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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 17 — Aug. 15, 2011
  • pp: 15990–15995
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Higher order modes in photonic crystal slabs

Roman Gansch, Stefan Kalchmair, Hermann Detz, Aaron M. Andrews, Pavel Klang, Werner Schrenk, and Gottfried Strasser  »View Author Affiliations


Optics Express, Vol. 19, Issue 17, pp. 15990-15995 (2011)
http://dx.doi.org/10.1364/OE.19.015990


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Abstract

We present a detailed investigation of higher order modes in photonic crystal slabs. In such structures the resonances exhibit a blue-shift compared to an ideal two-dimensional photonic crystal, which depends on the order of the slab mode and the polarization. By fabricating a series of photonic crystal slab photo detecting devices, with varying ratios of slab thickness to photonic crystal lattice constant, we are able to distinguish between 0th and 1st order slab modes as well as the polarization from the shift of resonances in the photocurrent spectra. This method complements the photonic band structure mapping technique for characterization of photonic crystal slabs.

© 2011 OSA

1. Introduction

The properties of photonic crystals (PCs) [1

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]

,2

2. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997). [CrossRef]

] offer unique ways for the control of light. Because of their compatibility with standard semiconductor processing, two-dimensional (2D) PC structures are the most commonly used [3

3. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284(5421), 1819–1821 (1999). [CrossRef] [PubMed]

5

5. A. Benz, Ch. Deutsch, G. Fasching, K. Unterrainer, A. M. Andrews, P. Klang, W. Schrenk, and G. Strasser, “Active photonic crystal terahertz laser,” Opt. Express 17(2), 941–946 (2009). [CrossRef] [PubMed]

]. An important class of 2D-PCs is the photonic crystal slab (PCS) [6

6. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]

8

8. S. Kalchmair, H. Detz, G. D. Cole, A. M. Andrews, P. Klang, M. Nobile, R. Gansch, C. Ostermaier, W. Schrenk, and G. Strasser, “Photonic crystal slab quantum well infrared photodetector,” Appl. Phys. Lett. 98(1), 011105 (2011). [CrossRef]

], which additionally confines photons in the out-of-plane direction with a dielectric slab. Applications of PCSs include micro cavities [9

9. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

], lasers [10

10. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003). [CrossRef] [PubMed]

,11

11. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]

] and detectors [12

12. J. Yang, M. Seo, I. Hwang, S. Kim, and Y. Lee, “Polarization-selective resonant photonic crystal photodetector,” Appl. Phys. Lett. 93(21), 211103 (2008). [CrossRef]

,13

13. K. T. Posani, V. Tripathi, S. Annamalai, N. R. Weisse-Bernstein, S. Krishna, R. Perahia, O. Crisafulli, and O. J. Painter, “Nanoscale quantum dot infrared sensors with photonic crystal cavity,” Appl. Phys. Lett. 88(15), 151104 (2006). [CrossRef]

].

The resonances of a PCS in the mid-infrared region (MIR) can be directly measured by fabricating it from a quantum well infrared photodetector (QWIP) [8

8. S. Kalchmair, H. Detz, G. D. Cole, A. M. Andrews, P. Klang, M. Nobile, R. Gansch, C. Ostermaier, W. Schrenk, and G. Strasser, “Photonic crystal slab quantum well infrared photodetector,” Appl. Phys. Lett. 98(1), 011105 (2011). [CrossRef]

,15

15. B. F. Levine, “Quantum‐well infrared photodetectors,” J. Appl. Phys. 74(8), R1–R81 (1993). [CrossRef]

,16

16. H. Schneider and H. C. Liu, Quantum Well Infrared Photodetectors (Springer, 2007).

]. In this letter, we present an investigation of PCS-QWIP structures where higher order slab modes are excited and demonstrate a method to identify the order of the slab mode as well as the polarization. This complements characterization methods of PCs such as the photonic band structure mapping technique [4

4. S. Schartner, S. Golka, C. Pflügl, W. Schrenk, A. M. Andrews, T. Roch, and G. Strasser, “Band structure mapping of photonic crystal intersubband detectors,” Appl. Phys. Lett. 89(15), 151107 (2006). [CrossRef]

,17

17. V. N. Astratov, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60(24), R16255–R16258 (1999). [CrossRef]

,18

18. V. Lousse, W. Suh, O. Kilic, S. Kim, O. Solgaard, and S. Fan, “Angular and polarization properties of a photonic crystal slab mirror,” Opt. Express 12(8), 1575–1582 (2004). [CrossRef] [PubMed]

], which provides information about the PC modes, but not about the slab modes.

2. Design and fabrication

The layer sequence of the used structure is grown by molecular beam epitaxy (MBE) with a 2 µm thick Al0.85Ga0.15As sacrificial layer, a 500 nm thick GaAs Si doped (carrier concentration 2 × 1018 cm−3) bottom contact layer, a 1.4 µm thick active region and a 100 nm thick GaAs doped (carrier concentration 2 × 1018 cm−3) top contact layer. The contact layers are used for collection and lateral transport of the photocurrent to the gold contacts. The QWIP consists of 26 periods of 45 nm Al0.30Ga0.70As barriers and 4.5 nm GaAs quantum wells. The optical intersubband transition from the bound ground state to the first excited state, here a quasi-bound state, is designed to operate at a wavelength of 8 µm. Electrons in the bound state are supplied by Si delta doping the QW with an equivalent sheet carrier density of 4 × 1011 cm−2 per period.

The PC with a hexagonal lattice (lattice constant a) of holes (radius r) in a dielectric medium is fabricated by anisotropic deep etching through the active zone into the sacrificial layer. After mesa etching and SiN isolation deposition, the extended top and bottom gold contact pads, which are annealed to obtain ohmic contacts for efficient extraction of the generated photocurrent, are deposited. The last process step is selective underetching with a 24% HCl solution to remove the sacrificial AlGaAs layer creating a free standing PCS of the air-bridge type with a final slab thickness of d = 2 µm (Fig. 1(a)
Fig. 1 (a) Scanning electron microscope (SEM) image of a cleaved PCS with mode profiles for the 0th and 1st order TE modes of the slab wave guide. (b) Normalized photocurrent spectra of a standard QWIP (dashed line) at 45° wedged illumination and a PCS-QWIP at surface normal incidence (solid line).
).

The QWIP is operated at 50 K, well below the background limited performance temperature TBLIP = 69 K. In a standard QWIP, normal incident light does not generate photocurrent due to intersubband transitions requiring an electric field in the growth direction [16

16. H. Schneider and H. C. Liu, Quantum Well Infrared Photodetectors (Springer, 2007).

]. Therefore, the photocurrent spectrum of the standard QWIP is obtained by a 45° wedged sample measurement. In the PCS-QWIP, normal incident light can couple into the radiating modes of the PCS, which are located above the light line [19

19. K. Inoue and K. Ohtaka, Photonic Crystals (Springer, 2004).

]. Incident photons with insufficient energy to excite electrons from the bound into the quasi-bound state cannot generate photocurrent. Therefore, all resonances of the PCS are designed above the QWIP peak responsivity (1250 cm−1), where photons can excite electrons into the continuum. Due to the transition probability becoming smaller, the generated photocurrent diminishes for higher energetic photons, but the long lifetime of photons in the PCS modes enhances the generated spectral photocurrent. The spectral dependence of the photocurrent is obtained with a broadband glowbar source and a Fourier transform infrared (FTIR) spectroscopy setup. We have previously shown that the PCS enhances the performance of the QWIP [8

8. S. Kalchmair, H. Detz, G. D. Cole, A. M. Andrews, P. Klang, M. Nobile, R. Gansch, C. Ostermaier, W. Schrenk, and G. Strasser, “Photonic crystal slab quantum well infrared photodetector,” Appl. Phys. Lett. 98(1), 011105 (2011). [CrossRef]

]. In the range from 1500 to 3000 cm−1, where the QWIP shows a low responsivity (Fig. 1(b), dashed line), sharp peaks are measured at the PCS resonances (Fig. 1(b), solid line).

3. Photonic band structure calculation

To design the response peaks of the PCS-QWIP, the photonic band structure of the PC has to be modeled. The band structure is calculated with the revised plane wave expansion method (RPWEM) [20

20. S. Shi, C. Chen, and D. W. Prather, “Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs,” Appl. Phys. Lett. 86(4), 043104 (2005). [CrossRef]

,21

21. V. Zabelin, “Numerical investigation of two-dimensional photonic crystal optical properties, design and analysis of photonic crystal based structures,” PhD thesis, (École Polytechnique Fédérale de Lausanne, 2009), http://library.epfl.ch/theses/?nr=4315.

] with an effective refractive index approximation to account for wave guiding in the slab. In an ideal 2D-PC, the modes can be separated into pure transversal electric (TE) and transversal magnetic (TM) polarized waves. Due to the missing translational invariance in the out-of-plane direction, the PCS modes are not purely TE or TM. However, since they strongly resemble TE and TM modes of an ideal 2D-PC, they are classified as TE-like and TM-like [22

22. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001). [CrossRef]

]. In contrast to the pure TE modes in an ideal 2D-PC, the TE-like modes of a PCS generate photocurrent in the QWIP, because of mode-mixing [17

17. V. N. Astratov, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60(24), R16255–R16258 (1999). [CrossRef]

] and a non-zero electric field component along the growth direction.

For the band structure calculation the PCS is split into a 2D-PC and a uniform slab wave guide problem. The effective refractive index neff of the slab wave guide is then introduced as frequency dependent permittivity of the PC material. Depending on the ratio of slab thickness to PC lattice constant, the slab-to-PC ratio d/a, higher order modes of the slab (1st order, 2nd order, etc.) are excited and additional response peaks are contributed to the photocurrent spectrum of the PCS-QWIP. Since higher order modes have a lower effective refractive index, the PCS resonances are located at higher frequencies compared to the fundamental mode.

The effective refractive index for TE and TM modes of a slab waveguide consisting of the slab material ns surrounded by air cladding material nc = 1 can be solved analytically (Fig. 2
Fig. 2 Analytical solution to the effective refractive index neff for TM (solid) and TE (dashed) modes of a slab waveguide. Depending on the slab-to-PC ratio d/a higher order slab modes have to be considered when designing PCS devices (gray axes insets). A measured peak (indicated by arrow, see Fig. 1(b) in the photocurrent spectrum can correspond to several slab modes.
). As an approximation for the heterostructure of the slab, an average refractive index of ns = 3.12 is used. For large filling factors F the perturbation due to the PC air holes can be neglected for this calculation. By introducing the slab normalized frequency kd=d/λ=ωd/2πc0 (analogous to the PC normalized frequency ka) the cutoff frequency of the m th order mode is given by

kd=m12ns2nc2=0.17m       m
(1)

With the effective refractive index for the 0th order and 1st order slab mode the band structure of the TM-like (Figs. 3(a)
Fig. 3 TM-like photonic band structure between the Γ-K symmetry points of a PCS for the (a) 0th order and (b) 1st order slab mode. Reducing the slab-to-PC ratio from d/a = 0.63 (filled circles) to d/a = 0.45 (hollow circles) shifts the resonances towards higher PC normalized frequencies. Modes inside the gray area are guided modes, whereas modes on the outside are radiating. The boundary between radiating and guided modes is marked by the light line.
and 3(b)) and TE-like (not shown) PCS modes are computed with the RPWEM for d/a = 0.45 and d/a = 0.63. For increasing slab-to-PC ratio, the PCS approaches the ideal two-dimensional PC with infinite extension into the third dimension (d/a→∞). Reducing the slab-to-PC ratio shifts the resonances of the PCS towards higher PC normalized frequencies. The effective refractive index of higher order slab modes is more sensitive to a change in d/a. Therefore, the PCS resonances of higher order slab modes exhibit a larger shift, as indicated by the arrows in Fig. 3. Since the effective refractive index of TE slab modes differs from TM slab modes, the polarization can also be determined. From the magnitude of the resonance shift in the photocurrent spectra we deduce the slab mode, as well as the polarization of measured resonances.

4. Experimental results and discussion

Variation of the slab-to-PC ratio can be achieved by either changing the slab thickness or the lattice constant. For different slab thicknesses, various samples would have to be grown. Variation of the lattice constant can be achieved on the same sample, which also means that processing variations will be the same for all devices. Therefore, we used the latter approach, fabricating a series of PCSs with lattice constants ranging from a = 3.2 to 4.4 µm (d/a = 0.63 to 0.45) and a hole radius of r = 0.22a (F = 0.82, measured with SEM) from the same QWIP sample with a slab thickness of d = 2 µm.

Using normal incident light, the shift of the resonances in the Γ-point can be observed in the measured spectra with PC normalized frequency axes (Fig. 4
Fig. 4 Measured spectra for normal incident light of a series of PCSs with varying lattice constant a and constant slab thickness d. The overlaid frames illustrate the peak shift due to a changing slab-to-PC ratio d/a. The 1st order TM-like modes (dotted frame) shift stronger than 0th order TM-like modes (dashed frame), whereas 0th order TE-like modes (dash-dotted frame) exhibit the smallest peak shift.
). Since the QWIP absorption stays constant, it will shift on the PC normalized frequency axis when changing the lattice constant. Comparing the shift magnitude of the PCS resonances with the RPWEM results, the pronounced peaks can be distinguished into 3 groups according to the 0th order TM-like, 1st order TM-like and 0th order TE-like PCS modes.

The coupling efficiency into the PCS depends on the in-plane symmetry-matching of the PCS mode to the incident wave [22

22. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001). [CrossRef]

], hence not all theoretically possible modes of the band structure are measured with the PCS-QWIP.

Detailed comparison of the RPWEM simulation to the measured spectra (Fig. 5
Fig. 5 Comparison of the measured peak shift (symbols) to the RPWEM simulation (lines) by variation of d/a. Symmetry unmatched modes with low coupling efficiency and hence less pronounced peaks are not plotted. The axis break on the left shows the simulated resonances for an ideal 2D-PC (d/a = ∞).
) shows good agreement of the location of the peaks as well as the magnitude of their shift. For the lowest 1st order TM-like PCS mode at a = 3.8 µm the calculated resonance is located at ka = 0.698 and the measured peak is at 0.706 (1% error). The calculated slope is Δ(ka) / Δa = 0.114 µm−1, whereas the measured value is 0.118 µm−1 (4% error). At higher frequencies the simulation fits better to the measurement, whereas for lower frequencies the offset becomes larger. This can be attributed to an increasing error of the effective refractive index approximation and the evanescent field of the slab mode coupling into the substrate at low frequencies.

5. Conclusion

We have presented a method to identify the slab mode order of PCS resonances by using a QWIP photodetector for direct resonance frequency measurement. The slab wave guide in the out-of-plane direction shifts the PCS modes to higher frequencies, compared to the ideal 2D-PC. Variation of the slab-to-PC ratio by processing a series of devices with different lattice constants allowed us to deduce the order of the slab mode as well as the polarization of resonances by analyzing the magnitude of their shift. The locations of the peaks as well as the shift magnitude in the measured photocurrent spectra are in good agreement with the results of RPWEM simulations with an effective refractive index approximation for the slab.

Using the RPWEM as a fast and reliable design tool, we envision fine-tuning of PCS detectors by post-fabrication thinning (e.g. etching of the top contact layer) or thickening (e.g. deposition of a dielectric material) of the slab to achieve a desired resonance shift. Further, the presented identification technique and knowledge about higher order slab modes will aid in the design of advanced PCS devices.

Acknowledgments

The authors acknowledge the support by the Austrian Science Fund (FWF): F2503-N17, the PLATON project within the Austrian NANO Initiative and the “Gesellschaft für Mikro- und Nanoelektronik” GMe.

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]

2.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997). [CrossRef]

3.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284(5421), 1819–1821 (1999). [CrossRef] [PubMed]

4.

S. Schartner, S. Golka, C. Pflügl, W. Schrenk, A. M. Andrews, T. Roch, and G. Strasser, “Band structure mapping of photonic crystal intersubband detectors,” Appl. Phys. Lett. 89(15), 151107 (2006). [CrossRef]

5.

A. Benz, Ch. Deutsch, G. Fasching, K. Unterrainer, A. M. Andrews, P. Klang, W. Schrenk, and G. Strasser, “Active photonic crystal terahertz laser,” Opt. Express 17(2), 941–946 (2009). [CrossRef] [PubMed]

6.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]

7.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]

8.

S. Kalchmair, H. Detz, G. D. Cole, A. M. Andrews, P. Klang, M. Nobile, R. Gansch, C. Ostermaier, W. Schrenk, and G. Strasser, “Photonic crystal slab quantum well infrared photodetector,” Appl. Phys. Lett. 98(1), 011105 (2011). [CrossRef]

9.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

10.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003). [CrossRef] [PubMed]

11.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]

12.

J. Yang, M. Seo, I. Hwang, S. Kim, and Y. Lee, “Polarization-selective resonant photonic crystal photodetector,” Appl. Phys. Lett. 93(21), 211103 (2008). [CrossRef]

13.

K. T. Posani, V. Tripathi, S. Annamalai, N. R. Weisse-Bernstein, S. Krishna, R. Perahia, O. Crisafulli, and O. J. Painter, “Nanoscale quantum dot infrared sensors with photonic crystal cavity,” Appl. Phys. Lett. 88(15), 151104 (2006). [CrossRef]

14.

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansión method,” Phys. Rev. B 73(23), 235114 (2006). [CrossRef]

15.

B. F. Levine, “Quantum‐well infrared photodetectors,” J. Appl. Phys. 74(8), R1–R81 (1993). [CrossRef]

16.

H. Schneider and H. C. Liu, Quantum Well Infrared Photodetectors (Springer, 2007).

17.

V. N. Astratov, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60(24), R16255–R16258 (1999). [CrossRef]

18.

V. Lousse, W. Suh, O. Kilic, S. Kim, O. Solgaard, and S. Fan, “Angular and polarization properties of a photonic crystal slab mirror,” Opt. Express 12(8), 1575–1582 (2004). [CrossRef] [PubMed]

19.

K. Inoue and K. Ohtaka, Photonic Crystals (Springer, 2004).

20.

S. Shi, C. Chen, and D. W. Prather, “Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs,” Appl. Phys. Lett. 86(4), 043104 (2005). [CrossRef]

21.

V. Zabelin, “Numerical investigation of two-dimensional photonic crystal optical properties, design and analysis of photonic crystal based structures,” PhD thesis, (École Polytechnique Fédérale de Lausanne, 2009), http://library.epfl.ch/theses/?nr=4315.

22.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001). [CrossRef]

OCIS Codes
(040.5160) Detectors : Photodetectors
(230.5298) Optical devices : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: May 4, 2011
Revised Manuscript: June 10, 2011
Manuscript Accepted: June 16, 2011
Published: August 5, 2011

Citation
Roman Gansch, Stefan Kalchmair, Hermann Detz, Aaron M. Andrews, Pavel Klang, Werner Schrenk, and Gottfried Strasser, "Higher order modes in photonic crystal slabs," Opt. Express 19, 15990-15995 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-15990


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References

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]
  2. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386(6621), 143–149 (1997). [CrossRef]
  3. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284(5421), 1819–1821 (1999). [CrossRef] [PubMed]
  4. S. Schartner, S. Golka, C. Pflügl, W. Schrenk, A. M. Andrews, T. Roch, and G. Strasser, “Band structure mapping of photonic crystal intersubband detectors,” Appl. Phys. Lett. 89(15), 151107 (2006). [CrossRef]
  5. A. Benz, Ch. Deutsch, G. Fasching, K. Unterrainer, A. M. Andrews, P. Klang, W. Schrenk, and G. Strasser, “Active photonic crystal terahertz laser,” Opt. Express 17(2), 941–946 (2009). [CrossRef] [PubMed]
  6. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]
  7. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]
  8. S. Kalchmair, H. Detz, G. D. Cole, A. M. Andrews, P. Klang, M. Nobile, R. Gansch, C. Ostermaier, W. Schrenk, and G. Strasser, “Photonic crystal slab quantum well infrared photodetector,” Appl. Phys. Lett. 98(1), 011105 (2011). [CrossRef]
  9. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]
  10. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003). [CrossRef] [PubMed]
  11. S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007). [CrossRef]
  12. J. Yang, M. Seo, I. Hwang, S. Kim, and Y. Lee, “Polarization-selective resonant photonic crystal photodetector,” Appl. Phys. Lett. 93(21), 211103 (2008). [CrossRef]
  13. K. T. Posani, V. Tripathi, S. Annamalai, N. R. Weisse-Bernstein, S. Krishna, R. Perahia, O. Crisafulli, and O. J. Painter, “Nanoscale quantum dot infrared sensors with photonic crystal cavity,” Appl. Phys. Lett. 88(15), 151104 (2006). [CrossRef]
  14. L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansión method,” Phys. Rev. B 73(23), 235114 (2006). [CrossRef]
  15. B. F. Levine, “Quantum‐well infrared photodetectors,” J. Appl. Phys. 74(8), R1–R81 (1993). [CrossRef]
  16. H. Schneider and H. C. Liu, Quantum Well Infrared Photodetectors (Springer, 2007).
  17. V. N. Astratov, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B 60(24), R16255–R16258 (1999). [CrossRef]
  18. V. Lousse, W. Suh, O. Kilic, S. Kim, O. Solgaard, and S. Fan, “Angular and polarization properties of a photonic crystal slab mirror,” Opt. Express 12(8), 1575–1582 (2004). [CrossRef] [PubMed]
  19. K. Inoue and K. Ohtaka, Photonic Crystals (Springer, 2004).
  20. S. Shi, C. Chen, and D. W. Prather, “Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs,” Appl. Phys. Lett. 86(4), 043104 (2005). [CrossRef]
  21. V. Zabelin, “Numerical investigation of two-dimensional photonic crystal optical properties, design and analysis of photonic crystal based structures,” PhD thesis, (École Polytechnique Fédérale de Lausanne, 2009), http://library.epfl.ch/theses/?nr=4315 .
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