1. Introduction

Over recent years, there has been a great deal of interest in photonic crystal structures [1]. A photonic crystal is a periodic, usually dielectric, structure that exhibits transmission gaps for electromagnetic (EM) radiation within certain frequency and wavevector bands due to interference between multiple Bragg reflections. These structures possess unique EM band structures related to the specific structure geometry and refractive index contrast. The transmission properties of photonic crystals have been compared to the electronic bands of solid-state materials. However, there is a critical difference between the two cases: a scalar field approximation is typically adequate for solid-state calculations, whereas a full vectorial approach is fundamentally important for photonic crystals.

In principle, a photonic band gap can exist in any structure with periodicity in one or more dimensions; however, due in part to the difficulty in micromachining complex three-dimensional (3D) photonic crystal structures appropriate for optical wavelengths, much of the work to date has focused on two-dimensional (2D) periodic dielectric structures. Although many striking features of 3D structures have been predicted [2], there is also a great wealth of interesting physical phenomena accessible in 2D photonic crystals. Among the possible applications for 2D structures are novel waveguides [3] and photonic crystal circuit elements [4].

One type of 2D photonic crystal which has received significant attention is the photonic crystal slab [5]. These structures consist of a 2D periodic plane and an extruded or nonperiodic dimension on the order of the lattice constant. They typically possess a band gap for radiation with wavevector and polarization in the plane of periodicity and have several potential applications such as the possibility of guiding light through designed defects or channels [6]. Theoretical calculations of the band structure in photonic crystal slabs [7, 8], and experimental transmission studies at optical wavelengths have been performed [5, 9].

On the other hand, less attention has been paid to bulk 2D photonic crystals. A bulk crystal (as opposed to a slab) is one in which the structure is much thicker or longer in the nonperiodic dimension than the lattice constant. These bulk structures exhibit a well-defined band gap for EM waves with polarizations both parallel and perpendicular to the plane of periodicity (transverse magnetic (TM) and transverse electric (TE) waves, respectively), although the band gap center frequency, depth and spectral width depend on polarization [10, 11].

In addition to their unusual power transmission properties, photonic crystals also impart nontrivial dispersive phases to transmitted waves. Phase effects are known to exist in one-dimensional (1D)photonic crystals such as optical thin film stacks or dielectric mirrors [12]. Furthermore, it has been shown that the dispersion relation for waves propagating in the plane of symmetry of a 2D photonic crystal exhibits polarization-dependent properties within the band gap [13]. Since the transmission profile of the band gap depends on polarization, this result is not surprising. Somewhat more unexpected is that important phase effects also exist for frequencies away from the band gap, in the transparent regions. Similar phase effects occur in the transparent regions of absorptive systems; the essential element for these effects is merely a system with a frequency-dependent transmission profile [14, 15]. Thus, the band gaps of a photonic crystal should generate similar dispersive effects, even in transparent spectral regions. Furthermore, since bulk 2D photonic crystals have polarization-dependent band gap characteristics, their transparent region phase delay properties should be birefringent.

It is also possible to understand this birefringence through the cumulative effect of Bragg reflections from multiple curved interfaces. Since the dielectric-air interfaces are not flat, the Fresnel coefficients should depend on polarization even at normal incidence, thus producing birefringence.

Studies of birefringence in photonic crystals have been performed in photonic band gap waveguides [16, 17, 18], photonic crystal lasing cavities [19], and nonlinear fiber Bragg gratings [20, 21]. The polarization properties of 2D photonic crystals have been theoretically discussed [22], in a paper suggesting a birefringent behavior characteristic of a waveplate. Recently, we have designed, constructed and experimentally tested a half waveplate using the dispersive birefringent properties of a bulk 2D photonic crystal away from its band gap [23]. In this paper, we present extensive experimental measurements of the birefringence above and below the fundamental band gap for different sets of bulk 2D hexagonal photonic crystals using microwave radiation. We also construct and experimentally demonstrate quarter waveplates using 2D bulk photonic crystals.

2. Experimental setup

Although bulk photonic crystals are difficult to construct at the scales appropriate for optical wavelengths, microwave-scale photonic crystals can be relatively easily fabricated by stacking and gluing lattices of acrylic pipes [10]. Structures of different air-filling fractions (AFF) can be produced by changing the ratio of the inner to outer diameters of the rods. We used this technique to construct bulk hexagonal crystals for this work with component rods of outer diameter 1/2 inch and inner diameters of 1/4 inch (AFF = 0.32) and 3/8 inch (AFF = 0.60).

Since Maxwell’s equations are scale-invariant, results obtained in the microwave spectral region can be applied directly to structures designed for any other wavelengths (e.g., optical and infrared radiation). Furthermore, without any scaling, these results could be relevant for electron beam devices at microwave and millimeter wavelengths such as traveling-wave tubes and backward-wave oscillators.

The methods we employed to make transmission measurements have also been described in a previous paper [10]. In summary, an HP 8720A vector network analyzer (VNA) along with polarization-sensitive receiver and transmitter horn antennae were used to make power transmission and phase measurements. The crystal was placed 1.6 m away from the transmitter horn inside a microwave-shielded box with an aperture with dimensions 14 cm × 17 cm. These distances were chosen so that the microwave wavefronts had a planar spatial profile with a well-defined linear polarization at the location of the crystal. The crystal was oriented so that the microwaves were incident in the ΓM direction [11]. The VNA was calibrated with the horn antennae separated as described and the crystal removed from the path; in this way, it was possible to subsequently measure the phase delay through the crystal without explicitly subtracting the effects of free-space propagation.

 

Fig. 1. Experimentally measured TM (green line) and TE (blue line) indices of refraction for hexagonal crystals with 16 layers and AFFs 0.60 (a) and 0.32 (b). The curves are shown on both sides of the fundamental band gap (shaded areas).

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In a previous work, the center frequency, width, and depth of the fundamental band gap for light incident in the ΓM direction on a hexagonal structure with AFF 0.60 were reported [10]. The transmission of radiation in the band gap was found to decrease exponentially with propagation distance, as expected [5]. The specific characteristics of the band gap were shown to depend on the polarization of the radiation; the TE gap is shallower, narrower and centered at approximately 10.5 GHz, whereas the TM gap is deeper, wider and centered at roughly 11 GHz. The structure with an AFF of 0.32 has similar band gap properties, although its fundamental gap is significantly shallower and narrower for both polarizations, and shifted roughly 1 GHz lower in frequency.

3. Results and discussion

We measured the phase delays of TE and TM waves transmitted through both the 0.60 and 0.32 AFF crystals in spectral regions above and below the fundamental band gap. Using this phase information, the frequency-dependent index of refraction n(ω) was calculated for each polarization using the relationship

where ω is the angular frequency, c is the speed of light, Δϕ is the calibrated phase delay in radians, and L is the path length through the crystal (dependent on the number of layers). For a hexagonal structure, the path length L is given by the geometric formula

where N is the number of layers in the crystal, and r is the outer radius of a component rod.

Typical results for the indices of refraction are shown for 16-layer crystals of AFF 0.60 and 0.32 in Fig. 1 (a) and Fig. 1 (b), respectively. The data for crystals with different numbers of layers and the same filling fraction are similar. From these curves, it is clear that the index is substantially larger for TE waves over the plotted spectral regions. In addition, the curves are typically flatter (i.e., less dispersive) below the shaded band gap region, and steeper above it. Typically, the TE curve is also steeper relative to the TM curve, especially for the crystals with a larger AFF. For the 0.60 AFF crystal, the TM curve does not resume the same baseline above the band gap; rather, it remains fairly flat, but is lower by roughly 3%.

 

Fig. 2. Experimentally measured differences between the indices of refraction (purple solid lines) for 4- (a), 8- (b), and 16-layer (c) crystals with AFF 0.60. Data are shown on both sides of the fundamental TM and TE band gap (shaded areas). First (red dashed lines), second (green dotted lines), third (blue short dashed line), and fourth (cyan short dotted line) order quarter wave condition (QWC) curves are also displayed.

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A further observation about the measured indices of refraction begins by noting that one can compute an effective index of refraction n̄ of a composite structure using a weighted average based on the filling fractions of its components. This value is the index one would expect for a structure of randomly distributed material with that composition. Performing this simple calculation for our crystals, we find that n̄ = 1.41 for an AFF of 0.32 and n̄ = 1.24 for an AFF of 0.60 (the published value for the index of acrylic at these wavelengths is 1.61 [24]). In comparison, if we average the low frequency limits of the experimental TE and TM indices plotted in Fig. 1, we find n_av = 1.42 and n_av = 1.27 for the AFFs 0.32 and 0.60, respectively, in reasonably good agreement with the results of the simple weighted average. Evidently, the geometry of the hexagonal crystal defines an optic axis (parallel to the rods), and splits the indices of refraction for polarizations parallel and perpendicular to this axis. In the low frequency limit, the indices are separated on opposite sides of the effective average index of a structure with randomly distributed material of the same overall filling fraction.

The large birefringence of this structure suggests that photonic crystals may be useful for the construction of compact waveplates. If the difference between the indices of refraction at some frequency is such that the relative phase shift of TM and TE waves is $\frac{\pi }{2}\left(2m+1\right)$ over the optical path length, the crystal will act as an m ^th-order quarter waveplate. We express this condition as

where m can take any integral value greater than or equal to 0.

 

Fig. 3. Experimentally measured differences in the indices of refraction (solid lines) for 4- (a), 8- (b), and 16-layer (c) crystals with AFF 0.32. Data are shown on both sides of the fundamental TM and TE band gap (shaded areas). First (red dashed lines), second (green dotted line), and third (blue short dashed line) order quarter waveplate condition (QWC) curves are also displayed.

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Using our experimental data for the TM and TE indices of refraction, we have calculated their differences for crystals with AFF 0.60 over spectral regions away from the band gaps. These data are plotted for 4-, 8-, and 16-layer crystals in Figs. 2 (a), 2 (b), and 2 (c) along with calculated quarter wave conditions (QWCs) of Eq. 3. It is worth noting that significant birefringence occurs even in the 4-layer case; this is because the crystal is so strongly anisotropic. The intersection of a QWC curve with an experimental index difference predicts the existence of a quarter waveplate at that frequency. Quarter waveplate intersections occur for the 4-, 8-, and 16-layer 0.60 AFF crystal in regions outside the band gap at 15 GHz, 17.5 GHz, and 16.5 GHz, respectively (see Fig. 2). We also performed the same procedure on experimental data for the 0.32 AFF crystals to determine their birefringence. The results along with the relevant quarter waveplate conditions are plotted in Fig. 3.

In order to verify the operation of our photonic crystal as a quarter waveplate at certain frequencies, we examined its effect on linearly polarized incident radiation. Our antennae are highly linearly polarized for emission and detection; when crossed at 90° the detected power is suppressed by ≥ 35 dB relative to the parallel configuration. We aligned our photonic crystal optic axis at 45° to the emitter horn and measured the transmission as a function of frequency for four different receiver horn orientations. A true quarter waveplate will convert the incident linear polarization to pure circular, which should then give the same power transmission for all receiver horn angles. We performed this measurement for the 4-, 8-, and 16-layer crystals of AFF 0.60, with the results shown in Fig. 4.

 

Fig. 4. Experimentally measured power transmission vs. frequency with the receiver horn oriented at 0°, 45°, 90°, and 135° from the transmitter horn; for 4- (a), 8-(b), and 16-layer (c) crystals with AFF 0.60. At 0°, the receiver horn antenna is perpendicular to the optic axis of the crystal.

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In Fig. 4 (a), all 4 traces have nearly the same value near 15 GHz, indicating that this crystal behaves as a quarter waveplate at this frequency, in agreement with the predictions from the phase data. Similarly, Fig. 4 (b) displays equal transmission near 17.5 GHz, also in agreement with the phase data. Finally, Fig. 4 (c) shows the predicted quarter waveplate behavior in the neighborhood of 16.5 GHz, but does not have the second expected equal transmission point near 18 GHz. This is caused by the close proximity of a higher-order (double frequency) band gap, which attenuates the radiation of one polarization but not the other.

To further verify the quarter waveplate operation of our photonic crystal, we fixed the position of the crystal and rotated the transmitting and receiving horns around the axis of propagation, keeping their relative angle constant (this is equivalent to fixing the horns and rotating the crystal). For both parallel and perpendicular relative horn configurations, the field (amplitude) transmission should vary sinusoidally with the rotation angle, with a period of $\frac{\pi }{2}$ . Labeling θ = 0 as the angle at which the transmitting horn is perpendicular to the optic axis of the crystal, the transmission for the two horn configurations should behave as

 

Fig. 5. Experimentally measured amplitude transmission at 17.4 GHz vs. horn rotation angle, for an 8-layer crystal with AFF 0.60. The horns are rotated together with a fixed angle between them of 0 (closed green squares) and 90° (open blue circles). Also plotted are the functions $\frac{1}{\sqrt{2}}\sin 2\theta$ and $\sqrt{{\cos }^4\theta +{\sin }^4\theta }$ .

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where |E‖/E ₀| and |E⊥/E ₀| are the field transmission amplitude ratios for parallel and perpendicular horn configurations, respectively. As shown in Fig. 5, the data are in good agreement with these expressions for both horn configurations (no fitting parameters were used). This demonstrates that the equal power transmission values measured in Fig. 4 cannot be caused by depolarization of the incident light, but indeed to quarter waveplate behavior of the photonic crystal.

4. Conclusions

We have measured substantial birefringence in the transparent region of 2D photonic crystals with different number of layers and air-filling fractions. Based on these measurements, we designed and demonstrated the operation of photonic crystal quarter waveplates. We believe the large birefringence characteristic of photonic crystals has important potential for application. Extremely compact modular optical elements such as polarization discriminators, polarizing beamsplitters and optical diodes may be possible with photonic crystal birefringence. Photonic crystals could also lead to much more compact waveplates than are currently available in the optical regime. Finally, these systems should be compatible with nonlinear optical systems based on selectively-defective periodic dielectric structures, also known as photonic crystal circuits, which have been proposed as candidates for all-optical switches [4].