## Nonreciprocal waveguide Bragg gratings

Optics Express, Vol. 13, Issue 8, pp. 3068-3078 (2005)

http://dx.doi.org/10.1364/OPEX.13.003068

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### Abstract

The use of a complex short-period (Bragg) grating which combines matched periodic modulations of refractive index and loss/gain allows asymmetrical mode coupling within a contra-directional waveguide coupler. Such a complex Bragg grating exhibits a different behavior (e.g. in terms of the reflection and transmission spectra) when probed from opposite ends. More specifically, the grating has a single reflection peak when used from one end, but it is transparent (zero reflection) when used from the opposite end. In this paper, we conduct a systematic analytical and numerical analysis of this new class of Bragg gratings. The spectral performance of these, so-called nonreciprocal gratings, is first investigated in detail and the influence of device parameters on the transmission spectra of these devices is also analyzed. Our studies reveal that in addition to the nonreciprocal behavior, a nonreciprocal Bragg grating exhibits a strong amplification at the resonance wavelength (even with zero net-gain level in the waveguide) while simultaneously providing higher wavelength selectivity than the equivalent index Bragg grating. However, it is also shown that in order to achieve nonreciprocity in the device, a very careful adjustment of the parameters corresponding to the index and gain/loss gratings is required.

© 2005 Optical Society of America

## 1. Introduction

1. X. Daxhelet and M. Kulishov, “Theory and practice of long-period gratings: when a loss becomes a gain,” Opt. Lett. **28**, 686–688 (2003). [CrossRef] [PubMed]

2. M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, and D.V. Plant, “Tunable waveguide transmission gratings based on active gain control,” IEEE J. Quantum Electron. **40**, 1715–1724 (2004). [CrossRef]

3. L. Poladian, “Rresonance mode expansions and exact solutions for nonuniform gratings,” Physical Review E **54**, 2963–2975 (1996). [CrossRef]

3. L. Poladian, “Rresonance mode expansions and exact solutions for nonuniform gratings,” Physical Review E **54**, 2963–2975 (1996). [CrossRef]

*ß*=Δ

*n*exp(

_{o}*j*2

*πz*Λ). This perturbation can be practically realized using a combination of an index grating (real grating) and a gain/loss grating (imaginary grating) of the form exp(±

*j2πz*/

*Λ*)=cos(

*2πz*/

*Λ*)±

*j*sin(

*2πz*/

*Λ*). Since the original suggestion by Poladian [3

3. L. Poladian, “Rresonance mode expansions and exact solutions for nonuniform gratings,” Physical Review E **54**, 2963–2975 (1996). [CrossRef]

4. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. **29**, 451–453 (2004). [CrossRef] [PubMed]

*j2πz*/

*Λ*) can also be used “to break the space-time reversibility in (two) co-propagating mode interaction within a grating-assisted co-directional coupler”. This original work has been followed by another more detailed study on the same topic [5

5. M. Greenberg and M. Orenstein, “Unidirectional complex gratings assisted couplers,” Opt. Express **12**, 4013–4018 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-4013. [CrossRef] [PubMed]

## 2. Theoretical model of NRBG

*Λ*). This perturbation can be expressed as follows:

*n*and Δ

_{DC}*n*are respectively the constant and modulated perturbation to the refractive index and Δ

_{AC}*α*and Δ

_{DC}*α*are respectively the constant and modulated perturbation to the loss/gain, and

_{AC}*k*

_{0}=2

*π*/

*λ*where

*λ*is the wavelength in vacuum. Eq. (1) is a general description of a complex grating and it takes into account the following: the DC and AC terms can have different amplitudes; the real and imaginary gratings can also exhibit different amplitudes; and the phase difference between the two gratings is not necessarily π/2. Specifically, the term

*Δz*represents the additional phase shift between the cosine (real) and the sine (imaginary) gratings. Notice that Eq. (1) does not include the effect of gain or loss saturation. It is important to emphasize that ideal non-reciprocal behavior is achieved when (i) the phase difference between the index and gain/loss gratings is exactly

*π*/2 (

*Δz*=

*0*) and (ii) both gratings exhibit an identical strength (Δ

*n*Δ

_{AC}*α*/

_{AC}*k*

_{0}Δ=Δ). In this case, the refractive index perturbation is described by a purely imaginary function that has only single-sided spatial components. As mentioned in the introduction, this is the fundamental requirement to ensure a non-reciprocal behavior in the grating structure.

*β*. The perturbation given by Eq. (1) will induce power transference from the forward-propagating mode (+

*β*) into the backward-propagating mode (-

*β*) at the phase-matching wavelength. The evolution of the forward-propagating,

*A*(

*z*), and backward-propagating,

*B*(

*z*), mode amplitudes within the slowly varying envelope approximation is determined by the following equations [6, 7

7. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

*jα*, and σ and

*α*are proportional to the non-modulated real and imaginary part of Eq. (1), Δ

*β*=

*β*-

*π*/

*Λ*and the coupling coefficients are:

*κ*and

_{n}*κ*are proportional to the overlap between the spatial mode distributions of each waveguide and the AC component of Eq.(1). It is important to point out that

_{α}*α*represents the gain/loss coefficient experienced by the propagating fields. Therefore, the gain/loss associated with the intensities is given by 2

*α*. The gain or loss experienced by the propagating modes can be modeled by either taking

*α*>0 for loss or

_{i}*α*<0 for gain. In order to simplify Eqs. (2) and (3), it is possible to define a complex propagation constant

_{i}*β*̃=

*β*+σ+

*jα*and a complex phase-matching factor:

*β*̃-

*π*/Λ. Eq. (2) and (3) can then be re-written as follows:

*z*=0), the coupling coefficients are:

*κ*is cancelled out, i.e.,

_{12}*κ*-

_{n}*κ*=0.

_{α}*z*=0 and

*L*. The solutions for the coupling equations (6) and (7) are obtained by using 2×2 transfer matrices

**M**[7

7. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

*A*(

*L*) and

*B*(

*L*) with the complex amplitudes

*A*(

*0*) and

*B*(

*0*) in the following way:

*γ*=(

*κ*

_{12}

*κ*

_{21}-

^{2})

^{1/2}. It is important to point out that the matrix elements given by Eq. (13) contain complex parameters. The only real parameter in (13) is the position

*z*along the grating. If a signal is injected from the left side of the grating,

*A*(0)=1 and therefore

*B*(

*L*)=0, (see Fig. 1) then the complex amplitude of the reflected signal is given by

*B*(0)=-

*M*

_{21}/

*M*

_{22}and the complex amplitude of the transmitted signal is given by

*A*(

*L*)=

*M*

_{11}-

*M*

_{12}

*M*

_{21}/

*M*

_{22}. On the other hand, if the signal is launched from the right side of the grating, i.e.

*B*(

*L*)=1 and therefore

*A*(0)=0, then the complex amplitude of the reflected signal is given by

*A*(

*L*)=

*M*

_{12}/

*M*

_{22}and the complex amplitude of the transmitted signal is given by

*B*(0)=1/

*M*

_{22}.

## 3. Spectral and dispersion characteristics of NRBGs

*n*=1.55 (the propagation constant is then given by

_{eff}*β*=

*k*

_{0}

*n*). It is assumed here that the effective index is equal to the group index of the propagation mode. The grating period which allows coupling between the forward- and backward-propagating modes at a resonant wavelength of 1550 nm is

_{eff}*Λ*=0.5 µm. In all the cases, the spectra are calculated assuming a grating of length

*L*=5 mm.

*κ*=

_{n}L*π/2*,

*κ*=0) and Fig. 2(b) ideal NRBG (

_{α}*κ*=

_{n}*κ*,

_{α}*Δz*=0) for

*κ*=(

_{21}L*κ*+

_{α}*κ*)

_{n}*L*=

*π*. It is important to point out that in our calculations, we neglect any contributions from the DC component in Eq. (1), which means that we assume that σ=0 and

*α*=0.

*ideal*NRBG takes the following form:

*z*=0 so that the boundary condition are

*A*(0)=1 and

*B*(L)=0 (see Fig.1), we can infer from Eq. (12) that the transmitted light is

*A*(

*L*)=

*M*=exp(

_{11}*jβ*̃

*L*) and the reflected light is

*B*(0)=-

*M*/

_{21}*M*(which is the case showed in Fig. 1(b). However, if the optical signal is injected into the right end of the grating,

_{22}*B*(

*L*)=1 and

*A*(

*0*)=0, then there is complete light transmission:

*B*(0)=

*1*/

*M*=exp(

_{22}*jβ*̃

*L*) with zero reflection

*A*(

*L*)=0. In other words, as expected, the mode coupling process is strongly asymmetric in this structure.

## 3.1 Amplification mechanism

*κ*=

_{n}L*π*/2 is only moderately strong, i.e. it provides a reflectivity slightly higher than 80% at the resonance wavelength. However, combination of this weak index grating with the complementary grating of gain/loss (of the same strength

*κ*=

_{α}L*π*/2) results in surprisingly strong reflection combined with amplification of almost 10 dB. It is important to point out that the spectra shown in Fig. 2 were computed for zero DC amplification in the waveguide (

*α*=0). Equal segments of gain and loss in the imaginary grating imply no net gain or loss along the grating. However, the power conservation law is not violated here since the complex grating is an active structure and power is supplied to maintain optical gain.

*κ*=

_{n}*κ*and

*κ*=

_{α}*κ*), its imaginary counterpart is defined by the following equation:

*Δn*} of Eq. (15) is positive, it represents loss while when it is negative, it represents gain. At the phase matching condition (Δ

*β*=0) and no DC contribution, Eqs. (2) and (3) are greatly simplified, and for

*A*(0)=

*A*and

_{0}*B*(

*L*)=0, these equations have the following solutions:

*A*(

*z*)=

*A*and

_{0}*B*(

*z*)=-

*2*

*jκA*(

_{0}*z*-

*L*). The electric field distribution inside the NRBG is then given by the following relation:

*E*

^{+}(z) and E

^{-}(z) are respectively the forward and backward propagating fields inside the device. As a result, the field intensity inside the NRBG,

*E*(

*z*)

*E**(

*z*), is found to be

*κL*. Figure 3 shows the field intensity inside the grating with respect to the gain grating periodicity.

## 3.2 Frequency selectivity

*κ*=

_{n}L*π*). In the NRBG, equal contributions from the real and imaginary parts result in a total grating strength of

*κ*

_{21}

*L*=

*κ*+

_{n}L*κ*=2

_{α}L*κ*=2

_{n}L*π*(here

*κ*

_{12}

*L*=0). The corresponding bandwidths are estimated as the difference between the wavelength at peak reflectivity and the first zero at either side around this peak reflectivity. The respective bandwidths Δ

*λ*(for the NRBG) and Δ

_{nr}*λ*(for the IBG) are given by the following expressions:

_{c}## 3.3 Dispersion Characteristics of NRBGs

*n*/

_{eff}L*c*, where

*c*is the speed of light in vacuum. This time is also marked by the blue dotted line in Fig. 2(c). At wavelengths of zero reflection, the phase undergoes discontinuities, which translate into the observed sharp jumps (delta functions) in the group delay characteristic.

## 3.4 Effect of net gain in the waveguide

8. H. Kogelnik and C.V. Shank, “Coupled mode theory of distributed feedback lasers,” J.Appl.Phys. **43**, 2327–2335 (1972). [CrossRef]

^{-4}corresponds to a gain peak of approximately 2

*α*=4.05 cm

^{-1}(35 dB/cm) (for a perfect matching gain grating, operating at 1550 nm). This gain level can be achieved relatively easily with current integrated technologies using semiconductor gain media [9

9. D.R. Zimmerman and L.H. Spiekman, “Amplifiers for the masses: EDFA, EDWA, and SOA Amplets for metro and access applications,” IEEE J. Lightwave Technol. **22**, 63–71 (2004). [CrossRef]

9. D.R. Zimmerman and L.H. Spiekman, “Amplifiers for the masses: EDFA, EDWA, and SOA Amplets for metro and access applications,” IEEE J. Lightwave Technol. **22**, 63–71 (2004). [CrossRef]

## 4. Robustness of the NRBG design

*κ*=

_{n}*κ*), and they are shifted exactly a quarter of the period with respect to each other (Δ

_{α}*z*=0). Our objective now is to evaluate the impact of deviations from these ideal conditions on the grating’s spectral characteristics.

## 4.1 Grating amplitude misbalance

*κ*and

_{n}*κ*, affects the transmission spectra (as compared with those of the ideal NRBG). It is important to point out that in our simulations, we assume perfect phase matching between the index and the gain gratings, i.e. Δ

_{α}*z*=0. In all the plots in Fig. 5, the solid (red) curves correspond to the case of an ideal grating or 0% difference between the respective amplitudes (the difference is given by 2(

*κ*-

_{n}*κ*)/(

_{α}*κ*+

_{n}*κ*)) in the grating amplitudes), the dashed (blue) curves correspond to a 1% difference, and the dash-dotted (brown) curves correspond to a 5% difference. As we can see, the grating amplitude imbalance of 5% leads to i) small reduction of amplified reflection (~1.2 dB) for the wave incident from the left side (Fig. 5a); ii) approximately 2 dB deviation from perfect 100% transmission (Fig. 5(c)); iii) and slight increase in reflection from right to left from zero to -18 dB (Fig. 5(e)). If one neglects sharp peaks on the plots for the group delay (Fig. 5(b), Fig. 5(d), Fig. 5(f)), the reflected and both transmitted waves exhibit practically the same deviation from the ideal non-dispersive behavior.

_{α}## 4.2 Grating position deviation

*z*, from the ideal position of real and imaginary gratings (ideally Δ

*z*=0). Fig. 6 presents simulations for amplitude-matching gratings (

*κ*=

_{n}*κ*) with different values of position imbalance: Δ

_{α}*z*/

*Λ*=0% (solid, red), Δ

*z*/

*Λ*=1% (dash, blue) and Δ

*z*/

*Λ*=5% (dot, magenta). Unlike the amplitude imbalance, the 5% position deviation leads to i) the shift in the resonance wavelength of the reflected light in Fig. 6(a) accompanied by ~2 dB increase in peak amplification; ii) ~5 dB deviation from perfect 100% transmission (Fig. 6(c)); iii) and substantial increase in reflection from right to left from zero to -4 dB (Fig. 6(e)). The position imbalance also produces strong dispersion especially within the reflection band, as it can be seen in Fig. 6(b) and Fig. 6(f). Notice that the position deviation generally imposes a stricter tolerance on the grating design, than the amplitude imbalance.

## 5. Conclusion

## Acknowledgments

## References and links

1. | X. Daxhelet and M. Kulishov, “Theory and practice of long-period gratings: when a loss becomes a gain,” Opt. Lett. |

2. | M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, and D.V. Plant, “Tunable waveguide transmission gratings based on active gain control,” IEEE J. Quantum Electron. |

3. | L. Poladian, “Rresonance mode expansions and exact solutions for nonuniform gratings,” Physical Review E |

4. | M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. |

5. | M. Greenberg and M. Orenstein, “Unidirectional complex gratings assisted couplers,” Opt. Express |

6. | R. Kashyap, |

7. | T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. |

8. | H. Kogelnik and C.V. Shank, “Coupled mode theory of distributed feedback lasers,” J.Appl.Phys. |

9. | D.R. Zimmerman and L.H. Spiekman, “Amplifiers for the masses: EDFA, EDWA, and SOA Amplets for metro and access applications,” IEEE J. Lightwave Technol. |

**OCIS Codes**

(230.1950) Optical devices : Diffraction gratings

(230.3120) Optical devices : Integrated optics devices

(250.4480) Optoelectronics : Optical amplifiers

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 22, 2005

Revised Manuscript: April 6, 2005

Published: April 18, 2005

**Citation**

Mykola Kulishov, Jacques Laniel, Nicolas Bélanger, José Azaña, and David Plant, "Nonreciprocal waveguide Bragg gratings," Opt. Express **13**, 3068-3078 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3068

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### References

- X. Daxhelet, M. Kulishov, �??Theory and practice of long-period gratings: when a loss becomes a gain,�?? Opt. Lett. 28, 686-688 (2003). [CrossRef] [PubMed]
- M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, D.V. Plant, �??Tunable waveguide transmission gratings based on active gain control,�?? IEEE J. Quantum Electron. 40, 1715-1724 (2004). [CrossRef]
- L. Poladian, �??Rresonance mode expansions and exact solutions for nonuniform gratings,�?? Physical Review E 54, 2963-2975 (1996). [CrossRef]
- M. Greenberg, M. Orenstein, �??Irreversible coupling by use of dissipative optics,�?? Opt. Lett. 29, 451-453 (2004). [CrossRef] [PubMed]
- M. Greenberg, M. Orenstein, �??Unidirectional complex gratings assisted couplers,�?? Opt. Express 12, 4013-4018 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-4013.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-4013.</a> [CrossRef] [PubMed]
- R. Kashyap, Fiber Bragg Gratings (SanDiego, CA: Academic, 1999, ch.4).
- T. Erdogan, �??Fiber grating spectra,�?? J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]
- H. Kogelnik, C.V. Shank, �??Coupled mode theory of distributed feedback lasers,�?? J.Appl.Phys. 43, 2327-2335 (1972). [CrossRef]
- D.R. Zimmerman, L.H. Spiekman, �??Amplifiers for the masses: EDFA, EDWA, and SOA Amplets for metro and access applications,�?? IEEE J. Lightwave Technol. 22, 63-71 (2004). [CrossRef]

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