## Unidirectional complex gratings assisted couplers

Optics Express, Vol. 12, Issue 17, pp. 4013-4018 (2004)

http://dx.doi.org/10.1364/OPEX.12.004013

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### Abstract

We present a novel concept which enables the realization of unidirectional and irreversible grating assisted couplers by using gain-loss modulated medium to eliminate the reversibility. Employing a matched periodic modulation of both refractive index and loss (gain) we achieve a unidirectional energy transfer between the modes of the coupler which translates to light transmission from one waveguide to another while disabling the inverse transmission. The importance of self coupling coefficients is explored as well and a feasible implementation, where the real and imaginary perturbations are implemented in different waveguides is presented.

© 2004 Optical Society of America

## 1. Introduction

*p*(

*x*,

*y*,

*z*)=Δ(

*x*,

*y*)

*f*(

*z*), where Δ(

*x*,

*y*) is a constant lateral profile and

*f*(

*z*) is a longitudinal perturbation function. It can be shown [6] that, to the first approximation, the inter-mode power conversion is proportional to the Fourier transform of the longitudinal perturbation function

*f*(

*z*):

*P*

_{m,n}is the power converted from compound mode

*n*to

*m*and

*β*

_{m},

*β*

_{n}are the modal propagation constants. Two compound modes are coupled if the perturbation is purely sinusoidal, with the period matched to the difference of their propagation constants. The power conversion is symmetric in this case, since the Fourier transform of any real function (transparent optics) is either symmetric or antisymmetric:

*F*{

*sin*(Δ

*βz*)}=(

*δ*(

*Ω*-Δ

*β*)-

*δ*(

*Ω*+Δ

*βz*))/

*2*

*i*, where Ω stands for a spatial frequency. Such perturbation will convert an initially excited compound mode

*m*to

*n*and vice-versa, after the same propagation distance. A perturbation with a single spatial frequency is the complex function:

*f*(

*z*)=

*exp*{±

*i*Δ

*βz*}, that provides a unidirectional and time irreversible mode conversion [7

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

*m*to

*n*, while the ‘-’ one will match only the conversion from

*n*to

*m*. The realization of

*exp*{±

*i*Δ

*βz*}=

*cos*(Δ

*βz*)±

*isin*(Δ

*βz*), is by a simultaneous synchronized modulation of the real and imaginary (absorption/gain) parts of refractive index.

## 2. Analysis

*h*=5µm, cladding index

*n*

_{clad}=1.445 and core index is

*n*

_{core}=1.455, all the calculations performed at the wavelength of

*λ*=1.55µm. The directional coupler can be described by the notation

*w1*×

*d*×

*w2*where

*w1*,

*w2*are the widths of the waveguides and

*d*is their separation. The analysis is for the first (zero order) and second compound modes of the GAC. The slowly varying mode amplitudes

*c*

_{1},

*c*

_{2}for a complex single sideband perturbation function

*f*(

*z*)=

*exp*{

*i*Δ

*βz*} can be derived from the coupled equations [6]:

*K*

_{mn}with

*m,n*=1,2 are the coupling coefficients given by:

*β*, 2Δ

*β*can be neglected, since their average over a period is zero to the first approximation. Averaging Eqs. (2) and (3) over one period results in:

*0*)=C01,

*c*

_{2}(

*0*)=

*C*

_{02}the formal solution is:

*c*

_{1}is constant for any choice of the initial conditions. However, if

*c*

_{1}(

*0*) is non-zero the amplitude of the second compound mode grows linearly with the propagation distance. We define this case as “conversion”. On the other hand when

*c*

_{1}(

*0*)=

*0*the power in both compound modes remains constant and there is no power transfer (“no-conversion”). For both

*c*

_{1}(

*0*),

*c*

_{2}(

*0*) non-zero, the total solution for

*c*

_{2}(

*z*) is the interference of the initial field of the second mode

*C*

_{02}with the field which flows to the second mode. The total power in the system grows with the propagation but the conservation law is not violated since we are dealing with an “active” system and energy should be supplied to maintain optical gain. The reasons for the peculiar overall power growth are discussed in [7

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

*K*

_{11},

*K*

_{22}which can be defined as self-coupling coefficients of the compound modes. To elucidate their impact, we performed a numerical solution of Eqs. (2) and (3) shown for the case of “conversion” in Fig. 1. The values for a period and inter-modal coupling coefficients were chosen Λ=500µm,

*K*

_{12}=

*K*

_{21}=

*K*=1.15·10

^{-3}µm

^{-1}, respectively. The self-coupling coefficients varied in each simulation according to: 1.

*K*

_{11}=

*K*

_{22}=0; 2.

*K*

_{11}=

*K*,

*K*

_{22}=0; 3.

*K*

_{11}=0,

*K*

_{22}=

*K*; and

*4*.

*K*

_{11}=

*K*

_{22}=

*K*. It follows that

*K*

_{11}can increase slightly the conversion efficiency for positive value of

*K*

_{11}(or decrease for negative) due to the dependence of the derivative of

*c*

_{2}on this coupling coefficient mediated by

*c*

_{1}. By a similar reasoning, in the case of “no-conversion”

*K*

_{11}has no influence. The oscillations of

*c*

_{1}(not shown) are mainly attributed to the term proportional to

*K*

_{12}

*exp*{

*i2*Δ

*βz*}, so they have halved periodicity. The value of

*K*

_{22}is in charge of the undesirable power oscillations of the second compound mode.

*K*

_{22}is multiplied by

*c*

_{2}which in both cases (conversion and no-conversion) is much larger than

*c*

_{1}, thus it is the critical parameter which enhances undesirable oscillations in the system.

_{eff1}=1.449849 and n

_{eff2}=1.447640 corresponding to a difference period of Λ≈705µm. The overlap integrals between the single and compound modes are 99.8% and 99.4% for the first and second modes respectively. In order to achieve a substantial coupling coefficient and consequently short device length, the gratings should be located where both compound modes have significant power. This optimization for a given structure can be performed using Eq. (4). Our analysis showed the best position of the perturbation is within the core of the waveguides, but slightly decentered to compensate for the asymmetry. It is possible to eliminate

*K*

_{22}which is responsible for the undesirable oscillations if the perturbation is located only in the core of the larger waveguide (

*w1*=

*6µm*). With the perturbation strength of Δ=0.001 the inter-modal coupling coefficient is

*K*

_{12}=2.7510

^{-3}µm

^{-1}.

*K*

_{22}is negligible since the second mode has only a small fraction of power in this waveguide.

8. W. P. Huang and C. L. Xu “Simulation of Three-Dimensional Optical Waveguides by a Full-Vector Beam Propagation Metod,” IEEE J. Quantum Electron. **29**, 2639–2649 (1993). [CrossRef]

*K*

_{11}which enhances the conversion efficiency; 2. The period for simulations by the SBPM can be slightly out of resonance. (The dependence of the system on detuning from resonant condition is out of scope of current paper).

*w1*=

*6µm*waveguide) the power of the second mode is growing constantly (contrary to the case of real gratings where the power is exchanged periodically from one compound mode to the other). The longer the device is, the more power is converted to the second waveguide (until saturation). However if light is incident onto the

*w2*=

*3µm*waveguide, almost no power (less than -26dB) is transferred to the

*w1*=

*6µm*guide.

## 3. Separation of real and imaginary perturbations

*f*(

*z′*)=

*exp*{-

*i*Δ

*βz′*},

*z′*=-

*z*, thus power is back converted from the second to the first mode. However – it is obvious that the original initial conditions (e.g., power only in waveguide 1) – will not be regenerated here in contrary to all conventional time reversal optical elements.

## 6. Summary

^{-1}(~twice of a reasonable value) – but it was done only to reduce simulation time – reducing further the gain will result only in increasing the device length by the same factor; The accurate matching between the modulation depth of the real and imaginary gratings is achievable since (at least) the gain-loss gratings is actively controlled by injection and thus can be tuned to the proper value.

## References and links

1. | T.L. Koch, E.G. Burkhard, F.G. Stortz, T.J. Bridges, and T. Sizer, II, Vertically grating-coupled ARROW structures for III–V integrated optics,” IEEE J. Quantum Electron. |

2. | B. Little and T. Murphy, “Design rules for maximally flat wavelength-insensitive optical power dividers using Mach-Zehnder structures,” IEEE Photon. Technol. Lett. |

3. | R.C. Alferness, T.L. Koch, L.L. Buhl, F. Storz, F. Heismann, and M.J.R. Martyak, “Grating-assisted InGaAs/InP verstical codirectional coupler filter,” Appl. Phys. Lett. |

4. | G.R. Hill, “Wavelength domain optical network techniques,” in Proc. IEEE |

5. | L. Poladian, “Resonanse mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E |

6. | D. Marcuse, |

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

8. | W. P. Huang and C. L. Xu “Simulation of Three-Dimensional Optical Waveguides by a Full-Vector Beam Propagation Metod,” IEEE J. Quantum Electron. |

**OCIS Codes**

(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers

(130.3120) Integrated optics : Integrated optics devices

(250.4480) Optoelectronics : Optical amplifiers

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 27, 2004

Revised Manuscript: August 11, 2004

Published: August 23, 2004

**Citation**

Maxim Greenberg and Meir Orenstein, "Unidirectional complex grating assisted couplers," Opt. Express **12**, 4013-4018 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-4013

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### References

- T.L. Koch, E.G. Burkhard, F.G. Stortz, T.J. Bridges and T. Sizer, II, Vertically grating-coupled ARROW structures for III-V integrated optics,�?? IEEE J. Quantum Electron. QE-23, 889-897, (1987). [CrossRef]
- B.Little and T.Murphy, "Design rules for maximally flat wavelength-insensitive optical power dividers using Mach-Zehnder structures," IEEE Photon. Technol. Lett. 9, 1607-1609, (1997). [CrossRef]
- R.C. Alferness, T.L. Koch, L.L.Buhl, F.Storz, F.Heismann and M.J.R. Martyak, "Grating-assisted InGaAs/InP verstical codirectional coupler filter," Appl. Phys. Lett. 55, 2011-2013, (1989) [CrossRef]
- G.R. Hill, "Wavelength domain optical network techniques," in Proc. IEEE 77, 121-132, (1989).
- L.Poladian, "Resonanse mode expansions and exact solutions for nonuniform gratings," Phys. Rev. E 54, 2963-2975, (1996). [CrossRef]
- D.Marcuse, Theory of Dielectric Optical Waveguides Sec. Ed., Academic Press, Boston San Diego, New York, (1974).
- M.Greenberg, M.Orenstein, "Irreversible coupling by use of dissipative optics," Opt. Lett. 29, 451-453, (2004). [CrossRef] [PubMed]
- W. P. Huang, C. L. Xu "Simulation of Three-Dimensional Optical Waveguides by a Full-Vector Beam Propagation Metod," IEEE J. Quantum Electron. 29, 2639-2649 (1993). [CrossRef]

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