Propagation through apodized gratings

Propagation through apodized gratings

C. Martijn de Sterke

School of Physics, University of Sydney, NSW 2006, Australia, and AustralianPhotonics Cooperative Research Centre, Australian Technology Park, Eveleigh,

1430, Australia

[email protected]

Abstract: It is shown that light propagation in an apodized fiberBragg grating with a Kerr nonlinearity approximately obeys a nonlinearSchrodinger-like equation, but with extra terms because the eigenstatesof the grating vary with position. It is shown that propagation throughsuch a grating leads to field enhancement, and to a nontrivial phaseshift; an approximate expression for the reflectivity is also found.

c©1998 Optical Society of AmericaOCIS codes: (060.2340) Fiber optics components; (050.2770) Gratings

References and links

1. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics 33, E. Wolf, ed. (Elsevier,Amsterdam, 1994). 203-260.

2. C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrodinger equation,”in press Phys. Rev. E .

3. P. S. Cross and H. Kogelnik, Opt. Lett. 1, 43-45 (1977).4. B. Malo, D. C. Johnson, F. Bilodeau, J. Albert and K. O. Hill, Elect. Lett. 31, 223-225 (1995).5. F. Ouellette, Opt. Lett. 12, 847-849 (1987).6. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg grating solitons in the nonlinearSchrodinger limit: theory and experiment,” submitted to J. Opt. Soc. Am. B

7. L. Poladian, Phys. Rev. E 48 4758-4767 (1993).8. D. Marcuse, Theory of dielectric optical waveguides, 2nd Ed. (Academic, San Diego, 1991).

1. Introduction

It is well known that light propagation through a uniform grating is governed by aset of coupled mode equations for the forward and backward propagating mode am-plitudes. However, propagation through a uniform grating can also approximately bedescribed by the nonlinear Schrodinger equation (NLSE) when the pulse spectrum isnarrow enough to neglect cubic grating dispersion, and the pulse intensity is not toohigh [2]. This simplified description is useful since results established for the NLSE, suchas the existence of solitons, can be applied to light propagation through a grating.

Many grating applications call for nonuniform gratings [3, 4, 5], such as apodizedgratings. In apodized gratings the grating strength does not jump at the edges, but in-creases gradually until it reaches its maximum value [3, 4, 6]. The advantage of suchgratings is that the reflection outside the photonic band gap is strongly reduced com-pared to that of uniform gratings. Pulse propagation through tapered gratings is alsoimportant in recent nonlinear grating experiments, in which short, intense light pulseare incident on an apodized grating and form a Bragg grating soliton [6].

Light propagation through nonuniform gratings can be described by the coupledmode equations with slowly varying coefficients [7]. However, the NLSE description isnot so simple, because the field is expressed in the grating’s eigenfunctions which varywith position if the grating is not uniform [1, 2]. This dependence is ignored if onesimply varies the NLSE coefficients. Issue like this also arise in propagation through a

(C) 1998 OSA 23 November 1998 / Vol. 3, No. 11 / OPTICS EXPRESS 405#6299 - $15.00 US Received September 22, 1998; Revised October 20, 1998

tapered waveguide. There the mode varies with position, and may change significantlyover the structure’s length. Such problems are treated using local coupled mode theory,which accounts for changes of the basis functions [8]. Here, light propagation throughnonuniform gratings is treated similarly, though now the grating’s eigenstates are vary-ing. This leads to an NLS-like equation for the field envelope [Eq. (10)], but with extraterms due to variations in the eigenstates. We also show that the intensity inside a grat-ing is enhanced by a factor v−1 compared to that outside. Here v is the instantaneousvelocity in units of the group velocity in the grating’s absence [Eq. (7)]. For brevity weconsider only apodized gratings in which the Bragg frequency constant, as this is thetype of grating used in recent experiments by Eggleton et al.[6].

2. Forward propagation

The electric field in a grating satisfies the coupled mode equations [1]

i

V

∂E±∂t±i∂E±∂z+κ(z)E∓ + Γ(|E±|

2 + 2|E∓|2)E±=0, (1)

where E± are the forward and backward mode amplitudes, V is the group velocity inthe grating’s absence, Γ is a nonlinear coefficient that is proportional to the nonlin-ear refractive index, and κ is the grating strength. For tapered gratings, κ = κ(z) bydefinition [7]. Though Γ also depends on z, this variation is weak and is ignored here.

We derive the approximate nonlinear Schrodinger results using a multiple scalesanalysis. Since this has been discussed before [1, 2], we leave the treatment here brief.The key is that the E± are written as the linear eigenfunctions of Eqs. (1) but with aslowly varying amplitude. By substituting such an ansatz into Eqs. (1) one finds thatin a uniform grating the amplitude evolves according to the NLSE [1, 2].

The dispersion relation of a grating can be written as

Q = κγv, Ω± = ±V κγ, (2)

where γ = 1/√1− v2, and −1 < v < +1 is the group velocity dΩ+/dQ in units of V .

The eigenstates ϕ± associated with these eigenvalues are also well known [1, 2]. Notethat Ω2 = Q2 + κ2, which is the usual way the dispersion relation is written. Since ina tapered grating κ = κ(z), then also v = v(z) and γ = γ(z). Denoting the vector with

elements E± as ~E , we write

~E =(µa(z1, z2; t1, t2)ϕ+(z1, z2) + µ

2b(z1, z2; t1, t2)ϕ−(z1, z2)

+µ3c(z1, z2; t1, t2)ϕ−(z1, z2))e−iΩ+t0 eiΦ(z0), (3)

where

Φ(z) =

∫ z00

Q(z′)dz′, (4)

is the rapidly varying contribution to the phase, and a, b and c are envelopes yet to bedetermined. Here z = z0 + µz1 + µ

2z2 and similarly for t, where z0 and t0 vary on thelength and time scales in the problem, respectively, whereas z1, t1 and z2, t2 describephenomena on increasingly longer scales. In the analysis below these are all taken tobe independent. Parameter µ 1 tracks the size of the various terms [1, 2]. The novelelement here is that Q and ϕ± vary slowly to account for the grating taper.

We substitute Eq. (3) into Eqs. (1) and collect terms with equal powers of µ.The equation at order µ is then satisfied. At order µ2 two simultaneous equations result

i

(∂a

∂z1±∂a

∂t1

)±i

2

1

1± v

dv

dz1a+

2κ

1± vb = 0. (5)


The middle terms in these equations are new and are due to the grating variationsthrough dv/dz1. Eliminating b from Eqs. (5) it is found that, to this order, a satisfies

v∂a

∂z1+∂a

∂t1+1

2

dv

dz1a = 0. (6)

The general solution of (6) is

a(z1, t1) =f [t1 −

∫ z10 dz

′/v(z′)]√v(z)

, (7)

where f is an arbitrary function. The numerator simply expresses that the wave prop-agates at its instantaneous velocity. However, the denominator is due to changes in thegrating eigenstates and leads to increases in the wave intensity when propagation. Thiswas pointed out earlier but followed from an ad hoc argument [6]; here it is provenrigorously. This enhancement is clearly important in nonlinear grating experiments [6].Returning to Eqs. (5) we find by eliminating the ∂a/∂t1 terms that

b = −i

2κγ2∂a

∂z1+iv

4κ

dv

dz1a. (8)

We now take the multiple scales analysis to order µ3. It is straightforward,though tedious, to show that this leads to two coupled equations for the envelopes, thatare of the same general form of Eqs. (5). Eliminating c from these equations gives

v∂a

∂z2+∂a

∂t2+1

2

dv

dz2a+1

γ

∂b

∂z1−γv

2

dv

dz1b−iΓ

2(3 − v2) |a|2a = 0 (9)

Since, to this order, ∂/∂z = (∂/∂z0)+(∂/∂z1)+(∂/∂z2), we find by combining Eqs. (6)and (9) that envelopes a and b satisfy an equation of form of Eq. (9), but with ∂/∂zireplaced by ∂/∂z, and similarly for t. Using then Eq. (8), leads to the final equation fora. Because of Eq. (7) it is preferable to use A ≡

√v a rather than a, leading to

i(At+ vAz)+Azz

2κγ3−(1 + v2)v′

2κγvAz −

v′′

4κγvA−γ(4v2 − 3)(v′)2

8κv2A+3− v2

2vΓ|A|2A = 0.

(10)Here v′ ≡ dv/dz, and Az ≡ ∂A/∂z, etc. Since 1/(κγ3) is the quadratic grating dis-persion, Eq. (10) reduces to the NLSE for a uniform grating [2], for which v′ = 0 andv′′ = 0. Note the strong v−1 velocity dependence of the nonlinear term in Eq. (10) dueto the velocity-dependent amplitude of the envelope [see Eq. (7)].

We now find the simplest nontrivial solutions to Eq. (10). We take the fieldweak enough to neglect the nonlinear term. We further consider CW solutions withfrequency ∆ > 0 with respect to the Bragg frequency. Choosing Ω+ = ∆, then the Atterm in Eq. (10) vanishes. Recall further that to lowest order A is constant in the limitwe are considering. We therefore drop terms in Eq. (10) that enter at level µ3 and thatcontain spatial derivatives of A, as these are small. We are therefore left with

ivAz −v′′

4κγvA−

γ(4v2 − 3)(v′)2

8κv2A = 0. (11)

Since the second and third terms in Eq. (11) are real, the lowest order effect of the taperon the CW signal is a phase change that can be calculated by quadrature. To determineits importance we consider the particular apodized grating with [6]

κ(z) = κ sin2(πz/L) (0 ≤ z ≤ L), (12)


Note that an apodized grating usually has a uniform section between the tapers. Since insuch a section v′ = 0, it does not affect Eq. (11) and we neglect it here. We take κL = 15[6], and calculate arg(A). Fig. 1(a) shows the results for ∆/κ = 1.512, for which theminimum velocity within the grating is v = 0.8. Fig. 1(b) is for ∆/κ = 1.25, for whichthis velocity is 0.6. The black lines are exact results, the red lines follow from Eq. (11).The agreement is clearly good in Fig. 1(a), though in Fig. 1(b) the oscillations that aremissed are substantial. By considering the exact results it can also be ascertained thatthe variations in the modulus of A are much smaller than those in the phase. Thoughthe results in Fig. 1 are for κL = 15, those at others apodizations can be found fromthe scaling arg(A) ∝ 1/(κL).

Figure 1. Argument of A vs position in a tapered grating. Black lines are exact,red lines follow by integrating Eq. (11). In (a) ∆/κ = 1.512, in (b) ∆/κ = 1.25.

Of course the most relevant phase is that at z = L. Fig. 2 shows the phaseat z = L as a function of the normalized detuning ∆/κ, for κL = 15 as in Figs. 1.The black line gives results following by quadrature from Eq. (11). The dots give exactresults obtained from Eqs. (1). Finally, the red line is the analytic approximation

arg[A(L)] = −π2

16

(κ

∆

)31

κL

(1 +25

32

(κ

∆

)2), (13)

obtained from Eq. (11) by expanding in κ/∆. The agreement between the exact resultsand Eq. (11) is good, except at the smallest detunings. This is due to the gratingreflection that was neglected thus far.

Figure 2. Absolute value of the argument of envelope A at z = L vs the normal-ized detuning ∆/κ for an apodized grating with κL = 15. The black line follows fromEq. (11), the dots are exact results, and the red line is analytic approximation (13).


The phase discussed here is due purely to changes in the eigenstates. Thoughit is included implicitly in Eqs. (1), in the present treatment it appears explicitly.

3. Contradirectional propagation

In Section 2 we considered forward propagation, neglecting reflections. Here we estimatethe reflectivity of an apodized grating. This is a key consideration in Bragg gratingsoliton experiments, as the apodization is applied to reduce reflection [6]. The derivationis similar to that in Section 2, except that backward propagating modes are includedexplicitly. The calculation also only needs to be taken to order µ2. We thus take

~E=

[(a+ϕ+ + b+ϕ−

)eiΦ+

(a−ϕ

′+ + b−ϕ

′−

)e−iΦ

]e−iΩt, (14)

where the eigenstates ϕ′± are those for backward propagation. This ansatz is substitutedinto Eqs. (1) and terms at equal powers of µ are collected. This results in two coupledequations for the envelopes a± and b±. Eliminating b− leads to

va−z − a−t +v′

2a− =

(a+tγ+γ

2v′a+ + 2iκγvb+

)e2iΦ. (15)

We now assume that the reflectivity is small, and thus the forward component is takento be unaffected by the backward propagating component. Thus the relation betweena+ and b+ is still given by Eq. (8). Using this, the definitions of A, and again takingΩ+ = ∆, so that any time derivative vanishes, Eq. (15) reduces to

A−z =A+z

γe2iΦ, (16)

where A± =√v a±. But (11) is an approximate relation between A+z and A. We thus

easily find an approximate expression for A−z

A−z = −iA+

[v′′

4κγv+γ(4v2 − 3)(v′)2

8κv2

]e2iΦ (17)

Now since the reflection is taken weak, according to Eq. (7) A+ is constant to lowestorder. We thus find a simple expression for the reflectivity R of an apodized grating

R =

∣∣∣∣∣∫ L0

[v′′

4κγv+γ(4v2 − 3)(v′)2

8κv2

]e2iΦdz

∣∣∣∣∣2

. (18)

which can be evaluated by quadrature. As a general observation note that the expo-nential is generally rapidly varying, and so the reflectivity is small unless ∆/κ→ 1. Ofcourse discontinuities in v(z) or v′(z), caused by discontinuities in the grating strengthor its first derivative κ(z) and κ′(z), can also lead to substantial reflections.

Fig. (3)(a) shows the reflectivity as a function of the relative detuning ∆/κ foran apodized grating with κL = 15, as in our previous examples. The black line is theexact result obtained from Eqs. (1), whereas the red line corresponds to approximateresult (18). Clearly, for detuning such that ∆ < 1.3κ the agreement is quite good. Atlarger detunings, the exact results indicate resonances (for example around ∆ = 1.46κ),which the present approximate treatment misses. Nonetheless, for modest detuningsEq. (18) is a good approximation. Fig. (3)(b) is similar to Fig. (3)(a) except that κL =100, corresponding to a smoother taper. Clearly, the agreement between the exact resultand approximation (18) is excellent.


Figure 3. Reflectivity vs normalized detuning ∆/κ for two apodized gratings. In(a) κL = 15, in (b) κL = 100. Black lines are exact results, red lines follow fromapproximation (18).

Expressions for the reflectivity of nonuninform gratings have been found pre-viously using the WKB method [7]. However, there only the reflectivity due to thephotonic band gap, and due to jumps in the grating parameters were considered. Herewe consider the more subtle effects that occur wholly outside the photonic band gap.

4. Discussion and Conclusions

In this paper the propagation of light through a tapered grating has been approachedin the spirit of a local coupled mode expansion. Unusually, here the local modes are theeigenfunctions of a tapered grating. It should be noted that this is an initial investigation,and that the method needs to be studied more completely before its features can beevaluated. Nonetheless, a rigorous derivation of the enhanced electric field strengthinside a tapered grating, and simple expressions for the phase evolution of the fieldinside the grating, and for the grating reflectivity were obtained.

Even though the method described here reduces the calculations to quadrature,the resulting integrals still needs to be performed numerically. It is likely that for gratingsof the form (12) they can be written in terms of hyperelliptic functions, though this isnot practical here. However, it would seem likely that good approximations can beobtained using a perturbation method, such as the method of steepest descent.

In conclusion, a novel method for the description of light propagation throughtapered gratings has been presented. The method leads to a NLSE, with additionalterms that are due to the fact that the eigenstates of the grating change with position.It is concluded that the propagation through a taper leads to a nontrivial phase shift.A simple, approximate expression for the grating reflectivity is also found.

Acknowledgements

I am grateful to John Sipe and Leon Poladian for pertinent discussions. This work wassupported by the Australian Research Council.


Propagation through apodized gratings

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