## Propagation through apodized gratings

Optics Express, Vol. 3, Issue 11, pp. 405-410 (1998)

http://dx.doi.org/10.1364/OE.3.000405

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### Abstract

It is shown that light propagation in an apodized fiber Bragg grating with a Kerr nonlinearity approximately obeys a nonlinear Schrödinger-like equation, but with extra terms because the eigenstates of the grating vary with position. It is shown that propagation through such a grating leads to field enhancement, and to a nontrivial phase shift; an approximate expression for the reflectivity is also found.

© Optical Society of America

## 1. Introduction

3. P. S. Cross and H. Kogelnik, Opt. Lett. **1**, 43–45 (1977). [CrossRef] [PubMed]

4. B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. **31**, 223–225 (1995). [CrossRef]

5. F. Ouellette, Opt. Lett. **12**, 847–849 (1987). [CrossRef] [PubMed]

3. P. S. Cross and H. Kogelnik, Opt. Lett. **1**, 43–45 (1977). [CrossRef] [PubMed]

4. B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. **31**, 223–225 (1995). [CrossRef]

7. L. Poladian, Phys. Rev. E **48**4758–4767 (1993). [CrossRef]

*v*

^{-1}compared to that outside. Here

*v*is the instantaneous velocity in units of the group velocity in the grating’s absence [Eq. (7)]. For brevity we consider only apodized gratings in which the Bragg frequency constant, as this is the type of grating used in recent experiments by Eggleton

*et al*. [6].

## 2. Forward propagation

_{±}are the forward and backward mode amplitudes,

*V*is the group velocity in the grating’s absence, Γ is a nonlinear coefficient that is proportional to the nonlinear refractive index, and

*κ*is the grating strength. For tapered gratings,

*κ*=

*κ*(

*z*) by definition [7

7. L. Poladian, Phys. Rev. E **48**4758–4767 (1993). [CrossRef]

*z*, this variation is weak and is ignored here.

*ϵ*

_{±}are written as the linear eigenfunctions of Eqs. (1) but with a slowly varying amplitude. By substituting such an

*ansatz*into Eqs. (1) one finds that in a uniform grating the amplitude evolves according to the NLSE [1, 2].

*γ*= 1/√1 -

*v*

^{2}, and -1 <

*v*< +1 is the group velocity

*d*Ω

_{+}/

*dQ*in units of

*V*. The eigenstates

*φ*

_{±}associated with these eigenvalues are also well known [1, 2]. Note that Ω

^{2}=

*Q*

^{2}+

*κ*

^{2}, which is the usual way the dispersion relation is written. Since in a tapered grating

*κ*=

*κ*(

*z*), then also

*v*=

*v*(

*z*) and

*γ*=

*γ*(

*z*). Denoting the vector with elements

*ϵ*

_{±}as

*ϵ*⃗, we write

*a*,

*b*and

*c*are envelopes yet to be determined. Here

*z*=

*z*+

_{0}*μz*

_{1}+

*μ*

^{2}

*z*

_{2}and similarly for

*t*, where

*Z*

_{0}and

*t*

_{0}vary on the length and time scales in the problem, respectively, whereas

*z*

_{1},

*t*

_{1}and

*z*

_{2},

*t*

_{2}describe phenomena on increasingly longer scales. In the analysis below these are all taken to be independent. Parameter

*μ*≪ 1 tracks the size of the various terms [1, 2]. The novel element here is that

*Q*and

*φ*

_{±}vary slowly to account for the grating taper.

*μ*. The equation at order

*μ*is then satisfied. At order μ

^{2}two simultaneous equations result

*dv*/

*dz*

_{1}. Eliminating

*b*from Eqs. (5) it is found that, to this order,

*a*satisfies

*f*is an arbitrary function. The numerator simply expresses that the wave propagates at its instantaneous velocity. However, the denominator is due to changes in the grating eigenstates and leads to increases in the wave intensity when propagation. This was pointed out earlier but followed from an

*ad hoc*argument [6]; here it is proven rigorously. This enhancement is clearly important in nonlinear grating experiments [6]. Returning to Eqs. (5) we find by eliminating the

*∂a*/

*∂t*

_{1}terms that

*μ*

^{3}. It is straightforward, though tedious, to show that this leads to two coupled equations for the envelopes, that are of the same general form of Eqs. (5). Eliminating

*c*from these equations gives

*∂*/

*∂z*= (

*∂*/

*∂z*

_{0})+(

*∂*\

*∂z*

_{1}) + (

*∂*/

*∂z*

_{2}), we find by combining Eqs. (6) and (9) that envelopes

*a*and

*b*satisfy an equation of form of Eq. (9), but with

*∂*/

*∂z*

_{i}replaced by

*∂*/

*∂z*, and similarly for

*t*. Using then Eq. (8), leads to the final equation for

*a*. Because of Eq. (7) it is preferable to use

*A*= √

*v*

*a*rather than

*a*, leading to

*v*' =

*dv*/

*dz*, and

*A*=

_{z}*∂A*/

*∂z*,

*etc*. Since 1/(

*κγ*

^{3}) is the quadratic grating dispersion, Eq. (10) reduces to the NLSE for a uniform grating [2], for which

*v*' = 0 and

*v*

^{-1}= 0. Note the strong

*v*

^{-1}velocity dependence of the nonlinear term in Eq. (10) due to the velocity-dependent amplitude of the envelope [see Eq. (7)].

_{+}= Δ, then the

*A*term in Eq. (10) vanishes. Recall further that to lowest order

_{t}*A*is constant in the limit we are considering. We therefore drop terms in Eq. (10) that enter at level

*μ*

^{3}and that contain spatial derivatives of

*A*, as these are small. We are therefore left with

*v*' = 0, it does not affect Eq. (11) and we neglect it here. We take

*κ*̃

*L*= 15 [6], and calculate arg(

*A*). Fig. 1(a) shows the results for Δ/

*κ*̃ = 1.512, for which the minimum velocity within the grating is

*v*= 0.8. Fig. 1(b) is for Δ/

*v*̃ = 1.25, for which this 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 are missed are substantial. By considering the exact results it can also be ascertained that the variations in the modulus of A are much smaller than those in the phase. Though the results in Fig. 1 are for

*κ*̃

*L*= 15, those at others apodizations can be found from the scaling arg(

*A*) ∝ 1/(

*v*̃

*L*).

*z*=

*L*. Fig. 2 shows the phase at

*z*=

*L*as a function of the normalized detuning Δ/

*v*̃, for

*v*̃

*L*= 15 as in Figs. 1. The black line gives results following by quadrature from Eq. (11). The dots give exact results obtained from Eqs. (1). Finally, the red line is the analytic approximation

*/Δ. The agreement between the exact results and Eq. (11) is good, except at the smallest detunings. This is due to the grating reflection that was neglected thus far.*κ ˜

## 3. Contradirectional propagation

*μ*

_{2}. We thus take

*φ'*

_{±}are those for backward propagation. This

*ansatz*is substituted into Eqs. (1) and terms at equal powers of

*μ*are collected. This results in two coupled equations for the envelopes

*a*

_{±}and

*b*

_{±}. Eliminating

*b*

_{-}leads to

*a*

_{+}and

*b*

_{+}is still given by Eq. (8). Using this, the definitions of

*A*, and again taking Ω

_{+}= Δ, so that any time derivative vanishes, Eq. (15) reduces to

*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*

_{+}is constant to lowest order. We thus find a simple expression for the reflectivity

*R*of an apodized grating

*κ*̃ → 1. Of course discontinuities in

*v*(

*z*) or

*κ*'(

*z*), caused by discontinuities in the grating strength or its first derivative

*κ*(

*z*) and

*κ*'(

*z*), can also lead to substantial reflections.

*κ*̃ for an apodized grating with

*κ*̃

*L*= 15, as in our previous examples. The black line is the exact result obtained from Eqs. (1), whereas the red line corresponds to approximate result (18). Clearly, for detuning such that Δ < 1.3

*κ*̃ the agreement is quite good. At larger detunings, the exact results indicate resonances (for example around Δ = 1.46

*κ*̃), which the present approximate treatment misses. Nonetheless, for modest detunings Eq. (18) is a good approximation. Fig. (3)(b) is similar to Fig. (3)(a) except that

*κ*̃

*L*= 100, corresponding to a smoother taper. Clearly, the agreement between the exact result and approximation (18) is excellent.

7. L. Poladian, Phys. Rev. E **48**4758–4767 (1993). [CrossRef]

## 4. Discussion and Conclusions

## Acknowledgements

## References and links

1. | C. M. de Sterke and J. E. Sipe, “Gap solitons,” in |

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

3. | P. S. Cross and H. Kogelnik, Opt. Lett. |

4. | B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, Elect. Lett. |

5. | F. Ouellette, Opt. Lett. |

6. | B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg grating solitons in the nonlinear Schrödinger limit: theory and experiment,” submitted to J. Opt. Soc. Am. B |

7. | L. Poladian, Phys. Rev. E |

8. | D. Marcuse, |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(060.2340) Fiber optics and optical communications : Fiber optics components

**ToC Category:**

Focus Issue: Bragg solitons and nonlinear optics of periodic structures

**History**

Original Manuscript: September 24, 1998

Published: November 23, 1998

**Citation**

Martijn de Sterke, "Propagation through apodized gratings," Opt. Express **3**, 405-410 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-11-405

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### References

- C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics 33, E. Wolf, ed. (Elsevier, Amsterdam, 1994) 203-260.
- C. M. de Sterke and B. J. Eggleton, "Bragg solitons and the nonlinear Schr"odinger equation," Phys. Rev. E (to be published).
- P. S. Cross and H. Kogelnik, Opt. Lett. 1, 43-45 (1977). [CrossRef] [PubMed]
- B. Malo, D. C. Johnson, F. Bilodeau, J. Albert and K. O. Hill, Elect. Lett. 31, 223-225 (1995). [CrossRef]
- F. Ouellette, Opt. Lett. 12, 847-849 (1987). [CrossRef] [PubMed]
- B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, "Bragg grating solitons in the nonlinear Schr"odinger limit: theory and experiment," submitted to J. Opt. Soc. Am. B.
- L. Poladian, Phys. Rev. E 48 4758-4767 (1993). [CrossRef]
- D. Marcuse, Theory of dielectric optical waveguides, 2nd Ed. (Academic, San Diego, 1991).

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