cleveland State University 

Department of Physics 

Cleveland, Ohio 44115 

Summary 

 The dynamic response of a fiber optic Bragg grating to mechanical vibrations is examined both 

theoretically and experimentally. The theoretical expressions describing the consequences of changes in 

the grating’s reflection spectrum are derived for partially coherent beams in an interferometer. The 

analysis is given in terms of the dominant wavelength, optical bandwidth, and optical path difference of 

the interfering signals. Changes in the reflection spectrum caused by a periodic stretching and 

compression of the grating were experimentally measured using an unbalanced Michelson interferometer, 

a Michelson interferometer with a non-zero optical path difference. The interferometer’s sensitivity to 

changes in dominant wavelength of the interfering beams was measured as a function of interferometer 

unbalance and was compared to theoretical predictions. The theoretical analysis enables the user to 

determine the optimum performance for an unbalanced interferometer. 

Introduction

 Fiber optic Bragg gratings (FOBG) have become a useful tool for sensing applications.1,2 They have 

been successfully used in the validation of the structural integrity of bridges and aircraft components,3–5

temperature and pressure measurements in various applications,6–8 and detection of leaks.9

 In addition to 

their use in static and quasi-static situations, dynamic changes in various system parameters have been 

measured using an FOBG.10,11 In each of these applications, changes in a parameter of the system under 

observation such as strain or temperature causes a corresponding change in the wavelength of light 

reflected by the grating. A suitable detector detects the changes in the wavelength and generates 

corresponding signals that are further analyzed by signal processing electronics. 

 This paper provides a theoretical foundation for analysis of optical signals reflected by FOBG’s under 

dynamic excitation. In particular, the performance of an unbalanced interferometer used as a spectrometer 

is analyzed, and the optical path difference producing maximal sensitivity to changes in the wavelength 

reflected by an FOBG is derived. An unequal optical pathlength interferometer was chosen because of its 

high sensitivity12 as well as its compact size and low weight. These features make unbalanced 

interferometers especially attractive for in-flight health monitoring of aerospace vehicles and their 

components. 

 The paper also provides a detail derivation of equations used to optimized the design of an 

unbalanced interferometer for the analysis of spectrally encoded signals. The equations are derived 

assuming the signals have a Gaussian power spectral density, and are consistent with a previously 
NASA/TM2002-211565 2 

reported analysis (Ref. 13). An experiment was devised to verify the theoretical dependence of the optical 

pathlength deference on the detected intensity. A preliminary version of this experiment has been reported 

elsewhere.14 The experimental data are compared to the predicted results and possible causes of the small 

discrepancy between the two are discussed. 

Interference of Two Optical Beams in an Interferometer 

 The light intensity produced by the interference of two electromagnetic beams depends on the 

intensities of the beams and their optical path difference, as well as on the dominant wavelength and 

bandwidth of each of the beams. Consider a beam entering an interferometer and having a dominant 

wavelength λ0 and full-width-at-half-maximum optical bandwidth ∆λ. After being equally divided by a 

beam splitter, the two component beams, each of intensity I0, traverse two different paths, of length l1 and 

l2, before being recombined at the detector. The optical path difference (unbalance) of the interferometer 

is n∆l = n(l1–l2). The beams exiting the interferometer interfere at a detector with the intensity 

( )















∆ 















I = I + n∆l γ n l λ

π

0

2 2 1 cos 0 , (1) 

where γ(n∆l) is the fringe visibility or fringe contrast reduction factor due to the spread in wavelength of 

each of the beams. We assume the spectral power density I(ν) of each beam is Gaussian14 and write the 

expression for I(ν) using conventional notations15 

































∆

− − ∆ =

2

0 exp 2 ln 2 2 ln 2 ( )

ν

ν ν

π ν

I ν , (2) 

where ν0 = c/λ0, c is the speed of light, and 

2

0

λ

λ

ν

∆ ∆ = c . (3) 

The fringe visibility factor is the modulus of the complex degree of coherence, i.e., the Fourier 

transformation of the spectral power density function,15

() () [ ] () ( ) 



















 



 ∆ = I = − = − ∫

∞ 2

0 2 ln 2 exp 2 exp

π ν τ

γ τ I ν I ν j πντ dν . (4) 

Using eq. (3) in eq. (4) we obtain 

( )











































 ∆ = −

2

2

0

2 ln 2 exp τ

λ

π λ

γ τ c , (5) 
NASA/TM2002-211565 3 

and substituting cτ = n∆l, the fringe visibility factor can be converted from the time domain to the spatial 

domain, 

( )













































∆ ∆ ∆ = −

2

2

0

2 ln 2

n l exp n l

λ

π λ

γ . (6) 

The intensity of the interference pattern at the detector then becomes 





























∆ ∆ = + ∆ −









































































2

0 2

0

2 ln 2 exp

0

2 I 2I 1 cos n l n l

λ

π λ

λ

π . (7) 

Figures 1 and 2 show I/I0 as a function of the optical path difference for the ranges from 0 mm to 5 mm 

and 2.10 mm to 2.15 mm, respectively for λ0 = 1300 nm and ∆λ = 0.3 nm. These values are typical of the 

reflection spectrum of a generic optical communication wavelength FOBG. Note that Fig. 1 contains 

oscillations that are too fast to resolve. Those oscillations of a cosine nature are clearly seen in Fig. 2. 

 As the parameters of the interfering beams change, the intensity of eq. (7) recorded by the detector 

changes as 

( ) ( ) λ λ λ λ ⋅ ∆

∂∆

∂ ⋅ +

∂

∂ ⋅ ∆ +

∂ ∆

∂ = d I d I d n l

n l

I dI 0

0

. (8) 

If the optical path difference of the interferometer and the optical bandwidth of the beams remain constant 

while the dominant wavelength changes, the coefficient 













































−





















































+















= ∆ ∆ ∆ ∆ ∆ ∆ ∆

∂

∂

2

2

0 0

2

0 0 0

2

0

0

0 2 ln 2 2 ln 2

2 2 1

4 sin n l n l cos n l exp n l n

I

I l

λ

π λ

λ

π λ π

λ

π

λ

π

λ λ λ (9) 

gives the sensitivity of the intensity recorded by the detector to changes in dominant wavelength of the 

beams. The first term in square brackets in eq. (9) is the derivative of the rapid oscillations in Fig. 2 and 

the second term is the derivative of the much more slowly varying overall modulation in Fig. 1. Figures 3 

and 4 show the wavelength sensitivity function ( )0 0 0

λ λ S ≡ ∂ I I ∂ as a function of the optical path 

difference n∆l for the values of λ0 and ∆λ used in Figs. 1 and 2. Since the second term in square brackets 

in eq. (9) is much smaller than the first term, the wavelength sensitivity function may be approximated by 

( )













































∆ ∆ ⋅ − ∆ ⋅

















∆ = ∆

2

0 2

0

2 ln 2 exp

0 0

2 sin

0

4

n l n l S n l n l

λ

π λ

λ λ

π

λ

π

λ . (10) 
NASA/TM2002-211565 4 

If we had assumed instead that the optical path difference and dominant wavelength remained constant 

while the optical bandwidth varied, a similar expression for the bandwidth sensitivity function 

( ) λ λ = ∂ ∂∆ ∆ 0 S I I could be derived. The sensitivity function 

0 λ S in eq. (10) possesses both 

rapid oscillatory dependence on n∆l and slower linear and exponential dependence. In order to more 

easily compare with the experimental results of Sec. 3, we consider the envelope of the sensitivity 

function, which we call the envelope sensitivity function, 













































∆ ∆ ⋅ − ∆ ∆ = ⋅

2

0 2

0

2 ln 2 exp 2

0

( ) 4 n l n l E n l

λ

π λ

λ

π λ . (11) 

The function λ0

E is graphed as a function of n∆l in Fig. 5 for λ0 = 1300 nm and ∆λ = 0.3 nm. It is clear 

from Fig. 5 that in order to be sensitive to dynamic changes in the dominant wavelength of the signal 

entering the interferometer, the interferometer must have a nonzero unbalance or optical path difference. 

 Setting the derivative of Eλ0

(n∆l) with respect to the optical path difference to zero, the path 

difference 

λ

λ

π ∆

∆ =

2

2 ln 2 0

max

n l

 (12) 

produces the maximum sensitivity of a given interferometer to changes in the dominant wavelength of the 

component beams. For example, for λ0 and ∆λ as in Figs. 1 to 4, we obtain n∆lmax = 2.111 mm. The 

dependence of Eλ0

(n∆l) and n∆lmax on the optical bandwidth is shown in Fig. 6 for =1300 nm. The figure 

has plots of three curves that describe Eλ0

(n∆l) and locations of its maximum that correspond to n∆lmax

for three different values of ∆λ. It could be also observed from eq. (12) that for a given wavelength λ0 a 

decrease in the optical bandwidth ∆λ leads to an increase in the value of n∆lmax. Because the bandwidth 

of optical signal is related to its temporal coherence, highly coherent optical signals would require the 

interferometer, in order for Eλ0

(n∆l) to reach its maximum, to have a very large unbalance n∆lmax. 

Practical considerations, in this case, should be used for the design of an appropriate interferometer. 

Since Eλ0

(n∆l) in Figs. 5 and 6 is slowly varying near n∆lmax, it is not necessary to operate the 

interferometer at exactly this unbalance in order to achieve good sensitivity. Rather, one can operate the 

interferometer within a relatively wide range of unbalance about n∆lmax and still obtain near-maximal 

sensitivity. This range of unbalance depends on the amount of deviation from the maximum sensitivity 

∆λ

2ln 2 1

max 4

e

E = (13) 

that can be tolerated. We define the fractional deviation of Eλ0

(n∆l) from Emax as 
NASA/TM2002-211565 5 

( )

max

max 0

E

E E n l

F

− ∆

= λ

. (14) 

To determine the range of unbalance for a given F we rewrite eq. (11), for simplicity, as: 





 



 = ⋅ − 2 2

0

E (x) x exp a x λ , (15) 

where 

2

0

4

λ

π

n l

x

∆ = ⋅ and = ⋅∆λ

8 ln 2

1

a . 

The Taylor series expansion of eq. (15) about 2

1

0 a

x = yields 

( ) ( ) ( ) ( ) ... 2

0 0

2

2 2

0

0 0

+ 















⋅ −  + 

 



 = + −

x x dx

x x d E

dx

dE E x E x x x , (16) 

where x0 is the value of x at which the curve Eλ0

(n∆l) reaches its maximum 

a e

E

2

1

max = . 

The resultant Taylor series in eq. (16) is approximated by retaining the first two nonzero terms, and after 

some algebraic simplifications we arrive at 

( ) 













 



 



 = − − 2

2

2

1 1 2

2

1

a

a

x

a e

E x . (17) 

Thus, for x in the vicinity of x0, the shape of the curve that describes Eλ0

(x) in Figs. 5 and 6 resembles a 

parabola. Substituting eq. (17) into eq. (14) and rearranging the terms, we obtain 

( ) ( )

( )

( )

2

0

2

0

2

2

0

0

2

1 2

x

x x

a

a x

E x

E x E x

F −

 = 

 



 = − − = . 

Replacing x and x0 by the original argument n∆l and n∆lmax, respectively, we yield 

max max n∆l − n∆l ≤ F n∆l (18) 

for the range of allowable unbalances about n∆lmax. If for example a sensitivity of the interferometer to 

within 1% of maximum is desired, an unbalance to within ±10% of the optimal value is acceptable. 
NASA/TM2002-211565 6 

Experiment 

 The apparatus of Fig. 7 was used to measure the sensitivity envelope Eλ0

(n∆l) of an unbalanced 

Michelson interferometer. A narrow portion of the ~30 nm wide spectrum emitted by a superluminescent 

diode with the central wavelength of 1300 nm was reflected by a commercially available 

telecommunications-grade FOBG and entered the interferometer consisting of a cube beam-splitter and 

two mirrors, one of which was mounted on a translation stage. At the beginning of the experiment, the 

mirrors were positioned at slightly different distances from the respective nearest faces of the beam 

splitter. The FOBG was affixed to a lead zirconate titanate piezoelectric element (PZT) and a 1 KHz 

periodic voltage applied to the PZT generated vibrations that stretched and compressed the grating. The 

dominant wavelength reflected by the FOBG changed as a function of time in response to the periodic 

stretching and compression. The resulting time dependent light intensity exiting the interferometer was 

recorded by a PIN diode and was sent to an electronic spectrum analyzer where the amplitude of the PIN 

diode current at the PZT driving frequency was recorded. 

 The amplitude of the detected signal at the PZT frequency was measured for a number of different 

values of interferometer unbalance. Changes in the optical path difference were produced by moving the 

mirror mounted on the translation stage. Each movement consisted of two steps. First, the mirror was 

moved approximately 250 μm from its previous position. Next, the mirror was moved about this new 

position several times with an increment of about 0.1 μm in order determine the maximum amplitude of 

the PIN diode current at the PZT driving frequency. This maximum occurs at the quadrature point of the 

interferometer, and is proportional to Eλ0

(n∆l). The measured wavelength sensitivity function is shown in 

Fig. 8. It is in good qualitative agreement with Fig. 5. Two similarly-shaped maximum sensitivity humps 

in Fig. 8 separated by a point of zero sensitivity are due to the fact that, in the process of changing the 

optical path difference (n∆l), the movable mirror translated through a point corresponding to a zero 

optical path difference. This point is denoted in Fig. 8 as Z0. An obvious difference between Fig. 5 and 

the data of Fig. 8, however, is that although Fig. 5 predicts a linear variation of the sensitivity function 

near the zero path difference point, the variation in Fig. 8 appears to be roughly quadratic. In addition, 

the experimental data is more rapidly varying in the vicinity of the relative maximum than is the 

theoretical prediction. In particular, an optical path length 10% greater or less than the optimal values in 

Fig. 8 produces a sensitivity from 7% to 8% less than maximum, whereas the decrease in the theoretical 

sensitivity curve of eq. (11) is only 1%. 

 To analyze the data of Fig. 8, we assumed the spectrum reflected by the Bragg grating was Gaussian 

and the optical bandwidth of the reflected spectrum was independent of the dominant wavelength. This 

last assumption has been found to be accurate for static measurements,16 and we presume it is also the 

case for dynamic excitations of 1 KHz. The dominant wavelength of the reflected spectrum of the 

unstretched grating was determined using an optical spectrum analyzer to be λ0 = 1310.2 nm ± 0.1 nm, 

and agreed with the value λ0 = 1310.17 nm ± 0.005 nm measured at the time of fabrication by the grating 

manufacturer. The optical bandwidth measured by the manufacturer at that time was ∆λ = 0.192 nm ±

0.002 nm. The value of max n∆l corresponding to this optical bandwidth via eq. (12) is 3.351 mm. 

 In Fig. 8, the distances between the zero sensitivity point Z0 and points of maximum sensitivity to the 

left and to the right of that point are about 1950 μm ± 25 μm and 2100 μm ± 25 μm, respectively. 

Because of the double pass optical configuration of the Michelson interferometer, these distances are 

equal to half the optical path difference required to achieve maximum sensitivity of the interferometer to 

changes in dominant wavelength. Thus, the corresponding values of the optimal path difference n∆lmax

are about 3.9 mm ± 0.05 mm and 4.2 mm ± 0.05 mm and the corresponding values of optical bandwidth 

via eq. (12) are ∆λ = 0.165 nm ± 0.002 nm and 0.153 nm ± 0.002 nm, respectively. The average of these 
NASA/TM2002-211565 7 

two values of the optical bandwidth is 17% below that quoted by the manufacturer. A least-squares fit of 

the results of Fig. 8 to the functional form of eq. (11) was not attempted. 

Discussion 

 The good qualitative agreement between the theoretical sensitivity function and the experimental data 

underscores the correctness of our approach. The relatively small difference between our value of the 

optical bandwidth obtained experimentally, along with our theoretical prediction of the maximum 

sensitivity (eq. (12)), and the value provided by the manufacturer illustrates the basic validity of our 

theory. There are, however, a number of possible sources of error that could have contributed to the 17% 

difference. In particular, the interferometer was not locked at the quadrature position. As a result, slow 

thermal drifts of the optical path difference or building vibrations could have introduced errors into the 

measurements. A solution to this would be construction of an unbalanced interferometer on a chip with 

integrated compensation. Also, in addition to assuming the reflection spectrum of the FOBG was 

Gaussian and the optical bandwidth was independent of dominant wavelength for dynamic excitations, we 

also ignored the variation in the ~30 nm wide superluminescent diode spectrum over the ~0.2 nm wide 

bandwidth of the FOBG reflection spectrum. This effect, however, is expected to be small. 

 The slowness of variation of the theoretical sensitivity curve in the vicinity of the relative maximum 

works to one’s advantage when designing an FOBG sensor system. The value of optimum optical 

difference n∆lmax obtained experimentally (Fig. 8) varies roughly from 16% to 25% of the unbalance 

n∆lmax = 3.351 mm, the value predicted by eq. (12) using the FOBG manufacturer’s optical bandwidth 

measurement. As a result, one can use eq. (12) as a reasonable estimate for setting the interferometer 

unbalance since small deviations of the actual optimal unbalance point from the prediction of eq. (12) 

produce correspondingly small decreases in the sensitivity of the interferometer to changes in the 

dominant wavelength of the input signal. 

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microsystems with photonic element and their applications to aerospace platform,” in Smart 

Structures and Devices, D.K. Sood, R.A. Lawes, and V.V. Varadan, (eds.), Proc. SPIE 4235, 

407–418 (2001). 

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interferometric wavelength-shift detection,” Electron. Lett. 28, 236–238 (1992). 

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dynamic pressure sensing,” in Sensory Phenomena and Measurement Instrumentation for Smart 

Structures and Materials, E. Udd, (ed.), Proc. SPIE 4328, 151–159 (2001). 

15. J.W. Goodman, Statistical Optics, Wiley, New York, 1985. 

16. S.M. Melle, K. Liu, and R.M. Measures, “Practical fiber optic Bragg grating strain gauge system,” 

Appl. Opt. 32, 3601–3609 (1993). 

0.0 1.0 2.0

Optical path difference, nl, mm

3.0 4.0 5.0

Realtive intensity, I/I0 0

1

2

3

4

Figure 1.—Intensity of two interfering optical beams of 

 dominant wavelength 0 = 1300 nm and FWHM optical 

 bandwidth ∆= 0.3 nm as a function of optical path

 difference.
NASA/TM2002-211565 9 

2.09 2.10 2.11 2.12 2.13

Optical path difference, nl, mm

2.14 2.15

Realtive intensity, I/I0 0

1

2

3

4

Figure 2.—Fine structure of Figure 1 for n∆l between 

 2.10 mm and 2.15 mm

0.0 1.0 2.0

Optical path difference nl, mm

3.0 4.0 5.0

Wavelength sensitivity

function, S

0 (n

l) 106

–10

–5

5

10

0

Figure 3.—Interferometer sensitivity to changes in 

 dominant wavelength as a function of optical path

 difference for 0 = 1300 nm and ∆= 0.3 nm.
NASA/TM2002-211565 10 

2.09 2.10 2.11

Optical path difference, nl, mm

2.12 2.13 2.14 2.15

Wavelength sensitivity

function, S

0 (n

l) 106

–10

–5

5

10

0

Figure 4.—Fine structure of the wavelength sensitivity

 function of Figure 3 for n∆l between 2.10 mm and 

 2.15 mm.

0.0 1.0 2.0

Optical path difference, nl, mm

3.0 4.0 5.0

Wavelength envelope sensitivity

function, E

0 (n

l) 106

0

4

8

10

6

2

Figure 5.—Wavelength sensitivity envelope E0

 as a function of the optical path difference for 

 0 = 1300 nm and ∆= 0.3 nm.
NASA/TM2002-211565 11 

0.0 1.0 2.0

Optical path difference, nl, mm

3.0 4.0 5.0

Wavelength envelope sensitivity

function, E

0 (n

l) 106

0

2

6

12

4

8

10

= 0.25 nm

= 0.30 nm

= 0.35 nm

Figure 6.—Wavelength sensitivity envelope E0

 as a function of the optical path difference for 

 0 = 1300 nm and ∆= 0.25, 0.30, and 0.35 nm.

Broadband

light

source Stationary

mirror

Moveable

mirror

Cube beam splitter

V = Vo cost

Beam

collimating

optics

Spectrum

analyzer

FO coupler PZT actuator

FO Bragg grating

Figure 7.—Experimental apparatus for measuring the

 wavelength sensitivity envelope function.
NASA/TM2002-211565 12

Z0

0 1000 2000 3000

Micrometer readout, microns

80007000600050004000

Spectral power amplitude,

nVrms2

0

500

600

700

800

200

300

400

100

Figure 8.—Measured power at the PZT driving fre-

 quency as a function of the micrometer readout

 of the movable arm of the interferometer.
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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18

Detection, Evaluation, and Optimization o f Optical Signals Generated by Fiber

Optic Bragg Gratings Under Dynamic Excitations

Grigory Adamovsky, John Lekki, and James A. Lock

Photonics; Gratings; Spectra; Optical measuring instruments; interferometers; Bragg

gratings; Fiber optics; Optical sensors; Wavelength detection; Unbalanced interferometer

Unclassified -Unlimited

Subject Categories: 06, 19, and 35 Distribution: Nonstandard

Grigory Adamovsky and John Lekki, NASA Glenn Research Center; James A. Lock, Cleveland State University,

Cleveland, Ohio 44115. Responsible person, Grigory Adamovsky, organization code 5520, 216–433–3736.

The dynamic response of a fiber optic Bragg grating to mechanical vibrations is examined both theoretically and experimentally. The theoretical expressions describing the consequences of changes in the grating’s reflection spectrum are

derived for partially coherent beams in an interferometer. The analysis is given in terms of the dominant wavelength,

optical bandwidth, and optical path difference of the interfering signals. Changes in the reflection spectrum caused by a

periodic stretching and compression of the grating were experimentally measured using an unbalanced Michelson

interferometer, a Michelson interferometer with a non-zero optical path difference. The interferometer’s sensitivity to

changes in dominant wavelength of the interfering beams was measured as a function of interferometer unbalance and

was compared to theoretical predictions. The theoretical analysis enables the user to determine the optimum performance

for an unbalanced interferometer.