## Short and robust directional couplers designed by shortcuts to adiabaticity |

Optics Express, Vol. 22, Issue 16, pp. 18849-18859 (2014)

http://dx.doi.org/10.1364/OE.22.018849

Acrobat PDF (888 KB)

### Abstract

We propose short and robust directional couplers designed by shortcuts to adiabaticity, based on Lewis-Riesenfeld invariant theory. The design of directional couplers is discussed by combining invariant-based inverse engineering and perturbation theory. The error sensitivity of the coupler is minimized by optimizing the evolution of dynamical invariant with respect to coupling coefficient/input wavelength variations. The proposed robust coupler devices are verified with beam propagation simulations.

© 2014 Optical Society of America

## 1. Introduction

7. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev. **3**, 243–261 (2009). [CrossRef]

8. E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. **62**, 117–169 (2013). [CrossRef]

6. S.-Y. Tseng, “Counterdiabatic mode-evolution based coupled-waveguide devices,” Opt. Express **21**, 21224–21235 (2013). [CrossRef] [PubMed]

9. 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]

11. S. Martínez-Garaot, S.-Y. Tseng, and J. G. Muga, “Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity,” Opt. Lett. **39**, 2306 (2014). [CrossRef] [PubMed]

12. X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A **86**, 033405 (2012). [CrossRef]

13. S.-Y. Tseng and Y.-W. Jhang, “Fast and robust beam coupling in a three waveguide directional coupler,” IEEE Photon. Technol. Lett. **25**, 2478–2481 (2013). [CrossRef]

14. A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. **14**, 093040 (2012). [CrossRef]

16. X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A **88**, 033406 (2013). [CrossRef]

7. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev. **3**, 243–261 (2009). [CrossRef]

6. S.-Y. Tseng, “Counterdiabatic mode-evolution based coupled-waveguide devices,” Opt. Express **21**, 21224–21235 (2013). [CrossRef] [PubMed]

## 2. Model, dynamical invariant and optimzation

*β*

_{+}(

*z*) and

*β*

_{−}(

*z*). 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, the changes in the guided-mode amplitudes in the individual waveguides

**A**= [

*A*

_{+},

*A*

_{−}]

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

^{T}*d*

**A**

*/dz*= −

*i*

**H**

_{0}(

*z*)

**A**, that is, where Ω ≡ Ω(

*z*) (real) is the coupling coefficient, and Δ ≡ Δ(

*z*) = (

*β*

_{−}(

*z*) −

*β*

_{+}(

*z*))/2 describes the degree of mismatch between the waveguides. Replacing the spatial variation

*z*with the temporal variation

*t*, Eq. (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. In the following, we shall apply such quantum-optics analogy to describe the coupling dynamics of the directional coupler, and design the waveguide parameters Ω and Δ such that the coupler is robust against the variations of coupling coefficient and input wavelength.

**H**(

_{0}*z*) given by (1), a dynamical invariant

**I**(

*z*) may be parameterized as [17

17. X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A **83**, 062116 (2011). [CrossRef]

*θ*≡

*θ*(

*z*) and

*β*≡

*β*(

*z*) are

*z*-dependent angles,

*κ*is an arbitrary constant with units of

*mm*

^{−1}. Using the invariance condition,

*∂*

_{t}**I**+ (1/

*i*)[

**I**,

**H**

_{0}] = 0 [18

18. H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. **10**, 1458–1473 (1969). [CrossRef]

*λ*

_{±}= ±Ω/2. According to Lewis-Riesenfeld theory [18

18. H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. **10**, 1458–1473 (1969). [CrossRef]

*c*

_{±}are

*z*-independent amplitudes, and Lewis-Riesenfeld phases

*γ*

_{±}(

*z*) must be the solution of

**H**

_{1}=

*δ*Ω(

*z*)

*σ*/2, where

_{x}*σ*is the Pauli spin matrix and

_{x}*δ*is the amplitude of the relative error. Let the ideal, unperturbed Hamiltonian be

**H**

_{0}given by Eq. (1). In the error-free case, we consider the family of protocols that results in perfect state transfer from |2〉 to |1〉: the unperturbed solution, |Ψ(

*z*)〉 =

*e*

^{iγ+ (z)}|

*ϕ*

_{+}(

*z*)〉, satisfying Eq. (9). We shall consider

*H*

_{1}as perturbation, and define the coupling coefficient-error sensitivity as [14

14. A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. **14**, 093040 (2012). [CrossRef]

*P*

_{1}is the probability to be in the state |1〉 at

*z*=

*L*, i.e.

*P*

_{1}≈ 1 −

*q*

_{Ω}

*δ*

^{2}. By using perturbation theory, keeping the second order, we obtain with

*m*(

*z*) = 2

*γ*

_{+}(

*z*) −

*β*(

*z*). The minimum of

*q*

_{Ω}is achieved when

*q*

_{Ω}is nullified, which gives the maximal robustness with respect to coupling coefficient variations. Following [14

14. A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. **14**, 093040 (2012). [CrossRef]

*q*

_{Ω}= sin

^{2}(

*nπ*)/(4

*n*

^{2}) by imposing

*m*(

*z*) =

*n*(2

*θ*− sin2

*θ*), so that

*q*

_{Ω}= 0 is achieved, if

*n*= 1, 2, 3.... In the case of

*n*= 1, we have which results in

*q*

_{Ω}= 0. As an example, we choose smooth function satisfying Eqs. (9) and (10). Substituting Eqs. (13) and (14) into Eqs. (3) and (4), we can solve for to achieve optimal coupling dynamics with respect to coupling coefficient variations.

*λ*[5

5. G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett. **16**, 515–517 (2004). [CrossRef]

**H**

_{2}=

*δσ*/2, where

_{z}*σ*is the Pauli spin matrix and

_{z}*δ*is a constant frequency shift. Similarly, we can define the wavelength-error sensitivity and finally obtain [16

16. X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A **88**, 033406 (2013). [CrossRef]

*m*(

*z*) = 2

*γ*

_{+}(

*z*) −

*β*(

*z*). To nullify

*q*

_{Δ}, i.e.

*q*

_{Δ}= 0, we may assume

*m*(

*z*) = 2

*θ*+ 2

*α*sin(2

*θ*), with free parameter

*α*[16

16. X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A **88**, 033406 (2013). [CrossRef]

*M*= 1 + 2

*α*cos(2

*θ*). Combining Eqs. (14) and (18), we solve for the design parameters as follows which finally gives

*q*

_{Δ}= 0 with the parameter

*α*= −0.206 and the smallest value of required coupling coefficient, Ω

^{max}= 14.784.

## 3. Numerical simulations

*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). This weakly-guided coupled waveguide structure can be well-described by the scalar and paraxial wave equation, and the evolution of the guided modes in the coupler can be accurately described by Eq. (1). The BPM code solves the the scalar and paraxial wave equation using the finite difference scheme with the transparent boundary condition [20

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

*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 coupler design, the default width for the right and left waveguides is chosen to be

*W*=

_{R}*W*= 2

_{L}*μ*m. Through BPM simulations, we verified that the relation between the mismatch Δ and width difference

*δW*can be approximated by a linear relation [1

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

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

_{0}exp[−

*γ*(

*D*−

*D*

_{0})] [21]. We also assume that the exponential relation can be used to obtain an estimation of the coupling coefficient in the asymmetric coupler [1

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

*D*(

*z*) and width difference

*δW*(

*z*) are then adjusted along the propagation direction to satisfy the designed set of Ω(

*z*) and Δ(

*z*) functions.

### 3.1. Ω-coupler

*L*=1 mm. Figure 2(a) shows the calculated coupling coefficient Ω and detuning Δ. Using the exponential relation between Ω and

*D*and the linear relation between Δ and

*δW*, we obtain the corresponding waveguide parameters

*W*,

_{R}*W*, and

_{L}*D*as a function of

*z*as shown in Fig. 2(b). The geometry of the designed directional coupler is shown in Fig. 3(a). We also design an adiabatic coupler of the same length using the linearly tapered mismatch profile [2

2. T. A. Ramadan, R. Scarmozzino, and R. M. Osgood, “Adiabatic couplers: design rules and optimization,” J. Lightwave Technol. **16**, 277–283 (1998). [CrossRef]

22. D. R. Rowland, Y. Chen, and A. W. Snyder, “Tapered mismatched couplers,” J. Lightwave Technol. **9**, 567–570 (1991). [CrossRef]

*z*= 0 with a input wavelength of 1.55

*μ*m, and the BPM results are shown in the same figures. Power is coupled completely to the upper waveguide for the Ω-coupler in Fig. 3(a) and the resonant coupler in Fig. 3(c). For the adiabatic coupler in Fig. 3(b), the power coupling is incomplete because the adiabatic condition has not been met. For the chosen design parameters, we plot the conversion efficiency as a function of the device length

*L*(interaction length) for the three coupler designs in Fig. 4. It is clear that the adiabatic coupler length needs to satisfy

*L*≥ 1.5 mm in order to have minimized coupling between the supermodes. Even with the increased device length, we still observe oscillatory behavior in the coupling efficiency, which is characteristic of adiabatic devices due to finite coupling between the supermodes. So, complete power coupling can only happen at specific device lengths. On the other hand, the Ω-coupler exhibits optimized tolerance to device length (interaction length) variations around the designed length of 1 mm. The resonant coupler exhibits the expected sinusoidal behavior with device length change.

*δD*in Fig. 1(a)) in BPM simulations and show the resulting coupling efficiencies as a function of

*δD*in Fig. 5. The Ω-coupler design exhibits the desired flatness around

*δD*= 0 as a result of our optimization of the perturbation theory result, see Eq. (12), by using invariant-based inverse engineering. The invariant based design clearly shows better tolerance to coupling coefficient variations at a shorter length than a conventional tapered mismatched adiabatic coupler and the resonant coupler.

*n*from −75 % to +75 % in our BPM simulations and compare the coupling efficiencies of the three devices in Fig. 6. The Ω coupler shows flat response against refractive index difference variations, while the adiabatic coupler and the resonant coupler show the expected oscillatory behavior. So far, we have examined the robustness of the Ω-coupler against device length variations as well as against coupling coefficient variations resulting from waveguide spacing errors and refractive index variations. Overall, the Ω-coupler shows the desired robustness against errors in Ω in Eq. (1) at a short device length, as a result of our optimization.

### 3.2. Δ-coupler

*L*=1 mm. Figures 7(a) and (b) shows the calculated Ω, Δ and the corresponding waveguide parameters. Figure 8(a) shows the geometry of the designed device. Again, we also design an adiabatic coupler using the linearly tapered mismatch profile with a constant coupling coefficient such that the area under the Ω curves in Figs. 7(a) and (c) are equal, and the maxima and minima of the Δ curves in the same figures are equal. The resulting waveguide parameters and the corresponding waveguide geometry of the adiabatic coupler are shown in Fig. 7(d) and Fig. 8(b), respectively. We excite the lower waveguides by their unperturbed mode at

*z*= 0 with a input wavelength of 1.55

*μ*m, and the BPM results are shown in the same figures. Power is coupled completely to the upper waveguide for the Δ-coupler in Fig. 8(a), and the coupling efficiency is 0.96 for the adiabatic coupler in Fig. 8(b). The power coupling is incomplete in the adiabatic coupler because its coupling efficiency exhibits similar oscillatory behavior as a function of the device length as shown in Fig. 4. For the chosen design parameter, we find that the device length needs to satisfy

*L*≥ 1.7 mm in order to have minimized coupling between the supermodes.

*μ*m to 2

*μ*m with 0.05

*μ*m steps in BPM simulations, and material dispersion in the waveguide is not considered. The resulting coupling efficiency spectra for the Δ-coupler and the adiabatic coupler in Fig. 8, as well as for the resonant coupler in Fig. 3(c), are shown in Fig. 9. The spectrum of the Δ-coupler design exhibits the desired flatness around the zero-detuning wavelength of 1.55

*μ*m. The adiabatic coupler exhibits broadband response as compared to the resonant coupler, although still narrower than the optimized Δ-coupler. The oscillation with wavelength is also characteristic of an adiabatic coupler. The spectrum of the resonant coupler shows the expected sinusoidal variation [21]. The robustness of the Δ-coupler is superior to that of the adiabatic coupler and the resonant coupler as expected, and the robustness is achieved at a shorter length than a conventional tapered mismatched adiabatic coupler. In fact, more stable schemes can be further designed by cancelling higher orders of the perturbations approximation [15

15. D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. **111**, 050404 (2013). [CrossRef] [PubMed]

## 4. Discussion and conclusion

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

2. T. A. Ramadan, R. Scarmozzino, and R. M. Osgood, “Adiabatic couplers: design rules and optimization,” J. Lightwave Technol. **16**, 277–283 (1998). [CrossRef]

## Acknowledgments

## References and links

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

2. | T. A. Ramadan, R. Scarmozzino, and R. M. Osgood, “Adiabatic couplers: design rules and optimization,” J. Lightwave Technol. |

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

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

5. | G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett. |

6. | S.-Y. Tseng, “Counterdiabatic mode-evolution based coupled-waveguide devices,” Opt. Express |

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

8. | E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. |

9. | 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 |

10. | S.-Y. Tseng and X. Chen, “Engineering of fast mode conversion in multimode waveguides,” Opt. Lett. |

11. | S. Martínez-Garaot, S.-Y. Tseng, and J. G. Muga, “Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity,” Opt. Lett. |

12. | X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A |

13. | S.-Y. Tseng and Y.-W. Jhang, “Fast and robust beam coupling in a three waveguide directional coupler,” IEEE Photon. Technol. Lett. |

14. | A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. |

15. | D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. |

16. | X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A |

17. | X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A |

18. | H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. |

19. | C.-L. Chen, |

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

21. | K. Okamoto, |

22. | D. R. Rowland, Y. Chen, and A. W. Snyder, “Tapered mismatched couplers,” 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: June 13, 2014

Revised Manuscript: July 17, 2014

Manuscript Accepted: July 18, 2014

Published: July 28, 2014

**Citation**

Shuo-Yen Tseng, Rui-Dan Wen, Ying-Feng Chiu, and Xi Chen, "Short and robust directional couplers designed by shortcuts to adiabaticity," Opt. Express **22**, 18849-18859 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-18849

Sort: Year | Journal | Reset

### References

- A. Syahriar, V. M. Schneider, and S. Al-Bader, “The design of mode evolution couplers,” J. Lightwave Technol.16, 1907–1914 (1998). [CrossRef]
- T. A. Ramadan, R. Scarmozzino, and R. M. Osgood, “Adiabatic couplers: design rules and optimization,” J. Lightwave Technol.16, 277–283 (1998). [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]
- R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett.4, 1135–1138 (1992). [CrossRef]
- G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett.16, 515–517 (2004). [CrossRef]
- S.-Y. Tseng, “Counterdiabatic mode-evolution based coupled-waveguide devices,” Opt. Express21, 21224–21235 (2013). [CrossRef] [PubMed]
- S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev.3, 243–261 (2009). [CrossRef]
- E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys.62, 117–169 (2013). [CrossRef]
- 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.-Y. Tseng and X. Chen, “Engineering of fast mode conversion in multimode waveguides,” Opt. Lett.37, 5118–5120 (2012). [CrossRef] [PubMed]
- S. Martínez-Garaot, S.-Y. Tseng, and J. G. Muga, “Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity,” Opt. Lett.39, 2306 (2014). [CrossRef] [PubMed]
- X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A86, 033405 (2012). [CrossRef]
- S.-Y. Tseng and Y.-W. Jhang, “Fast and robust beam coupling in a three waveguide directional coupler,” IEEE Photon. Technol. Lett.25, 2478–2481 (2013). [CrossRef]
- A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys.14, 093040 (2012). [CrossRef]
- D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett.111, 050404 (2013). [CrossRef] [PubMed]
- X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A88, 033406 (2013). [CrossRef]
- X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A83, 062116 (2011). [CrossRef]
- H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys.10, 1458–1473 (1969). [CrossRef]
- C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).
- K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations (Wiley, 2001). [CrossRef]
- K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).
- D. R. Rowland, Y. Chen, and A. W. Snyder, “Tapered mismatched couplers,” J. Lightwave Technol.9, 567–570 (1991). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.