TECHNICAL NOTE

Distributed measurement of the complexmodulation of a photoinduced Bragg gratingin an optical fiber

Mark Froggatt

A method of measuring the complex modulation of a Bragg grating is derived from a one-dimensionalmodel of light propagating in an optical fiber. Interference fringes between the Bragg grating and areference air-gap reflector are measured, and a Fourier transform of the interference fringes generatedas a laser is swept through the wavelength is used to compute the complex modulation function of theBragg grating over a restricted domain. Supporting data, taken by temperature tuning a distributedfeedback diode laser, are shown.

1. Introduction

Much of the analytical research concerning Bragggratings has been directed at determining the spec-tral response of a known grating profile.1 This re-search is based on the coupled-mode theory andcomes primarily from early research byMarcuse2 andHill et al.3,4 In some early research it was noted thatfor weakly reflective gratings the problem of deter-mining the reflected spectrum from the grating struc-ture could be a simple Fourier transform.4 In themethod described here an inverse Fourier transformis applied to the spectral response of a fiber-opticsystem to determine the complex modulation of aBragg grating. The coupled equations for a Bragggrating in an optical fiber are rederived by the use ofsome simplifying assumptions and are found to be inagreement with Hill.4 An experiment is then per-formed to confirm the theory and validate the as-sumptions.The advent of side-written Bragg gratings5 and

point-by-point written gratings6 makes the ability tomeasure the spatial location and distribution ofBragg gratings attractive as a means of evaluatingthe writing process. Researchers have used low-coherence path-matching interferometry7 to measurethe spatial distribution of Bragg gratings in opticalfiber. The method described here uses a highly co-herent ~meters of coherence length! tunable source.

The author is with Langley Research Center, National Aeronau-tics and Space Administration, Hampton, Virginia 23681-0001.Received 19 January 1996; revised manuscript received 24 May

1996.

5162 APPLIED OPTICS y Vol. 35, No. 25 y 1 September 1996

2. Theory

The analysis begins with the one-dimensional scalarwave equation for an electric fieldE in amediumwitha spatially varying relative index of refraction εr~x!.The wave number b is determined primarily by theindex of refraction of thematerial andmodified by thegeometry of the single-mode waveguide. For sim-plicity, b is assumed to vary directly with the changein the relative index of refraction of the core, giving

d2Edz2

1 b2F1 1Dεr~z!

εrGE 5 0. (1)

A number of simplifications have been made to arriveat this equation. The waveguide has been assumedto be weakly guiding and lossless, and the grating isassumed to couple no light to radiative modes.Single-mode propagation is assumed, and birefrin-gence effects have been neglected. The radial andthe angular variations of the electric field are notdescribed.The electric field is assumed to be two counterpro-

pogating waves with slowly varying complex ampli-tudes. The wave number of the field will differ fromthe central wave number b by a small amount j. Itfollows that the complex amplitudes c6 must alsovary with j:

E 5 c1~z, j!exp@i~b 1 j!z#

1 c2~z, j!exp@2i~b 1 j!z#. (2)

The Bragg grating with a central reflectivity at b isrepresented by a cosine with a slowly varying phaseand amplitude. A complex function k~x! is used to

describe this variation, giving

Dεr~z!εr

5 k~z!exp~i2bz! 1 k*~z!exp~2i2bz!. (3)

Substituting Eqs. ~2! and ~3! into Eq. ~1!, applying theslowly varying wave approximation, and collectingphase-matched terms into coupled differential equa-tions with the method described by Yariv8 give

dc1~z, j!

dz5

~b 1 j!

2ik~z!c2~z, j!exp~22ijz!, (4)

dc2~z, j!

dz5 2

~b 1 j!

2ik*~z!c1~z, j!exp~2ijz!. (5)

This set of equations with k~z! set to a constant ap-pears in an article by Lam and Garside.1 Equations~4! and ~5! are then integrated over z with the bound-ary conditions chosen as

c1~z02, j! 5 1, (6)

c2~`, j! 5 0, (7)

giving

c1~x, j! 5 11~b 1 j!

2i *z02

x

k~z!c2~z, j!exp~22ijz!dz,

(8)

c2~x, j! 5~b 1 j!

2i *x

`

k*~z!c1~z, j!exp~2ijz!dz. (9)

One can use this pair of integral equations to com-pute the backscattered and forward scattered lightfrom an arbitrary Bragg grating in an optical fiber,using only the slowly varying wave approximation.For weakly reflecting gratings such as those withreflectivities of less than 10%, one can decouple Eq.~9! from Eq. ~8! by assuming that the forward-traveling wave is not affected by the grating andtherefore that c1~x, j! 5 1 for all x and j. An ap-proximation to Eq. ~9! can then be written as

c2~z02, j! 5

~b 1 j!

2i *2`

`

k*~z!exp~2ijz!dz. (10)

For one to obtain the necessary phase information,the reflected light must interfere with a referencebeam. The reference beam is created by introducinga wavelength-independent reflector ~such as an air–glass interface! into the fiber as shown in Fig. 1. Inthis case, Eq. ~10!must bemodified to incorporate thereference reflection in the expression for the back-ward traveling light:

c2~z02, j! 5 ~1 2 a2!1y2 ~b 1 j!

2i *z02

`

k*~z!exp~2ijz!dz

1 a exp~2ijz0!, (11)

where a is the magnitude of the reference reflectionand z0 is the location of the reference reflection. Thereflector is assumed to be in front of the Bragg grat-ing. Multiple-path reflections between the reflectorand the grating have been neglected. The power atthe detector is computed by finding the magnitudesquared of c2~z0

2, j!:

c2~z02, j!c2*~z0

2, j! 5 a2 1 ~12 a2!~b 1 j!2

4

3 *z02

`

k*~z!exp~2ijz!dz *z02

`

k~z9!exp~22ijz9!dz9

2 a~1 2 a2!1y2 ~b 1 j!

2i *z02

`

k~z!exp@22ij~z 2 z0!#dz

1 a~1 2 a2!1y2 ~b 1 j!

2i *z02

`

k*~z!exp@2ij~z 2 z0!#dz.

(12)

The intensity versus wavelength is Fourier trans-formed to produce a function describing k~z!:

F$cc*% 5 k̃Sx2D 5 a2d~x! 1 ~1 2 a2!b2

8

3 *2`

`

k*Sz9 1x2Dk~z9!dz9 1 a~1 2 a2!1y2 b

2i

3 Fk*Sz0 1x2D 2 kSz0 2

x2DG , (13)

where the spatial variable has been changed to x toavoid confusion of variables in the transformationsand z0

2 has been extended to 2` without changingthe value of the integral because k~z! is zero for allz # z0

2. The second term in the Fourier transformis the correlation of the Bragg-grating modulationfunction with its complex conjugate. Note that if theregion over which the Bragg grating exists is of lengthL, the value of the second term will be zero for all uxu .2L. Let the Bragg grating exist only for L , z 2 z0 ,2L and evaluate the above expression for 2L , x , 4L:

k̃Sx2D 5 a~1 2 a2!1/ 2b

2ik*Sz0 1

x2D . (14)

The Fourier transform of the reflected spectrum froma reference reflector and a Bragg grating has beenshown to be approximately the complex modulation

Fig. 1. Schematic of the optical system used for Bragg gratingcomplex modulation measurement.

1 September 1996 y Vol. 35, No. 25 y APPLIED OPTICS 5163

function of the Bragg grating over a particular regionof the transform.

3. Experiment

When the above result is used, it should be possible tomeasure the spatially varying amplitude of Bragggratings written into a single-mode optical fiber. Anoptical system consisting of a Bragg grating and areference reflector as shown in Fig. 1 along with adetector and tunable laser was assembled to measurec2~z0

2, j!c2*~z02, j!. Once measured, this function

was transformed by means of fast Fourier transformto produce the spatial variation of the Bragg-gratingamplitude.Five 3-mm gratings with peak reflectivities at 1310

nm were written into a hydrogen-loaded optical fiber;0.6 cm apart. The gratings were written holo-graphically with a quadrupled Nd:YAG laser. Thefiber was cleaved ;17 cm from the gratings. Thelaser was a standard cable-TV telecommunicationsdistributed feedback laser with an internal TE cooler.This TE cooler and the associated internal tempera-ture sensing resistor were used to control the tem-perature of the laser diode and thereby control thewavelength. The detector was a standard InGaAsPIN detector with a FC connector. The reflector wasformed with a finger splice without index-matchinggel.The temperature tuning of the laser was achieved

by stepping the thermal set-point voltage from a tem-perature of 35–15 °C. The reflected intensity wassampled as the wavelength swept through values be-tween 1311 and 1309.7 nm. A total of 2048 intensi-ties was measured. This sampling rate determinesthe range overwhich gratings can bemeasured and inthis case gives a range of 0.925 m. The gratings

Fig. 2. Detected reflected power as a function of wave numberfrom a fiber-optic system comprised of five 3-mm Bragg gratingslocated 17 cm behind an air-gap reflector.

Fig. 3. Amplitude plot of the Fourier-transformed interferencespectrum from a fiber-optic Bragg grating–reflector comprising five3-mm Bragg gratings located 17 cm behind an air-gap reflector.

5164 APPLIED OPTICS y Vol. 35, No. 25 y 1 September 1996

were written over a length of ;2.5 cm, and thereforethe first 2.5 cm of the transformation include effects ofthe correlation term in Eq. ~13!. The raw intensitydata from the sweep are shown in Fig. 2, and thetransformed data are shown in Fig. 3. The slowlyincreasing bias in Fig. 2 is due to an increase in thelaser output power as its operating temperature wasdecreased. An expanded view of the section of fiberwith the gratings is shown in Fig. 4. The expectedshape of k~z! is also shown in Fig. 4 and is based onthe assumption of a top-hat UV beam 3 mm in diam-eter and a 0.6-cm translation of the fiber betweenwritings. Although the signal-to-noise ratio is only 15dB, each of the five peaks does appear to be visible witha spacing of ;0.6 cm. Unfortunately, there is nomeans readily available to verify this measurementindependently, other than knowledge of the procedureby which the gratings were written into the fiber.

4. Conclusion

A method of determining the complex modulation ofa Bragg grating as a function of distance has beenderived. The method was used to measure the am-plitude of the complex modulation function as a func-tion of the length of a multiple-Bragg-grating systemimposed in a fiber.

I thank John Deaton for providing the gratingsdescribed here.

References1. D. K. W. Lam and B. K. Garside, “Characterization of single-

mode optical fiber filters,” Appl. Opt. 20, 440–445 ~1981!.2. D. Marcuse, “Coupled mode theory of round optical fibers,” Bell

Sys. Tech. J. 52, 817–842 ~1973!.3. K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photo-

sensitivity in optical fiber waveguides: application to reflectionfilter fabrication,” Appl. Phys. Lett. 32, 647–649 ~1978!.

4. K. O. Hill, “Aperiodic distributed-parameter waveguides for in-tegrated optics,” Appl. Opt. 13, 1853–1856 ~1974!.

5. G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragggratings in optical fibers by a transverse holographic method,”Opt. Lett. 14, 823–825 ~1989!.

6. B. Malo, K. O. Hill, F. Bilodeau, D. C. Johnson, and J. Albert,“Point-by-point fabrication of micro-Bragg gratings in photosen-sitive fiber using single excimer pulse refractive index modifi-cation techniques,” Electron. Lett. 29, 1668–1669 ~1993!.

7. P. Lambelet, P. Y. Fonjallaz, R. P. Salathe, Ch. Zimmer, andH.H.Gilgen, “Bragg grating characterization by optical low-coherencereflectometry,” IEEE Photon. Technol. Lett. 5, 565–567 ~1993!.

8. A. Yariv, Quantum Electronics ~Wiley, New York, 1989!, pp.608–610.

Fig. 4. Expanded view of the grating region showing the fivegrating peaks separated by ;0.6 cm.