SPIE Proceedings [SPIE Information Technologies 2000 - Boston, MA (Sunday 5 November 2000)] Optical Devices for Fiber Communication II - Detuning factors in the synthesis of optical

Detuning Factors in the Synthesis of Optical Fiber Gratingsusing the Gel'fand-Levitan-Marchenko Inverse Scattering

Method

Jinho Bae and Joohwan Chun

Scientific Computing Laboratory,Department of Electrical Engineering and Computer Science,

Korea Advanced Institute of Science and Technology,373-1 Kusong-dong, Yusong-gu, Taejon 305-701, South Korea

ABSTRACTWe present necessary conditions of detuning factor in the coupled mode equations for the synthesis of optical fibergratings using the Gel'fand-Levitan-Marchenko inverse scattering method. From the presented conditions in thecoupled mode equations for the short and long period fiber gratings, we show that the use of Gel'fand-Levitan-Marchenko inverse scattering method for the long period fiber gratings is very restrictive. We also propose somemethods to synthesize short period fiber gratings using the Gel'fand-Levitan-Marchenko inverse scattering method.

Keywords: coupled mode equations, coupling coefficient, detuning factor, GLM ISM

1. INTRODUCTIONSynthesis of optical fiber grarings using the Gel'fand-Levitan-Marchenko inverse scattering method (GLM ISM) isa technique to find coupling coefficients from scattering data. The GLM ISM is used to synthesize the optical fibergratings modeled as coupled mode equations. Song and Shin1'2 used the method for design of corrugated waveguidefilters, Winick3 applied the method to the directional couplers, Peral et al.4'5 proposed an iterative solution to theGLM coupled equations, and Skaar et al.6 designed a fiber-optic bandpass filter using the method by Peral et al..4'5

Techniques of manufacturing optical fiber gratings using ultraviolet irradiation are developing rapidly in recentyears. Optical fiber gratings, which may be used in optical communication applications7 such as the wavelengthdivision multiplexers (WDM),8 dispersion compensators,9'1° band-stop rejection filters," and equalizer,'2'4 aswell as in the fiber sensor applications.'5 The optical fiber gratings are classified into two; the short period fibergratings which have the forward and backward coupled modes in the core, and the long period fiber gratings thatonly have forward coupled modes in the core as well as in cladding. Optical fiber grating structures are fabricatedby exposing a section of a single mode optical fiber to KrF excimer laser light of wavelength 248 nm through aphase mask for the short period fiber gratings and an amplitude mask for the long period fiber gratings. Opticalfiber gratings can be analyzed using the coupling coefficients and the detuning factor that can be obtained from theindices and grating periods. Synthesis of optical fiber gratings is by determining the coupling coefficients and thedetuning factors (or the indices and grating periods) using a synthesis method such as the numerical optimizationmethods,'3"4 GLM ISM,'6 Fourier transform,'6 and the inverse scattering algorithm using the layer peelingmethod.

In these synthesis methods, the GLM ISM can determine the coupling coefficients along the z-direction fromscattering data, but the detuning factor cannot be computed. However, to synthesis the optical fiber gratings,detuning factor is a very important parameter. In this paper, we explain difficulties with the GLM ISM to determinethe synthesis parameters, and propose some methods to solve the presented problems. We aslo show that the GLMISM cannot be used for the synthesis of long period fiber gratings.

Optical Devices for Fiber Communication II, Michel J. F. Digonnet, Osman S. Gebizlioglu,Roger A. Greenwell, Dennis N. Horwitz, Dilip K. Paul, Editors, Proceedings of SPIE

Vol. 4216 (2001) © 2001 SPIE · 0277-786X/01/$15.00182

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 12/10/2013 Terms of Use: http://spiedl.org/terms

dA(A,z) = +j(A,z)A(,z)e_____________ —j2ö(A,z)zdz

O<x<7171 x < 271

2. ANALYSIS MODEL FOR OPTICAL FIBER GRATINGSA section of optical fiber gratings is illustrated in Figure 1 , where n0 is the unperturbed refractive index of thecore, and nj is the refractive index of the cladding. The length of the kth grating will be denoted as Ak . Dependingon the periodicity length A larger or shorter than the coupled light wavelength, it can be classified as short periodor long period fiber gratings, respectively. Backward reflection coupled modes exist in short period fiber gratings,while the forward propagating cladding coupled modes exist in long period fiber gratings. Optical fiber gratings withgrating period A(z) can be modeled by a simple refractive index modulation as follows'8"9:

I n(z) = nco + /.n(z)(1 + cos()z)) for gratings with sinusoidal fringes1 n(z) = nco + n(z)(1 + square()z)) for gratings with square shaped fringes

where n(z) is the induced index change, and sqare(x) is defined by sqare(x) ={

HAk """U

Figure 1. A section of the optical fiber gratings.

The total fields in the optical fiber gratings can be derived as a superposition of the two mode fields, whichhave amplitudes A(), z) and B(A, z), respectively. In the short period fiber gratings A(A, z) = A(.\., z) andB(A, z) = A(A, z) are slowly varying amplitudes in the +z and —z directions, respectively. Then, to begin with,a pair of coupled mode equations are

dA(A,z)—K(;\, z)A) (A, z)e12",dz

where '0', z) and 5(A, z) are the coupling coefficients and the detuning factor, respectively, which varies along thez-directions and the wavelength, the index profile, and the grating length, and A is wavelength. The couplingcoefficients ic(A, z) and the detuning factor ö(\, z) can be written as follows:

(2)

ict, z) — irzn(z) for gratings with sinusoidal fringes—n°)(A,z)A(z){

K(A, z)— 4n(z)—

n°)(A,z)A(z)for gratings with square shaped fringes

0 (co)

ö(\, z) —'eff(A' z)- _x,

(3)

(4)

Proc. SPIE Vol. 4216 183


czn(z)/cj)t, z) =

AC (for gratings with sinusoidal fringes),

C) (co)1 1eff (A, z) 2irn(A, z)8(A,z) —A A

where n) (A, z) is the effective index of the fundamental core mode along the wavelength and z-direction, and A(z)is the grating length which is a function of z. A pair of coupled mode equations for long period fiber gratings arethen

dAco(A, z) = —jK(A, z)A (A, z)ei2A,

= —j(A, Z)Aco(A, z)e_i2X, (5)

where Aco(A, z) = A(A,z) and A(A, z) = B(A,z) are slowly varying amplitudes for the fundamental core mode andthe ith cladding mode in the +z-directions, respectively, K(A, z) and ö(A, z) that couple the fundamental core modeand ith cladding mode are as follows:

where z) are the effective indices of ith cladding mode along the wavelength and z-direction, and C is theoverlap integral18 by fundamental core mode and ith cladding mode. Assuming that we know the i(A, z) and 6(A, z),reflectivity and transmission for the short period gratings and bar transmission and cross transmission for the longperiod gratings can be obtained from (2) and (5).18_20

(6)

(7)

Section 1

Section 2

Figure 2. Grating structure for the optical fiber gratings.

Figures 3 and 4 show simple examples for the short and long period fiber gratings having the piecewise uniformgratings with square shaped fringes as shown in Figure 2. In these figures, the parameters of the fiber used are:

co 1.449, cl = 1.444, aco 4.5pm, ad 62.5 1am,

where aco and act are core and cladding radius, respectively.

2.1. Short period fiber gratingsFigure 3(a) shows the resonant wavelength points of each section by ö = 0 (see down arrow points in Figure 3(a)).The detuning factors along the wavelength, and the coupling coefficients of each sections as shown in Figure 2 arepresented as Figure 3(b) along the wavelength between 1547 nm and 1549 nm. Slope of the detuning factor of eachsection along the wavelength are the same, but the resonant wavelength of each section has a different wavelength

Proc. SPIE Vol. 4216184


point. From Figure 3(b), we can see that the coupling coefficients vary independently along the wavelength becausemeasure or calculated (usable) band for the short period fiber gratings is very narrow. However, the couplingcoefficients are varied as z-direction. The parameters in Figure 2 to fabricate the short period fiber gratings havingthe spectrum as shown in Figure 3(c) are:

1. Section1

2. Section2

Ln1 = 0.000125, A = 535nm, L1 = 0.321cm.

Zn1 = 0.000135, A = 535.lO7nm, L1 = 0.3210642cm.

650

6401547 154-75 1548 1545.5 1549VV9v8Irigth [nmj

(b)

The reflection magnitude shown in Figure 3(c) is calculated using the fundamental matrix method.18'2°

700

690

680

- C of &ction (0 � z L )670• • of 8ctiorI2 (L1 � z -'—2)

660

(a)

(c)

Figure 3. (a) Detuning factor v.5. wavelength (sectioni and section2). (b) Coupling coefficients of each section(section1 and section2) along the wavelength. (c) Reflection magnitude.



2.2. Long period fiber gratingsFigure 4(a) shows the effective index profiles (core and cladding modes) of each section in the wavelength bandbetween 1510 nm and 1580 nm. An usable band for the long period fiber gratings is very wide, so we can see theeffective indices are varied as the wavelength. The coupling coefficients in (6) are also varied along the wavelengthand z-direction. The detuning factor for the long period fiber gratings are shown in Figure 4(b). From this figure,we can see that the slope of each sections (sectioni and section2) is a different as the wavelength, and the detuningfactors as z-direction has also different value. The parameters for the long period fiber gratings of each section toobtained these in Figure 4 are:

1 . Section1

2. Section2

Ln1 = 0.000125, A = 412.1255 161856um.

An1 = 0.00025, A — 391.11773016785um.

1 .4465

1.446

1 .4.455

I .4.45

1 .4445

1 .4435

1 .443

1 .4425 - 1540 1560\N'vI8ngth [rmm]

Figure 4. (a) Effective indices of each section (section1v.5. wavelength (section1 and section2).

and section2) along the wavelength. (b) Detuning factor

3. PROBLEMS IN SYNTHESIS OF OPTICAL FIBER GRATINGS USING GLM ISMAND CONDITIONS TO SOLVE THE THESE PROBLEMS

In (1) and (5), coupling coefficient ,c(A, z) is the parameter to control the bandwidth and the strength of the reflectivityand the transmission, and detuning factor ö(À, z) can find the position of dip (or peak) in the spectrum by ö(À, z) = 0as a basic equation of phase matching condition for the optical fiber gratings. The GLM ISM proposed by refs. (1)- (6) is the method to compute ic(z) ,c(A, z) with 5(A) ó(À, z) computed by the wavelength. The coupled modeequations used in refs. (4) are as follows:

dA(A,z) = —JK(z)B(A, z)ei2o())z,dzdB(),z) = z)e_i2ô(dz (8)

However, in the use of the GLM ISM to synthesize of the optical fiber gratings, it has some difficulties and constraintsas follows:

. Long period fiber gratings

*-.. Ecr ndx in ctkri 1 -— — Ecr irdx ri cticn-i2_____ Iaddrg irdx r ctkn1 (LP04)

. c: cktddirig ridx ri cticr2 (LF'0)

1.444

1 520

(a)

1 580

(b)



1. The coupling coefficient in (6) is a function of the.wavelength and z, so the approximation as ic(A,z) K(z)is not valid.

2. The detuning factor in (7) is a function of the wavelength and z as shown in Figure 4, so the approximationas 5(A, z) is also very difficult.

3. For an exact synthesis, because long period fiber gratings exist the multiple mode coupling between thefundamental core mode (LP01) and the multiple cladding modes (LP01 ,. . . , LPON),18'19'2' the multiplemode coupling must be considered in GLM ISM. However, the ordinary GLM ISM just uses two modecoupling between the fundamental core mode (LP01) and ith cladding mode (LP0).

Because of the above three reasons, the use of the GLM ISM for the long period fiber gratings is very difficult.

. Short period fiber gratings

1 . The coupling coefficient in (3) is a function of z as shown in Figure 3(b) , so we can approximate asz) R ic(z).

2. The detuning factor in (4) is a function of the wavelength and z as shown in Figure 3(a), so the approxi-mation as 5(A, z) ö(Á) is very difficult.

3. Short period fiber gratings exist two mode coupling between the forward core mode and the backwardcore mode.

From above contents, we can use the GLM ISM to synthesize the short period fiber gratings, if we can solvethe problem with the detuning factor.

Because the use of the GLM ISM for the long period fiber gratings is very difficult, we shall restrict our attentionto the conditions for the short period fiber gratings. In general, 6(,\, z) used of the GLM ISM is a constant alongz-direction, i.e., fleff,co(A, z), nefl,ci(A, z), and A(z) are constant along the z-direction. This condition means theoptical fiber gratings having the index profile with zero-dc index change and uniform grating length, physically, forexample, can be regarded as the uniform period and index short period fiber gratings.

In the uniform period and index short period fiber gratings, we can easily obtain the coupling coefficient usingthe GLM ISM, because of ö(Á, z) 5(A). But it is difficult to synthesize the nonuniform short period fiber gratingsas example of Figure 2 using the GLM ISM. In this paper, we propose methods to synthesize the nonuniform shortperiod fiber gratings using the GLM ISM as follows:

General methods

This method is proposed by many researchers."3'4 For example, in Figure 2, we assume that ö(Á) = (A, z) =ö2(A, z), thereafter it calculates the coupling coefficients K(z) with ö(À) using the GLM ISM. The conditionc5(A) = öi (A, z) = o2ç, z) has to be fixed by the control of the grating index and period, however, fabricationof nonuniform short period fiber gratings with ö(À) =ö(A, z) = ö2(.\, z) is impossible, practically.

Oversampling method

The short period fiber gratings have the forward and backward coupled modes. To introduce the basic conceptof the short period fiber gratings based on the lattice filter structure, we analyze the two-layer case whose latticefilter structure is depicted in Figure 5(a) and is interpreted as the lattice filter structure illustrated in Figure 5(b).22The indices of refraction are k, (k = 1,2), the length of the layer are dk, (k = 1,2). The parameters r, (i = 0, 1, 2)are the reflection coefficients, and z"2 = (k = 1, 2) are the phase delay terms of layers, where A is

the wavelength, and j /4. From Figure 5(b) , we can obtain the transmission and reflection transfer functions



E°ni

E° E3

,d1 d2

(a)

E (0) to 1í-J:-"i i/2

ru

1 z112

r2

2

2E3

E°*—n—--—- I zh/2k *__1-;_ Iz1'2 e*-_r-_- E3

(b)

Figure 5. (a) Schematic diagram of two layers. (b) Lattice filter structure.

as functions of Zk from (2) by setting the boundary condition of E3 =0:

(3) —1/2 —1/2

T(z) = = _0t1t2z1z2 _ , (9)E° 1 + r0r1z1 + r1r2z2 + r0r2z1 z2

(0) —1 —1 —1 —1

R(z) = :!:__ = O + r1z1 + r0r1r2z2 + r2z1 z2(10)

E° 1 + rorizj + r1r2z1 + ror2zjz

Here, the oversampling means the insertion of virtual layers having a small uniform delay, z such that the actual delayZk becomes the integer multiple of z. The size of oversampled data should be sufficiently large to avoid the phaseerror generation to find the reflection coefficients correctly. With the oversampled data, (9) and (10) are changed asbelow

E3 M t(0v) —M/2TI'\ — __t__— i=oi (11'_) — 1O M •'E 1 + >:i=1az(O) —M

R(z) = Lj=O i(12)

E° 1+>i1azHere z1 = Zml 2 = Zm2 where m1 , m2 >> 1 are integers and M = m1 + m2 . Parameter t0 is the transmissioncoefficient at the ith virtual interface, and a and b2 are the coefficients of the denominator and the numeratorpolynomial, respectively. Here new phase delay z can be used to fix ö()) in the GLM ISM. If grating structure hasthe uniform period and index short period fiber gratings, the phase delay is the constant along z =exp(_j27rJ-.But the use of oversampled method to synthesize the short period fiber gratings needs excessive computational cost.

4. CONCLUSIONSWe presented the difficulties in the synthesis of the optical fiber gratings using the GLM ISM, and proposed a generaland oversampled methods to solve the problems in the case of the short period gratings. We presented that it is avery difficult to use the GLM ISM for the long period fiber gratings.



REFERENCES1. G. Song and S. Shin, "Design of corrugated waveguide filters by the gel'fand-levitan-marchenko inverse-scattering

method," J. Opt. Soc. Am. A, vol. 2, no. 11, pp. 1905—1915, 1985.2. G. Song and S. Shin, "Inverse scattering problem for the coupled-wave equations when the reflection coefficient

is a rational function," Proc. IEEE, vol. 71, pp. 266-268, 1983.3. K. A. Winick, "Design of grating-assisted waveguide couplers with weighted coupling," J. Lightwave Technol.,

vol. 9, no. 11, pp. 1481—1492, 1991.4. E. Peral, J. Capmany, and J. Marti, "Iterative solution to the gel'fand-levitan-marchenko coupled equations

and application to synthesis of fiber gratings," J. Quantum Electron., vol. 32, no. 12, pp. 2078—2084, 1996.5. E. Peral, J. Capmany, and J. Marti, "Design of fiber grating dispersion compensatiors using a novel iterative

solution to the gel'fand-levitan-marchenko coupled equations," Electron. Lett., vol. 32, no. 10, pp. 918—919,1996.

6. J. Skaar, B. Sahigren, P.-Y. Fonjallaz, H. Storoy, and R. Stubbe, "High-reflectivity fiber-optic bandpass filterdesigned by use of the iterative solution to the gel'fand-levitan-marchenko equations," Opt. Lett., vol. 23, no.12, pp. 933—935, 1998.

7. P. A. Krug, J. Chow, and B. J. Eggleton, "Applications for fiber bragg gratings in communications," OFC'96,pp. 116—116, 1996.

8. T. Mizuochi, K. Shimizu, and T. Kitayama, "All-fiber add/drop multiplexing of 6 x 10 gbit/s using a photo-induced bragg grating filter for wdm networks," OFC'96, pp. 116—117, 1996.

9. F. Ouellette, "Dispersion cancellation using linearly chirped bragg grating filters in optical waveguides," OpticsLetters, vol. 12, no. 10, pp. 847—849, 1987.

10. K. 0. Hill, E. Bilodeau, B. Malo, T. Kitagawa, S. Theriault, D. C. Johnson, and J. Albert, "Chirped in-fiberbragg gratings for compensation of optical-fiber dispersion," Optice Letters, vol. 19, no. 17, pp. 314—316, 1994.

11. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T Erdogan, and J. E. Sipe, "Long-period fibergratings as band-rejection filters," J. Lightwave Technol., vol. 14, no. 1, pp. 58—64, 1996.

12. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, "Long-period fiber-grating-based gainequalizers," Optices Lett., vol. 21, no. 5, pp. 336—338, 1996.

13. J. Bae, J. Chun, and S. Lee, "Two methods for synthesizing the long period fiber gratings with the invertederbium gain spectrum," Japanese J. Appi. Phys. part , vol. 38, no. 7B, pp. L819—L822, 1999.

14. J. Bae, J. Chun, and S. Lee, "Equalization of the non-fiat erbium gain spectrum using the multiport latticefilter model," OFC'1OOO, vol. WE7, pp. 80—83, 2000.

15. K. P. Koo and A. D. Kersey, "Bragg grating-based laser sensors systems with interferometeric interrogation andwavelength division multiplexing," J. Lightwave Technol., vol. 13, no. 7, pp. 336—338, 1996.

16. K. A. Winick and J. E. Roman, "Design of corrugated waveguide filters by fourier-transform techniques," J.Quantum Electron., vol. 26, no. 11, pp. 1911—1929, 1990.

17. R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniformfiber bragg gratings," J. Quantum Electron., vol. 35, no. 8, pp. 1105—1115, 1999.

18. T. Erdogan, "Fiber grating spectra," J. Lightwave Technol., vol. 15, no. 8, pp. 1277—1294, 1997.19. A. Yariv, Optical Electronics in Modern communications, Oxford Univ. Press, New York, 1996.20. M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via a fundamental

matrix approach," Appl. Optics, vol. 26, no. 16, pp. 3474—3478, 1987.21. J. Bae, J. Chun, S. Lee, and Y. Hong, "Multiport lattice filter model for long period fiber gratings," in Proc.

SPIE (Photonics West), to appear, 2000.22. J. Bae, J. Chun, and S. Lee, "Analysis of the fiber bragg gratings using the lattice filter model," Japanese J.

Appl. Phys. part 1, vol. 39, no. 4A, pp. 1752—1756, 2000.



Documents

SPIE Proceedings [SPIE Information Technologies 2000 - Boston, MA (Sunday 5 November 2000)] Optical Devices for Fiber Communication II - Detuning factors in the synthesis of optical