## Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission

Optics Express, Vol. 13, Issue 9, pp. 3567-3578 (2005)

http://dx.doi.org/10.1364/OPEX.13.003567

Acrobat PDF (192 KB)

### Abstract

A recently proposed concept suggests that a matched periodic modulation of both the refractive index and the gain/loss of the media breaks the coupling symmetry of the two co-propagating modes and allows only a unidirectional coupling from the *i*-th mode to *j*-the mode but not the opposite. This concept has been used to design a ring resonator coupled through a complex grating composed of both real (index) and imaginary (loss/gain) parts according to Euler relation: *Δn*=*n*_{0}
exp(-*jkx*)=*n*_{0}
(cos(*kx*)−*j* sin(*kx*)). Such asymmetrical coupling allows light to be coupled into the ring without letting it out. We present a detailed theoretical analysis of the ring resonator in the linear regime, and we investigate its linear temporal dynamics. Three possible states of the complex grating leads to the possibility of developing a dynamic optical memory cell where, for example, a data modulated train of optical pulses can be stored. This data can be accessed without destroying it, and can also be erased thus permitting the storage of a new bit. Finally, the ring can be used for pulse retiming.

© 2005 Optical Society of America

## 1. Introduction

1. A. Rebane and J. Feinberg, “Time-resolved holography,” Letters to Nature **351**, 378–380 (1991). [CrossRef]

2. A. Armitage, S. Skolnik, A.V. Kavokin, D.M. Whittaker, V.N. Astratov, G.A. Gehring, and J.S. Roberts, “Polariton-induced optical asymmetry in semiconductor microcavities,” Phys. Rev. B **58**, 15367–15370 (1998). [CrossRef]

3. G.S. Agarwal and S.D. Gupta, “Reciprocity relations for reflected amplitudes,” Opt. Lett. **27**, 1205–1207 (2002). [CrossRef]

4. R.J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. **67**, 717–754 (2004). [CrossRef]

5. V.S.C. M. Rao, S. D. Gupta, and G.S. Agarwal, “Study of asymmetric multilayered structures by means of nonreciprocity in phases,” J. Opt. B: Quantum Semiclass. Opt. **6**, 555–562 (2004). [CrossRef]

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

7. M. Kulishov, J.M. Laniel, N. Bélanger, J. Azana, and D.V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express **13**, 3068–3078 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-3068. [CrossRef] [PubMed]

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

9. 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. Spectral characteristics of switchable Asymmetric GACC

*β*

_{1}and

*β*

_{2}. It will be assumed that

*β*

_{1}>

*β*

_{2}. For simplicity and clarity, we limit the analysis to the situation where the two guides of the GACC are far from synchronism and weakly coupled, i.e. the only mode interaction mechanism is the one induced by the grating perturbation.

*Λ*is the grating period,

*Δn*

_{DC}and

*Δn*

_{AC}are respectively the constant and modulated perturbation to the refractive index,

*Δα*

_{DC}and

*Δα*

_{AC}are respectively the constant and modulated perturbation to the field amplitude gain/loss and

*k*

_{0}=2

*π*/

*λ*where

*λ*is the wavelength in vacuum.

*z*=0 and the signal at Ports C and D corresponding to

*z*=

*L*.

*M*

_{ij}are the elements of the transmission matrix

**M**and the bar over the electric fields

*E*

_{j}indicates that they are represented in the Fourier domain. The transfer matrix Eq. (3) is obtained through a similar process as the one used in Ref. [5

5. V.S.C. M. Rao, S. D. Gupta, and G.S. Agarwal, “Study of asymmetric multilayered structures by means of nonreciprocity in phases,” J. Opt. B: Quantum Semiclass. Opt. **6**, 555–562 (2004). [CrossRef]

*ã*

_{±}=cos(

*γ̃*

*L*)±

*jΔβ̃*

*L*sinc(

*γ̃*

*L*) and

*ψ*

_{±g}=exp(±

*jπL*/

*Λ*).

*T*

_{avg}and

*ψ*

_{avg}represent the amplitude and phase transmission coefficients of the A-GACC averaged on its two guides, i.e.

*T*

_{avg}=(

*T*

_{1}

*T*

_{2})

^{1/2}and

*ψ*

_{avg}=(

*ψ*

_{1}

*ψ*

_{2})

^{1/2}. Throughout this paper,

*T*

_{i}and

*ψ*

_{i}are amplitude transfer coefficients for a waveguide (or subcase) labeled by the index “

*i*”.

*T*

_{i}refers to the magnitude of the transmission in amplitude:

*ψ*

_{i}refers to a phase factor:

*κ*

_{n}and

*κ*

_{α}which are respectively proportional to the overlap between the real and imaginary AC parts of Eq. (1) and the spatial distribution of the propagating modes of the both guides. The complex phasematch condition is defined by Δ

*β̃*=(

*β̃*

_{1}-

*β̃*

_{2})/2-

*π*/

*Λ*, where the complex propagation constants are

*β̃*

_{i}=

*β*

_{i}+

*σ*

_{i}+

*jα*

_{i}=

*β*

_{i}′+

*jα*

_{i}((

*i*=1, 2). The coefficients

*σ*

_{i}and

*α*

_{i}are proportional to the overlap between the spatial mode distributions and the DC components of Eq. (1), respectively

*Δn*

_{DC}and

*Δα*

_{DC}/

*k*

_{0}. The effect of gain or loss can be modeled by either taking

*α*

_{i}>0 for loss or

*α*

_{i}<0 for gain. The prime added to

*β*′

_{i}indicates that it is the perturbed propagation constant including the DC contribution of the grating to the refractive index. Things become more complicated with

*α*

_{i}because it transforms the propagation constant

*β̃*

_{i}from a real to a complex quantity. On the other hand,

*γ̃*

^{2}is still a real quantity if the DC gain/loss experienced by each guide is the same, e.g.,

*α*

_{1}=

*α*

_{2}. This parameter

*γ̃*is then real except near the phase match condition (Δ

*β̃*=0) where it could be purely imaginary when the gain/loss grating is stronger than the index grating (|

*κ*

_{α}|>

*κ*

_{n}).

*ω*

_{0}=

*k*

_{0}

*c*of the light injected into the device:

*n*

_{eff,i}and

*n*

_{g,i}are the effective and group indices of each guide, respectively. Using Eq. (6), it is now possible to evaluate the frequency dependence of the previously determined parameters. For instance, the phase transmission coefficient of each guide (Eq. (5)) could be written as follows:

*ψ*

_{i,0}is simply a constant phase factor. However

*τ*

_{i}is a meaningful parameter representing the group delay across the guide #“

*i*”. A typical time

*τ*could also be associated with the phase match condition:

*τ*is, the broader the bandwidth of the A-GACC will be. The A-GACC described at Eq. (3) becomes ideal if the magnitude of the imaginary part of the modulation is equal to the real part, i.e. when

*κ*

_{n}=

*κ*and

*κ*

_{α}=

*Sκ*with

*S*equaling +1 or -1. One interesting feature that can extend the functionality of the A-GACC is the possibility of switching the phase relation between the real and imaginary grating. It requires controlling the sign

*S*of the AC gain/loss grating in Eq. (1) (

*S*=sgn(Δ

*α*

_{AC})). Such control provides two distinct states:

*S*=±1. A third state would be the one where the imaginary grating is switched off, or

*S*=0. This particular state is still described by the general transfer matrix (Eq. (3)) where only

*γ̃*is modified by setting

*κ*

_{α}=0. For such a case, the A-GACC becomes a standard GACC.

**M**

_{S}as the transfer matrix for

*S*=±1:

*S*=+1 or -1. The case of

*S*=+1 implies that

*κ*

_{n}=

*κ*

_{α}which means that both Δ

*n*

_{AC}and Δ

*α*

_{AC}in Eq. (1) have the same sign, namely positive. In one period of this grating along the propagation direction, a local maximum of the index of refraction is followed by a maximum of gain, a minimum of index of refraction, and finally a maximum of loss. According to Eq. (9), this A-GACC couples the light injected into Port A to both output Ports C and D, but it prevents any light injected into Port B from being coupled into Port D. This device is said to be asymmetrical because its coupling behavior between the two guides is reversed when the light is launched into Ports C and D. In such a case, the light injected into those Ports sees a grating with a maximum of loss just after a maximum of index of refraction, which corresponds to the state

*S*=-1. Therefore, the symmetry of the coupler is now reversed. It is now the light launched into guide #1 through Port D that can not couple at all in its cross state while the other input does (Port C).

*n*

_{eff,1}=1.519375 and

*n*

_{eff,2}=1.500000. For the sake of simplicity, we assume that the effective indices are frequency independent, so they are equal to their respective group indices (

*n*

_{g,i}=

*n*

_{eff,i}). The grating length is

*L*=25 mm and its period is

*Λ*=80 µm which satisfies the phase-matching condition at the central wavelength of 1.55 µm. The grating strength is

*κL*=π/2 or (

*κ*

_{n}+

*κ*

_{α})

*L*=π, where

*κ*

_{n}+

*κ*

_{α}is the coupling coefficient from guide #1 to guide #2. This grating strength is chosen so that if the imaginary part of the grating is removed, the resulting symmetrical GACC provides complete signal coupling from one guide to the other at the resonance wavelength.

*α*

_{1}=

*α*

_{2}=0.

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

9. 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]

## 3. Spectral characteristic and operation of an A-GACC coupled to a ring resonator

### 3.1 Trapping light in the ring resonator

*S*=+1. The signal will then propagate from Port C to Port B where it will not be able to leave the ring because of the asymmetric transmission. Therefore, the signal will then be trapped in the ring.

*L*

_{3}), propagation constant (

*β*

_{3}), loss coefficient (

*α*

_{3}), transmission in amplitude (

*T*

_{3}) and phase (

*ϕ*

_{3}). The amplitude transmission coefficient of the complete ring is given by:

*T*

_{1}and

*T*

_{2}. Under the assumption that neither the guide #2 nor the #3 has chromatic dispersion, the phase contributions for one roundtrip in the ring can be written as a linear phase term in angular frequency:

*τ*

_{r}is the group delay required to perform one roundtrip in the ring. Assuming an ideal A-GACC in state

*S*=+1 (see Eq. (9)) and no initial signal initially trapped in the ring, (the input at Port B is zero) the output at Port D can easily be obtained:

*N*:

*n*)” on the input field indicates how many roundtrips the signal has experienced in the ring. Therefore, before calculating its Fourier transforms

*N*windows of

*τ*

_{r}duration and subsequently propagating them.

### 3.2 Releasing the trapped light

*S*=0 and -1. The grating with no imaginary modulation (

*S*=0) reduces the A-GACC to a standard GACC. For such a case, it is possible to transfer the signal from Port B to Port D and practically extract the signal from the ring (i.e. without leaving a signal in Port C). According to Eq. (3), a complete transfer to Port D is possible only if

*ã*

_{−}equals zero. The only mean to achieve this, assuming both guides experience the same DC loss, is at the phase-match condition (i.e. when Δ

*β̃*=0) with

*κL*=

*π*/2 (or other solutions of cos(

*κL*)=0). However, the entire signal cannot exactly fulfill the phasematch condition. While a major portion of the signal is extracted, the rest performs a few roundtrips in the ring before coming out at Port D as weak distorted signals. It is then assumed that the ring will switch back to

*S*=+1 after

*τ*

_{r}. Based on this assumption, the signal extracted from the ring is the following:

*S*=+1 described by Eq. (9) will be referred as the “injector” state.

*S*=-

*1*reverses the asymmetric transmission of the A-GACC. For such case, it is possible to extract the signal from Port B to Port D without removing it from the ring. Using Eq. (9), the signal at Port C is:

*S*=-

*1*state. The signal trapped in the ring is then attenuated at each roundtrip: it acquires a phase shift, including time delay and even dispersion. An exact reconstruction of the signal at Port D has to sum over all the roundtrips performed in the ring from the moment the A-GACC has switch from

*S*=+

*1*to

*S*=-

*1*:

*P*indicates the duration the grating remains in state

*S*=-1 in terms of the maximum number of roundtrips allowed in the ring (the actual time is

*Pτ*

_{r}). For such a state, the functionality is more like a signal duplicator than a signal extractor. For the rest of the paper, this grating state will be labeled as the “duplicator” state.

## 4. Temporal dynamics of the A-GACC coupled to the ring resonator

### 4.1 Propagation within the ring resonator

*ϕ*is the pulse envelope,

*t*

_{R}is the time between each pulse. The fields at the output of the A-GACC can be obtained through the application of Eq. (12) and Eq. (13). This requires calculating the signal in the spectral domain using the Fourier transform of Eq. (17):

*n*

_{g,1}

*n*

_{m}is the number of roundtrips in the ring undergone by the

*m*

^{th}pulse in the input train. The signal at Port C can be obtained by taking the inverse Fourier transform of Eq. (20):

*τ*

_{avg}=(

*τ*

_{1}+

*τ*

_{2})/2 and

*ϕ*

_{inj}is the convolution of one pulse with the grating spectrum:

*n*

_{m}

*τ*

_{r}. The effects induced by the limited bandwidth of the grating are given by Eq. (22). They are not obvious at first glance because they depend on the bandwidth of the pulse itself.

*sinc*-transfer function. Let

*G*(

*t*;

*t*

_{0},

*δω*) be the complex amplitude of a Gaussian pulse with an energy normalized to 1 (∫|

*G*(

*t*)|

^{2}

*dt*=1), a FWHM of 2(ln2)

^{1/2}

*t*

_{0}and a carrier angular frequency of

*ω*

_{0}+

*δω*. Under the approximation of a narrowband pulse close to the phasematch condition, the

*sinc*function can also be approximate by a Gaussian:

*sinc*(

*x*)≅exp(-

*x*

^{2}/6). The transfer of the input Gaussian pulse switched through the A-GACC according to Eq. (22) can now be solved analytically:

_{inj}=(1+

*τ*

^{2}/3

^{1/2}and its carrier angular frequency shifts by

*δω*

_{inj}=-

*θτ*/3Γ

^{2}

*α*

_{1}=

*α*

_{2}, which implies a real phasematch condition Δ

*β*̃.

### 4.2 Access to the trapped signal

*T*

_{2}

*ψ*

_{2}. As for the injected signal, the duplication and extraction is only considered here for a train of identical pulses when they have been injected at Port A.

*P*+1 pulse trains. Since the signals are only duplicated, they continue to undergo roundtrips within the ring resonator. The modifications on the pulse shape are grouped in

*ϕ*

_{dup}:

_{out}=[1+(1+

*f*)

*τ*

^{2}/3

^{1/2}and

*δω*

_{out}=-(1+

*f*)

*θτ*/3

*T*

_{out}and

*f*equal 1. On the other hand, if the extractor state is used, and the whole pulse spectrum is close to the phasematch such as

*κL*≫Δ

*β̃*

*L*, they are instead given by

*T*

_{out}=sinc(

*κL*)/2 and

*f*=3[1-

*κL*cot(

*κL*)]/

*κ*

^{2}

*L*

^{2}. As for the study of the injection, the results presented have been simplified under the assumption of equal gain/loss in guide #1 and #2.

*τ*

^{2}/3

^{1/2}while for the extractor state the pulse broadening is given by [1+(1+

*f*)

*τ*

^{2}/3

^{1/2}. For the case of the grating strength

*κL*=π/2, the factor is evaluated to

*f*=12/

*π*

^{2}≈1.22. Therefore, the extracted signal is slightly longer than the duplicated one. This can be explained by the fact that an A-GACC has a broader bandwidth than a standard GACC. The duplicator state also delivers more power to the output pulse and allows repeating the same series of pulses with a period

*τ*

_{r}.

## 5. Application of an A-GACC coupled to a ring resonator

*t*

_{R}, the time delay associated with the propagation in guide #1

*τ*

_{1}and the time it takes to make one full circulation in the ring

*τ*

_{r}. For a fixed switchable A-GACC structure, the only adjustable parameter left is

*t*

_{R}. Considering that the pulse duration

*t*

_{0}is much shorter than these times, the dynamics will therefore be set by the relation between

*t*

_{R}and

*τ*

_{r}. Two interesting time regimes arise from such considerations: the one for which

*t*

_{R}≪

*τ*

_{r}and the one for which

*t*

_{R}≈

*τ*

_{r}. The former operation regime will be referred as the “memory regime” and the latter as the “retiming regime”. These two regimes are investigated in the next two subsections.

### 5.1 Memory cell

*m*pulses undergo the same number of roundtrips

*n*

_{m}during the injection state. As it was described in Section 2, the signal can now be accessed either by extracting it or by duplicating it and be returned in the photonic circuit. The switching must occur when the signal is performing a roundtrip in the ring before it reaches Port B or after it leaves Port C.

*t*

_{0}=3.6 ps and the time between each pulse of

*t*

_{R}=25 ps. The summation over the index

*m*will be done from 0 to 5 (total of six pulses). Each pulse within the train can carry a bit of information encoded by pulse amplitude, phase or, in digital case, by missing some pulses in the train, where the pulse presence is a logical unit, and its absence is a logical zero (e.g., RZ coding schemes). The physical parameters of the A-GACC are the same as the ones used for Fig. 2, except for the DC loss, which is 0.6 dB/cm (which is state of art now for planar waveguides) in all three guides. The length of guide #3 is assumed to be 35 mm and its constant of propagation is the same as guide #2.

*τ*

_{r}inside the ring. The choice of material should be carefully considered when designing an A-GACC that would allow such operation.

*κL*=π/2. The signal extracted will not be as strong as for the duplication state since it will not undergo amplification during the extraction. It is important to mention that the extraction process is not a perfect one. If the grating strength is adjusted to

*κL*=π/2, the maximum of extraction will be reached for the wavelength associated to perfect phasematch. Even though the extraction state allows the clearing of the signal from the ring, one must understand that small remnants of the distorted pulse train will be left in the ring and will decay due to the ring losses.

### 5.2 Pulse retiming

*t*

_{R}is slightly lower/higher than the time of the single full circulation in the ring

*τ*

_{r}. Each pulse then undergoes one more/less roundtrip in the ring under the injection state than its neighbors, which means

*n*

_{m+1}=

*n*

_{m}+1. In this case, the next coupled pulse will be added right before/after the previous injected pulse that at that time will have completed a roundtrip inside the ring. A careful adjustment of

*t*

_{R}can provide pulse retiming within a broad time range.

*t*

_{R}is 280 ps and

*t*

_{0}is 1.8 ps where the field inside the ring is shown as a function of time. Computation has been done for losses that are half the value used for the memory cell, i.e., 0.3 dB/cm. One can see that the number of pulses accumulates in the ring from one to six as expected. Unfortunately, each pulse has decreased amplitude. This is due to the fact that each time a new pulse is added to the signal train, the previous pulses have suffered losses proportional to the number of roundtrips they have made. After accumulating six retimed pulses, the train is duplicated into output Port D by switching the grating. This produces the blue pulse train in Fig. 4. Practically all pulses in the train have amplitudes much higher than the input pulse in green. The retimed train will be amplified (more than seven times) at the moment of releasing it from the ring into Port D. Obviously, the pulse amplitudes will be more equalized if losses in the ring are reduced.

## 6. Conclusions

## Acknowledgments

## References and links

1. | A. Rebane and J. Feinberg, “Time-resolved holography,” Letters to Nature |

2. | A. Armitage, S. Skolnik, A.V. Kavokin, D.M. Whittaker, V.N. Astratov, G.A. Gehring, and J.S. Roberts, “Polariton-induced optical asymmetry in semiconductor microcavities,” Phys. Rev. B |

3. | G.S. Agarwal and S.D. Gupta, “Reciprocity relations for reflected amplitudes,” Opt. Lett. |

4. | R.J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. |

5. | V.S.C. M. Rao, S. D. Gupta, and G.S. Agarwal, “Study of asymmetric multilayered structures by means of nonreciprocity in phases,” J. Opt. B: Quantum Semiclass. Opt. |

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

7. | M. Kulishov, J.M. Laniel, N. Bélanger, J. Azana, and D.V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express |

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

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

10. | M. Greenberg, “Unidirectional mode devices based on irreversible mode coupling,” MSc. Thesis, Israel Institute of Technology, Haifa, March, 2004. |

11. | R.R.A. Syms, S. Makrimichalou, and A.S. Holms, “High-speed optical signal processing potential of grating-coupled waveguide filters,” Appl. Opt. |

**OCIS Codes**

(200.4740) Optics in computing : Optical processing

(210.4680) Optical data storage : Optical memories

(230.3120) Optical devices : Integrated optics devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 4, 2005

Revised Manuscript: April 27, 2005

Published: May 2, 2005

**Citation**

Mykola Kulishov, Jacques Laniel, Nicolas Bélanger, and David Plant, "Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission," Opt. Express **13**, 3567-3578 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-9-3567

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### References

- A. Rebane, J. Feinberg, �??Time-resolved holography,�?? Letters to Nature 351, 378-380 (1991). [CrossRef]
- A. Armitage, S. Skolnik, A.V. Kavokin, D.M. Whittaker, V.N. Astratov, G.A. Gehring, J.S. Roberts, �??Polariton-induced optical asymmetry in semiconductor microcavities,�?? Phys. Rev. B 58, 15367-15370 (1998). [CrossRef]
- G.S. Agarwal, S.D. Gupta, �??Reciprocity relations for reflected amplitudes,�?? Opt. Lett. 27, 1205-1207 (2002). [CrossRef]
- R.J. Potton, �??Reciprocity in optics,�?? Rep. Prog. Phys. 67, 717-754 (2004). [CrossRef]
- V.S.C. M. Rao, S. D. Gupta, G.S. Agarwal, �??Study of asymmetric multilayered structures by means of nonreciprocity in phases,�?? J. Opt. B: Quantum Semiclass. Opt. 6, 555-562 (2004). [CrossRef]
- L. Poladian, �??Resonance mode expansions and exact solutions for nonuniform gratings,�?? Phys. Rev. E 54, 2963-2975 (1996). [CrossRef]
- M. Kulishov, J.M. Laniel, N. Bélanger, J. Azana, D.V. Plant, �??Nonreciprocal waveguide Bragg gratings,�?? Opt. Express 13, 3068-3078 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-3068">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-3068</a>. [CrossRef] [PubMed]
- 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]
- M. Greenberg, �??Unidirectional mode devices based on irreversible mode coupling,�?? MSc. Thesis, Israel Institute of Technology, Haifa, March, 2004.
- R.R.A. Syms, S. Makrimichalou, A.S. Holms, �??High-speed optical signal processing potential of grating-coupled waveguide filters,�?? Appl. Opt. 30, 3762-3769 (1991). [CrossRef] [PubMed]

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