1. Introduction

The fundamental phenomenon of optical refraction describes, when a light beam is transmitted through an interface between two media having different refractive indices, its propagation path is altered depending on the difference in the directional properties of the optical refractive index. For all known naturally occurring media, the refractive index assumes only positive values and as a result the incident beam and the refracted beam are on the opposite sides of the normal to the interface. In the last ten years or so, negative index materials (NIM) have attracted great interest as well as heated debates [1–13]. Some of the most striking consequences of the negative index of refraction include the reversal of the direction of the phase velocity with respect to that of the group velocity, the reversal of the Casimir force, the negative refraction, and sub-diffraction-limit imaging, i.e. the perfect lens [1–7]. A complete understanding of the properties of negative index structures requires the re-evaluation of some well-known principles of the electromagnetic wave theory. Application of the usual positive media formula to the analysis of negative index structures may sometimes lead to erroneous conclusions. However, it is clear that fundamental to all NIM is that, as the energy propagates forward, the accumulated optical phase along the light path decreases rather than increases as it does in normal positive index materials (PIM). Recognizing this decreasing optical phase along the energy propagation direction, one can show from the Huygens-Fresnel principle that the refracted beam bends toward the same side of the normal as the incident beam, resulting in the negative refraction. It should be noted that not all negative refraction requires NIM. For example, uniaxial optical crystals can be arranged in special orientations so that negative refraction occurs naturally [14]. In anisotropic media, the direction of energy flow is in general different from that of phase propagation except for the directions that are parallel to the crystal axes, although they are naturally not anti-parallel. An often misunderstood point about NIM negative refraction is that Snells law of refraction in general applies to the directions of phase propagation, not the directions of energy velocity. Only in isotropic media where the direction of energy velocity is in parallel (e.g. PIM case) or anti-parallel (e.g. NIM case) to the direction of phase propagation, one can use the direction of energy velocity in Snells law of refraction to infer the index of refraction. In those cases involving birefringent crystals, for example, Snells law of refraction in general does not apply to the directions of light energy velocity and the negative refraction resulted from the birefringence should not be used to infer the negative index of refraction [14]. On the other hand, however, the anomalous spectral phase behavior, as first predicted theoretically by Mandatori, et. al. [15, 16], does exhibit an effective negative index. By effective negative index here we note that, at the center frequency where the anomalous spectral phase occurs, increasing the thickness of the structure within some limits will result in a decrease of the accumulated optical phase. The simplest Mandatori structure involves only a single layer of birefringent material and can be reduced to a birefringent filter model, where a piece of anisotropic medium is sandwiched between two parallel polarizers as shown in Fig. 1. The realization that ordinary birefringent filters can provide a negative group index as well as effective negative index of refraction is somewhat surprising. However, as we will show later, naturally occurring birefringent media usually suffer a low index contrast Δn/n, which renders the predicted anomalous phase behavior barely noticeable or useful.

 

Fig. 1. A single-layer Mandatori structure can be modeled as a birefringent filter with parallel polarizers.

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In this study, we have achieved large birefringence by using form birefringent structures, making compact Mandatori structures viable. High index contrast Δn/n of 0.7 or higher can be achieved with surface-treated form birefringent discs. With a reduced model of the single-layer Mandatori structure, we derived and experimentally verified the negative group index and its dependence on the degree of birefringence, the structure thickness and the orientation of the birefringent layer. We also identified the relationship between design parameters, such as index contrast, birefringent layer thickness and azimuth angle, and performance parameters, such as the center operating frequency, bandwidth, and the transmission throughput. In addition, we also experimentally verified the predicted anomalous phase behavior in one-dimensional, multilayer Mandatori form birefringent structures.

2. Model of single-layer Mandatori structure with large birefringence

The simplest Mandatori structure involves only a single birefringent layer of material and reduces to a simple birefringent filter as shown in Fig. 1. Such filters made from natural birefringent materials do not exhibit noticeable or useful anomalous phase behavior. Consequently, they do not give rise to a useful negative group index and effective negative index of refraction. When birefringent metamaterials with large index contrast Δn/n are used, free space or waveguide devices with specified effective negative index and negative group velocity in the specified frequency range can be designed with reasonable throughputs. Indeed, such filters are capable of tunable negative group delays and effective negative indices at specific frequencies as shown in Fig. 2. While the accumulated phase of the transmitted field in a normal PIM always increases with the propagation distance, these structures can exhibit an effective negative index behavior [15]. As shown in Fig. 2(a), within the birefringent medium thickness range of d ₁ and d ₂, the accumulated optical phase of the transmitted field decreases rather than increases with the thickness of the material that the light is passing through. This anomalous phase behavior also exhibits itself in the spectral domain phase for a given thickness d. PIMs always exhibit a group delay (the negative spectral phase slope in Fig. 2(b)). However, for light in the frequency range of ω ₁ and ω ₂, these structures can exhibit a negative group delay (the positive spectral phase slope in Fig. 2(b)).

 

Fig. 2. (a) Accumulated optical phase v.s. thickness d that light passes through. The decreasing optical phase with thickness infers effective negative index of refraction. (b) Spectral phase v.s. optical frequency after light passes through a fixed thickness of the medium. Conventionally the negative slope of the spectral phase corresponds to a group delay. The anomalous spectral phase behavior exhibits a positive slope, implying a negative group delay.

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The relationship between the device performance parameters such as the maximum negative group delay and design parameters such as the index contrast Δn/n can be simplified in the case of single-layer structure. Given the birefringent index of refraction n_x, n_y (n_x > n_y), the thickness d of the birefringent layer and the field splitting ratio α = |E_x|/|E_y|, which is determined by the azimuthal angle between the birefringent layer and the polarizers, the center operating frequencies are found to be,

where m is a positive odd integer, c is the speed of light in vacuum, and Δn = n_x-ny. The group index of refraction at the center frequency is,

the bandwidth of the anomalous phase behavior is,

and the transmission throughput is,

for α < 1.

Given Δn, α, and d, performance parameters can be calculated using these formula. It can be seen from Eq. (2) that, in order for the group index to be negative, α has to be smaller than 1. This is in agreement with the condition given by Mandatori, et. al., that the field component with the larger index of refraction has to have the smaller amplitude. For naturally birefringent materials, the index contrast, Δn/n_y, is typically less than 0.01. Accordingly using naturally occurring birefringent materials, the group index is negative only if 0.99 < α < 1. This would result in a transmission throughput of only 9.8×10^-5, or an insertion loss of more than 40 dB, which is of little practical interest. However, for a large birefringence of, for example, Δn/n_y = 1, the group index can be negative for α as small as 0.5, resulting a transmission throughput of more than 25%, or a 6 dB insertion loss, which is acceptable for many applications.

3. Large birefringence via form birefringent metamaterial structures

As mentioned in section 1, the large birefringence that is needed for practical applications of a negative group delay and an effective negative index can be achieved by using form birefringence. Form birefringence refers to the anisotropic electromagnetic response of a material due to its structural design. Examples such as sub-wavelength lines patterns with varying fill-factors serve this purpose. The host material from which to fabricate a form birefringent element can be a low cost isotropic material such as a plastic or ceramic, which ideally has low loss and a high intrinsic dielectric constant. The simplest structural pattern that induces form birefringence is a sub-wavelength period grating, i.e., an array of parallel lines separated by air. Given the refractive index of the host material n and a volume filling ratio r of the lines (r ≤ 1), the index for the TE polarization having the electric field component parallel to the lines is given by the volume mixing formula [17],

and the index for TM polarization, i.e., the electric field component perpendicular to the lines is [17],

It can be easily verified that n_x = n_y and the maximum index contrast occurs when the filling ratio is 0.5. The maximum index contrast is,

For a moderate host material index of n = 1.6, corresponding to low cost ABS plastics in X-band, the maximum index contrast can be 0.11. Figure 3(a) shows an ABS form birefringent disc designed for X-band operation. The lines and spacings are each 500 μm. The discs are fabricated by using a Stratasys FDM Titan rapid prototyping machine. Injection molding process can be used for large volume production at low cost of these high index contrast birefringent structures. Figure 3(b) shows a measurement of TE and TM refractive indices of the ABS form birefringent disc from the complex transmission spectrum of a 5mm thick sample. At 10 GHz, the index difference Δn is 0.14, yielding a 0.11 index contrast, which is in good agreement with form birefringence mixing formula. For the measured index curves of TE and TM modes in Fig. 3(b), the large variation of the index of refraction below 9 GHz is attributed to the X-band waveguide dispersion and a weak multi-path interference due to the finite difference between the indices of refraction of the disc and air. These non-ideal effects do not affect strongly the measurements of anomalous phase, which are between 9GHz to 12 GHz.

The index contrast can be further increased by surface treating the form birefringent discs using metallic paint. Figure 4 shows the measurement of the surface-treated ABS form birefringent disc. Index contrasts as high as 0.7 have been easily and reproducibly achieved. The increased dispersion for the surface-treated form-birefringent discs is due to the increased absorption introduced during the process. Since there is no equivalent bulk material for the surface treated form-birefringent disc, no direct comparison between the measurement results of index and the form-birefringence mixing formula can be made.

 

Fig. 3. (a) Form-birefringent ABS disc fabricated using 3-D rapid prototype machine. (b) The measured birefringence and its frequency dependence.

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Fig. 4. Enhanced birefringence of a surface-treated form-birefringent ABS disc. Index contrast of Δn/n = 0.7.

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4. Experimental results of negative group index

The experimental setup for measuring the negative group index of the Mandatori structure is shown in Fig. 5. A vector network analyzer (VNA, Agilent Technology N5230A) is configured with X-band free space architecture. The complex transmission spectrum S ₂₁ is used to calculate the group index. The X-band horn antennas are linearly polarized. The birefringent layer is rotated in respect to the horn antennas to form the azimuth angle θ that determines the electrical field splitting ratio by α = cot² θ for single layer birefringent structures. The measurement of the azimuth angle is from the displacement of markers on the edge of the circular birefringent disc with respect to the incident polarization orientation and has an estimated error of ±3 degrees and a through calibration was performed with no DUT (device under test) in between the horns and used as the reference. The output beam from the launching horn antenna has been allowed to propagate freely in air for over 2 feet in distance. The 4-inch disc intercepts only a small portion of this expanded wave front and the accepted finite angle spectrum of plane waves is expected to be very narrow, essentially approximating a single spatial frequency plane wave.

 

Fig. 5. Experimental setup utilizing a free-space X-band architecture with a vector network analyzer (VNA).

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Figure 6 shows the anomalous spectral phase behavior in the calibrated transmission spectrum of single-layer structure. Different traces correspond to different thicknesses of the form-birefringent layer. The azimuth angle is adjusted to be close to and slightly larger than 45 degrees, which according to Eq. (2) gives rise to the most significant anomalous spectral phase behavior as seen in Fig. 6. The index difference Δn can be estimated from these curves by identifying the center frequency of the positive spectral phase slope and using Eq. (1). The results are summarized in Table 1. This is in excellent agreement with the direct measurement of the frequency dependent index and index difference for a number of different values of Δn, thereby validating the basis of the anomalous phase phenomenon and the formulae presented in Section 2. Figure 7(a) shows the spectral phase of the calibrated transmission spectrum for a 9 cm thick ABS form birefringent layer. Different traces in Fig. 7(a) correspond to different azimuth angle θ, which determine different splitting ratios α. Around 45 degrees where the splitting ratio α is close to 1, the slope of the spectral phase changes sign as predicted. According to the theory and Eq. (2), when θ is less than 45 degrees, α is larger than 1 and the slope of the spectral phase is always negative, corresponding to a positive group index. When θ is close to but larger than 45 degrees, more energy is distributed to the slower axis (larger index of refraction) of the birefringent layer and α is close to but smaller than 1. According to Eq. (2), the slope of the spectral phase will change sign and give rise to negative group index. The experimental results agree very well with these predictions for both untreated and treated ABS structures. A phase change close to 0.75π is observed for θ = 50 degrees within a frequency band from 10.8 GHz to 11.4 GHz, corresponding to a calibrated negative group delay of 0.625 ns. Since the transmission is calibrated to air and it takes 0.3 ns for the RF wave to transmit through a 9 cm distance in air, this means an absolute negative group delay of 0.325 ns has been experimentally observed in the frequency domain. Accordingly, the corresponding measured group index is n_g = -1.1. Figure 7(b) shows the intensity responses and as expected, the throughput T is a function of the splitting ratio which changes appreciably with the azimuth angle. For moderate fast light effects, obtained for example with θ = 80 degrees, an 8 dB insertion loss is incurred, which is still acceptable. The apparent gain toward the lower frequency end is a consequence of the waveguiding effect of the DUT, which improves slightly the directivity of the beam from the launching horn antenna. This effect diminishes at higher frequencies.

Table 1. Center frequency dependence on thickness and the inferred index difference

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Fig. 6. Thickness dependence of the center operating frequency of the anomalous spectral phase. Different curves correspond to single-layer birefringent structures with different thicknesses. The shift of the center operating frequency agrees very well with the theoretical prediction.

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Fig. 7. (a) Azimuthal angle dependence of the anomalous spectral phase. The group index is tunable from positive values to negative values by the rotation of the birefringent layer. The sign change happens when the azimuth angle is close to 45 degrees. (b) The intensity transmission spectra.

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The birefringent structure gives rise to not only fast light phenomena but also slow light phenomena as is evident in Fig. 7(a). When θ < 45 degrees, the splitting ratio α is larger than 1 and the field component with the larger index of refraction has larger amplitude than that with the smaller index. Under this condition as discussed previously, the group index of refraction will not be negative. However, it can now be significantly larger than 1, exhibiting a strong slow light effect. This is most significantly shown in Fig. 8, where the spectral phase of the transmission spectrum with surface-treated ABS form-birefringent discs is plotted. Different traces again correspond to different azimuthal angles and the fixed birefringent layer thickness of 2.5 cm. The surface-treated ABS form-birefringent discs had an index contrast of 0.7. Owing to the increased index contrast, the total thickness of the birefringent layer for an operation center frequency around 10 GHz is only 2.5 cm. For the azimuthal angle of 40 degrees, the accumulated spectral phase changes from -2.5 radians at 9.9 GHz to -7 radians at 10.2 GHz, corresponding to a calibrated delay of 2.4 ns and a slow light group index of n_g = 29.6. As typical for passive slow light media, a strong insertion loss of 25 dB is associated with this large slow light delay.

 

Fig. 8. Angle dependence of the anomalous spectral phase for a high Δn, 2.5 cm thick, surface treated form-birefringent ABS structure. The large slow light group delay with a bandwidth of 0.3 GHz around the center frequency of 10 GHz exhibits a group index of 29.6.

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Periodic multilayer birefringent structures have also been fabricated and tested. Such multilayer structures under certain conditions are capable of huge field concentrations at least in theory [18]. Figure 9 shows a typical transmission spectral phase spectrum of such structures close to the band-edge frequency, which also exhibits anomalous phase behavior, confirming again the theoretical prediction of Mandatori, et. al. [15].

 

Fig. 9. Typical transmission spectral phase spectra of a periodic multilayer birefringent structure: (a) Simulation. (b) Measurements. Both show the predicted anomalous spectral phase behavior.

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5. Conclusion

We experimentally verified the anomalous phase in single-layer as well as multi-layer birefringent structures as predicted by Mandatori, et. al. Using form birefringent structures, large Δn/n as much as 0.7 have been realized in surface-treated ABS form-birefringent discs. Such large birefringence is not readily available in naturally occurring media but is critical to make compact structures viable and useful. The design formula for single-layer structures for group delay and group index control are given here and verified by our experiments. The predicated as well as the observed anomalous phase behavior bears a close resemblance to the anomalous dispersion that is associated with for example the energy dissipation. To certain degree, the anomalous phase behavior realized by birefringence can be considered as an engineered absorption line with controllable bandwidth and strength. The Kramers-Kronig relation then ensures the anomalous phase behavior in the same way as it forces the anomalous dispersion within an absorption line. With the availability of tunable negative group indices, such simple structures can play an important role in both practical applications and in the understanding and exploitation of fast/slow light phenomena.