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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 13 — Jun. 28, 2004
  • pp: 3035–3045
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Recirculation-enhanced switching in photonic crystal Mach-Zehnder interferometers

T. P. White, C. Martijn de Sterke, R. C. McPhedran, T. Huang, and L. C. Botten  »View Author Affiliations


Optics Express, Vol. 12, Issue 13, pp. 3035-3045 (2004)
http://dx.doi.org/10.1364/OPEX.12.003035


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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

Mach-Zehnder interferometers (MZIs) are used extensively in optical systems as filtering and switching devices and in phase measurement and detection applications. Their wide range of uses and functionality has led to the fabrication of MZI devices in fibers [1

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

], planar dielectric waveguides [2

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

], rib waveguides [3

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

], and more recently two-dimensional photonic crystal (PC) waveguides [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]

, 5

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

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]

]. Although the morphology of all MZIs is the same regardless of type of waveguide in which they are formed, the operational characteristics can be strongly influenced by the waveguide and cladding properties.

In Section 3 we propose two specific examples of recirculating MZIs using 2D 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]

] that has the required low-reflectance properties. In this case, the perfectly reflecting cladding is provided by the photonic bandgap of the PC waveguide walls, which reflect light at all frequencies within the bandgap, regardless of incident angle. While it has been proposed that the strong reflections and associated interference observed in many PC based devices may limit their practical use in optical systems [8

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]

], these effects can also be used to great advantage. We recently proposed a novel PC-based filter that uses the strong reflection of waveguide modes to produce high-Q resonant transmission features [9

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]

], and a similar approach was used to design the wide bandwidth Y-junctions [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]

] used here.

The PC MZI designs that we investigate in Section 3 are analyzed using rigorous 2D 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

We consider first the simple MZI illustrated in Fig. 1(a), consisting of single input and output waveguides, and two waveguide junctions 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]

], or in passive devices, the output can be used to determine φ [1

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

].

Fig. 1. (a) Schematic of a simple MZI with single input and output waveguides joined by arms A 1 and A 2 via Y-junctions Y 1 and Y 2. (b) Schematic of the propagating modes in each of the waveguide sections in (a). The green and red curves show the even and odd superpositions of the individual waveguide modes.

|ψ±=12(|ψ1±|ψ2).
(1)

In an ideal structure, 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.

Since mode |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.

We now follow the propagation of a mode Ψ with unit energy 〈Ψ|Ψ〉 entering an unbalanced MZI from the input guide and consider the effect of odd mode reflection at Y 2. After passing through Y 1, the resulting field

|Ψ(0)(1)=|ψ+

can be written in terms of |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

Ψ(L)(1)=(eiβLψ1+ei(βL+φ)ψ2)/2=eiχ(cos(φ2)ψ+isin(φ2)ψ),
(2)

where 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),

T=eiχcos(φ2)2=cos2(φ2)=11+tan2(φ2).
(3)

In the PC structure, however, the reflected odd field component, -iexp() 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

Ψ(0)(2)=ieiχsin(φ2)(eiβL|ψ1ei(βL+φ)ψ2)2
=ie2iχsin(φ2)(cos(φ2)ψisin(φ2)ψ+).
(4)

Once again the odd field component is reflected back into the interferometer, while the even component is transmitted through the junction, in this case back into the input waveguide.

Repeating the above calculation for each pass of the odd mode through the interferometer, and summing the amplitudes of the fields transmitted into the output waveguide as a geometric series, we derive an expression for the overall transmitted intensity

T=4sin2(χ)cos2(φ2)sin4(φ2)1+4sin2(χ)cos2(φ2)sin4(φ2).
(5)

The reflected intensity (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 = (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

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]

] in the limit when we restrict the treatment to only include propagating modes and prescribe appropriate values for the various scattering matrices that accord with the above analysis.

Fig. 2. (a) Transmission contour plot as a function of λ and φ, calculated with Eq. (5) for a PC-based recirculating MZI with the geometry shown in (d), and parameters described in the text. The red and green lines correspond to the transmission plots in (b) and (c) at λ=1.5496µm and φ=7π/4, respectively. The dashed curve in (b) shows the transmission of a conventional MZI as a function of φ, given by Eq. (3). (d) Generic recirculating PC MZI formed in a square lattice of rods with the parameters given in the text. The red cylinders, with radius a′, are used to generate φ in Section 3.1. The magenta and blue dashed-dotted curves in (a) show φ as a function of λ when a′=0.20d for Lφ=8d and Lφ=16d respectively. (Video file showing transmission as a function of λ and φ 1.9Mb)

Any calculation involving Eq. (5) requires numerical input of the wavelength variation of 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.

Figures 2(a)–(c) show the TM transmission of a recirculating MZI of length L=36d=19.9µm, 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=0.18d, and refractive index n cyl=3.4, in a background of air (nb=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 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 2π. The slight tilt of the valleys from vertical is determined by the dispersion slope of β, which appears in χ in Eq. (5).

We first consider the properties of Eq. (5) as 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=2π. 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=1.5496µm, 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/2,3π/2,5π/2 …, for which

T=4cot4(φ2)1+4cot4(φ2)=11+tan4(φ2)4.
(6)

Note that although the specific waveguide geometry determines the wavelengths at which this condition occurs, when it is satisfied, the phase response is entirely independent of the waveguide properties.

Comparing Eqs. (3) and (6), we see that the recirculating MZI response is quartic in tan(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 0.32π, whereas a conventional MZI would require a change of 0.59π. Although cos4(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]

], we find that at least 6 identical conventional MZIs would be required to obtain a superior switching performance to that of a single recirculating MZI. Note, also, that the transmission given by Eq. (6) has a fourth-order variation near both 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.

If the wavelength is varied while φ is fixed, the transmission properties are very different from the examples of Fig. 2(b). In this case, T is a function of (λ), and Eq. (5) closely resembles the expression for the reflectance of a Fabry-Perot etalon [13

13. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

] with finesse

𝓕=πcos(φ2)1cos(φ2).
(7)

Thus, the transmission behavior of the recirculating MZI is essentially the reverse of a Fabry-Perot etalon, exhibiting resonant reflection, rather than resonant transmission. This is seen in Fig. 2(a), where horizontal lines of constant φ intersect the reflection resonances. As φ approaches 0 or 2π, 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=7π/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.

In practice, 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

Fig. 3. (a) Y-junction used in the PC MZI design of Section 3.2. (b) Transmission through the junction as a function of wavelength. The 98% transmission bandwidth is approximately 36nm.

3.1. Dielectric rod square lattice PC MZI

The MZI design shown in Fig. 2(d) is formed in the square lattice of dielectric cylinders with the parameters described in Section 2. This PC lattice exhibits a bandgap for TM polarized light in the wavelength range 2.249d≤λ≤3.306d. A waveguide formed by removing a single line of cylinders supports a single propagating mode of even symmetry in the wavelength range 2.249dλ≤3.205d. Experimental results for similar dielectric rod PC waveguide structures have recently been reported [15

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

].

Fig. 4. Comparison of the transmission of the PC MZIs calculated with the rigorous numerical calculations (solid curves) and the semianalytic results of Eq. (5) (dashed curves). (a) TM transmission of the PC MZI shown in Fig. 2(d) with L=36d, Ly=5d, a′=0.20d and Lφ=8d (magenta) and Lφ=16d (blue). (b) TE transmission of the PC MZI shown in Fig. 5(d) with L=37d, Ly=11d, a′=0.30d and Lφ=3d (magenta) and Lφ=7d (blue).

Since β and hence Δβ are functions of wavelength, φ is also wavelength dependent in this example. Figure 4(a) shows the transmission of two MZIs with Lφ=8d and Lφ=16d respectively for a′=0.2d and L=36d. 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 Lφ=8d structure, since φLφΔβ. The parameters L and Lφ 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 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

Fig. 5. (a) Transmission contour plot calculated with Eq. (5) for a PC MZI with the geometry shown in (d), and parameters described in the text. The red and green lines correspond to the transmission plots in (b) and (c) at λ=1.5595µm and φ=π/4, respectively. The dashed curve in (b) is the response of a conventional MZI, as given by Eq. (3). (d) Generic PC MZI formed in a triangular lattice of air holes with the parameters described in the text. The red cylinders, with radius a′ are used to generate φ. The magenta and blue dashed-dotted curves in (a) show φ as a function of λ when a′=0.30d for Lφ=3d and Lφ=7d respectively. (Video file showing transmission as a function of λ and φ 1.9Mb)

Figure 5(d) shows the MZI design in the air hole triangular lattice. The optimum coupled Y-junction in this PC has a length Ly=11d, as shown in Fig. 3. As in Section 3.1, the radii of Lφ/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).

There are some clear differences between the contour plots in Figs. 5(a) and 2(a). First, the low transmission valleys run in the opposite diagonal direction because of the different dispersion slopes of the waveguide modes in the two PC structures. Second, in Figs. 5(a) the tilt and the separation of the resonances changes significantly at longer wavelengths, whereas in Fig. 2(a), the valleys remain almost parallel and equally spaced. This is also a result of the different dispersion curves. The modes of the MZI in Fig. 2(d) are well away from cutoff, so the dispersion slope is relatively constant, resulting in the regular appearance of Fig. 2(a). In contrast, the modes of the MZI in Fig. 5(d) are closer to cutoff and the resulting strong dispersion distorts the transmission profile. This is also seen in Fig. 5(c) where the transmission is plotted as a function of wavelength for fixed φ=π/2. The spacing of the resonances decreases with increasing wavelength, while the resonances become narrower, maintaining a constant finesse.

Despite the differences in the transmission properties of the two examples here, the quartic response with 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=1.5595µm, indicated by the red line in Fig. 5(a).

4. Discussion and conclusion

Examples of specific device designs exhibiting these characteristics have been presented for 2D 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]

] with two input and output ports would not display these properties, since modes of both odd and even symmetry can be transmitted through such junctions. Although we have not explicitly studied the radiation losses of such devices, as with any cavity within a 2D 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 3D 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]

]. Alternatively, radiation losses could be reduced by tuning the cavities to minimize coupling into radiation modes above and below the waveguides [18

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]

].

The semi-analytic model derived in Section 2 gives insight into the physics of MZIs and provides a powerful design tool for rapidly predicting the properties of relatively complicated devices knowing only the dispersion properties of a single waveguide. An example of this was given in Section 2 where we derived the requirements for a recirculating MZI with a transmission response that has a quartic phase dependence, and a correspondingly steep response curve linking a flat high transmission region to a flat low transmission region. A device with such a response could provide enhanced switching performance in both linear and nonlinear applications.

Acknowledgments

This work was produced with the assistance of the Australian Research Council under its ARC Centres of Excellence Program.

References and links

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 1.3µm,” Electron. Lett. 27, 118–120 (1991). [CrossRef]

3.

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

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]

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]

12.

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

13.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

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]

15.

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

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]

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]

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

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