## Counterdiabatic mode-evolution based coupled-waveguide devices |

Optics Express, Vol. 21, Issue 18, pp. 21224-21235 (2013)

http://dx.doi.org/10.1364/OE.21.021224

Acrobat PDF (1884 KB)

### Abstract

The goal in designing mode-evolution based devices is to realise short and high-fidelity components. The counterdiabatic protocol in coherent quantum state control can be used to cancel unwanted coupling between adiabatic modes in mode evolution but is not directly realisable in the coupled-waveguide system. By finding alternative coupled-mode equations that links to the same interaction picture dynamical equation as the counterdiabatic protocol via unitary transformations, we have derived a universal formalism for the design of short and high-fidelity mode-evolution based coupled-waveguide devices. Starting from a traditional adiabatic device design, the counterdiabatic protocol leads to a high-fidelity device, with its evolution following the adiabatic modes exactly even when the adiabatic condition is violated. Tolerance analysis shows that the countera-diabatic devices combine the advantages of adiabatic and resonant devices. The formalism is used to design asymmetric waveguide couplers.

© 2013 OSA

## 1. Introduction

1. R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett. **4**, 1135–1138 (1992). [CrossRef]

2. X. Sun, H.-C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. **34**, 280–282 (2009). [CrossRef] [PubMed]

3. M. R. Watts, H. A. Haus, and E. P. Ippen, “Integrated mode-evolution-based polarization splitter,” Opt. Lett. **30**, 967–969 (2005). [CrossRef] [PubMed]

4. M. V. Berry, “Histories of adiabatic quantum transitions,” Proc. R. Soc. Lond. A **429**, 61–72 (1990). [CrossRef]

15. S. K. Korotky, “Three-space representation of phase-mismatch switching in coupled two-state optical systems,” IEEE J. Quantum Electron. **22**, 952–958 (1986). [CrossRef]

16. S. Ibáñez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt, “Multiple Schrödinger pictures and dynamics in shortcuts to adiabaticity,” Phys. Rev. Lett. **109**, 100403 (2012). [CrossRef]

## 2. Theoretical analysis

### 2.1. Coupled-mode equations

17. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. **9**, 919–933 (1973). [CrossRef]

*β*

_{1}and

*β*

_{2}. The refractive index or geometry of the two waveguides are allowed to vary along the propagation direction

*z*. Light is coupled into the device at

*z*= 0 and out at

*z*=

*L*. Under the scalar and paraxial approximation and assuming weak coupling [18], the changes in the guided-mode amplitudes in the individual waveguides [

*A*

_{1},

*A*

_{2}]

*with propagation distance is described by coupled-mode equations as where*

^{T}*κ*(real) is the coupling coefficient, and Δ = (

*β*

_{1}−

*β*

_{2})/2 describes the degree of mismatch between the waveguides. Replacing the spatial variation

*z*with the temporal variation

*t*, (1) is equivalent to the time-dependent Schrödinger equation (

*h̄*= 1) describing the interaction dynamics of a two-state system driven by a coherent laser excitation, and

**H**

_{0}is the Hamiltonian.

**H**

_{0}can be written in terms of Pauli spin matrices as

**H**

_{0}=

*κσ*+ 0

_{x}*σ*+ Δ

_{y}*σ*and mapped to a real three dimensional space as a rotation vector Ω

_{z}_{0}= [

*κ*, 0, Δ]

*. When the input state coincides with Ω*

^{T}_{0}, adiabatic evolution is achieved with

**H**

_{0}by slowly varying

*κ*and Δ so that the system state follows Ω

_{0}collinearly to a final state with motion on the

*x*−

*z*plane. If the adiabatic condition is not satisfied, the system state cannot follow Ω

_{0}, resulting in low conversion efficiency. Also, we note that the adiabatic following is an approximation. Even if the adiabaticity criterion is satisfied, efficiency is generally close to, but less than 1.

### 2.2. Counterdiabatic term

9. M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A **107**, 9937–9945 (2003). [CrossRef]

*a*〉 that diagonalises

_{n}**H**

_{0}so that

**U**= ∑|

*a*〉〈

_{n}*a*|. The counterdiabatic term thus obtained is with

_{n}*κ*(

_{a}*z*) = (

*κ̇*Δ − Δ̇

*κ*)/(

*κ*

^{2}+ Δ

^{2}). The counterdiabatic protocol Hamiltonian is thus

**H**

_{0}+

**H**

*=*

_{cd}*κσ*+ (1/2)

_{x}*κ*+ Δ

_{a}σ_{y}*σ*, corresponding to a rotation vector

_{z}*x*−

*z*plane [15

15. S. K. Korotky, “Three-space representation of phase-mismatch switching in coupled two-state optical systems,” IEEE J. Quantum Electron. **22**, 952–958 (1986). [CrossRef]

### 2.3. Unitary transformation

*κ*+ Δ

_{s}σ_{x}*, eliminating the need for non-zero imaginary part in the coupling coefficient. For the coupled-waveguide system, the required transformation can be obtained by observing that a rotation around the*

_{s}σ_{z}*z*-axis will bring Ω

*back to the*

_{cd}*x*−

*z*plane. Write where

*ϕ*=

*tan*

^{−1}(

*κ*/2

_{a}*κ*) and

*z*-axis rotation [6

6. M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nature Phys. **8**, 147–152 (2012). [CrossRef]

*= Δ −(1/2)*

_{s}*ϕ̇*. It can be regarded as a Schrödinger picture Hamiltonian, and together with (3), link to the same interaction picture Hamiltonian. Casted in the new basis, it can now be realised using the coupled-waveguide system and guarantees the same fast and high-fidelity mode evolution described by

**H**

_{0}+

**H**

*.*

_{cd}### 2.4. The counterdiabatic coupler

## 3. Numerical analysis

### 3.1. Adiabatic following

**H**

_{0}are

*κ*=

*κ*

_{0}sin

*θ*and Δ =

*κ*

_{0}cos

*θ*[1

1. R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett. **4**, 1135–1138 (1992). [CrossRef]

12. X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett. **105**, 123003 (2010). [CrossRef] [PubMed]

*A*

_{1}(

*z*)|

^{2}, |

*A*

_{2}(

*z*)|

^{2}] follows approximately the eigenmode power evolution given by [cos

^{2}(

*θ*/2), sin

^{2}(

*θ*/2)] in the limit

*κ*

_{0}

*L*→

**∞**.

*θ*, for illustrative purposes, we choose a linear function

*θ*=

*πz/L*, where

*L*is the device length. With this choice of taper function, we obtain the adiabatic condition as

*κ*

_{0}= 1 mm

^{−1}, corresponding to the adiabatic condition of

**H**

_{0}and using [

*A*

_{1}(0),

*A*

_{2}(0)]

*= [1, 0]*

^{T}*as the input. The results of fractional power evolution for different device lengths are shown in Fig. 2 (dash-dotted lines). The solid lines are the theoretical eigenmode power evolution given by [cos*

^{T}^{2}(

*θ*/2), sin

^{2}(

*θ*/2)]. It is clear that as the device length increases, the power evolution follows the eigenmode more closely, and the coupling efficiency at the output is closer to 1. At shorter lengths, some large ripples can be seen, indicating scattering of power from one eigenmode into the other. Small ripples can still be seen even for the 50 mm device.

### 3.2. Counterdiabatic following

*κ*+ Δ

_{s}σ_{x}*. Solving (1) using [*

_{s}σ_{z}*A*

_{1}(0),

*A*

_{2}(0)]

*= [1, 0]*

^{T}*as the input, the results are also shown in Fig. 2 (solid lines). The fractional power evolution coincides with the eigenmode completely for every device length, demonstrating high-fidelity power coupling predicted by the theory. To illustrate the magnitude of coupling and waveguide mismatch required to implement the counterdiabatic formalism, we show the required*

^{T}*κ*and Δ

_{s}*for the four device lengths considered in Fig. 3 (solid lines). Also shown in the figure are the associated*

_{s}*κ*=

*κ*

_{0}sin

*θ*and Δ =

*κ*

_{0}cos

*θ*of the original adiabatic devices (dashed lines). To implement the counterdiabatic formalism in a coupled waveguide system shown in Fig. 1, one designs the waveguide separation as a function of

*z*to realise the coupling coefficient

*κ*, and the waveguide widths are adjusted as a function of

_{s}*z*to achieve the required mismatch Δ

*. From Fig. 3(a), we can see that when the device length is short, the required additional coupling and mismatch are also increased throughout the device; thus putting a limit on the physically realisable shortest length of the counterdiabatic coupler depending on the chosen waveguide platform. For longer device lengths in Figs. 3(b)–3(d), additional coupling and mismatch at the beginning and the end of the device are sufficient to cancel the ripples seen in Figs. 2(b)–2(d).*

_{s}*κ*

_{0}, the adiabatic condition can be satisfied at shorter device lengths. However, even with the increased coupling, the device still works under the adiabatic approximation. That is, the fidelity will be close to but not equal to 1, and the ripples seen in the dash-dotted lines in Fig. 2 will remain. On the other hand, the counterdiabatic formalism works to cancel all unwanted coupling between adiabatic modes in mode evolution so that the fidelity is always perfect.

### 3.3. Comparison with adiabatic and resonant design schemes

**H**

*might seem a daunting task in quantum system state control because the level of control needed over laser pulse parameters in order to implement the time-dependent*

_{cd}*κ*, the implementation of the counterdiabatic formalism in coupled-waveguide system is straightforward with modern fabrication technologies. Apparently,

_{a}*κ*and Δ

_{s}*depend on*

_{s}**H**

_{0}, and some control of the waveguide parameters over the propagation distance is required to implement the counterdiabatic design. We know that adiabatic designs requires less precise control over waveguide parameters, while on the opposite end, resonant coupling techniques require very precise definition of the coupling length and coupling coefficient [18]. So the question that naturally arises is: how does the counteradiabatic coupler compare with the adiabatic coupler and the resonant coupler in terms of robustness to waveguide parameter variations?

**H**

*in (2) itself drives the system along the eigenstates of*

_{cd}**H**

_{0}exactly [12

12. X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett. **105**, 123003 (2010). [CrossRef] [PubMed]

**H**

*are the same as a resonantly coupled device [8*

_{cd}8. R. G. Unanyan, L. P. Yatsenko, K. Bergmann, and B. W. Shore, “Laser-induced adiabatic atomic reorientation with control of diabatic losses,” Opt. Commun. **139**, 48–54 (1997). [CrossRef]

**H**

_{0}, (b) the counterdiabatic coupler described by

**H**

_{0}+

**H**

*, and (c) the resonant coupler described by*

_{cd}**H**

*. The design parameters are fixed for all three cases at every device length. The coupling efficiencies from waveguide 1 to waveguide 2 subject to*

_{cd}*±*50% variations in coupling coefficients uniformly along the couplers for all three cases are shown in Fig. 5. The adiabatic coupler in Fig. 5(a) shows good tolerance to coupling coefficient variations when the device length is long enough. And the ripples seen in the figure indicates less than 1 fidelity due to coupling between the adiabatic modes. As expected, the tolerance of the resonant coupler in Fig. 5(c) is inferior to the other two couplers, and its characteristics can be described by the square of a sinusoidal function [18]. The results for the counterdiabatic coupler are shown in Fig. 5(b). At the short length extreme, its tolerance is the same as the resonant coupler. This is expected because

**H**

*is dominant. As the device length increases, the tolerance of counterdiabatic couplers exceeds that of resonant couplers and is comparable to the adiabatic couplers; at the same time, the ripples seen in adiabatic couplers are reduced, indicating higher fidelity than adiabatic couplers. For even longer device lengths, the tolerance of counterdiabatic couplers becomes the same as adiabatic couplers because*

_{cd}**H**

_{0}is dominant now. The level of waveguide parameter engineering required to implement the counterdiabatic scheme is on the same level of resonant designs in the short device length limit. For the intermediate length, the engineering requirement is relaxed to the level of adiabatic designs, and the fidelity is improved. For the longer device length, the engineering requirement is the same as adiabatic devices. So, the counterdiabatic coupler provides a high-fidelity and robust transition from the short but not robust resonant coupler to the long but robust adiabatic coupler. Of course, the price to pay is the extra coupling strength and mismatch needed to implement the counterdiabatic term

**H**

*.*

_{cd}## 4. Device design examples and beam propagation simulations

### 4.1. Simulation model

19. K. Kawano and T. Kitoh, *Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations* (Wiley, 2001). [CrossRef]

20. Y. Bai, Q. Liu, K. P. Lor, and K. S. Chiang, “Widely tunable long-period waveguide grating couplers,” Opt. Express **14**, 12644–12654 (2006). [CrossRef] [PubMed]

*μ*m thick SiO

_{2}(

*n*=1.46) on a Si (

*n*= 3.48) wafer is used for the bottom cladding layer, the core consists of a 2.4

*μ*m layer of BCB (

*n*= 1.53), and the upper cladding is epoxy (

*n*= 1.50). The device is simulated at 1.55

*μ*m input wavelength and the TE polarization. Subsequent waveguide design and BPM simulations are performed on the 2D structure obtained using the effective index method. For the chosen waveguide system, the relation between the mismatch Δ and width difference

*δW*can be approximated by a linear relation [21

21. A. Syahriar, V. M. Schneider, and S. Al-Bader, “The design of mode evolution couplers,” J. Lightwave Technol. **16**, 1907–1914 (1998). [CrossRef]

*κ*and waveguide separation

*D*in a symmetric coupler is well fitted by the exponential relation

*κ*=

*k*

_{0}exp[−

*γ*(

*D*−

*D*

_{0})] [18]. We also assume that the exponential relation can be used to obtain an estimation of the coupling coefficient in the asymmetric coupler [21

21. A. Syahriar, V. M. Schneider, and S. Al-Bader, “The design of mode evolution couplers,” J. Lightwave Technol. **16**, 1907–1914 (1998). [CrossRef]

### 4.2. *L* = 5 *mm devices*

*κ*=

*κ*

_{0}sin(

*πz/L*) and Δ =

*κ*

_{0}cos(

*πz/L*) with

*κ*

_{0}= 1 mm

^{−1}and

*L*= 1 mm. As explained in section 3.1, this design, even though not being the best possible taper function, minimise the coupling between the eigenmodes because the separation of the eigenvalues is the maximum value allowed by the system throughout the evolution. We excite the lower waveguide by its unperturbed mode at

*z*= 0, and the BPM result is shown in Fig. 6(a). The corresponding waveguide power evolution obtained by the overlap integral of the propagating field and the unperturbed waveguide modes is shown in Fig. 6(b). The result agrees very well with the coupled-mode equation solution shown by the dash-dotted lines in Fig. 2(a). Next, we design the corresponding counterdiabatic coupler using the parameters shown by the solid lines in Fig. 3(a). The design result is shown in Fig. 7(a). We note that, as illustrated in Fig. 3, the coupling coefficient and mismatch of the adiabatic coupler and the corresponding counterdiabatic coupler are different, thus their device geometries are different in Fig. 6 and Fig. 7. We again excite the lower waveguide by its unperturbed mode at

*z*= 0, and the BPM result and the corresponding waveguide power evolution are shown in Fig. 7. The power evolution in Fig. 7(b) again agrees very well with the coupled-mode equation solution shown by the solid lines in Fig. 2(a). The counterdiabatic coupler in Fig. 7 is the optimised design corresponding to the adiabatic coupler in Fig. 6, showing the properties of an ideal adiabatic coupler by following the theoretical eigenmode power evolution.

### 4.3. *L* = 500*μm devices*

*L*= 500

*μ*m, and the coupling coefficient and mismatch of the adiabatic coupler still obey

*κ*=

*κ*

_{0}sin(

*πz/L*) and Δ =

*κ*

_{0}cos(

*πz/L*) with

*κ*

_{0}= 1 mm

^{−1}. The resulting adiabatic coupler and the corresponding counterdiabatic coupler are shown in Fig. 8. We also show the BPM results in the couplers by exciting the lower waveguides by their unperturbed modes at

*z*= 0. Clearly, this adiabatic coupler design in Fig. 8(a) is a bad one, and the coupling efficiency to the upper waveguide is low due to violation of the adiabatic condition

**H**

*, with the additional coupling strength evident from the closer waveguide spacing in the counterdiabatic coupler in Fig. 8(b) as compared to the adiabatic coupler in Fig. 8(a). Started with a bad adiabatic design, the counterdiabatic formalism has successfully generated a short and high-fidelity counterdiabatic coupler.*

_{cd}## 5. Discussion and conclusion

2. X. Sun, H.-C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. **34**, 280–282 (2009). [CrossRef] [PubMed]

21. A. Syahriar, V. M. Schneider, and S. Al-Bader, “The design of mode evolution couplers,” J. Lightwave Technol. **16**, 1907–1914 (1998). [CrossRef]

13. T.-Y. Lin, F.-C. Hsiao, Y.-W. Jhang, C. Hu, and S.-Y. Tseng, “Mode conversion using optical analogy of shortcut to adiabatic passage in engineered multimode waveguides,” Opt. Express **20**, 24085–24092 (2012). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett. |

2. | X. Sun, H.-C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. |

3. | M. R. Watts, H. A. Haus, and E. P. Ippen, “Integrated mode-evolution-based polarization splitter,” Opt. Lett. |

4. | M. V. Berry, “Histories of adiabatic quantum transitions,” Proc. R. Soc. Lond. A |

5. | S. Guérin, S. Thomas, and H. R. Jauslin, “Optimization of population transfer by adiabatic passage,” Phys. Rev. A |

6. | M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nature Phys. |

7. | B. T. Torosov, S. Guérin, and N. V. Vitanov, “High-fidelity adiabatic passage by composite sequence of chirped pulses,” Phys. Rev. Lett. |

8. | R. G. Unanyan, L. P. Yatsenko, K. Bergmann, and B. W. Shore, “Laser-induced adiabatic atomic reorientation with control of diabatic losses,” Opt. Commun. |

9. | M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A |

10. | M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B |

11. | M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor. |

12. | X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett. |

13. | T.-Y. Lin, F.-C. Hsiao, Y.-W. Jhang, C. Hu, and S.-Y. Tseng, “Mode conversion using optical analogy of shortcut to adiabatic passage in engineered multimode waveguides,” Opt. Express |

14. | S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev. |

15. | S. K. Korotky, “Three-space representation of phase-mismatch switching in coupled two-state optical systems,” IEEE J. Quantum Electron. |

16. | S. Ibáñez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt, “Multiple Schrödinger pictures and dynamics in shortcuts to adiabaticity,” Phys. Rev. Lett. |

17. | A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. |

18. | K. Okamoto, |

19. | K. Kawano and T. Kitoh, |

20. | Y. Bai, Q. Liu, K. P. Lor, and K. S. Chiang, “Widely tunable long-period waveguide grating couplers,” Opt. Express |

21. | A. Syahriar, V. M. Schneider, and S. Al-Bader, “The design of mode evolution couplers,” J. Lightwave Technol. |

22. | S.-Y. Tseng and M.-C. Wu, “Mode conversion/splitting by optical analogy of multistate stimulated Raman adiabatic passage in multimode waveguides,” J. Lightwave Technol. |

**OCIS Codes**

(000.1600) General : Classical and quantum physics

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

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: August 12, 2013

Manuscript Accepted: August 19, 2013

Published: September 3, 2013

**Citation**

Shuo-Yen Tseng, "Counterdiabatic mode-evolution based coupled-waveguide devices," Opt. Express **21**, 21224-21235 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-21224

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### References

- R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett.4, 1135–1138 (1992). [CrossRef]
- X. Sun, H.-C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett.34, 280–282 (2009). [CrossRef] [PubMed]
- M. R. Watts, H. A. Haus, and E. P. Ippen, “Integrated mode-evolution-based polarization splitter,” Opt. Lett.30, 967–969 (2005). [CrossRef] [PubMed]
- M. V. Berry, “Histories of adiabatic quantum transitions,” Proc. R. Soc. Lond. A429, 61–72 (1990). [CrossRef]
- S. Guérin, S. Thomas, and H. R. Jauslin, “Optimization of population transfer by adiabatic passage,” Phys. Rev. A65, 023409 (2002). [CrossRef]
- M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nature Phys.8, 147–152 (2012). [CrossRef]
- B. T. Torosov, S. Guérin, and N. V. Vitanov, “High-fidelity adiabatic passage by composite sequence of chirped pulses,” Phys. Rev. Lett.106, 233001 (2011). [CrossRef]
- R. G. Unanyan, L. P. Yatsenko, K. Bergmann, and B. W. Shore, “Laser-induced adiabatic atomic reorientation with control of diabatic losses,” Opt. Commun.139, 48–54 (1997). [CrossRef]
- M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A107, 9937–9945 (2003). [CrossRef]
- M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B109, 6838–6844 (2005). [CrossRef]
- M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor.42, 365303 (2009). [CrossRef]
- X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett.105, 123003 (2010). [CrossRef] [PubMed]
- T.-Y. Lin, F.-C. Hsiao, Y.-W. Jhang, C. Hu, and S.-Y. Tseng, “Mode conversion using optical analogy of shortcut to adiabatic passage in engineered multimode waveguides,” Opt. Express20, 24085–24092 (2012). [CrossRef] [PubMed]
- S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev.3, 243–261 (2009). [CrossRef]
- S. K. Korotky, “Three-space representation of phase-mismatch switching in coupled two-state optical systems,” IEEE J. Quantum Electron.22, 952–958 (1986). [CrossRef]
- S. Ibáñez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt, “Multiple Schrödinger pictures and dynamics in shortcuts to adiabaticity,” Phys. Rev. Lett.109, 100403 (2012). [CrossRef]
- A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron.9, 919–933 (1973). [CrossRef]
- K. Okamoto, Fundamentals of Optical Waveguides(Academic, 2006).
- K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations (Wiley, 2001). [CrossRef]
- Y. Bai, Q. Liu, K. P. Lor, and K. S. Chiang, “Widely tunable long-period waveguide grating couplers,” Opt. Express14, 12644–12654 (2006). [CrossRef] [PubMed]
- A. Syahriar, V. M. Schneider, and S. Al-Bader, “The design of mode evolution couplers,” J. Lightwave Technol.16, 1907–1914 (1998). [CrossRef]
- S.-Y. Tseng and M.-C. Wu, “Mode conversion/splitting by optical analogy of multistate stimulated Raman adiabatic passage in multimode waveguides,” J. Lightwave Technol.28, 3529–3534 (2010).

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