## Mode conversion using optical analogy of shortcut to adiabatic passage in engineered multimode waveguides |

Optics Express, Vol. 20, Issue 21, pp. 24085-24092 (2012)

http://dx.doi.org/10.1364/OE.20.024085

Acrobat PDF (2332 KB)

### Abstract

A shortcut to adiabatic mode conversion in multimode waveguides using optical analogy of stimulated Raman adiabatic passage is investigated. The design of mode converters using the shortcut scheme is discussed. Computer-generated planar holograms are used to mimic the shaped pulses used to speed up adiabatic passage in quantum systems based on the transitionless quantum driving algorithm. The mode coupling properties are analyzed using the coupled mode theory and beam propagation simulations. We show reduced device length using the shortcut scheme as compared to the common adiabatic scheme. Modal evolution in the shortened device indeed follows the adiabatic eigenmode exactly amid the violation of adiabatic criterion.

© 2012 OSA

## 1. Introduction

1. S. Berdague and P. Facq, “Mode division multiplexing in optical fibers,” Appl. Opt. **21**, 1950–1955 (1982). [CrossRef] [PubMed]

2. M. Greenberg and M. Orenstein, “Multimode add-drop multiplexing by adiabatic linearly tapered coupling,” Opt. Express **13**, 9381–9387 (2005). [CrossRef] [PubMed]

3. J. B. Park, D.-M. Yeo, and S.-Y. Shin, “Variable optical mode generator in a multimode waveguide,” IEEE Photon. Technol. Lett. **18**, 2084–2086 (2006). [CrossRef]

4. T. Ando, T. Murata, H. Nakayama, J. Yamauchi, and H. Nakano, “Analysis and measurement of polarization conversion in a periodically loaded dielectric waveguide,” IEEE Photon. Technol. Lett. **14**, 1288–1290 (2002). [CrossRef]

5. S. Longhi, “Adiabatic passage of light in coupled optical waveguides,” Phys. Rev. E **73**, 026607 (2006). [CrossRef]

6. S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B **76**, 201101(R) (2007). [CrossRef]

7. G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett. **92**, 011106 (2008). [CrossRef]

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

9. F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett. **95**, 261102 (2009). [CrossRef]

10. E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Opt. Commun. **258**, 30–34 (2006). [CrossRef]

11. E. Paspalakis, M. Protopapas, and P. L. Knight, “Time-dependent pulse and frequency effects in population trapping via the continuum,” J Phys. B **31**, 775–794 (1998). [CrossRef]

12. R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, “Coherent properties of a tripod system coupled via a continuum,” Phys. Rev. A **61**, 043408 (2000). [CrossRef]

13. H. Theuer, R. G. Unanyan, C. Habscheid, K. Klein, and K. Bergmann, “Novel laser controlled variable matter wave beamsplitter,” Opt. Express **4**, 77–83 (1999). [CrossRef] [PubMed]

15. F. Vewinger, M. Heinz, R. Garcia Fernandez, N. V. Vitanov, and K. Bergmann, “Creation and measurement of a coherent superposition of quantum states,” Phys. Rev. Lett. **91**, 213001 (2003). [CrossRef] [PubMed]

16. A. A. Rangelov and N. V. Vitanov, “Achromatic multiple beam splitting by adiabatic passage in optical waveguides,” Phys. Rev. A **85**, 055803 (2012). [CrossRef]

17. F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. **34**, 2405–2407 (2009). [CrossRef] [PubMed]

18. S.-Y. Tseng and M.-C. Wu, “Adiabatic mode conversion in multimode waveguides using computer-generated planar holograms,” IEEE Photon. Technol. Lett. **22**, 1211–1213 (2010). [CrossRef]

20. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. **70**, 1003–1025 (1998). [CrossRef]

21. G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A **80**, 013417 (2009). [CrossRef]

22. G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A **80**, 043408 (2009). [CrossRef]

23. 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]

24. M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor. **42**, 365303 (2009). [CrossRef]

**H**

_{0}(

*t*) initially in an eigenstate |Ψ

*(0)〉, the adiabatic theorem states that it will follow the same instantaneous eigenstate |Ψ*

_{n}*(*

_{n}*t*)〉

*closely*, as long as the evolution of the Hamiltonian is slow enough [25

25. M. H. S. Amin, “Consistency of the adiabatic theorem,” Phys. Rev. Lett. **102**, 220401 (2009). [CrossRef] [PubMed]

**H**(

*t*), associated with any chosen

**H**

_{0}(

*t*), that drive the instantaneous eigenstates |Ψ

*(*

_{n}*t*)〉

*exactly*. The Hamlitonian

**H**(

*t*) can be found as [23

23. 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]

24. M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor. **42**, 365303 (2009). [CrossRef]

**H**(

*t*), elaborate coupling schemes might be required to realize optical analogies of such systems. In this paper, we propose a optical realization of the SHAPE scheme using CGPH in multimode waveguides. The designed mode converter could find applications in MDM systems.

## 2. Shortcut to adiabatic passage in multimode waveguides

*N*≥ 3 forward-propagating modes, we consider three distinct modes |Ψ

_{1}〉, |Ψ

_{2}〉, and |Ψ

_{3}〉, coupled by a CGPH. The CGPHs are multiplexed long-period gratings which couple the guided modes depending on the grating shape and periodicity [26

26. S.-Y. Tseng, S. K. Choi, and B. Kippelen, “Variable-ratio power splitters using computer-generated planar holograms on multimode interference couplers,” Opt. Lett. **34**, 512–514 (2009). [CrossRef] [PubMed]

_{1}〉 and |Ψ

_{2}〉 and modes |Ψ

_{2}〉 and |Ψ

_{3}〉 coupled by gratings Λ

_{12}and Λ

_{23}with coupling coefficients

*κ*

_{12}and

*κ*

_{23}. Figure 1 shows the level scheme of STIRAP, the schematic of a CGPH loaded multimode waveguide, and amplitude profiles of the three guided modes. When only Λ

_{12}and Λ

_{23}are present, the evolution of mode amplitudes

*A*(

_{i}*i*= 1, 2, 3) obey the coupled mode equations Replacing the spatial variation

*z*with the temporal variation

*t*, (3) is used to describe the probability amplitudes of a three-level atomic system driven by two laser pulses shown in Fig. 1(a) using the Schrödinger equation (

*h*̄ = 1) under the rotating-wave approximation, in which

**A**represents the probability amplitudes of the states being populated,

*κ*is the Rabi frequency of the pulse coupling states |Ψ

_{mn}*〉 and |Ψ*

_{m}*〉. Solving for the eigenmodes |Ψ*

_{n}*〉, |Ψ*

_{D}_{+}〉, and |Ψ

_{−}〉, we can find a dark eigenmode as When the two spatially variable coupling coefficients

*κ*

_{12}(

*z*) and

*κ*

_{23}(

*z*) are applied in a counterintuitive scheme to mimic the laser pulses in STIRAP, |Ψ

*〉 can be used to convert |Ψ*

_{D}_{1}〉 to |Ψ

_{3}〉. If the device length is not sufficiently long, unwanted couplings happen among |Ψ

*〉, |Ψ*

_{D}_{+}〉, and |Ψ

_{−}〉, resulting in low conversion efficiency.

*〉 exactly. Replace*

_{D}*t*with

*z*in (2), let

*h*̄ = 1, and substitute |Ψ

*〉, |Ψ*

_{D}_{+}〉, and |Ψ

_{−}〉 in to (2), we obtain with The shortcut here is thus to add a grating Λ

_{13}coupling |Ψ

_{1}〉 and |Ψ

_{3}〉 to the original CGPH implementing the STIRAP scheme

**H**

_{0}(

*z*) in (3) as shown in Fig. 1(b). We also note that grating Λ

_{13}is 90° out of phase with gratings Λ

_{12}and Λ

_{23}due to the presence of

*i*in (5). To implement

**H**(

*z*) =

**H**

_{0}(

*z*) +

**H**

_{1}(

*z*) in a coupled PLC waveguide system would be difficult, because the waveguides representing |Ψ

_{1}〉 and |Ψ

_{3}〉 are not adjacent to each other. In a multimode waveguide, by designing a CGPH to implement

**H**(

*z*) with the addition of a grating Λ

_{13}coupling modes |Ψ

_{1}〉 and |Ψ

_{3}〉 using the coupling coefficient in (6), we can realize optical analogy of the SHAPE scheme as shown in Fig 1(b). In the following, we use a numerical example to demonstrate SHAPE in an engineered multimode waveguide and compare it with STIRAP in the same system.

## 3. Numerical results

*μ*m wide, five-moded polymer waveguide similar to the one in [19] for mode conversion from |Ψ

_{1}〉 to |Ψ

_{3}〉 via |Ψ

_{2}〉. The CGPHs used to implement the SHAPE and STIRAP schemes are designed at the zero-detuning wavelength

*λ*

_{0}=1.55

*μ*m and the TE polarization. The maximum effective index modulation is assumed to be Δ

*n*= 0.003. Figure 2 shows Gaussian shaped coupling coefficients

*κ*

_{12}(

*z*) =

*k*

_{12}exp[−(

*z*

_{0}/

*z*− 2)

^{2}/

*c*

^{2}] and

*κ*

_{23}(

*z*) =

*k*

_{23}exp[−(

*z*+

*z*

_{0}/2)

^{2}/

*c*

^{2}], with

*k*

_{12}and

*k*

_{23}directly proportional to Δ

*n*, chosen arbitrarily to mimic the counterintuitive optical pulses used in STIRAP. The parameter

*c*is chosen to be the same as the delay

*z*

_{0}to minimize nonadibatic coupling [20

20. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. **70**, 1003–1025 (1998). [CrossRef]

*L*(mm) as

*c*=

*z*

_{0}= 3

*L*/20. To implement SHAPE, the coupling coefficient

*κ*

_{13}(

*z*) corresponding to the additional Λ

_{13}is calculated using (6) and also shown in Fig. 2.

**H**

_{1}(

*z*)) in a mode converter at 5 mm length using the beam propagation method (BPM), which solves the wave equation governing light propagation in the CGPH loaded multimode using a finite difference scheme. We design a CGPH using the method outlined in [26

26. S.-Y. Tseng, S. K. Choi, and B. Kippelen, “Variable-ratio power splitters using computer-generated planar holograms on multimode interference couplers,” Opt. Lett. **34**, 512–514 (2009). [CrossRef] [PubMed]

**H**(

*z*) =

**H**

_{0}(

*z*) +

**H**

_{1}(

*z*). Figure 3 shows the calculated CGPH pattern. The CGPH pattern is used as an effective index perturbation to the multimode waveguide. Figure 4 shows the calculated beam propagation using mode |Ψ

_{1}〉 as the input from the left hand side. According to the SHAPE scheme, |Ψ

_{1}〉 is converted to |Ψ

_{3}〉 at the output on the right hand side in a 5 mm device. As a comparison, the beam propagation in a 5 mm STIRAP mode converter designed according to

**H**

_{0}(

*z*) in (3) is shown in Fig. 5. Complex mode coupling is evident, and the conversion fails at such a short distance. In Fig. 6(a), the corresponding modal power evolution along the propagation distance for different modes is shown for the SHAPE case. As shown in (4), when the system evolution follows the dark eigenmode |Ψ

*〉, no component of |Ψ*

_{D}_{2}〉 is excited. From Fig. 6(a), it is clear that |Ψ

_{2}〉 is not involved in the conversion, and the modal power evolution indeed follows what is expected from the STIRAP process when the adiabatic criterion is satisfied, but at a much shorter device length. The smooth conversion curve shows the good device length tolerance which is characteristic of adiabatic devices. For comparison, the modal power evolution for the STIRAP case is shown in Fig. 6(b). Coupling occurs among the eigenmodes due to the breakdown of adiabaticity, resulting in the excitation of |Ψ

_{2}〉 and lowered conversion efficiency at the output. Clearly, the phenomenon of SHAPE can be observed in an engineered multimode waveguide using BPM, and it provides a shortcut to the STIRAP scheme.

*κ*

_{12}and

*κ*

_{23}defined in Fig. 2 and plot the normalized modal power at the output of the converter as a function of the device length in Fig. 7(a). It is clear that as the device length decreases, adiabaticity breaks down, |Ψ

_{2}〉 is excited, and the conversion efficiency deteriorates due to coupling among the eigenmodes. The excitation of |Ψ

_{2}〉 indicates that the mode converter no longer follows the adiabatic pathway described in (4). To obtain a conversion efficiency ≥ 99% using STIRAP, the minimum device length is 11.4 mm in this numerical example. Next, we consider the SHAPE case. We note that the maximum value of coupling coefficient corresponding to Δ

*n*= 0.003 is 1.594 mm

^{−1}for this polymer waveguide platform in our numerical example. So, we cap the maxima of

*κ*

_{13}(

*z*) at 1.594 mm

^{−1}in our simulation to account for physical realizability in fabrication and to avoid additional scattering loss resulting from large effective index modulation. In Fig. 7(b), we show the normalized modal power at the output of the converter as a function of the device length for SHAPE. Complete conversion can be observed at shorter lengths then STIRAP because the system follows |Ψ

*〉 exactly without the excitation of |Ψ*

_{D}_{2}〉. For the same 99% conversion efficiency using SHAPE, the device length can be reduced to 3.9 mm in this numerical example, corresponding to a 65% reduction in device length as compared to STIRAP. As evidenced by the excitation of |Ψ

_{2}〉, coupling among eigenmodes occurs at lengths below 3.9 mm because of the limitation we put on the maximum value of

*κ*

_{13}; otherwise, complete conversion can be achieved at arbitrarily shorter length.

## 4. Conclusion

## Acknowledgment

## References and links

1. | S. Berdague and P. Facq, “Mode division multiplexing in optical fibers,” Appl. Opt. |

2. | M. Greenberg and M. Orenstein, “Multimode add-drop multiplexing by adiabatic linearly tapered coupling,” Opt. Express |

3. | J. B. Park, D.-M. Yeo, and S.-Y. Shin, “Variable optical mode generator in a multimode waveguide,” IEEE Photon. Technol. Lett. |

4. | T. Ando, T. Murata, H. Nakayama, J. Yamauchi, and H. Nakano, “Analysis and measurement of polarization conversion in a periodically loaded dielectric waveguide,” IEEE Photon. Technol. Lett. |

5. | S. Longhi, “Adiabatic passage of light in coupled optical waveguides,” Phys. Rev. E |

6. | S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B |

7. | G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett. |

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

9. | F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett. |

10. | E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Opt. Commun. |

11. | E. Paspalakis, M. Protopapas, and P. L. Knight, “Time-dependent pulse and frequency effects in population trapping via the continuum,” J Phys. B |

12. | R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, “Coherent properties of a tripod system coupled via a continuum,” Phys. Rev. A |

13. | H. Theuer, R. G. Unanyan, C. Habscheid, K. Klein, and K. Bergmann, “Novel laser controlled variable matter wave beamsplitter,” Opt. Express |

14. | R. G. Unanyan, B. W. Shore, and K. Bergmann, “Preparation of an N-component maximal coherent superposition state using the stimulated Raman adiabatic passage method,” Phys. Rev. A |

15. | F. Vewinger, M. Heinz, R. Garcia Fernandez, N. V. Vitanov, and K. Bergmann, “Creation and measurement of a coherent superposition of quantum states,” Phys. Rev. Lett. |

16. | A. A. Rangelov and N. V. Vitanov, “Achromatic multiple beam splitting by adiabatic passage in optical waveguides,” Phys. Rev. A |

17. | F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. |

18. | S.-Y. Tseng and M.-C. Wu, “Adiabatic mode conversion in multimode waveguides using computer-generated planar holograms,” IEEE Photon. Technol. Lett. |

19. | 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. |

20. | K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. |

21. | G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A |

22. | G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A |

23. | 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. |

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

25. | M. H. S. Amin, “Consistency of the adiabatic theorem,” Phys. Rev. Lett. |

26. | S.-Y. Tseng, S. K. Choi, and B. Kippelen, “Variable-ratio power splitters using computer-generated planar holograms on multimode interference couplers,” Opt. Lett. |

**OCIS Codes**

(000.1600) General : Classical and quantum physics

(090.1760) Holography : Computer holography

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: July 23, 2012

Revised Manuscript: September 26, 2012

Manuscript Accepted: September 27, 2012

Published: October 5, 2012

**Citation**

Tzung-Yi Lin, Fu-Chen Hsiao, Yao-Wun Jhang, Chieh Hu, and Shuo-Yen Tseng, "Mode conversion using optical analogy of shortcut to adiabatic passage in engineered multimode waveguides," Opt. Express **20**, 24085-24092 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-24085

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### References

- S. Berdague and P. Facq, “Mode division multiplexing in optical fibers,” Appl. Opt.21, 1950–1955 (1982). [CrossRef] [PubMed]
- M. Greenberg and M. Orenstein, “Multimode add-drop multiplexing by adiabatic linearly tapered coupling,” Opt. Express13, 9381–9387 (2005). [CrossRef] [PubMed]
- J. B. Park, D.-M. Yeo, and S.-Y. Shin, “Variable optical mode generator in a multimode waveguide,” IEEE Photon. Technol. Lett.18, 2084–2086 (2006). [CrossRef]
- T. Ando, T. Murata, H. Nakayama, J. Yamauchi, and H. Nakano, “Analysis and measurement of polarization conversion in a periodically loaded dielectric waveguide,” IEEE Photon. Technol. Lett.14, 1288–1290 (2002). [CrossRef]
- S. Longhi, “Adiabatic passage of light in coupled optical waveguides,” Phys. Rev. E73, 026607 (2006). [CrossRef]
- S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B76, 201101(R) (2007). [CrossRef]
- G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008). [CrossRef]
- S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev.3, 243–261 (2009). [CrossRef]
- F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009). [CrossRef]
- E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Opt. Commun.258, 30–34 (2006). [CrossRef]
- E. Paspalakis, M. Protopapas, and P. L. Knight, “Time-dependent pulse and frequency effects in population trapping via the continuum,” J Phys. B31, 775–794 (1998). [CrossRef]
- R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, “Coherent properties of a tripod system coupled via a continuum,” Phys. Rev. A61, 043408 (2000). [CrossRef]
- H. Theuer, R. G. Unanyan, C. Habscheid, K. Klein, and K. Bergmann, “Novel laser controlled variable matter wave beamsplitter,” Opt. Express4, 77–83 (1999). [CrossRef] [PubMed]
- R. G. Unanyan, B. W. Shore, and K. Bergmann, “Preparation of an N-component maximal coherent superposition state using the stimulated Raman adiabatic passage method,” Phys. Rev. A63, 043401 (2001). [CrossRef]
- F. Vewinger, M. Heinz, R. Garcia Fernandez, N. V. Vitanov, and K. Bergmann, “Creation and measurement of a coherent superposition of quantum states,” Phys. Rev. Lett.91, 213001 (2003). [CrossRef] [PubMed]
- A. A. Rangelov and N. V. Vitanov, “Achromatic multiple beam splitting by adiabatic passage in optical waveguides,” Phys. Rev. A85, 055803 (2012). [CrossRef]
- F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett.34, 2405–2407 (2009). [CrossRef] [PubMed]
- S.-Y. Tseng and M.-C. Wu, “Adiabatic mode conversion in multimode waveguides using computer-generated planar holograms,” IEEE Photon. Technol. Lett.22, 1211–1213 (2010). [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).
- K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys.70, 1003–1025 (1998). [CrossRef]
- G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A80, 013417 (2009). [CrossRef]
- G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A80, 043408 (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]
- M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor.42, 365303 (2009). [CrossRef]
- M. H. S. Amin, “Consistency of the adiabatic theorem,” Phys. Rev. Lett.102, 220401 (2009). [CrossRef] [PubMed]
- S.-Y. Tseng, S. K. Choi, and B. Kippelen, “Variable-ratio power splitters using computer-generated planar holograms on multimode interference couplers,” Opt. Lett.34, 512–514 (2009). [CrossRef] [PubMed]

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