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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 13 — Jun. 30, 2003
  • pp: 1490–1496
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Coupled-mode theory and propagation losses in photonic crystal waveguides

S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, and R. Houdré  »View Author Affiliations


Optics Express, Vol. 11, Issue 13, pp. 1490-1496 (2003)
http://dx.doi.org/10.1364/OE.11.001490


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Abstract

Mode coupling phenomena, manifested by transmission “mini-stopbands”, occur in two-dimensional photonic crystal channel waveguides. The huge difference in the group velocities of the coupled modes is a new feature with respect to the classical Bragg reflection occurring, e.g., in distributed feedback lasers. We show that an adequate ansatz of the classical coupled-mode theory remarkably well accounts for this new phenomenon. The fit of experimental transmission data from GaAs-based photonic crystal waveguides then leads to an accurate determination of the propagation losses of both fundamental and higher, low-group-velocity modes.

© 2003 Optical Society of America

1. Introduction

Waveguides and other optical elements, such as ultra-small cavities and abrupt bends, based on photonic crystals, are promising candidates to implement miniature integrated optics. While it is generally desirable in integrated optics to make use of single-mode waveguides, such guides have various drawbacks when they rely on high-index contrast structures, be it photonic crystal or not. The main difficulty is to couple light into them from fibers or broader deep-ridge waveguides. They are also very sensitive to fabrication imperfections, in particular to roughness of the walls, that induces not only large propagation losses, but also unwanted reflection and polarization conversion as soon as the translation invariance is broken, e.g., at a bend. Hence, broader multimode waveguides have been considered, with the idea of information transport in their fundamental mode. A typical example is the photonic crystal channel waveguides (PCCW) formed by three missing rows in a triangular lattice of holes etched through a vertically monomode semiconductor heterostructure (Fig. 1(a)), that we name W3 (Fig. 1(b)).

Fig. 1. (a) Two-dimensional photonic crystal etched through a vertically monomode semiconductor heterostructure; (b) Micrograph of a three-missing-row PCCW (top view) with period a=260 nm in a GaAs heterostructure.

The periodicity of the photonic crystal waveguide boundaries is at the origin of Bragg coupling phenomena between guided modes, which occur at well-defined frequencies and wavevectors. In particular, the propagation of the fundamental mode is frustrated by diffraction induced by the corrugated boundaries, when its folded dispersion relation crosses that of a high-order mode with compatible symmetry [1

1. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B 63, 113311 (2001). [CrossRef]

,2

2. S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002). [CrossRef]

]. Mode coupling is then manifested by a dip or mini-stopband (MSB) in the transmission spectrum of the fundamental guided mode and energy is transferred to the high-order mode, which propagates backward with larger losses. We prefer the term mini-stopband to minigap, because there is no cancellation of the overall density-of-states of the PCCW, due to the presence of other guided modes.

This phenomenon, specific to multimode PCCWs, could lead to an original coupling configuration between photonic crystal elements [3

3. C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” App. Phys. Lett. 78, 1487–1489 (2001). [CrossRef]

]. However its quantitative modeling has been, until now, numerical and relatively heavy, and for this reason, the impact of propagation losses could not be studied easily. We show in this contribution that an adequate generalization of the well-known coupled-mode theory, blended with plane-wave expansion simulations, now very classical for planar photonic crystals, results in simple and efficient modeling of these transmission mini-stopbands. We apply this analysis to the determination of propagation losses in three-missing-rows photonic-crystal waveguides W3, carved in a GaAs-based heterostructure.

2. Theory

2.1. Loss-less case

We consider a channel waveguide carved into a planar photonic crystal consisting of a triangular array of air holes etched through a semiconductor heterostructure by Reactive Ion Etching (RIE). We focus here on canonical waveguides, i.e. defined by an integer number of missing rows, but any periodic line defect could be imagined (see [1

1. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B 63, 113311 (2001). [CrossRef]

,2

2. S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002). [CrossRef]

]). Fig. 2(a) shows the dispersion relation of a three-missing-row waveguide calculated by the 2D plane wave expansion method in a supercell [4

4. H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996). [CrossRef]

], where the vertical index structure is replaced by a homogeneous matrix, whose dielectric constant is taken as the square of the effective index of the vertical guided mode. The degree of validity of this 2D approach has been largely assessed [5

5. M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002). [CrossRef]

]. For our GaAs-based heterostructure, the background dielectric constant is ε eff =11 and the air-filling factor is f=0.37, which leads to a photonic band gap in TE polarization covering the reduced frequency range u=a/λ=0.22-0.3.

The modes of W3 are easily classified and numbered by following the various branches: The fundamental mode, for which the scalar magnetic map Hz(x,y) is represented in Fig. 2(c), basically resembles a traditional index-guided mode. Its dispersion relation is folded at the Brillouin zone edge due to the periodicity. The high-order modes follow at higher frequencies. The highest modes #4, #5 and #6, which have their cut-off frequency inside the photonic bandgap of the crystal, are purely Bragg-guided modes. They have a very low group velocity (or high group index ng =c/νg ) when they cross the lower-order modes. If symmetry allows, anti-crossings instead of crossings occur between modes of different order, labeled “a” and “b”, when they obey the first-order Bragg condition: βa0 +βb0 =2π/a [1

1. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B 63, 113311 (2001). [CrossRef]

], giving rise to a mini-gap. As an example, Fig. 2(b) shows the local picture of the mini-gap associated to the coupling between mode #1 and mode #5, whose magnetic field patterns are represented in Figs. 2(c) and (d).

Fig. 2. (a) Dispersion relations of the W3 waveguide folded into the first Brillouin zone. The waveguide supports up to 6 modes; (b) Local picture of the mini-gap associated to the coupling of the index-guided fundamental mode and the Bragg-guided 5th order mode; (c)–(d) magnetic field patterns associated to the fundamental mode and to the 5th order mode.

We model the coupling of these two modes, around the central reduced frequency u0 of the anticrossing, by the apparently classical coupled-mode formulation:

ddy(A(y,u)B(y,u))=(iδa(u)iκabiκabiδb(u))(A(y,u)B(y,u))
(1)

In this equation, A(y,u) and B(y,u) are the amplitude of both modes “a” and “b” along the waveguide and κab is their coupling constant. δa(u) and δb(u), which are the detuning of the uncoupled modes with respect to the Bragg condition, can be expressed as a function of the reduced frequency u, assuming that the dispersion relations of the uncoupled modes are linear around the central frequency u0 (i.e., their group velocities or group indices nga and ngb are constant):

δa(u)=βa0βa(u)=2πanga(u0u)
(2a)
δb(u)=βb0βb(u)=2πangb(u0u)
(2b)

The electromagnetic basis for such Eqs. is the same as the usual coupled mode equations introduced by Kogelnik [6

6. H. Kogelnick and C.V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2328 (1972).

,7

7. See for example Tamir, Guided Wave Optoelectronics, Springer Verlag, Berlin, chaps 2,6 (1988).

]. Although they have widely been used in guided optics to describe the Bragg reflection occuring e.g. in distributed feedback lasers consisting in monomode corrugated waveguides, there have been few significant reports of their use to describe coupling between modes with very different group velocity. Multimode corrugated waveguides have been addressed in [8

8. P. Ferrand, R. Romestain, and J.C. Vial, “Photonic band-gap properties of a porous silicon periodic planar waveguide,” Phys. Rev. B 63, 115106 (2001). [CrossRef]

,9

9. E. Peral and A. Yariv, “Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,” J. Lightwave Tech. 17, 942–947 (1999). [CrossRef]

], but the authors have not considered in particular the role of the diminished group velocity of the high-order mode.

Δuab=a2π4κabnga+ngb
(3)

In practice, we need not calculate explicitly the coupling constant from the overlap integral of the mode profiles. Rather, we calculate the dispersion relations of the guide of interest using the plane wave expansion method in a supercell [4

4. H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996). [CrossRef]

] and extract the numerical value of the mini-gap spectral width using a sufficently thin k-space mesh for a very good accuracy. The group indices n ga and n gb of both modes are extracted from the slope of their dispersion relation, taken far from the coupling region. The value of the coupling constant κ ab can then be derived from the knowledge of these three parameters, using Eq. (3). For the W3 waveguide considered above, the parameters extracted from the dispersion diagram of Fig. 2 are n ga=3.3, n gb=33 and Δu ab=2.5 10-3, leading to κ ab=0.14 a -1.

2.2. Influence of propagation losses

When coming to real waveguides, the propagation losses αa and αb of both modes must be taken into account, in order to obtain an accurate model of the shape and intensity level of the transmission spectrum. The high-order mode, which has a much lower group velocity and a much larger penetration in the guide boundaries, experiences much larger propagation losses than the fundamental mode, with a realistic ratio of αba=10. For short waveguides (typically 30 rows), the losses of the fundamental mode will have a negligible effect, whereas they will become more visible for longer waveguides. The impact of the propagation losses is therefore studied separately in the following, where simulations are performed for a W3 waveguide of length 60a.

Taking into account the propagation losses only for the high-order mode with the value αb=0.1 µm-1, one obtains the spectra represented in Figs. 3(a) and (b), superimposed on the lossless spectra. The transmission and reflection spectra are essentially the bottom envelope of the spectrum calculated in the lossless case, washing out secondary oscillations. The transmission level of the fundamental mode still drops to zero at the central frequency, while the reflection of the high-order mode at the input is now well below 100%.

Adding now the propagation losses of the fundamental mode with a smaller value αa=0.01 µm-1, we obtain the spectra displayed in Figs. 3(c) and (d). The transmission of the fundamental mode at the output is 70%, while the reflection level of the high-order mode at the input is essentially dominated by its propagation losses and suffers negligibly from the losses of the fundamental mode that excited it.

Fig. 3. Influence of the propagation losses on the spectral shapes of the transmission spectrum of mode a at the output of the guide and of the reflection spectrum of mode b at the input of the guide, showing the disappearance of secondary lobes.

3. Experimental results and comparison with theory

3.1 Experimental procedure

Three W3 waveguides of different length 60 a, 120 a and 240 a (with a=260 nm), carved into a GaAs-based heterostructure have been measured using the internal source technique described in earlier work [10

10. D. Labilloy, H. Benisty, C. Weisbuch, T.F. Krauss, R. Houdré, and U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensionam photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997). [CrossRef]

] and illustrated in Fig. 4(a). The photoluminescence of InAs quantum dots embedded in the GaAs core of the heterostructure is used as an internal source to probe a photonic crystal structure. The source is created at the entrance of the waveguide at the focus of a laser diode and a spectral analysis of the signal collected at the cleaved edge after propagation through the PCCW is performed. Owing to the small internal collection angle dictated by the numerical aperture of the collection objective, the transmission of the fundamental mode is selectively measured, whatever the other excited guided modes.

3.2 Determination of the propagation losses

Figure 4(b) shows the experimental spectra. A MSB is observed around the frequency u0 =0.265, in good agreement with the frequency of the anticrossing in the dispersion diagram of Fig. 2. Note that the shift of the mini-stopband frequency towards the higher frequencies when the guide length increases is mainly an artefact due to the proximity effects during e-beam patterning (slightly larger holes for larger overall exposure area). From the fit of the three experimental spectra with the coupling constant κ ab and the losses αa and αb as the adjustable parameters, we obtain an accurate measurement of the losses of the fundamental mode and of the 5th order mode. We find αa=35 cm-1=1.5 dB/100 µm for mode #1 and αb=400 cm-1=17 dB/100 µm for mode #5, with an accuracy of 10%. The value of the coupling constant is κ ab=0.09 a -1, a little below the calculated value 0.14 a -1, maybe on account of a minute width variation along the guide. It is remarkable that the spectral width of the transmission dip, simultaneously with the transmission level of the fundamental mode on both sides of the dip, are very well reproduced by the model. The propagation losses of the high-order mode are responsible for the apparently surprising fact that the MSB takes a broader and broader appearance when the length of the waveguide increases, whereas the coupling coefficient is unchanged.

Fig. 4. (a) Experimental set-up for the selective measurement of the transmission of the fundamental guided mode; (b) Experimental transmission spectra for W3 waveguides of lengths indocated on the graph (coloured curves) and fit using the coupled-mode theory (black curves) with the propagation losses as the only adjustable parameters.

3.3 Comparison with other methods

4. Conclusion

References and Links

1.

S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B 63, 113311 (2001). [CrossRef]

2.

S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002). [CrossRef]

3.

C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” App. Phys. Lett. 78, 1487–1489 (2001). [CrossRef]

4.

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996). [CrossRef]

5.

M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002). [CrossRef]

6.

H. Kogelnick and C.V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2328 (1972).

7.

See for example Tamir, Guided Wave Optoelectronics, Springer Verlag, Berlin, chaps 2,6 (1988).

8.

P. Ferrand, R. Romestain, and J.C. Vial, “Photonic band-gap properties of a porous silicon periodic planar waveguide,” Phys. Rev. B 63, 115106 (2001). [CrossRef]

9.

E. Peral and A. Yariv, “Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,” J. Lightwave Tech. 17, 942–947 (1999). [CrossRef]

10.

D. Labilloy, H. Benisty, C. Weisbuch, T.F. Krauss, R. Houdré, and U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensionam photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997). [CrossRef]

11.

E. Schwoob, H. Benisty, S. Olivier, C. Weisbuch, C.J.M Smith, T.F. Krauss, R. Houdre, and U. Oesterle, “Two-mode fringes in planar photonic crystal waveguides with constrictions: a sensitive probe to propagation losses,” J. Opt. Soc. Am. B 19, 2403–2412 (2002). [CrossRef]

12.

M. Qiu, B. Jaskorzynska, M. Swillo, and H. Benisty, “Time-domain 2D modeling of slab-waveguide-based photonic-crystal devices in the presence of radiation losses,” Microwave and Opt. Tech. Lett. 34, 387–393 (2002). [CrossRef]

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(230.7380) Optical devices : Waveguides, channeled

ToC Category:
Research Papers

History
Original Manuscript: May 13, 2003
Revised Manuscript: June 11, 2003
Published: June 30, 2003

Citation
Segolene Olivier, H. Benisty, C. Weisbuch, C. Smith, T. Krauss, and R. Houdre, "Coupled-mode theory and propagation losses in photonic crystal waveguides," Opt. Express 11, 1490-1496 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-13-1490


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References

  1. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle and R. Houdré, �??Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,�?? Phys. Rev. B 63, 113311 (2001). [CrossRef]
  2. S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré and U. Oesterle, �??Transmission properties of two-dimensional photonic crystal channel waveguides,�?? Opt. Quantum Electron. 34, 171-181 (2002). [CrossRef]
  3. C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré and U. Oesterle, �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? App. Phys. Lett. 78, 1487-1489 (2001). [CrossRef]
  4. H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,�?? J. Appl. Phys. 79, 7483-7492 (1996). [CrossRef]
  5. M. Qiu, �??Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,�?? Appl. Phys. Lett. 81, 1163-1165 (2002). [CrossRef]
  6. H. Kogelnick and C.V. Shank, �??Coupled wave theory of distributed feedback lasers,�?? J. Appl. Phys. 43, 2328 (1972).
  7. See for example Tamir, Guided Wave Optoelectronics, Springer Verlag, Berlin, chaps 2,6 (1988).
  8. P. Ferrand, R. Romestain and J.C. Vial, �??Photonic band-gap properties of a porous silicon periodic planar waveguide,�?? Phys. Rev. B 63, 115106 (2001). [CrossRef]
  9. E. Peral and A. Yariv, �??Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,�?? J. Lightwave Tech. 17, 942-947 (1999). [CrossRef]
  10. D. Labilloy, H. Benisty, C. Weisbuch, T.F. Krauss, R. Houdré and U. Oesterle, �??Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensionam photonic crystals,�?? Appl. Phys. Lett. 71, 738-740 (1997). [CrossRef]
  11. E. Schwoob, H. Benisty, S. Olivier, C. Weisbuch, C.J.M Smith, T.F. Krauss, R. Houdre and U. Oesterle, �??Two-mode fringes in planar photonic crystal waveguides with constrictions: a sensitive probe to propagation losses,�?? J. Opt. Soc. Am. B 19, 2403-2412 (2002). [CrossRef]
  12. M. Qiu, B. Jaskorzynska, M. Swillo and H. Benisty, �??Time-domain 2D modeling of slab-waveguide-based photonic-crystal devices in the presence of radiation losses,�?? Microwave and Opt. Tech. Lett. 34, 387-393 (2002). [CrossRef]

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