## Recirculation-enhanced switching in photonic crystal Mach-Zehnder interferometers

Optics Express, Vol. 12, Issue 13, pp. 3035-3045 (2004)

http://dx.doi.org/10.1364/OPEX.12.003035

Acrobat PDF (459 KB)

### Abstract

We show that Mach-Zehnder interferometers (MZIs) formed from waveguides in a perfectly reflecting cladding can display manifestly different transmission characteristics to conventional MZIs due to mode recirculation and resonant reflection. Understanding and exploiting this behavior, rather than avoiding it, may lead to improved performance of photonic crystal (PC) based MZIs, for which cladding radiation is forbidden for frequencies within a photonic bandgap. Mode recirculation in such devices can result in a significantly sharper switching response than in conventional interferometers. A simple and accurate analytic model is presented and we propose specific PC structures with both high and low refractive index backgrounds that display these properties.

© 2004 Optical Society of America

## 1. Introduction

1. A. N. Chester, S. Martellucci, and A. M. V. Scheggi, eds., *Optical fiber sensors* (Martinus Nijhoff, Dordrecht, 1987). [CrossRef]

2. G. V. Treyz, “Silicon Mach-Zehnder waveguide interferometers operating at 1.3*µ*m,” Electron. Lett. **27**, 118–120 (1991). [CrossRef]

3. C. Rolland, R. S. Moore, F. Shepherd, and G. Hillier, “10 Gbit/s, 1.56*µ*m Multiquantum well InP/InGaAsP Mach-Zehnder Optical Modulator,” Electron. Lett. **29**, 471–472 (1993). [CrossRef]

4. E. A. Camargo, H. M. H. Chong, and R. M. De. La. Rue, “Photonic crystal Mach-Zehnder structures for thermo-optic switching,” Opt. Express **12**, 588–592 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-588. [CrossRef] [PubMed]

5. M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, H. Yukawa, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Two-dimensional photonic crystal Mach-Zehnder interferometers,” Appl. Phys. Lett.84 (2004). [CrossRef]

6. A. Martinez, A. Griol, P. Sanchis, and J. Marti, “Mach-Zehnder interferometer employing coupled-resonator optical waveguides,” Opt. Lett. **28**, 405–407 (2003). [CrossRef] [PubMed]

*D*photonic crystal waveguides with both high and low refractive index backgrounds. In both cases we use a novel junction type [7

7. T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part II: Applications,” Phys. Rev. E (2004). To be published. [CrossRef]

8. A. Lan, K. Kanamoto, T. Yang, A. Nishikawa, Y. Sugimoto, N. Ikeda, H. Nakamura, K. Asakawa, and H. Ishikawa, “Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,” Phys. Rev. B67 (2003). [CrossRef]

9. T. P. White, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Utracompact resonant filters in photonic crystals,” Opt. Lett. **28**, 2452–2454 (2003). [CrossRef] [PubMed]

7. T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part II: Applications,” Phys. Rev. E (2004). To be published. [CrossRef]

*D*numerical calculations, and the results are compared to those of Section 2. As predicted by the model, the recirculating PC MZI designs exhibit highly unconventional properties that could provide novel response characteristics for future MZI devices.

## 2. Modal analysis of recirculating MZIs

*Y*

_{1}and

*Y*

_{2}joined by waveguide arms

*A*

_{1}and

*A*

_{2}of length

*L*. An ideal Y-junction splits light equally between

*A*

_{1}and

*A*

_{2}. If an element is placed in

*A*

_{2}to introduce a phase difference

*φ*between the two arms of the interferometer, the intensity of light transmitted into the output depends on

*φ*. In a balanced MZI,

*φ*=0, so light entering from the input guide recombines in-phase at

*Y*

_{2}and is transmitted into the output guide - a balanced MZI transmits equally well at all wavelengths. Active devices such as switches and filters can be made by varying

*φ*to modulate the output [4

4. E. A. Camargo, H. M. H. Chong, and R. M. De. La. Rue, “Photonic crystal Mach-Zehnder structures for thermo-optic switching,” Opt. Express **12**, 588–592 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-588. [CrossRef] [PubMed]

*φ*[1

1. A. N. Chester, S. Martellucci, and A. M. V. Scheggi, eds., *Optical fiber sensors* (Martinus Nijhoff, Dordrecht, 1987). [CrossRef]

*A*

_{1}and

*A*

_{2}are separated sufficiently that cross talk between the two guides is negligible, in which case the propagation constants of |

*ψ*±〉 are equal to the propagation constant of an isolated waveguide,

*β*

_{+}=

*β*

_{-}=

*β*respectively. We return to this in Section 3.2.

*ψ*

_{in}〉 is symmetric with respect to

*A*

_{1}and

*A*

_{2}, it can only couple to the supermode with the same symmetry, |

*ψ*

_{+}〉, so for an ideal

*Y*

_{1}junction, mode |

*ψ*

_{in}〉 is fully transmitted into |

*ψ*

_{+}〉. When this mode propagates through

*A*

_{1}and

*A*

_{2}, the effect of

*φ*is to introduce an asymmetric field component, equivalent to transferring some of the energy into |

*ψ*

_{-}〉. As at

*Y*

_{1}, the |

*ψ*

_{+}〉 component of the field entering

*Y*

_{2}can couple into the output guide, but the |

*ψ*

_{-}〉 component cannot and must be either radiated or reflected.

*ψ*

_{-}〉 at

*Y*

_{2}. In dielectric waveguides, most of the light in |

*ψ*

_{-}〉 is radiated into the cladding and lost, and only a very small amount is reflected back into the arms of the interferometer. However, if light is unable to radiate into the cladding, any light in |

*ψ*

_{-}〉 must be reflected at

*Y*

_{2}, back into the interferometer. This odd mode reflection results in unique resonant behavior of recirculating MZIs with single-mode input and output waveguides.

*Y*

_{2}. After passing through

*Y*

_{1}, the resulting field

*ψ*

_{1}〉 and |

*ψ*

_{2}〉 using Eq. (1). Here the superscript (1) indicates that we are considering the fields on the first pass through the interferometer. The modes then propagate through each arm, essentially independently of each other, advancing in phase as they propagate. At

*Y*

_{2}, the combined field in

*A*

_{1}and

*A*

_{2}is

*χ*=

*βL*+

*φ*/2 and

*β*is the propagation constant of the single propagating mode of an isolated waveguide. At

*Y*

_{2}, the even field component is transmitted into the output guide, and the odd mode is reflected back into

*A*

_{1}and

*A*

_{2}. Note that in a conventional MZI the odd mode is radiated into the cladding and the transmission is simply given by the even mode term in Eq. (2),

*i*exp(

*iχ*) sin(

*φ*/2)|

*ψ*

_{-}〉, propagates back through the two arms, accumulating a further phase

*φ*so that the field arriving back at

*Y*

_{1}is

*R*=1-

*T*) can be derived in the same way by summing the field amplitudes transmitted from the interferometer back into the input waveguide. Observe that Eq. (5) is a function of both

*φ*and

*Lβ*=

*Lβ*(

*λ*), in contrast to the conventional MZI transmission (3), which is a function of

*φ*only. Thus, the dispersion properties of the waveguides are important in recirculating MZIs. Apart from the different functional forms of Eqs. (3) and (5), the additional dependence on wavelength and structure length leads to significantly different transmission properties. The explicit dependence on structure length results from the mode recirculation inside the device. An alternative derivation of Eq. (5) for PC based MZIs follows directly from the rigorous Bloch mode method [10

10. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part I: Theory,” Phys. Rev. E (2004). To be published. [CrossRef]

11. L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. Langtry, “Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics,” Opt. Express **12**, 1592–1604 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1592. [CrossRef] [PubMed]

*β*, a property that depends on the geometry of the waveguides. In the remaining sections of this paper, the dispersion properties of photonic crystal waveguides are used in the numerical calculations, however the transmission behavior discussed is generic to all recirculating MZIs satisfying the conditions of the above analysis. This is exemplified in the two PC examples presented in Section 3; the waveguide dispersion properties are very different, with opposite dispersion slopes and strong dispersion near cutoff in one case, but the transmission characteristics bear a close resemblance to each other.

*L*=36

*d*=19.9

*µ*m, calculated using Eq. (5). The wavelength dependence of

*β*was calculated for a single waveguide formed by removing one row of rods from the square PC lattice of period

*d*=570nm shown in Fig. 2(d). The PC consists of dielectric rods of radius

*a*=0.18

*d*, and refractive index

*n*

_{cyl}=3.4, in a background of air (

*n*=1). Figure 2(a) is a contour plot of transmission as a function of wavelength and phase difference, where the red areas correspond to high transmission, and the blue areas correspond to low transmission (high reflection). The transmission is high across much of the parameter space, and in particular it can be seen that

_{b}*T*=1 when

*φ*=0 and

*T*=0 when

*φ*=

*π*, as for a conventional MZI. The slightly tilted, low transmission ‘valleys’ correspond to strong resonances due to the multiple reflections of the odd mode. At the center of the valleys, the transmission drops to zero and the valleys become rapidly narrower as

*φ*approaches 0 or 2

*π*. The slight tilt of the valleys from vertical is determined by the dispersion slope of

*β*, which appears in

*χ*in Eq. (5).

*φ*is varied at a fixed wavelength, corresponding to traversing a vertical line across Fig. 2(a). For most wavelengths, a vertical line intersects one end of a low transmission valley, resulting in a transmission response with a strong resonant reflection, the width decreasing as the resonance position approaches

*φ*=0 or

*φ*=2

*π*. There are, however, wavelengths for which the sharp resonances are avoided. Of particular interest is the response of the MZI at the wavelengths where the diagonal valleys intersect the low transmission band at

*φ*=

*π*. The red curve in Fig. 2(b) shows the response as a function of

*φ*at one of these wavelengths,

*λ*=1.5496

*µ*m, corresponding to the red line in Fig. 2(a). The dashed curve on the same axes shows the transmission of a conventional MZI, given by Eq. (3). Comparing the two curves, it is seen that the recirculating MZI response exhibits a significantly sharper transition between ‘on’ and ‘off’ states. From Eq. (5), we find that this behavior occurs when

*β L*=

*π*/2,3

*π*/2,5

*π*/2 …, for which

*φ*/2), compared with the quadratic response of a conventional MZI. The change in

*φ*required to switch the transmission from 0.9 to 0.1 in the recirculating MZI is approximately 0.32

*π*, whereas a conventional MZI would require a change of 0.59

*π*. Although cos

^{4}(

*φ*/2) responses can be achieved with a pair of conventional MZIs, and higher order behavior occurs for a chain of such devices [12

12. G. P. Agrawal, *Fiber-optic communication systems*, 3rd ed. (John Wiley and Sons, New York, 2002). [CrossRef]

*T*=0 and

*T*=1, whereas a chain of MZIs always shows a quadratic variation in the region around

*T*=1. The steep, almost square response exhibited by the recirculating MZI is highly desirable for switching applications, and could be used to improve performance in both linear and nonlinear devices.

*φ*is fixed, the transmission properties are very different from the examples of Fig. 2(b). In this case,

*T*is a function of

*Lβ*(

*λ*), and Eq. (5) closely resembles the expression for the reflectance of a Fabry-Perot etalon [13] with finesse

*φ*intersect the reflection resonances. As

*φ*approaches 0 or 2

*π*, these bands become narrower, and the finesse increases according to Eq. (7). Fig. 2(c) shows the transmission as a function of

*λ*when

*φ*=7

*π*/4, for which the finesse given by Eq. (7) is

*𝓕*=38 and the fringe visibility is very close to 1. As

*φ*approaches

*π*, both the fringe visibility and the finesse decrease. These reversed Fabry-Perot transmission characteristics could be used in the design of a tunable notch rejection filter where the finesse and resonance position could be adjusted by varying

*β*and

*φ*.

*φ*is a function of wavelength, and so it would be difficult to design a device to exhibit the transmission spectra shown in Fig. 2(c). Any realistic measurement of transmission spectra would correspond to the transmission along a curved line in Fig. 2(a), the exact shape of such a curve being determined by the method used to induce

*φ*. This aspect is discussed further in Sections 3.1 and 3.2 below.

## 3. Photonic crystal recirculating MZI structures

4. E. A. Camargo, H. M. H. Chong, and R. M. De. La. Rue, “Photonic crystal Mach-Zehnder structures for thermo-optic switching,” Opt. Express **12**, 588–592 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-588. [CrossRef] [PubMed]

7. T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part II: Applications,” Phys. Rev. E (2004). To be published. [CrossRef]

5. M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, H. Yukawa, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Two-dimensional photonic crystal Mach-Zehnder interferometers,” Appl. Phys. Lett.84 (2004). [CrossRef]

14. R. Wilson, T. J. Karle, I. Moerman, and T. F. Krauss, “Efficient photonic crystal Y-junctions,” J. Opt. A **5**, S76–S80 (2003). [CrossRef]

### 3.1. Dielectric rod square lattice PC MZI

*λ*≤3.306

*d*. A waveguide formed by removing a single line of cylinders supports a single propagating mode of even symmetry in the wavelength range 2.249

*d*≤

*λ*≤3.205

*d*. Experimental results for similar dielectric rod PC waveguide structures have recently been reported [15

15. M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-*µ*m-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. **84**, 4298–4300 (2004). [CrossRef]

*λ*=1.55

*µ*m, that roughly defines the operating bandwidth of the MZI. We generate

*φ*by changing the radii of

*L*/

_{φ}*d*pairs of cylinders on either side of one waveguide from

*a*to

*a′*as indicated by the red cylinders in Fig. 2(d). This changes the modal propagation constant of that section of waveguide by Δ

*β*≪

*β*, introducing a total phase difference of

*φ*=Δ

*βL*. The change in radius is chosen to be small enough that there is no significant reflection produced by the change in the guide properties. We choose this approach for illustrative purposes only, as it is a convenient way of generating

_{φ}*φ*for our numerical formulation, which is based on a Bloch mode scattering matrix method [10

10. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part I: Theory,” Phys. Rev. E (2004). To be published. [CrossRef]

11. L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. Langtry, “Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics,” Opt. Express **12**, 1592–1604 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1592. [CrossRef] [PubMed]

*φ*could be varied by using, for instance, localized heating, as in Ref. [4

**12**, 588–592 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-588. [CrossRef] [PubMed]

*β*and hence Δ

*β*are functions of wavelength,

*φ*is also wavelength dependent in this example. Figure 4(a) shows the transmission of two MZIs with

*L*=8

_{φ}*d*and

*L*=16

_{φ}*d*respectively for

*a′*=0.2

*d*and

*L*=36

*d*. The solid curves are calculated using the full Bloch mode numerical calculation, and the dotted curves are the result of Eq. (5), calculated along the dashed-dotted curves in Fig. 2(b) to account for the wavelength dependence of

*φ*. As expected from Eq. (7), the resonances are considerably narrower for the

*L*=8

_{φ}*d*structure, since

*φ*≈

*L*Δ

_{φ}*β*. The parameters

*L*and

*L*used in Eq. (5) are not necessarily identical to those used to define the structure in Fig. 2(d). Since PCs behave as distributed reflectors, defining a structure length is somewhat ambiguous. We find that the best agreement in resonance position occurs when we take

_{φ}*L*in Eq. (5) to be the total length of the structure, including the junctions, as shown in Fig. 2(d), with a slight correction for the phase change introduced by reflection from the PC. Under these conditions, the agreement between the rigorous numerical result, and the semi-analytic model is excellent. In particular, the model predicts the resonance widths with very good accuracy, and with the adjusted parameter

*L*, the resonance positions are also closely matched.

### 3.2. Air hole triangular lattice PC MZI

*n*

_{cyl}=1) with period

*d*=350nm and radius

*a*=0.32

*d*in a dielectric of refractive index

*n*=3.4. An infinite lattice with these parameters exhibits a TE bandgap between 3.506

_{b}*d*≤

*λ*≤3.761

*d*. In contrast to the dielectric rod PC in the previous example,

*W*1 waveguides formed in triangular lattices of air holes typically support more than one mode at some frequencies in the bandgap. Since the oddmode resonance properties rely on the input and output guides being single-mode, the structure shown in Fig. 5(d) is designed to operate in the low-frequency region of the bandgap where a single mode is supported for 4.100

*d*≤

*λ*≤4.515

*d*. The mode has strong dispersion as it approaches cutoff near the low frequency end of this range which causes the MZI transmission properties to vary with wavelength more than the example in Section 3.1.

*L*=11

_{y}*d*, as shown in Fig. 3. As in Section 3.1, the radii of

*L*/

_{φ}*d*pairs of cylinders are varied in order to generate

*φ*, as indicated by the red cylinders in Fig. 5(d). Figure 5(a) is a contour plot of the transmission for an MZI of this type, calculated as a function of wavelength and

*φ*using Eq. 5. In comparing Eq. (5) with full numerical calculations of the realistic structure (Fig. 4(b)), we have found that the agreement is improved at longer wavelengths if

*β*is replaced with

*β*_ in the model. This slight modification makes sense since the resonances are driven by multiple reflections of the odd mode. In the example of Section 3.1,

*β*and

*β*_ were essentially equal, so such a modification makes little difference to the comparison in Fig. 4(a).

*φ*=

*π*/2. The spacing of the resonances decreases with increasing wavelength, while the resonances become narrower, maintaining a constant finesse.

*φ*, given by Eq. (6), is unchanged, as shown by the example in Fig. 5(c), where the transmission is plotted as a function of

*φ*for

*λ*=1.5595

*µ*m, indicated by the red line in Fig. 5(a).

*L*=3

_{φ}*d*and

*L*=5

_{φ}*d*respectively for

*a′*=0.30d and

*L*=37

*d*. The solid curves are the full numerical result, and the dotted curves are given by Eq. (5), calculated along the magenta (

*L*=3

*d*) and blue (

*L*=7

*d*) curves shown in Fig. 5(a), corresponding to the wavelength variation of

*φ*=Δ

*βL*. Once again we see that the semianalytic model predicts accurately the finesse of the resonances. The resonance positions line up very closely at the shorter wavelength end of the transmission region, however the agreement is poorer at longer wavelengths, again due to the strong modal dispersion close to cutoff.

## 4. Discussion and conclusion

*D*PCs with both high index inclusions in a low index background, and the inverted geometry. We note that alternative junction designs based on directional couplers [16

16. Y. Sugimoto, Y. Tanaka, N. Ikeda, T. Yang, H. Nakamura, K. Asakawa, K. Inoue, T. Maruyama, K. Miyashita, K. Ishida, and Y. Watanabe, “Design, fabrication, and characterization of coupling-strength-controlled directional coupler based on two-dimensional photonic-crystal slab waveguides,” Appl. Phys. Lett. **83**, 3236–3238 (2003). [CrossRef]

*D*PC, the out-of-plane losses must be considered and minimized to achieve a practical device. One way of achieving this would be to incorporate a 3

*D*PC layer above and below the waveguide layer, as proposed in [17

17. A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. **90**, 123,901 (2003). [CrossRef]

18. P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express **12**, 458–467 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-458. [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | A. N. Chester, S. Martellucci, and A. M. V. Scheggi, eds., |

2. | G. V. Treyz, “Silicon Mach-Zehnder waveguide interferometers operating at 1.3 |

3. | C. Rolland, R. S. Moore, F. Shepherd, and G. Hillier, “10 Gbit/s, 1.56 |

4. | E. A. Camargo, H. M. H. Chong, and R. M. De. La. Rue, “Photonic crystal Mach-Zehnder structures for thermo-optic switching,” Opt. Express |

5. | M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, H. Yukawa, S. J. Choi, J. D. O’Brien, and P. D. Dapkus, “Two-dimensional photonic crystal Mach-Zehnder interferometers,” Appl. Phys. Lett.84 (2004). [CrossRef] |

6. | A. Martinez, A. Griol, P. Sanchis, and J. Marti, “Mach-Zehnder interferometer employing coupled-resonator optical waveguides,” Opt. Lett. |

7. | T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part II: Applications,” Phys. Rev. E (2004). To be published. [CrossRef] |

8. | A. Lan, K. Kanamoto, T. Yang, A. Nishikawa, Y. Sugimoto, N. Ikeda, H. Nakamura, K. Asakawa, and H. Ishikawa, “Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,” Phys. Rev. B67 (2003). [CrossRef] |

9. | T. P. White, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Utracompact resonant filters in photonic crystals,” Opt. Lett. |

10. | L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part I: Theory,” Phys. Rev. E (2004). To be published. [CrossRef] |

11. | L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. Langtry, “Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics,” Opt. Express |

12. | G. P. Agrawal, |

13. | M. Born and E. Wolf, |

14. | R. Wilson, T. J. Karle, I. Moerman, and T. F. Krauss, “Efficient photonic crystal Y-junctions,” J. Opt. A |

15. | M. Tokushima, H. Yamada, and Y. Arakawa, “1.5- |

16. | Y. Sugimoto, Y. Tanaka, N. Ikeda, T. Yang, H. Nakamura, K. Asakawa, K. Inoue, T. Maruyama, K. Miyashita, K. Ishida, and Y. Watanabe, “Design, fabrication, and characterization of coupling-strength-controlled directional coupler based on two-dimensional photonic-crystal slab waveguides,” Appl. Phys. Lett. |

17. | A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. |

18. | P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(130.3120) Integrated optics : Integrated optics devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 2, 2004

Revised Manuscript: June 21, 2004

Published: June 28, 2004

**Citation**

Thomas White, C. de Sterke, R. McPhedran, T. Huang, and L. Botten, "Recirculation-enhanced switching in photonic crystal Mach-Zehnder interferometers," Opt. Express **12**, 3035-3045 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-13-3035

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### References

- A. N. Chester, S. Martellucci, and A. M. V. Scheggi, eds., Optical fiber sensors (Martinus Nijhoff, Dordrecht, 1987). [CrossRef]
- G. V. Treyz, ???Silicon Mach-Zehnder waveguide interferometers operating at 1.3 µm,??? Electron. Lett. 27, 118???120 (1991). [CrossRef]
- C. Rolland, R. S. Moore, F. Shepherd, and G. Hillier, ???10 Gbit/s, 1.56 µm Multiquantum well InP/InGaAsP Mach-Zehnder Optical Modulator,??? Electron. Lett. 29, 471???472 (1993). [CrossRef]
- E. A. Camargo, H. M. H. Chong, and R. M. De. La. Rue, ???Photonic crystal Mach-Zehnder structures for thermo-optic switching,??? Opt. Express 12, 588???592 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-588">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-588</a>. [CrossRef] [PubMed]
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