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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 2 — Jan. 27, 2003
  • pp: 125–133
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Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann  »View Author Affiliations


Optics Express, Vol. 11, Issue 2, pp. 125-133 (2003)
http://dx.doi.org/10.1364/OE.11.000125


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Abstract

We have experimentally measured the birefringence in bulk two-dimensional hexagonal photonic crystals in transparent spectral regions above and below the fundamental band gap. Data is presented for structures with different numbers of layers and two different air-filling fractions. We have used these data to design a photonic crystal quarter waveplate and provide independent experimental demonstrations of its operation.

© 2002 Optical Society of America

1. Introduction

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

2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

], 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

3. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537 (1999) [CrossRef] [PubMed]

] and photonic crystal circuit elements [4

4. S. Mingaleev and Y. Kivshar, “Nonlinear photonic crystals toward all-optical technologies,” Opt. & Phot. News 13, 48 (2002) [CrossRef]

].

One type of 2D photonic crystal which has received significant attention is the photonic crystal slab [5

5. E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983 (2000) [CrossRef] [PubMed]

]. 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

6. R. D. Meade, A. Devenyl, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band-gap materials - Low-loss bends and high Q-cavities,” J. Appl. Phys. 75, 4753 (1994) [CrossRef]

]. Theoretical calculations of the band structure in photonic crystal slabs [7

7. T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electronics Lett. 31, 1941 (1995) [CrossRef]

, 8

8. S. G. Johnson, J. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999) [CrossRef]

], and experimental transmission studies at optical wavelengths have been performed [5

5. E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983 (2000) [CrossRef] [PubMed]

, 9

9. E. Özbay, in Photonic Band Gap Materials, C. M. Soukoulis, ed. (Kluwer, Amsterdam, 1996), p. 41. [CrossRef]

].

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

12. H. A. Macleod, Thin-film Optical Filters, (Hilger, London, 1969).

]. 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

13. D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental observation of superluminal group velocities in bulk two-dimensional photonic band gap crystals,” to appear inIEEE J. Sel. Top. Quantum Electr. (2003). [CrossRef]

]. 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

14. R. W. Ditchburn, Light, 3rd ed., (Interscience, New York, 1976), Appendix XIX B.

, 15

15. C. Kittel, Introduction to Solid State Physics, 7th ed., (Wiley, New York, 1996), Chap. 11.

]. 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

16. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25, 1325 (2000) [CrossRef]

, 17

17. T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Phot. Tech. Lett. 13, 588 (2001) [CrossRef]

, 18

18. C. Kerbage, P. Steinvurzel, P. Reyes, P. S. Westbrook, R. S. Windeler, A. Hale, and B. J. Eggleton, “Highly tunable birefringent microstructured optical fiber,” Opt. Lett. 27, 842 (2002) [CrossRef]

], photonic crystal lasing cavities [19

19. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001) [CrossRef] [PubMed]

], and nonlinear fiber Bragg gratings [20

20. R. E. Slusher, S. Spalter, B. J. Eggleton, S. Pereira, and J. E. Sipe, “Bragg-grating-enhanced polarization instabilities,” Opt. Lett. 25, 749 (2000) [CrossRef]

, 21

21. S. Pereira, J. E. Sipe, R. E. Slusher, and Stefan Spälter, “Enhanced and suppressed birefringence in fiber Bragg gratings,” J. Opt. Soc. Am. B 19, 1509 (2002). [CrossRef]

]. The polarization properties of 2D photonic crystals have been theoretically discussed [22

22. L. Li, “Two-dimensional photonic crystals: Candidate for waveplates,” Appl. Phys. Lett. 78, 3400 (2001) [CrossRef]

], 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

23. D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental demonstration of photonic crystal waveplates,” to appear inAppl. Phys. Lett. (February 03, 2003).

]. 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

10. J. M. Hickmann, D. Solli, C. F. McCormick, R. Plambeck, and R. Y. Chiao, “Microwave measurements of the photonic band gap in a two-dimensional photonic crystal slab,” J. Appl. Phys. 92, 6918 (2002) [CrossRef]

]. 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.

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).

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

n(ω)=1+cΔϕωL,
(1)

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

L=[2+(N1)3]r,
(2)

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.

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 π2(2m+1) over the optical path length, the crystal will act as an m th-order quarter waveplate. We express this condition as

Δn(ω)=πcωL(m+12)
(3)

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.

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.

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 π2 . 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

EE0=cos4θ+sin4θ
(4)
EE0=12sin2θ
(5)
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 12sin2θ and cos4θ+sin4θ .

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

4. S. Mingaleev and Y. Kivshar, “Nonlinear photonic crystals toward all-optical technologies,” Opt. & Phot. News 13, 48 (2002) [CrossRef]

].

Acknowledgments

This work was supported by ARO grant number DAAD19-02-1-0276. We thank the UC Berkeley Astronomy Department, in particular Dr. R. Plambeck, for lending us the VNA. JMH thanks the support from Instituto do Milênio de Informação Quǎntica, CAPES, CNPq, FAPEAL, PRONEX-NEON, ANP-CTPETRO.

References and links

1.

E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173 (1994). [CrossRef]

2.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

3.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537 (1999) [CrossRef] [PubMed]

4.

S. Mingaleev and Y. Kivshar, “Nonlinear photonic crystals toward all-optical technologies,” Opt. & Phot. News 13, 48 (2002) [CrossRef]

5.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983 (2000) [CrossRef] [PubMed]

6.

R. D. Meade, A. Devenyl, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band-gap materials - Low-loss bends and high Q-cavities,” J. Appl. Phys. 75, 4753 (1994) [CrossRef]

7.

T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electronics Lett. 31, 1941 (1995) [CrossRef]

8.

S. G. Johnson, J. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751 (1999) [CrossRef]

9.

E. Özbay, in Photonic Band Gap Materials, C. M. Soukoulis, ed. (Kluwer, Amsterdam, 1996), p. 41. [CrossRef]

10.

J. M. Hickmann, D. Solli, C. F. McCormick, R. Plambeck, and R. Y. Chiao, “Microwave measurements of the photonic band gap in a two-dimensional photonic crystal slab,” J. Appl. Phys. 92, 6918 (2002) [CrossRef]

11.

S. Foteinopoulou, A. Rosenberg, M M. Sigalas, and C. M. Soukoulis, “In- and out-of-plane propagation of electromagnetic waves in low index contrast two dimensional photonic crystals,” J. Appl. Phys. 89, 824 (2001) [CrossRef]

12.

H. A. Macleod, Thin-film Optical Filters, (Hilger, London, 1969).

13.

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental observation of superluminal group velocities in bulk two-dimensional photonic band gap crystals,” to appear inIEEE J. Sel. Top. Quantum Electr. (2003). [CrossRef]

14.

R. W. Ditchburn, Light, 3rd ed., (Interscience, New York, 1976), Appendix XIX B.

15.

C. Kittel, Introduction to Solid State Physics, 7th ed., (Wiley, New York, 1996), Chap. 11.

16.

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25, 1325 (2000) [CrossRef]

17.

T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Phot. Tech. Lett. 13, 588 (2001) [CrossRef]

18.

C. Kerbage, P. Steinvurzel, P. Reyes, P. S. Westbrook, R. S. Windeler, A. Hale, and B. J. Eggleton, “Highly tunable birefringent microstructured optical fiber,” Opt. Lett. 27, 842 (2002) [CrossRef]

19.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001) [CrossRef] [PubMed]

20.

R. E. Slusher, S. Spalter, B. J. Eggleton, S. Pereira, and J. E. Sipe, “Bragg-grating-enhanced polarization instabilities,” Opt. Lett. 25, 749 (2000) [CrossRef]

21.

S. Pereira, J. E. Sipe, R. E. Slusher, and Stefan Spälter, “Enhanced and suppressed birefringence in fiber Bragg gratings,” J. Opt. Soc. Am. B 19, 1509 (2002). [CrossRef]

22.

L. Li, “Two-dimensional photonic crystals: Candidate for waveplates,” Appl. Phys. Lett. 78, 3400 (2001) [CrossRef]

23.

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental demonstration of photonic crystal waveplates,” to appear inAppl. Phys. Lett. (February 03, 2003).

24.

AIP Handbook, 3rd ed., edited by D. E. Gray (McGraw-Hill, New York, 1972), p. 5–132.

OCIS Codes
(230.5440) Optical devices : Polarization-selective devices
(260.2110) Physical optics : Electromagnetic optics
(350.4010) Other areas of optics : Microwaves

ToC Category:
Research Papers

History
Original Manuscript: December 19, 2002
Revised Manuscript: January 15, 2003
Published: January 27, 2003

Citation
D. Solli, C. McCormick, R. Chiao, and Jandir Hickmann, "Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates," Opt. Express 11, 125-133 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-2-125


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References

  1. E. Yablonovitch, �??Photonic crystals,�?? J. Mod. Opt. 41, 173 (1994). [CrossRef]
  2. E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  3. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, D. C. Allan, �??Single-mode photonic band gap guidance of light in air,�?? Science 285, 1537 (1999). [CrossRef] [PubMed]
  4. S. Mingaleev and Y. Kivshar, �??Nonlinear photonic crystals toward all-optical technologies,�?? Opt. & Phot. News 13, 48 (2002). [CrossRef]
  5. E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, �??Three-dimensional control of light in a twodimensional photonic crystal slab,�?? Nature 407, 983 (2000). [CrossRef] [PubMed]
  6. R. D. Meade, A. Devenyl, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, �??Novel applications of photonic band-gap materials - Low-loss bends and high Q-cavities,�?? J. Appl. Phys. 75, 4753 (1994). [CrossRef]
  7. T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin and T. J. Shepherd, �??Full 2-D photonic bandgaps in silica/air structures,�?? Electron. Lett. 31, 1941 (1995). [CrossRef]
  8. S. G. Johnson, J. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 60, 5751 (1999). [CrossRef]
  9. E. Ozbay, in Photonic Band Gap Materials, C. M. Soukoulis, ed. (Kluwer, Amsterdam, 1996), p. 41. [CrossRef]
  10. J. M. Hickmann, D. Solli, C. F. McCormick, R. Plambeck, and R. Y. Chiao, �??Microwave measurements of the photonic band gap in a two-dimensional photonic crystal slab,�?? J. Appl. Phys. 92, 6918 (2002). [CrossRef]
  11. S. Foteinopoulou, A. Rosenberg, M M. Sigalas, and C. M. Soukoulis, �??In- and out-of-plane propagation of electromagnetic waves in low index contrast two dimensional photonic crystals,�?? J. Appl. Phys. 89, 824 (2001). [CrossRef]
  12. H. A. Macleod, Thin-film Optical Filters, (Hilger, London, 1969).
  13. D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, �??Experimental observation of superluminal group velocities in bulk two-dimensional photonic band gap crystals,�?? to appear in IEEE J. Sel. Top. Quantum Electron. (2003). [CrossRef]
  14. R. W. Ditchburn, Light, 3rd ed., (Interscience, New York, 1976), Appendix XIX B.
  15. C. Kittel, Introduction to Solid State Physics, 7th ed., (Wiley, New York, 1996), Chap. 11.
  16. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, �??Highly birefringent photonic crystal fibers,�?? Opt. Lett. 25, 1325 (2000). [CrossRef]
  17. T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, �??Highly birefringent index-guiding photonic crystal fibers,�?? IEEE Phot. Tech. Lett. 13, 588 (2001). [CrossRef]
  18. C. Kerbage, P. Steinvurzel, P. Reyes, P. S.Westbrook, R. S. Windeler, A. Hale, and B. J. Eggleton, �??Highly tunable birefringent microstructured optical fiber,�?? Opt. Lett. 27, 842 (2002). [CrossRef]
  19. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, �??Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,�?? Science 293, 1123 (2001). [CrossRef] [PubMed]
  20. R. E. Slusher, S. Spalter, B. J. Eggleton, S. Pereira, and J. E. Sipe, �??Bragg-grating-enhanced polarization instabilities,�?? Opt. Lett. 25, 749 (2000). [CrossRef]
  21. S. Pereira, J. E. Sipe, R. E. Slusher, and Stefan Sp¨alter, �??Enhanced and suppressed birefringence in .ber Bragg gratings,�?? J. Opt. Soc. Am. B 19, 1509 (2002). [CrossRef]
  22. L. Li, �??Two-dimensional photonic crystals: Candidate for waveplates,�?? Appl. Phys. Lett. 78, 3400 (2001). [CrossRef]
  23. D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, �??Experimental demonstration of photonic crystal waveplates,�?? to appear in Appl. Phys. Lett. (February 03, 2003).
  24. AIP Handbook, 3rd ed., edited by D. E. Gray (McGraw-Hill, New York, 1972), p. 5-132.

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