PHOTONIC CRYSTALS

COUPLING AND PHASE ANALYSIS OF CAVITY STRUCTURES IN PHOTONIC CRYSTALS

  In this study, the coupling properties of cavities in a 2D dielectric photonic crystal are investigated and a Mach-Zehnder type interferometer based on   coupled cavity waveguides in a photonic crystal is developed.

  The photonic crystal is a hexagonal array of cylindrical alumina rods with radius r = 1.575 mm and with a dielectric constant of ε = 3.13 at microwave   frequencies. Lattice constant, a, is 7.0 mm. The transmission and phase spectra are measured using a network analyzer and a set of horn antennas in TM   polarization (electric field, E, perpendicular to the plane of 2D photonic crystal). The interfaces of the photonic crystal are normal to the ΓK direction.   Numerical simulations based on finite-difference-time-domain (FTDT) method are used to compare with experimental data.

  Figure 1 shows the confinement effects on the cavity mode in this photonic crystal which has a band gap extending from 12.8 GHz to 18.7 GHz. The   crystal is sufficiently large along the lateral directions so that the confinement is mainly characterized by the number of layers surrounding the cavity along   the longitudinal direction, which is the propagation direction of the probing signal. The cavity mode frequency located at f = 16.898 GHz appears to be   independent of the confinement strength. This agrees well with the transfer matrix results where no significant dependence of resonant frequency on   defect size is reported.

FIG. 1. Measured (solid lines) and simulated (dashed lines) transmission spectrum of a single cavity for 2 (grey) and 3 (black) number of confining layers. Top right: Schematics of the respective structures (Full lateral width is not visible).

If two or more cavities are present, the eigenmode of the single cavity splits into coupled modes. This phenomenon is investigated experimentally and described within the classical wave analog of the Tight-Binding (TB) approximation for photonic crystal structures. In this formulation, the eigenfrequencies of two coupled cavities are given by

where α1 and β1 are the first order coupling parameters. By measuring the split mode   frequencies ω1,2, and the single cavity frequency Ω,, the coupling constant  can be obtained. Figure 2 displays the coupling constant obtained from   experiments and simulations as a function of the distance between the two cavities. Since the coupling constant is the measure of the overlap of localized   cavity fields, it reflects the exponential decay of these fields, as expected.

  FIG. 2. Measured (solid circles) and simulated (open squares) Tight-Binding coupling constant as a function of   the distance between the cavities. Dashed line denotes the exponential fit.

  We now discuss how the phase of the electromagnetic field advances through single and multiple cavity   structures. A direct measurement of the phase shift through CCWs provides a clear and solid basis for   interpreting the operation of CCW based interferometer structures.

_The phase measurements are performed with an HP 8510C network analyzer. The phase of the transmitted _signal (S12 in the S-parameter   convention), is measured between [-π, +π],  as a function of frequency. This _raw data is then “unwrapped” by adding 2π at the +/-π jumps, to obtain the   phase spectra. Since the absolute phase is meaningless, the phase is _measured with respect to a calibration. We first perform the calibration in air, by   removing the photonic crystal between the antennas, and then measure _the relative phase shift induced by structure, including the cavity.

Figure 3 shows the measured phase change through single cavity, two coupled, and three coupled cavities, respectively, along with the corresponding transmission spectra. It is evident that the net phase shift through a cavity, and through each of the coupled cavities is equal to π.

The phase shift has its origin in that the cavity in a photonic crystal resembles a Fabry Perot cavity. The transmission maximum occurs at the frequency corresponding to a standing wave within the cavity. So when the frequency is changed across the resonance, the induced phase shift becomes π. When many cavities are coupled to form a CCW, each cavity acts as a Fabry-Perot resonator, and the phase difference between the neighboring maxima in the guiding band should be π. As a result, the total phase shift across the waveguide should become Nπ, N being the number of cavities.

Mach-Zehnder interferometer based on photonic crystal CCW structures

The relation between the acquired phase shift and the number of cavities in a CCW can be used to describe the operation of a CCW based Mach-Zehnder interferometer in a lucid way. For this purpose, we have constructed various MZI structures consisting of two CCW branches connected via Y-junctions to input   and output CCW ports. The input and output ports are along the ГK direction of the PC, each having 4 cavities. In the following, we denote the MZI   structures by the notation (m ´ n) according to the number of cavities only in the branches (i.e., the cavities in the input and output ports are not included   in the notation). _For instance, 7´5 denotes a MZI with 7 cavities on one branch and 5 on the other.

Figure 4 compares the transmission spectrum of a 7´7 MZI (left) to that of a 7´5 MZI (right). The 7´7 MZI is a symmetric structure with two equal CCW branches. Thus, the phase acquired upon propagating either branch is the same, leading to zero phase difference between the branches. In contrast, the transmission spectra of a 7´5 MZI exhibits a distinct dip at f = 16.88 GHz of about -60 dB as shown in Fig. 5. In this asymmetric structure the phase difference of the CCW branches, , advances from zero to 2π across the transmission band (since the number of cavities in the branches differ by 2), it takes the value π at a certain frequency, for which the beams in two branches interfere destructively.  This is confirmed by the simulation of these structures. Moreover, the computed field mode at f = 16.88 GHz, shows clearly that it interferes destructively at the output Y-junction of the 7´5 MZI, but propagates   in the 7´7 MZI.

  When the difference of the number of cavities between the branches is increased to four, two interference dips appear for  and  as advances from 0 to 4   π across the guiding band.  

 FIG.4. (a) Measured (solid lines) and simulated (dashed lines) transmission spectra of 7´7 and   7´5 MZI structures, depicted in the middle schematics. (b) The computed electric field profile   at f = 16.88 GHz for 7´7 and 7´5 MZI respectively. The bottom arrow indicates the input   port.

_FIG. 5. Phase (solid black) and transmission (solid gray) spectra of 7´5 (left) and 8´4 (right) MZI structures (see middle schematics). The lower figures   show the transfer function of Ref. 21. Horizontal dashed lines denote the odd-π multiples of the phase difference, whereas vertical dotted lines denote _the zeros of the transfer function.

  We’ve also included transfer function calculations according to a simple transfer model.[2] There, the transfer function of the MZI with input output   directional couplers is given by

  which depends only on the eigenfrequency of an isolated cavity f0, coupling constant κ, and the difference in the number of cavities ΔN. According to   this formulation, the transfer function exhibits a dip for ΔN = 2 and two dips for ΔN = 4 within the guiding band. We observe that both cases are in _good   agreement with the measured transmission dips and the odd-π phase difference positions.

     References:

  K. Guven and E. Ozbay, Phys. Rev. B 71, 085108 (2005).

  A. Martinez, A. Garcia, P. Sanchis, and J. Mari, J. Opt. Soc. Am. A 20, 147 (2003).

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