Modified coupled-mode model for thermally chirped polymer Bragg gratings

Modified coupled-mode model for thermally chirpedpolymer Bragg gratings

Ilai Sher,1,* Bongtae Han,2 and Avram Bar-Cohen2

1School of Engineering, Cranfield University, Cranfield, Bedfordshire, MK43 0AL, UK2Department of Mechanical Engineering, University of Maryland at College Park, College Park, Maryland 20742, USA

*Corresponding author: [email protected]

Received 30 October 2009; revised 5 March 2010; accepted 6 March 2010;posted 9 March 2010 (Doc. ID 119302); published 5 April 2010

A modified coupled-mode (CM) model is proposed for the optical behavior of thermally chirped Bragggratings. The model accounts for the axial gradient in the modulation wavenumber, which has been ig-nored in the classical CM model. The model is used to characterize the optical behavior of a polymethylmethacrylate-based polymer Bragg grating subjected to nonisothermal conditions. The validity of theproposed method is verified by comparing the results of the modified CMmodel with those obtained fromthe exact numerical solution. © 2010 Optical Society of America

OCIS codes: 050.1590, 060.3735.

1. Introduction

Chirped gratings, resulting from natural or inten-tional variations in the properties and geometriesof an otherwise periodic structure, have been exten-sively treated in the literature [1–14]. Analyses ofchirped gratings are usually based on the coupled-mode (CM) theory, first proposed by Kogelnik [1].The direct numerical integration of the CM

equations along chirped gratings, as suggested byKogelnik [1], has been shown to agree well with ex-perimental results of mildly chirped gratings. Honget al. [2] have developed asymptotic approximationsof the CM equations for linearly chirped gratings.Fukuzawa and Nakamura [3] and Livanos et al.[4] have suggested a chirped grating reflectivity cal-culation method, which is based on the approximateform of the CM theory. In the approximate form, useis made of an “effective” grating length—definedas the distance over which significant reflectanceoccurs. Their approximation was found to agreereasonably well with the results of the direct CMnumerical integration.

Yamada and Sakuda [5] have suggested a methodapplicable to mildly chirped gratings, or “almost per-iodic” gratings. Their method is a piecewise-uniformapproach, in which the grating is divided into a num-ber of segments of uniform periodicity, the CM equa-tions are solved for each segment, and the resultscombined in a series of matrices. Erdogan [6] hasconcluded that the approximation of the piecewise-uniform approach is in good agreement with thesolutions of the direct CM numerical integration,for a variety of mildly chirped gratings. A similar ap-proach was taken byWeller-Brophy and Hall [7], whoanalyzed chirped gratings using the Rouard method,approximating the grating as a stack of a finite num-ber of slabs and determining the effective reflectivityof each slab by the CM theory. At the asymptoticlimit, where the grating is represented by an infinitenumber of slabs, the Rouard approximation becomesexact. Sipe et al. [8] have suggested an “effectiveproperties” description of chirped gratings in the con-text of the CM equations. A later study by Poladian[9] made use of that “effective properties” descrip-tion to develop phase integral Wentzel–Kramers–Brillouin (WKB) approximate solutions of the CMequations for chirped gratings. These have beenfound to agree with the results of the direct CMnumerical integration.

0003-6935/10/112079-06$15.00/0© 2010 Optical Society of America

10 April 2010 / Vol. 49, No. 11 / APPLIED OPTICS 2079

The thermo-optic effects of intrinsic heating on theperformance of polymer fiber gratings have attractedrecent attention. Notable examples of such experi-mental studies include the work of Peng and Chu[10] on the photosensitivities of polymer optical fi-bers and use of fiber temperature changes to createtunable fiber gratings, the work of Liu et al. [11] on athermal stability test for polymer fibers, whichpointed to a potential compatibility of polymers withsuch applications, and a more recent study by Littleret al. [12], considering the effect of intrinsic heatingon the Bragg shift in an isothermal grating.Consideration was then extended by Kim et al. [13]

to gratings with axial temperature gradients, cre-ated by nonuniform intrinsic heating, leading to anontraditional, “thermally chirped” Bragg grating.When exposed to a temperature gradient, the fiberBragg grating (FBG) optical properties will be al-tered in two ways: by a thermo-optic effect (varied,nonuniform, shift in the refractive index unrelatedto the Bragg grating) and by a thermomechanicaleffect (varied, nonuniform, thermal expansion ofthe material, altering the period of the Bragg grat-ing). The nonuniformity of these effects along thegrating causes a chirp in the FBG. A model to pre-cisely characterize the optical behavior of such ther-mally chirped gratings must consider the axialgradient of the modulation wavenumber, which hasbeen ignored in the original CM model.In this study, a rigorous approach to modeling of

thermal effects on thermally sensitive Bragg grat-ings is suggested, in which the CM optical equationsare rederived for thermally chirped polymer Bragggratings.

2. Modeling

In this analysis the Bragg grating is treated as alarge number (N) of sequential elements, with reflec-tion occurring at the interfaces between elementsand propagation within the elements. Consideringthe jth element, the local jth wave amplitudes ofthe forward (F) and the backward (R) waves are re-lated to the (jþ 1)th amplitudes through the multi-plication of the complex transmission coefficient, thecomplex reflectivity matrix, and the propagation ma-trices [14], i.e.,

�Fj

Rj

�¼ 1

τj

�1 −ρj�ρj 1

��eikjΔz 00 e−ikjΔz

��Fjþ1

Rjþ1

�; ð1Þ

where the complex transmission coefficient and thecomplex reflectivity coefficient are given, respec-tively, by

τj ¼2nj

nj þ njþ1; ρj ¼

nj − njþ1

nj þ njþ1; ð2Þ

The complex refractive index for any element inthe grating can be expressed as the sum of the non-periodic and the periodic contributions to the refrac-tive index variation, i.e.,

nðzÞ ¼ n0ðzÞ þ δneiBðzÞz: ð3Þ

The modulation wavenumber, BðzÞ varies along thegrating, i.e., BðzÞ ¼ 2π

ΛðzÞ, where ΛðzÞ is the gratingperiod. The propagation wavenumber may also varyalong the grating, according to kðzÞ ¼ 2π

λ0 nðzÞ, whereλ0 is the wavelength in vacuum.

Subtracting the (jþ 1)th wave amplitudes vectorfrom both sides of Eq. (1) and allowing the elementsto become vanishingly thin, yields a simple differen-tial equation:

d

�FR

�¼ A

�FR

�; ð4Þ

where

A ¼�1 −

1τ e

ikΔz ρ�τ e

−ikΔz

−ρτ e

ikΔz 1 −1τ e

−ikΔz

�:

The first term of the Taylor series of τ, ρ, and eikΔz

can be expressed as

ρj ¼nj − njþ1

nj þ njþ1≈

−n0Δz2nþ n0Δz

;

τj ¼2nj

nj þ njþ1¼ 2n

2nþ n0Δz; eikΔz

≈ 1þ ikΔz; ð5Þ

where n0≡

dndz. Substituting these relations into

matrix A, and taking the first term in Δz, yields

A ¼�−

12n0n − ik −

12n0n�

12n0n −

12n0n þ ik

�Δz: ð6Þ

By assuming that the modulated variation of re-fractive index is small in comparison to its base value(i.e., δn ≪ n0ðzÞ) and that the chirp-induced spatialvariation of the base refractive index is much smallerthan the modulated spatial variation of the refrac-tive index (i.e., n0

0ðzÞ ≪ δnBðzÞ), the refractive indexterms can be evaluated as

n0ðzÞnðzÞ ¼ n0

0ðzÞ þ i½BðzÞ þ zB0ðzÞ�δneiBðzÞzn0ðzÞ þ δneiBðzÞz

≈ iδn

n0ðzÞ½BðzÞ þ zB0ðzÞ�eiBðzÞz; ð7Þ

where B0ðzÞ≡ dBðzÞdz . In Eq. (7), explicit notation is

used to emphasize z-dependent variables. Theseassumptions can be quantitatively justified for glassand polymer fibers [13] by inserting typical opticalproperties of glass and polymer (Table 1 [13]).

By substituting Eq. (7) in (6) and making an addi-tional assumption that the numerical value of δn=n0is far smaller than k=B, i.e., δnn0

≪kB, which can also be

quantitatively justified, Eq. (6) can be expressed as

2080 APPLIED OPTICS / Vol. 49, No. 11 / 10 April 2010

A ≈ −i

�kðzÞ 1

2δn

n0ðzÞ ½BðzÞ þ zB0ðzÞ�e−iBðzÞz−

12

δnn0ðzÞ ½BðzÞ þ zB0ðzÞ�eiBðzÞz −kðzÞ

�Δz: ð8Þ

Converting to the phase-shifted wave amplitudeparameters (i.e., f ≡ Fe

12iBðzÞz and r≡ Re−

12iBðzÞz) such

that jf j ¼ jFj and jrj ¼ jRj, serves to eliminate the ex-ponents from the above relations. Then, Eq. (4) cantake a final form as

ddz

�fr

�¼ a

�fr

�; ð9Þ

where

a¼−i

�−12½BðzÞþzB0ðzÞ�þkðzÞ 1

2δn

n0ðzÞ½BðzÞþzB0ðzÞ�−12

δnn0ðzÞ½BðzÞþzB0ðzÞ� 1

2½BðzÞþzB0ðzÞ�−kðzÞ�:

For an isothermal grating in which B0ðzÞ ¼ 0,Eq. (9) reverts to the original CM model (see Orfani-dis [14]). It is worth noting that the contribution ofzB0ðzÞ to the cross-diagonal terms in a is negligiblesince, typically, zB0ðzÞ ≪ BðzÞ. However, the zB0ðzÞcontribution to the diagonal terms is not negligibleclose to the Bragg condition, where − 1

2BðzÞ þ kðzÞ ≈ 0.

3. Analysis

The present analysis addresses a nonisothermalFBG, with an assumed axial temperature variation,representing a thermally chirped Bragg grating. Thetemperature effect is taken into account through themodulation and propagation wave numbers. Themodulation wavenumber varies along the axial di-rection as

BðzÞ ¼ 2πΛðzÞ ¼

2πΛ0½1þ αΔTðzÞ� ; ð10Þ

where Λ0 is the grating period at the reference tem-perature, α is the thermal expansion coefficient, andΔTðzÞ is the excess temperature, representing thedeviation from the reference temperature. The pro-pagation wavenumber varies along the axial direc-tion as

kðzÞ ¼ 2πλ0

nðzÞ ≈ 2πλ0

½n0 þ n0TΔTðzÞ�; ð11Þ

where n0 is the refractive index at the referencetemperature and n0

T is its derivative with respectto temperature, i.e., n0

T ≡dndT.

The optical behavior of such a nonisothermal poly-meric Bragg grating is determined by three differentmethods: the CM method, the thermally modifiedCM method (henceforth referred to as CMT), andthe exact method, which is a direct numerical calcu-lation based on Eq. (1) [or Eq. (4)] and does not con-tain any limiting assumptions. To obtain correctresults, the direct numerical solution must be per-

formed using extremely small steps (Δz), typically2 orders of magnitude smaller than the wavelength.To assure convergence, in the present study Δz forthe direct numerical solution was taken equal to5:3nm.

The boundary conditions set the amplitude of theforward wave at the grating outlet to unity and theamplitude of the reflective wave at the outlet to zero:

�FN

RN

�¼

�10

�: ð12Þ

Linear normalization of the solution can further beapplied to tune the results to different forward waveamplitudes at inlet.

For the CM and CMT methods, numerical compu-tations are performed in Δz steps (j):

�f jrj

�¼ e−ajΔz

�f jþ1

rjþ1

�; ð13Þ

For each step, matrix a is approximated as a con-stant, i.e., aj ¼ aðzþΔz=2Þ, and calculated by itsdefinition [see Eq. (9)], where for the CM methodB is set equal to zero. Equation (13) is then solvedusing the boundary conditions:

�f NrN

�¼

�10

�: ð14Þ

Further linear normalization of results, as noted forEq. (12), can be applied.

The numerical calculations of CM and CMT can beperformed in far fewer steps (largerΔz) than the “ex-act” method since they only need to be smaller thanthe relevant length scale of temperature change.

4. Results and Discussion

In subsequent sections, thermo-optic results are gi-ven for grated fibers made of glass and polymethylmethacrylate (PMMA) polymer, respectively, andchosen to represent an extreme variation in thethermo-optic material properties of the Bragg grat-ings. The relevant optical and thermal parametersof these two materials are given in Table 1 [13]. Itis to be noted that the values of the thermo-optic coef-ficient (dn=dT) and thermal expansion coefficient (α)of the PMMA polymer are much larger than thoseof glass.

In order to examine the effect of temperature non-uniformity on the optical behavior of a grating, theimpact of different temperature profiles on the powerand spectral characteristics of the backward/reflected light wave is evaluated with the aid of


the classic CM equations, the CMT equations, andthe exact numerical calculation, respectively.Figure 1 shows the individual and aggregated ef-

fects of the mechanisms responsible for the thermo-optic behavior of a thermally chirped glass Bragggrating, subjected to a steep exponential axial tem-perature variation, ΔT ¼ 85e−3:344

zL, dropping from

an excess temperature of 85K at the inlet to just3K at the outlet of the grating. The thermomechani-cal effect, due to the thermal expansion of the grat-ing, is captured in the variation of the grating pitchwavenumber (B) and its axial derivative (B0). Thetemperature sensitivity of the refractive index (n0

T)is embodied in the variation of the propagating lightwavenumber (k).Because of the small thermal expansion coefficient

in glass, however, it is observed that the thermome-chanical effect (B andB0) is quite small in comparisonto the thermo-optic effect, and that, therefore, thereare only minor differences between the predictions ofthe CMT and the CM methods. It is noted that themagnitude of the thermo-optic effect in glass is small,due to its small thermo-optic coefficient, n0

T (muchsmaller than that of PMMA). Nevertheless, in theabsence of the thermo-optic compensation that can

occur under small gradients in PMMA, the totalBragg shift for the glass fiber under the stated con-ditions reaches 0:1nm.

Figure 2 shows the individual and aggregated ef-fects of the mechanisms responsible for the thermo-optic behavior of a thermally chirped polymer Bragggrating, subjected to a steep exponential axial tem-perature variation, ΔT ¼ 35e−3:555

zL, dropping from

an excess temperature of 35K at the inlet to just1K at the outlet of the grating. It is clear from thefigure that the contributions of B and k to the reflec-tivity of a PMMA grating are almost symmetricalabout the Bragg wavelength, leading to significant“compensation” between these two mechanismsand only a modest net Bragg shift. However, the com-bined thermomechanical contribution, including theeffects of both the wavenumber, B, and the axial de-rivative of the wavenumber, B0, destroys this symme-try. Consequently, while the solution of the classicCM equations (without the B0 contribution) suggestsstrong thermo-optic compensation and a very modestBragg shift of 0:04nm in this PMMA grating (see alsoKim et al. [13]), inclusion of the B0 term in the CMT,derived specifically to deal with large axial tempera-ture gradients, yields a significant Bragg shift and

Fig. 1. (Color online) Contribution of thermo-optic effects to spec-tral variation (shift from Bragg wavelength) in reflected power fora 10mm glass grating. A 10mm glass grating under an exponen-tial temperature profile of ΔT ¼ 85° − 3 °C is considered. Calcula-tions are separated for the sole effects of k, B, B& B0, and thecombined effects models CM (k and B) and CMT (k and B& B0).Power is normalized to the incident light of unity.

Fig. 2. (Color online) Contribution of thermo-optic effects to spec-tral variation (shift from Bragg wavelength) in reflected power fora 10mmPMMAgrating, under an exponential temperature profileof ΔT ¼ 35° − 1 °C. The individual effects of k, B, B& B0, and thecombined effects models CM (k and B) and CMT (k and B&B0) areshown. Power is normalized to the incident light of unity.

Table 1. Physical Parameters for Glass and PMMA (from Kim [13]) at Standard Room Temperature

Glass PMMA

Refractive index (at reference temperature) n0 1.4567 1.4853Index modulation δn 7:123 × 10−5 7:244 × 10−5

Index derivative with respect to temperature n0T 8:6 × 10−6 K−1

−1:1 × 10−4 K−1

Thermal expansion coefficient α 0:55 × 10−6 K−1 73 × 10−6 K−1

Grating period (at reference temperature) Λ0 532:00nm 530:70nmBragg wavelength (λB ¼ 2n0Λ0) λB 1549:93nm 1576:50nm


strong spectral “ringing” in the reflectivity of thegrating.Figure 3 presents an example of a thermally

chirped Bragg grating, chosen to highlight the signif-icant effect of temperature nonuniformity on thethermo-optic behavior of polymer grating and onthe predictive inaccuracy of the classic CM model.The figure depicts the axial variation in the normal-ized reflected power, at the Bragg wavelength of1576:5nm [Fig. 3(a)] and at 0:1nm below the Braggwavelength [Fig. 3(b)], for a thermally chirped,10mm long, PMMA polymer FBG. The calculationswere performed for an assumed linear excess tem-perature profile, dropping from 5K above ambientat the inlet to 0K at the outlet of the grating.At the Bragg wavelength, all three techniques

yield a nearly monotonic increase in reflected poweras the grating inlet is approached, but while thedirect numerical solution and the CMT method yieldnearly identical results with a reflected power ofsome 9%, the classic CM method substantially over-predicts the reflected power with a normalized valueof 0.8. Examining Fig. 3(b), it may be seen that theclose agreement between the CMT approximationand the exact solution is maintained at a 0:1nmlower wavelength and reflects somewhat larger per-

iodicity than seen at the Bragg wavelength. But, theresults of the classic CM calculation display broadand unacceptable deviations—in the shape and mag-nitude of the axial power distribution. This compar-ison reveals that even a relatively modest axialtemperature variation (5K across a 10mm grating)can produce significant “thermal chirp” effects.

The spectral variations in the power reflected by aPMMA grating, operating under different tempera-ture profiles, i.e., a uniform temperature change ofΔT ¼ 35°, an exponential temperature profile from10K to 1K (ΔT ¼ 10e−2:303

zL), and a linear tempera-

ture profile decreasing from an excess temperatureof 10K at the inlet to zero at the outlet, are comparedin Fig. 4. It is evident that in a PMMA Bragg grating,even a relatively small deviation from uniformityin temperature causes a significant shift in the re-flected power and in the wavelength of the maximumreflectivity, while for a uniform temperature profilethe overall thermo-optic effect is minor (small shiftin reflected power versus wavelength behavior). Thisis due to the fact that in PMMA the effect of k and Bresult in Bragg shifts of opposite sign and nearly thesame magnitude, thus leading to nearly completecompensation. However, when the axial variationin the expansion of the grating is included in the de-termination of the Bragg shift, through the contribu-tion of B0 to the CM solution, a decidedly asymmetricand complex spectral distribution emerges from theCMT methodology.

It is interesting to note thatPMMAgratings appearto be more sensitive to a linearly varying than to anexponentially varying temperature distribution. Thiscanbeattributed to a larger average value of zB0ðzÞ fora linear temperature profile. These pronounced ef-fects of nonuniform temperature cannot be predictedby the original CMmodel since it ignores the effect ofthe axial gradient of themodulation wavenumber,B0,

Fig. 3. (Color online) Normalized backward wave power for athermally chirped 10mm PMMA grating (at ΔλB ¼ 0 and−1:0nm) under a linear temperature profile of ΔT ¼ 5° − 0 °C(start to end of grating). Calculations are according to exact nu-merical model, CM, and CMT. Power is normalized to the incidentlight of 100 units.

Fig. 4. (Color online) Effect of temperature nonuniformity onspectral variation (shift fromBragg wavelength) in reflected powerfor a 10mm PMMA grating. Power is normalized to the incidentlight of unity.


butarewell capturedby the thermallymodifiedmeth-odology (CMT model) presented in the previoussections.It is also interesting to note that previous methods

of accounting for chirped gratings (see the Introduc-tion) were largely based on the original CM model.Some methods [5–7] have utilized the CM model di-rectly, by applying it to discrete finite segments of thechirped grating. Asymptotically, as the segments be-come infinitely small, the method becomes, at theinfinite limit, identical to the exact numerical calcu-lation. Hence, exact results will only be obtainedfrom calculations that are as computationally inten-sive as the exact numerical method. Indeed, thesestudies report the method to be useful only for mildchirps. These are not compatible with realistic ther-mal chirps in polymer fibers. Applying this method tothe thermally chirped PMMA fibers would yield re-sults largely between those obtained from the CMmodel to those obtained from the exact numericalmethod, with their numerical accuracy convergingonly at the computational-intensivity limit of theexact method, hence offering no practical (or other)advantage over the latter. Other methods [8,9] havesuggested semianalytical approximate solutions ofthe CM equations for chirped gratings. These math-ematical approximations were shown in thosestudies to be successfully close to numerical calcula-tions of the CM equations (classical CM model) and,hence, are essentially simplifying approximations ofthe numerical CMmodel. As such, they do not offer toproduce the substantially different behavior pre-dicted by the CMT model in this study, which wasverified by exact numerical calculations.

5. Conclusions

A thermally modified coupled-mode model (CMT)was developed to characterize the optical behaviorof thermally chirped Bragg gratings. The CMTmodelenjoys the benefit of simplicity, as provided by theoriginal CM model yet predicts the optical behaviorof thermally chirped polymer Bragg gratings as accu-rately as the much more computationally intensiveexact numerical calculation. Use of the CMT modelhas revealed the importance of accounting for the ax-ial variation in the thermomechanical terms in the

Bragg shift calculation to capture the aperiodicaldistortion that can occur in nonisothermal polymergratings.

The proposed CMT method was used to character-ize the thermo-optic behavior of a PMMA FBG sub-jected to various nonuniform temperature profiles. Itwas demonstrated that even relatively small tem-perature nonuniformity can cause a significant shiftin the reflected spectrum from the classical solution.

References

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10. G. D. Peng and P. L. Chu, “Polymer optical fiber photosensi-tivities and highly tunable fiber gratings,” Fiber Integr.Opt. 19, 277–293 (2000).

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Modified coupled-mode model for thermally chirped polymer Bragg gratings