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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 17 — Aug. 23, 2004
  • pp: 4013–4018
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Unidirectional complex gratings assisted couplers

Maxim Greenberg and Meir Orenstein  »View Author Affiliations


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

In the coupler structure – few compound optical modes are supported (“compound mode” – a mode of the complete structure consisting of the two unequal waveguides within the embedding medium). The transfer of excitation from one compound mode to the other (“mode conversion”) can be achieved by introducing a periodical perturbation of the refractive index of the form: 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

6. D. Marcuse, Theory of Dielectric Optical WaveguidesSec. Ed., Academic Press, Boston San Diego, New York, (1974).

] that, to the first approximation, the inter-mode power conversion is proportional to the Fourier transform of the longitudinal perturbation function f(z):

Pm,n(z)0zf(z)exp{i(βmβn)z}dz
(1)

2. Analysis

We explore the mode evolution due to this single sideband perturbation in a GAC which can be implemented by standard technologies: e.g., core height h=5µm, cladding index nclad =1.445 and core index is ncore =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 c1 , c2 for a complex single sideband perturbation function f(z)=exp{iΔβz} can be derived from the coupled equations [6

6. D. Marcuse, Theory of Dielectric Optical WaveguidesSec. Ed., Academic Press, Boston San Diego, New York, (1974).

]:

c˙1=iexp{iΔβz}K11c1iexp{iΔβz}K12c2exp{iΔβz}
(2)
c˙2=iexp{iΔβz}K21c1exp{iΔβz}iexp{iΔβz}K22c2
(3)

where Kmn with m,n=1,2 are the coupling coefficients given by:

Kmn=ωε04P++Δ(x,y)Emt*Entdxdy
(4)

The long term contribution of the fast oscillating terms of frequencies Δβ, 2Δβ 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:

c˙1=0
(5)
c˙2=iK21c1
(6)

The unidirectionality of the coupler is evident since Eqs. (5) and (6) are not symmetric. For the initial condition of the input power in compound modes 1 and 2 (which for the asynchronous coupler translates to light incident on waveguides 1 and 2) c1(0)=C01, c2 (0)=C02 the formal solution is:

c1(z)=C01
(7)
c2(z)=iK21C01z+C02
(8)

The amplitude c1 is constant for any choice of the initial conditions. However, if c1 (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 c1 (0)=0 the power in both compound modes remains constant and there is no power transfer (“no-conversion”). For both c1 (0),c2 (0) non-zero, the total solution for c2 (z) is the interference of the initial field of the second mode C02 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]

].

Fig. 1. Influence of the self-coupling coefficients in the case of conversion. In each simulation the inter-modal coupling was K12=K21=K=1.15·10-3µm-1. and the self-coupling coefficients were 1. K11=K22=0; 2. K11=K, K22=0; 3. K11=0, K22=K; and 4. K11=K22=K. Calculations were performed by numerical integration of Eqs.(2) and (3).

The averaging of Eqs. (2) and (3) deemphasized some oscillating terms proportional to K11 , K22 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, K12 =K21 =K=1.15·10-3µm-1, respectively. The self-coupling coefficients varied in each simulation according to: 1. K11 =K22 =0; 2. K11 =K, K22 =0; 3. K11 =0, K22 =K; and 4. K11 =K22 =K. It follows that K11 can increase slightly the conversion efficiency for positive value of K11 (or decrease for negative) due to the dependence of the derivative of c2 on this coupling coefficient mediated by c1 . By a similar reasoning, in the case of “no-conversion” K11 has no influence. The oscillations of c1 (not shown) are mainly attributed to the term proportional to K12 exp{i2Δβz}, so they have halved periodicity. The value of K22 is in charge of the undesirable power oscillations of the second compound mode. K22 is multiplied by c2 which in both cases (conversion and no-conversion) is much larger than c1 , thus it is the critical parameter which enhances undesirable oscillations in the system.

It is worth noting that if modes are either symmetric or antisymmetric and the lateral profile of the perturbation is antisymmetric then it follows from Eq. (4) that the self-coupling coefficients will be zero. This can be a key design rule for eliminating the undesirable oscillations from the system.

In order to obtain the unidirectional inter-waveguide power transfer we should use highly asynchronous waveguides for the coupler to have the compound modes confined each to a single waveguide. In that case the compound modes provide good approximation for the modes of the single waveguides. Figures 2(a) and (b) shows the fundamental modes of the single waveguides and Figs. 2(c) and (d) shows the compound modes of the coupler 6µm×6µm×3µm which is far from synchronism. The effective indices of the compound modes are neff1=1.449849 and neff2=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 K22 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 K12 =2.7510-3µm-1. K22 is negligible since the second mode has only a small fraction of power in this waveguide.

Numerical simulations to validate the closed form results were performed with the scalar finite-difference algorithm – scalar beam propagation method (SBPM) [8

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]

]. Since the index contrast is small the results are almost identical to the full-vector calculations. The results are shown in Fig. 3 together with the results obtained by closed form solutions and direct numerical integration of Eqs. (2),(3). The small discrepancy between the results is due to: 1. In the closed form analytical solution we neglected the terms proportional to K11 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).

The unidirectional power conversion is length insensitive in the sense that if the first compound mode is launched into the system (light incident onto 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.

Fig. 2. (a) Fundamental mode of the single waveguide w=6µm, (b) fundamental mode of the single waveguide w=3µm. (c) and (d) first and second compound modes of the coupler structure 6µm×6µm×3µm. The overlap integrals between the powers of the corresponding single and compound modes are 99.8% and 98.4%.
Fig. 3. (Movies 834 KB and 157 KB) Modal power for 6µm×6µm×3µm coupler with the complex perturbation f(z)=exp{iΔβz} located in w=6µm waveguide, perturbation strength Δ=0.001. (a) Initial conditions c1 (0)=1, c2 (0)=0 and (b) Initial conditions c1 (0)=0, c2 (0)=1. Maximum relative power of the first mode does not exceed -26dB.

3. Separation of real and imaginary perturbations

The modulation of the real and the imaginary parts of the refractive index was performed simultaneously in the same spatial location. It is technologically challenging, so separation of the modulation – e.g., modulation of refractive index in one waveguide, whereas modulation of the loss/gain in the other – can make this device more feasible. The underlying equations will than take a form:

c˙1=i[K11rcos(Δβz)+iK11isin(Δβz)]c1i[K12rcos(Δβz)+iK12isin(Δβz)]c2exp{iΔβz}
(9)
c˙2=i[K21rcos(Δβz)+iK21isin(Δβz)]c1exp{iΔβz}i[K22rcos(Δβz)+iK22isin(Δβz)]c2
(10)

where Krmn , Kimn are coupling coefficients of the real and imaginary perturbations. For the asynchronous coupler, no single transverse perturbation profile can result in equal real and imaginary coupling constants. This problem is mitigated by tuning the depth of the real and imaginary perturbations to obtain Kr12 =Kr21 =Ki12 =Ki21 . Yet, significant oscillations appear since Kr11Ki11 , Kr22Ki22 , and moreover the self-coupling coefficients are larger than the inter-modal coupling. Fig. 4 depicts these characteristics for the 6µm×6µm×3µm coupler with the real and imaginary perturbations in different waveguides. Even though the unidirectionality of the coupler is still evident, we loose the device length insensitivity – it should now be tuned to the center of the beat period of the compound modes.

The time irreversibility of the complex coupling is exemplified by tracing the light which is back launched from the output ports of the coupler. The perturbation that the back reflected light is experiencing is the inverse transformation f(z′)=exp{-iΔβz′}, 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.

Fig. 4. Modal power for the 6µm×6µm×3µm coupler with the complex gratings f(z)=exp{iΔβz}, gratings depth Δ=0.001. Real and imaginary perturbations are located in different waveguides. (a) Initial conditions c1 (0)=1, c2 (0)=0. The power of the second mode exhibits significant oscillations. (b) Initial conditions c1 (0)=0, c2 (0)=1. Maximum relative power of the first mode does not exceed -30[dB]. Calculations were performed by SBPM.

6. Summary

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. QE-23, 889–897, (1987). [CrossRef]

2.

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]

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. 55, 2011–2013, (1989) [CrossRef]

4.

G.R. Hill, “Wavelength domain optical network techniques,” in Proc. IEEE 77, 121–132, (1989).

5.

L. Poladian, “Resonanse mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975, (1996). [CrossRef]

6.

D. Marcuse, Theory of Dielectric Optical WaveguidesSec. Ed., Academic Press, Boston San Diego, New York, (1974).

7.

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

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]

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

  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. QE-23, 889-897, (1987). [CrossRef]
  2. 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]
  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. 55, 2011-2013, (1989) [CrossRef]
  4. G.R. Hill, "Wavelength domain optical network techniques," in Proc. IEEE 77, 121-132, (1989).
  5. L.Poladian, "Resonanse mode expansions and exact solutions for nonuniform gratings," Phys. Rev. E 54, 2963-2975, (1996). [CrossRef]
  6. D.Marcuse, Theory of Dielectric Optical Waveguides Sec. Ed., Academic Press, Boston San Diego, New York, (1974).
  7. M.Greenberg, M.Orenstein, "Irreversible coupling by use of dissipative optics," Opt. Lett. 29, 451-453, (2004). [CrossRef] [PubMed]
  8. 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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