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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 26351–26364
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Experimental and theoretical analysis of THz-frequency, direction-dependent, phonon polariton modes in a subwavelength, anisotropic slab waveguide

Chengliang Yang, Qiang Wu, Jingjun Xu, Keith A. Nelson, and Christopher A. Werley  »View Author Affiliations


Optics Express, Vol. 18, Issue 25, pp. 26351-26364 (2010)
http://dx.doi.org/10.1364/OE.18.026351


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Abstract

Femtosecond optical pulses were used to generate THz-frequency phonon polariton waves in a 50 micrometer lithium niobate slab, which acts as a subwavelength, anisotropic planar waveguide. The spatial and temporal electric field profiles of the THz waves were recorded for different propagation directions using a polarization gating imaging system, and experimental dispersion curves were determined via a two-dimensional Fourier transform. Dispersion relations for an anisotropic slab waveguide were derived via analytical analysis and found to be in excellent agreement with all observed experimental modes. From the dispersion relations, we analyze the propagation-direction-dependent behavior, effective refractive index values, and generation efficiencies for THz-frequency modes in the subwavelength, anisotropic slab waveguide.

© 2010 OSA

1. Introduction

Terahertz-frequency phonon polariton generation, control and detection have received extensive attention in recent years due to their outstanding capabilities in terahertz (THz) spectroscopy, imaging and advanced signal processing [1

1. Y.-S. Lee, Principles of Terahertz Science and Technology (Springer, New York, 2009).

5

5. T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007). [CrossRef]

]. Phonon polariton waves result from the coupling of lattice vibrational waves and electromagnetic waves, and can be generated in ferroelectric crystals such as LiNbO3 (LN) via impulsive stimulated Raman scattering (ISRS) using femtosecond optical pulses [6

6. T. P. Dougherty, G. P. Wiederrecht, and K. A. Nelson, “Impulsive Stimulated Raman Scattering Experiments in The Polariton Regime,” J. Opt. Soc. Am. 9(12), 2179–2189 (1992). [CrossRef]

,7

7. Y. X. Yan, E. B. Gamble, and K. A. Nelson, “Impulsive Stimulated Scattering: General Importance in Femtosecond Laser Pulse Interactions with Matter, and Spectroscopic Applications,” J. Chem. Phys. 83(11), 5391–5399 (1985). [CrossRef]

]. The electromagnetic component of the phonon polariton wave can be coupled into free space and is a source for intense THz pulses [8

8. D. Auston and M. Nuss, “Electrooptic Generation and Detection of Femtosecond Electrical Transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988). [CrossRef]

12

12. K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle THz phonon-polariton waves in a planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95(10), 103304 (2009). [CrossRef]

]. THz waves generated in the sample do not propagate collinearly with the pump beam due to the large index-mismatch between optical and THz frequencies. Instead they generate a Cherenkov radiation pattern and propagate primarily in the lateral direction [13

13. D. H. Auston, K. P. Cheung, J. A. Valdmains, and D. A. Kleinman, “Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media,” Phys. Rev. Lett. 53(16), 1555–1558 (1984). [CrossRef]

, 14

14. J. K. Wahlstrand and R. Merlin, “Cherenkov radiation emitted by ultrafast laser pulses and the generation of coherent polaritons,” Phys. Rev. B 68(5), 054301 (2003). [CrossRef]

]. This lateral propagation facilitates coherent control of the THz wave, which can easily be made to interact with subsequent optical pulses, other THz waves, or patterned structures all in the same small crystal of LN. As a result, a LN slab can serve as a platform for THz processing because generation, propagation, detection, and control can be fully integrated in one sample [5

5. T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007). [CrossRef]

, 15

15. N. S. Stoyanov, T. Feurer, D. W. Ward, and K. A. Nelson, “Integrated diffractive terahertz elements,” Appl. Phys. Lett. 82(5), 674–676 (2003). [CrossRef]

]. Furthermore, when the sample thickness becomes comparable to or less than the THz wavelength, the strong evanescent field of the THz wave can interact with material deposited on the crystal surface. This opens the door for spectroscopic analysis and interfacing the LN slab with other optical or photoelectric devices.

Because the THz wave propagates almost perpendicular to the optical pump beam, it is possible to obtain time-resolved images of the electric field in the LN slab. As the THz wave propagates through the crystal, its electric field changes the refractive index through the electro-optic effect. The time-delayed probe pulses, which can be expanded to illuminate the whole crystal, experience a spatially dependant phase shift proportional to the refractive index change. Four methods have been introduced to convert this phase pattern to an amplitude image: Talbot imaging [2

2. R. M. Koehl, S. Adachi, and K. A. Nelson, “Real-Space Polariton Wave Packet Imaging,” J. Chem. Phys. 110(3), 1317–1320 (1999). [CrossRef]

], Sagnac interferometry [16

16. P. Peier, S. Pilz, F. Müller, K. A. Nelson, and T. Feurer, “Analysis of phase contrast imaging of terahertz phonon-polaritons,” J. Opt. Soc. Am. B 25(7), B70–B75 (2008). [CrossRef]

], polarization gating [16

16. P. Peier, S. Pilz, F. Müller, K. A. Nelson, and T. Feurer, “Analysis of phase contrast imaging of terahertz phonon-polaritons,” J. Opt. Soc. Am. B 25(7), B70–B75 (2008). [CrossRef]

,18

18. C. A. Werley, Q. Wu, K. H. Lin, C. R. Tait, A. Dorn, and K. A. Nelson, “A comparison of phase sensitive imaging techniques for studying THz waves in structured LiNbO3,” J. Opt. Soc. Am. B 27(11) 2350-2359 (2010). [CrossRef]

], and phase contrast imaging [17

17. Q. Wu, C. A. Werley, K. H. Lin, A. Dorn, M. G. Bawendi, and K. A. Nelson, “Quantitative phase contrast imaging of THz electric fields in a dielectric waveguide,” Opt. Express 17(11), 9219–9225 (2009). [CrossRef] [PubMed]

]. In a recent comparison [18

18. C. A. Werley, Q. Wu, K. H. Lin, C. R. Tait, A. Dorn, and K. A. Nelson, “A comparison of phase sensitive imaging techniques for studying THz waves in structured LiNbO3,” J. Opt. Soc. Am. B 27(11) 2350-2359 (2010). [CrossRef]

], an improved geometry for polarization gating was found to offer the best sensitivity and most reliable field quantification, while phase contrast imaging was best in situations requiring high spatial resolution. In this paper, we used the polarization gating system similar to that shown in [18

18. C. A. Werley, Q. Wu, K. H. Lin, C. R. Tait, A. Dorn, and K. A. Nelson, “A comparison of phase sensitive imaging techniques for studying THz waves in structured LiNbO3,” J. Opt. Soc. Am. B 27(11) 2350-2359 (2010). [CrossRef]

] to record a sequence of images. The full spatio-temporal evolution was extracted from the image sequence and double Fourier transformed to obtain the wave vector vs. frequency dispersion curves [e.g 17.]. The data collection and analysis were performed as a function of wave propagation direction to study the complex mode structure present in an anisotropic slab waveguide, which was found to be in excellent agreement with theory. From the dispersion relations we extract the mode and propagation-angle dependent effective refractive index (ERI) and discuss pumping efficiencies for THz phonon polariton waves in a LN waveguide.

2. Experimental section

The experiments were performed with a Ti:sapphire regenerative amplifier whose pulse duration was 120 fs, central wavelength was 800 nm, and repetition rate was 1 KHz. The laser pulses were divided into a pump beam (370 μJ per pulse) and probe beam (35 μJ per pulse). The vertically polarized pump beam was routed through a mechanical delay stage and then focused to a line on the sample by a 200 mm focal length cylindrical lens (about 1 TW/cm2). The probe was frequency-doubled to 400 nm in a BBO crystal and expanded to be larger than the sample. The probe beam is nearly collinear with the pump by using a dichroic mirror, so the second harmonic wave of the pump on the sample, whose wavelength is the same as probe, can be blocked with a razor blade on the focal plane of the imaging lens. Figure 1(a)
Fig. 1 (a) Overview diagram of the experimental setup. GTP1 and GTP2 are Glan-Taylor prisms, whose polarizations are at + 45° and −45° to z-axis respectively. BS: 400 nm beam splitter; CL: cylindrical lens; DM: dichroic mirror; RM: retroreflective mirror. QW1 and QW2 are zero order 400 nm quarter-wave plates with optic axes at + 45 o and parallel to z-axis respectively. The 800 nm pump (red) and 400 nm probe (blue) are nearly collinear when they arrive at the sample, a 50 μm thick LiNbO3 slab. (b) The pump geometry and coordinate system. The 800 nm pump beam (red) propagates through the crystal, orthogonal to the crystal surface, while the THz (green) is guided down the slab. (c) The cylindrical lens can be rotated by θ relative to the z-axis (the c crystallographic axis of the LN sample) in order to launch the THz wave in a 90°-θ direction.
shows a sketch of the experimental setup and the coordinate system. A quarter-wave plate (QW1) and a retroreflective mirror were used in a 4-f system. The mirror and lenses imaged the sample precisely back onto itself without magnification or inversion. The axis of QW1, which was the same as the first Glan-Taylor polarizer (GTP1), was at + 45° so it exchanged the ordinary and the extraordinary polarization components of the probe. In this way the spatially varying phase shift between the vertical and horizontal polarization components accumulated from the probe’s first pass through the sample was compensated after the second pass. The phase shift after the first pass resulted from the intrinsic birefringence of the LN slab, and self-compensation was necessary to correct for spatial inhomogeneities in the phase shift due to thickness variation, strain, or other imperfections in the slab. The phase shift electro-optically induced by the THz wave, however, was not compensated because the THz wave was launched only after the probe pulse had passed through the sample the first time. The THz-induced phase information was converted to amplitude information prior to detection with the camera by QW2 (oriented vertically) and GTP2 (oriented at −45°). In this geometry a positive field results in a positive amplitude change and vice versa [18

18. C. A. Werley, Q. Wu, K. H. Lin, C. R. Tait, A. Dorn, and K. A. Nelson, “A comparison of phase sensitive imaging techniques for studying THz waves in structured LiNbO3,” J. Opt. Soc. Am. B 27(11) 2350-2359 (2010). [CrossRef]

].

The pump geometry is shown in Fig. 1(b). Red lines represent the 800 nm pump beam and green the broadband THz waves generated when the pump is focused into the 50 μm thick LiNbO3 crystal slab. Because the center wavelength of the THz phonon polariton wave is about 100 μm, the slab acts as a sub-wavelength waveguide. As Fig. 1(c) shows, the THz wave propagation direction was changed by rotating the cylindrical lens. Because of the strong anisotropy of LN at THz frequencies, the nature and behavior of the waveguide modes change drastically as the propagation direction rotates relative to the optic axis.

3. Results

In an anisotropic waveguide, constraints relating to propagation in bulk anisotropic material and constraints relating to propagation in a waveguide both come into play. In bulk anisotropic material waves are divided into two normal modes, ordinary waves and extraordinary waves, which propagate through the material at different velocities [20

20. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge,1999).

]. In an isotropic waveguide there are also two uncoupled eigenmodes, the transverse electric (TE) and transverse magnetic (TM) modes, which propagate through the waveguide at different velocities [19

19. E. A. Bahaa, B. Saleh and M. C. Teich, Fundamentals of photonics (JOHN WILEY&SONS, 1991)

]. When θ = 0 or 90°, these modes map directly onto one another. For θ = 0° the TE mode is an extraordinary wave and the TM mode is an ordinary wave while for θ = 90° the opposite pairing holds. When θ ≠ 0, however, the high-symmetry configuration is broken and all the modes couple together. The new eigenmodes of the system are neither purely TE nor TM and also not purely ordinary or extraordinary. The coupling effects the mode profiles, dispersion curves, and effective refractive indices in a fundamental and significant way, as will be demonstrated experimentally (presented immediately below) and theoretically (the full analysis can be found in the appendix) in the remainder of this paper.

With the experimental system mentioned above, we measured the dispersion curves for different propagation directions by rotating the cylindrical lens and CCD camera together, which kept the THz wavefront aligned vertically in the images. In this manner the THz wave propagation direction was varied from 0 to 90 degrees relative to the c-axis as shown in Fig. 1(c). The polarization of the 800 nm pump light was not rotated and thus was parallel to the c-axis in all measurements. Because of the strong r 33 electro-optic coefficient in LN, this ensured efficient pumping of THz waves with a large component polarized along the optic axis [21

21. D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988). [CrossRef]

23

23. N. S. Stoyanov, T. Feurer, D. Ward, E. Statz, and K. Nelson, “Direct visualization of a polariton resonator in the THz regime,” Opt. Express 12(11), 2387–2396 (2004). [CrossRef] [PubMed]

]. Using the same data collection and analysis procedure as was used to generate Fig. 2(b), the dispersion curves were measured for different angles θ, some examples of which are shown in Fig. 3
Fig. 3 (a) Dispersion curves for θ = 20°. Blue dotted lines are calculated TE-like mode dispersion curves and green dashed lines are TM-like modes. Experimentally we see three TE-like modes and no TM-like modes. The white box in the lower right shows a blow-up of the region around an avoided crossing between the two lowest symmetrical modes. (b) Dispersion curves for θ = 50°. In this case, TE-like modes still predominate and TM-like modes are too weak to be observed. (c) Dispersion curves for θ = 70°. We can see both TE- and TM-like modes, and all of the first 7 modes are observed experimentally. (d) Dispersion curves for θ = 90°, in which only the TM modes are excited. All the experimental data agree well with the calculated curves.
. Overlaid on the experimental data are the theoretical solutions for a bound mode propagating in an anisotropic, dielectric slab waveguide with a thickness of 50 μm, an extraordinary index of 5.11, and an ordinary index of 6.8. The full derivation is presented in appendix A. For all modes and all angles, the data agree very well with theoretical predictions.

In Fig. 3(a) where θ = 20°, one set of modes is very TE-like, and one is strongly TM-like. Because the TE-like modes have their primary polarization component along the c-axis, they were pumped much more strongly than the TM-like modes, which were too weak to be observed clearly. Blue dotted lines are calculated TE-like modes and green dashed lines are TM-like modes. An interesting effect resulting from propagation in the anisotropic waveguide when θ ≠ 0 or 90° is visible in the region containing the white lines, a magnified view of which is shown in the lower right corner. Although the TM-like modes are not visible, we still see an avoided crossing when two modes with the same symmetry (symmetric or antisymmetric) cross. The avoided crossing, visible in experiment and predicted by theory, results from coupling between TE- and TM-like modes in the anisotropic waveguide. Here the two lowest symmetric modes, the lowest TE-like mode and the second TM-like mode, avoid each other.

Figure 3(b) shows dispersion curves for the case of θ = 50°. The three TE-like modes predominate, although their strength is reduced, and TM modes still cannot be observed. No avoided crossings occur between modes of the same symmetry within the bandwidth of the experiment. As θ increases, the velocity of the extraordinary wave approaches that of the ordinary wave [30

30. D. Marcuse and I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. 15(2), 92–101 (1979). [CrossRef]

], which means that the slopes of TE-like and TM-like modes tend to be more similar at higher frequencies. Figure 3(c) shows the results for θ = 70°, where both TM-like and TE-like modes can be seen clearly. The strength of the TE-like modes continues to decrease with increased θ and TM-like modes are finally pumped strongly enough to detect. Although some of the modes are weak, the first seven modes can be observed in the experiment, all of which agree with theoretical predictions. Continuing the trend, at high frequencies the slopes of the TM-like and TE-like modes become even more similar. Finally, Fig. 3(d) shows results for θ = 90°, where only TM modes can be observed.

Using the derivation in appendix A, we can calculate the E-field profile of THz waves as shown in Fig. 4
Fig. 4 Electric field profiles for the lowest symmetric and antisymmetric modes at 0.5 THz. Ex, Ey and Ez are represented by blue, green and red lines respectively. The discontinuities in Ey located at ± 25 μm occur because of the slab surfaces. (a)-(b) TE and TM profiles when θ is 0°. (c) and (d) The electric field profile when θ is 50°.
. Figures 4(a) and (b) show field profiles for TE and TM modes respectively at θ = 0°. The coordinate system in Fig. 4 is the same as in the appendix (see Fig. 8
Fig. 8 The geometry for the waveguide mode derivation. (a) An anisotropic slab of width 2 centered at y = 0 embedded in an isotropic cladding which extends to infinity. The bound wave propagates along x and extends infinitely along z. ε and μ are the permittivity and permeability in the different regions. (b) The coordinate system for the derivation is defined by the slab surface normal and the propagation direction, which differs from Fig. 1 where the coordinates are defined in the lab frame. θ is the angle between the z-axis and the optic axis of the crystal.
), where the axes are defined by the propagation direction of the wave and not by the lab frame as in Fig. 1. Blue, green and red lines represent electric field along x-axis, y-axis and z-axis respectively. The electric field along the y-axis, whose polarization is perpendicular to the surface of the slab, changes drastically at the slab surface ( ± 25 μm). As mentioned above, pure TE and TM modes only exist at 0 and 90 degrees. At any other angle the eigenmodes are superpositions of TE and TM modes and contain all three polarization components, as shown in Fig. 4 (c) and (d) where θ = 50°.

One can extract the group and phase effective refractive index (ERI) from dispersion curves like those shown in Fig. 3. The phase ERI can be retrieved directly from the dispersion curve, np=c/vp=ck/(2πf), and the group ERI can be retrieved from the slope of the dispersion curve, ng=c/vg=cΔk/(2πΔf). Here vp and vg are the phase and group velocities, and the wave vector, k, and frequency, f, correspond to the axes in Fig. 3. From the derivation in the appendix, we can calculate both phase and group ERI for all propagation directions of TE-like and TM-like modes in the LN slab waveguide. Based on the agreement between experimental data and theoretical predictions shown in Fig. 3, Fig. 5
Fig. 5 (a) The frequency- and mode- dependent phase ERI for TE-like (dotted blue) and TM-like (dashed green) modes when θ = 0°. (b) The phase ERI when θ = 70°. (c) and (d) are the same as (a) and (b), but for the group ERI.
gives the theoretically calculated phase and group ERI for different angles, modes and frequencies. The ERI values are important for phase-matching in THz generation and for many nonlinear as well as linear optical processes.

In Fig. 5 dotted blue lines are the calculated ERI for TE-like modes and dashed green lines are the ERI for TM-like modes. From Fig. 5(a) and (b), we can see that the phase ERI for both TE-like and TM-like modes transitions from 1 (the index of air) to the bulk effective index. For the TM-like waves the bulk index is always the ordinary index of refraction, no 6.8, while for the TE-like waves the bulk index is that for the extraordinary wave in the anisotropic material, and changes from ~5.1 at 0° to ~6.8 at 90°. At all angles, the low-frequency TM-like modes have most of their energy in the evanescent field in the air and have ERI values near unity. The ERI then transitions rapidly to bulk-like values at higher frequencies. Much like the phase ERI, the group ERI transitions from 1 to the bulk effective index (see Fig. 5 (c) and (d)). In contrast to the phase ERI, however, the group ERI rises well above the bulk values before approaching them asymptotically at high frequencies. In contrast to the phase index, where higher modes always have lower ERIs, the group ERI is usually higher for higher modes. Another difference is that the peak group index changes drastically with θ for both TE-like and TM-like modes, while the peak phase ERI for the TM-like modes is just the bulk value and insensitive to angle.

A useful way to display the ERI is with an index ellipse, which highlights the angle-dependant behavior. Figure 6
Fig. 6 Effective refractive index (phase ERI) ellipse for three TE modes at a wave vector β = 50 rad/mm in a 50 μm LN slab waveguide. The open symbols are experimental data and the solid lines are calculated results. The scale along the x-axis is the same as that along y.
follows the phase ERI for the first three TE-like modes at wave vector magnitude β = 50 rad/mm as a function of angle, tracing out the phase ERI ellipse. We measured data in the first quadrant, and because of the symmetry these results can also be used for 90° to 360°. Values over 70° were not recorded because the TE-like modes were too weak to be observed. Figure 6 shows the measured values for the first three TE-like modes as open symbols and the calculated values predicted by the derivation in the appendix as solid lines. The experimental data can be fit to an ellipse, where the long and short axes are 5.44 and 4.18 for the first mode, 3.36 and 2.59 for the second mode and 2.01 and 1.74 for the third mode. The value of the long axis represents the ERI for an ordinary wave (the TE mode is purely ordinary at 90°) and the short axis represents the ERI for an extraordinary wave (the TE mode is purely extraordinary at 0°).

4. Conclusions

We have measured the propagation properties of THz waves in a 50 μm LiNbO3 anisotropic slab waveguide using a self-compensating polarization gating imaging system. This system can detect the THz electric fields both temporally and spatially over a wide wavelength range. Using the system, we studied the propagation-direction-dependent behavior of waveguide modes and determined the dispersion curves and effective refractive index for THz waves. A general solution for waveguide modes in a uniaxial slab waveguide was derived and found to agree with the experimental data.

Dispersion is integral to many processes in THz science and generally in linear and nonlinear optics, including broadening of ultrashort pulses, walk-off between pump and probe pulses, phase-matching of parametric processes, and generation of optical solitons. Because dispersion in a waveguide is determined by both the intrinsic material dispersion and geometric dispersion, it is essential to understand waveguiding effects. The results presented here will facilitate the design of functional devices with new capabilities in the LiNbO3 platform for integrated THz experiments and processing.

Appendix A: The general solution to a uniaxial slab waveguide with isotropic cladding

Anisotropic slab waveguides were extensively studied in the 1970’s [24

24. S. Wang, M. L. Shah, and J. D. Crow, “Wave propagation in thin film optical waveguides using gyrotropic and anisotropic materials as substrates,” IEEE J. Quantum Electron. 8(2), 212–216 (1972). [CrossRef]

30

30. D. Marcuse and I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. 15(2), 92–101 (1979). [CrossRef]

]. In many cases, attention was focused on anisotropic films deposited on a substrate that was itself anisotropic because mode converters, polarization mode filters, and other devices of that time had such a geometry [26

26. W. K. Burns and J. Warner, “Mode dispersion in uniaxial optical waveguides,” J. Opt. Soc. Am. 64(4), 441–446 (1974). [CrossRef]

]. The case in this paper is somewhat simpler because the geometry is symmetric (see Fig. 8). An additional simplification is that the anisotropic core (the slab) is embedded within an isotropic cladding (air in our experiment). In the derivation of the waveguide dispersion curves and mode profiles presented below, we assume the experimentally relevant conditions that the crystal is uniaxial (like LiNbO3) and its optic axis is parallel to the slab surface. The slab is assumed to extend infinitely along x and z and both core and cladding have no magnetic response. The wave is assumed to propagate along the x-direction and extend infinitely along the z-direction. In the experiment the cylindrical lens generating the wave was rotated instead of the sample, so the derivation here is performed in the coordinate frame of the lens. Finally, to simplify the analysis we assume that the waves are harmonic in space and along the propagation direction: E(x,y,z,t)=E(y)exp[i(βxωt)], where β=kx is the propagation constant.

To simplify notation we define several important variables. The propagation constant, βkx, was defined above, and the wave vector orthogonal to the slab surface is defined both outside the crystal, iαkyout, and for the ordinary and extraordinary waves inside the crystal, κo,κekyoin,kyein. α is defined as imaginary because bound modes will have evanescent, decaying fields in the cladding. There are three relevant bulk dispersion curves which define the relationships between wave vector, frequency, and index, one for the cladding and one each for the ordinary and extraordinary waves in the uniaxial core. They are:

cladding:α2=β2k2nc2
(2a)
ordinary:κo2=k2no2β2
(2b)
extraordinary:κe2=k2ne2β2(cos2θ+ne2no2sin2θ)
(2c)

where k=ω/c is the wave vector in free space and nc,no,  andne are the cladding index, ordinary index in the slab, and extraordinary index in the slab respectively. These relations are used to eliminate α,κo,  andκefrom the equations which follow, so everything is expressed in terms of β and k.

For a specific pair of β and k, there are four possible plane waves in each region, two signs for ky and two polarizations. In the anisotropic medium, the polarizations correspond to the ordinary (later represented by o) and extraordinary (later represented by e) waves. In the cladding, any pair of orthogonal polarizations can be chosen, so for convenience we choose the TE polarization (represented below by v for vertical polarization) and the TM polarization (represented later by hfor horizontal polarization). We can write out the most general form of the waveguide mode solution as:

cladding:E(y)=A1vexp[αy]+A2hexp[αy]+A3v+exp[αy]+A4h+exp[αy]
(3a)
core:E(y)=B1e+exp[iκey]+B2o+exp[iκoy]+B3eexp[iκey]+B4oexp[iκoy]
(3b)
cladding,:E(y)=C1vexp[αy]+C2hexp[αy]+C3v+exp[αy]+C4h+exp[αy]
(3c)

where A i, B i, and C i are scalar constants and the +/− superscripts correspond to the sign of ky.

The polarizations in the expression above can be determined from the appropriate vector constraints. In the cladding:

v±=v=[001],​ ​ ​ ​ ​                k±×vh±[±ihxhy0][±iαβ0]
(4)

where hx and hy are the magnitudes of the components of the normalized polarization vector. In contrast to the isotropic cladding, where any orthogonal polarizations could be chosen, in the slab the polarizations are uniquely determined as the ordinary and extraordinary wave polarizations in bulk material. The ordinary wave will be orthogonal to the plane containing the crystal axis and the wave vector: ok×c. The extraordinary wave will be located in the plane of k and c. The displacement field will be given byDk×(k×c), and the electric field is given through the constitutive relation: E=R¯¯(θ)ε¯¯1R¯¯(θ)D where R¯¯is the rotation matrix for rotation around the y-axis. This yields:

o±[±oxoyoz][±κocosθβcosθκosinθ]
(5a)
e±[exeyez][[1εo1εe](β2+κe2)cos2θsinθ+κe2sinθ[cos2θεo+sin2θεe]βκesinθεo[1εo1εe]κe2sin2θcosθ+(β2+κe2)cosθ[sin2θεo+cos2θεe]]
(5b)

where ox,oy,oz,ex,ey,andez are the magnitudes of the components of the normalized polarization vectors.

With the dispersion curves (Eq. 2) and polarizations (Eqs. 4 & 5) of waves in the bulk material in hand, we can simplify the expressions in Eq. 3 for E(y). For bound solutions, we require that the electric field decays to zero as y±, so the terms in the cladding that are exponentially growing can be discarded. We now apply the symmetry condition that there is a reflection plane down the center of the sample, which eliminates half of the coefficients. In this situation, the solution must be made of symmetric and antisymmetric modes. Absorbing some constant factors into the coefficients, we have:

Symmetric:

cladding,y<: E(y)=[hxA2ihyA2A1]exp[α(y+)]
(6a)
core:E(y)=B1[excos(κey)ieysin(κey)ezcos(κey)]+B2[oxcos(κoy)ioysin(κoy)ozcos(κoy)]
(6b)
cladding,y>: E(y)=[hxA2ihyA2A1]exp[α(y)]
(6c)

Antisymmetric:

cladding,y<: E(y)=[hxA2ihyA2A1]exp[α(y+)]
(7a)
core:E(y)=B1[exsin(κey)ieycos(κey)ezsin(κey)]+B2[oxsin(κoy)ioycos(κoy)ozsin(κoy)]
(7b)
cladding,y>: E(y)=[hxA2ihyA2A1]exp[α(y)]
(7c)

Applying the symmetry conditions eliminated half the unknowns, so now we need only apply boundary conditions at one interface to solve for the coefficients. The boundary condition is that the tangential E and H fields must be continuous across the boundary [20

20. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge,1999).

]. Using Faraday’s law and the fact that /z=0, /x=iβ, and /t=iω for our functional form, E(x,y,z,t)=E(y)exp[i(βxωt)], we can express all the boundary conditions in terms of the electric field components:

Ez,clad=Ez,core
(8a)
Ez,clady=Ez,corey
(8b)
Ex,clad=Ex,core
(8c)
iβEy,cladEx,clady=iβEy,coreEx,corey
(8d)

The constant coefficients in functional form of the solutions (Eqs. 6 & 7) must be chosen so the above boundary conditions are satisfied at the interface (y=). They must be solved independently for the symmetric and antisymmetric modes. The four expressions above yield a set of homogeneous equations which can be recast in matrix notation.

Symmetric:

[10ezcos(κe)ozcos(κo)α0ezκesin(κe)ozκosin(κo)0hxexcos(κe)oxcos(κo)0hyβhxα(eyβ+exκe)sin(κe)(oyβ+oxκo)sin(κo)][A1A2B1B2]=[0000]
(9)

Antisymmetric:

[10ezsin(κe)ozsin(κo)α0ezκecos(κe)ozκocos(κo)0hxexsin(κe)oxsin(κo)0hyβhxα(eyβ+exκe)cos(κe)(oyβ+oxκo)cos(κo)][A1A2B1B2]=[0000]
(10)

The polarizations (Eqs. 4 & 5) and bulk dispersion curves (Eq. 2) can be used to remove all dependence on α,κo,  andκe, so for a given angle θ, the only variables are β and k. The determinant will be zero, i.e. the set of equations has a solution, only for β and k pairs that are on the waveguide dispersion curve, and finding all allowed pairs traces out these curves. Using the allowed pairs, the bulk dispersion curves, and one additional “normalization condition” such as B1+B2=1, all wave vectors and coefficients can be completely determined. The theoretical dispersion curves for several angles are plotted along with the experimental data in Fig. 3, selected electric field profiles are shown in Fig. 4, and the effective indices of refraction for two angles are shown in Fig. 5.

Acknowledgements

The authors would like to thank Alexei Maznev for his invaluable insights regarding anisotropic waveguide behavior and solutions. This work was supported by the National Basic Research Program of China (2010CB933801), the 111 Project (B07013), the National Natural Science Foundation of China (10604033), the Tianjin Natural Science Foundation (09JCYBJC15100), U.S. National Science Foundation Grant No. ECCS-0824185, and a National Science Foundation Graduate Research Fellowship (C.A.W.).

References and links

1.

Y.-S. Lee, Principles of Terahertz Science and Technology (Springer, New York, 2009).

2.

R. M. Koehl, S. Adachi, and K. A. Nelson, “Real-Space Polariton Wave Packet Imaging,” J. Chem. Phys. 110(3), 1317–1320 (1999). [CrossRef]

3.

N. S. Stoyanov, D. W. Ward, T. Feurer, and K. A. Nelson, “Terahertz polariton propagation in patterned materials,” Nat. Mater. 1(2), 95–98 (2002). [CrossRef]

4.

T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal coherent control of lattice vibrational waves,” Science 299(5605), 374–377 (2003). [CrossRef] [PubMed]

5.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007). [CrossRef]

6.

T. P. Dougherty, G. P. Wiederrecht, and K. A. Nelson, “Impulsive Stimulated Raman Scattering Experiments in The Polariton Regime,” J. Opt. Soc. Am. 9(12), 2179–2189 (1992). [CrossRef]

7.

Y. X. Yan, E. B. Gamble, and K. A. Nelson, “Impulsive Stimulated Scattering: General Importance in Femtosecond Laser Pulse Interactions with Matter, and Spectroscopic Applications,” J. Chem. Phys. 83(11), 5391–5399 (1985). [CrossRef]

8.

D. Auston and M. Nuss, “Electrooptic Generation and Detection of Femtosecond Electrical Transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988). [CrossRef]

9.

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76(18), 2505–2507 (2000). [CrossRef]

10.

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrashort laser pulses with tilted pulse fronts,” Appl. Phys. B 78, 593–599 (2004). [CrossRef]

11.

K. L. Yeh, M. C. Hoffmann, J. Hebling, and A. Keith, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90(17), 171121 (2007). [CrossRef]

12.

K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle THz phonon-polariton waves in a planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95(10), 103304 (2009). [CrossRef]

13.

D. H. Auston, K. P. Cheung, J. A. Valdmains, and D. A. Kleinman, “Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media,” Phys. Rev. Lett. 53(16), 1555–1558 (1984). [CrossRef]

14.

J. K. Wahlstrand and R. Merlin, “Cherenkov radiation emitted by ultrafast laser pulses and the generation of coherent polaritons,” Phys. Rev. B 68(5), 054301 (2003). [CrossRef]

15.

N. S. Stoyanov, T. Feurer, D. W. Ward, and K. A. Nelson, “Integrated diffractive terahertz elements,” Appl. Phys. Lett. 82(5), 674–676 (2003). [CrossRef]

16.

P. Peier, S. Pilz, F. Müller, K. A. Nelson, and T. Feurer, “Analysis of phase contrast imaging of terahertz phonon-polaritons,” J. Opt. Soc. Am. B 25(7), B70–B75 (2008). [CrossRef]

17.

Q. Wu, C. A. Werley, K. H. Lin, A. Dorn, M. G. Bawendi, and K. A. Nelson, “Quantitative phase contrast imaging of THz electric fields in a dielectric waveguide,” Opt. Express 17(11), 9219–9225 (2009). [CrossRef] [PubMed]

18.

C. A. Werley, Q. Wu, K. H. Lin, C. R. Tait, A. Dorn, and K. A. Nelson, “A comparison of phase sensitive imaging techniques for studying THz waves in structured LiNbO3,” J. Opt. Soc. Am. B 27(11) 2350-2359 (2010). [CrossRef]

19.

E. A. Bahaa, B. Saleh and M. C. Teich, Fundamentals of photonics (JOHN WILEY&SONS, 1991)

20.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge,1999).

21.

D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988). [CrossRef]

22.

A. S. Barker Jr and R. Loudon, “Dielectric properties and optical phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967). [CrossRef]

23.

N. S. Stoyanov, T. Feurer, D. Ward, E. Statz, and K. Nelson, “Direct visualization of a polariton resonator in the THz regime,” Opt. Express 12(11), 2387–2396 (2004). [CrossRef] [PubMed]

24.

S. Wang, M. L. Shah, and J. D. Crow, “Wave propagation in thin film optical waveguides using gyrotropic and anisotropic materials as substrates,” IEEE J. Quantum Electron. 8(2), 212–216 (1972). [CrossRef]

25.

D. P. G. Russo and J. H. Harris, “Wave propagation in anisotropic thin-film optical waveguides,” J. Opt. Soc. Am. 63(2), 138–145 (1973). [CrossRef]

26.

W. K. Burns and J. Warner, “Mode dispersion in uniaxial optical waveguides,” J. Opt. Soc. Am. 64(4), 441–446 (1974). [CrossRef]

27.

V. Ramaswamy, “Propagation in asymmetrical anisotropic film waveguides,” Appl. Opt. 13(6), 1363–1371 (1974). [CrossRef] [PubMed]

28.

S. Nemoto and T. Makimoto, “Further discussion of the relationship between phase and group indices in anisotropic inhomogeneous guiding media,” J. Opt. Soc. Am. 67(9), 1281–1283 (1977). [CrossRef]

29.

D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media-Part I: Optical axis not in plane of slab,” IEEE J. Quantum Electron. 14(10), 736–741 (1978). [CrossRef]

30.

D. Marcuse and I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. 15(2), 92–101 (1979). [CrossRef]

OCIS Codes
(130.3730) Integrated optics : Lithium niobate
(240.5420) Optics at surfaces : Polaritons
(320.7110) Ultrafast optics : Ultrafast nonlinear optics
(100.0118) Image processing : Imaging ultrafast phenomena
(040.2235) Detectors : Far infrared or terahertz

ToC Category:
Ultrafast Optics

History
Original Manuscript: October 8, 2010
Revised Manuscript: November 18, 2010
Manuscript Accepted: November 19, 2010
Published: December 1, 2010

Citation
Chengliang Yang, Qiang Wu, Jingjun Xu, Keith A. Nelson, and Christopher A. Werley, "Experimental and theoretical analysis of THz-frequency, direction-dependent, phonon polariton modes in a subwavelength, anisotropic slab waveguide," Opt. Express 18, 26351-26364 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-26351


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References

  1. Y.-S. Lee, Principles of Terahertz Science and Technology (Springer, New York, 2009).
  2. R. M. Koehl, S. Adachi, and K. A. Nelson, “Real-Space Polariton Wave Packet Imaging,” J. Chem. Phys. 110(3), 1317–1320 (1999). [CrossRef]
  3. N. S. Stoyanov, D. W. Ward, T. Feurer, and K. A. Nelson, “Terahertz polariton propagation in patterned materials,” Nat. Mater. 1(2), 95–98 (2002). [CrossRef]
  4. T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal coherent control of lattice vibrational waves,” Science 299(5605), 374–377 (2003). [CrossRef] [PubMed]
  5. T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007). [CrossRef]
  6. T. P. Dougherty, G. P. Wiederrecht, and K. A. Nelson, “Impulsive Stimulated Raman Scattering Experiments in The Polariton Regime,” J. Opt. Soc. Am. 9(12), 2179–2189 (1992). [CrossRef]
  7. Y. X. Yan, E. B. Gamble, and K. A. Nelson, “Impulsive Stimulated Scattering: General Importance in Femtosecond Laser Pulse Interactions with Matter, and Spectroscopic Applications,” J. Chem. Phys. 83(11), 5391–5399 (1985). [CrossRef]
  8. D. Auston and M. Nuss, “Electrooptic Generation and Detection of Femtosecond Electrical Transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988). [CrossRef]
  9. Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76(18), 2505–2507 (2000). [CrossRef]
  10. J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrashort laser pulses with tilted pulse fronts,” Appl. Phys. B 78, 593–599 (2004). [CrossRef]
  11. K. L. Yeh, M. C. Hoffmann, J. Hebling, and A. Keith, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90(17), 171121 (2007). [CrossRef]
  12. K.-H. Lin, C. A. Werley, and K. A. Nelson, “Generation of multicycle THz phonon-polariton waves in a planar waveguide by tilted optical pulse fronts,” Appl. Phys. Lett. 95(10), 103304 (2009). [CrossRef]
  13. D. H. Auston, K. P. Cheung, J. A. Valdmains, and D. A. Kleinman, “Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media,” Phys. Rev. Lett. 53(16), 1555–1558 (1984). [CrossRef]
  14. J. K. Wahlstrand and R. Merlin, “Cherenkov radiation emitted by ultrafast laser pulses and the generation of coherent polaritons,” Phys. Rev. B 68(5), 054301 (2003). [CrossRef]
  15. N. S. Stoyanov, T. Feurer, D. W. Ward, and K. A. Nelson, “Integrated diffractive terahertz elements,” Appl. Phys. Lett. 82(5), 674–676 (2003). [CrossRef]
  16. P. Peier, S. Pilz, F. Müller, K. A. Nelson, and T. Feurer, “Analysis of phase contrast imaging of terahertz phonon-polaritons,” J. Opt. Soc. Am. B 25(7), B70–B75 (2008). [CrossRef]
  17. Q. Wu, C. A. Werley, K. H. Lin, A. Dorn, M. G. Bawendi, and K. A. Nelson, “Quantitative phase contrast imaging of THz electric fields in a dielectric waveguide,” Opt. Express 17(11), 9219–9225 (2009). [CrossRef] [PubMed]
  18. C. A. Werley, Q. Wu, K. H. Lin, C. R. Tait, A. Dorn, and K. A. Nelson, “A comparison of phase sensitive imaging techniques for studying THz waves in structured LiNbO3,” J. Opt. Soc. Am. B 27(11) 2350-2359 (2010). [CrossRef]
  19. E. A. Bahaa, B. Saleh and M. C. Teich, Fundamentals of photonics (JOHN WILEY&SONS, 1991)
  20. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge,1999).
  21. D. H. Auston and M. C. Nuss, “Electrooptic generation and detection of femtosecond electrical transients,” IEEE J. Quantum Electron. 24(2), 184–197 (1988). [CrossRef]
  22. A. S. Barker and R. Loudon, “Dielectric properties and optical phonons in LiNbO3,” Phys. Rev. 158(2), 433–445 (1967). [CrossRef]
  23. N. S. Stoyanov, T. Feurer, D. Ward, E. Statz, and K. Nelson, “Direct visualization of a polariton resonator in the THz regime,” Opt. Express 12(11), 2387–2396 (2004). [CrossRef] [PubMed]
  24. S. Wang, M. L. Shah, and J. D. Crow, “Wave propagation in thin film optical waveguides using gyrotropic and anisotropic materials as substrates,” IEEE J. Quantum Electron. 8(2), 212–216 (1972). [CrossRef]
  25. D. P. G. Russo and J. H. Harris, “Wave propagation in anisotropic thin-film optical waveguides,” J. Opt. Soc. Am. 63(2), 138–145 (1973). [CrossRef]
  26. W. K. Burns and J. Warner, “Mode dispersion in uniaxial optical waveguides,” J. Opt. Soc. Am. 64(4), 441–446 (1974). [CrossRef]
  27. V. Ramaswamy, “Propagation in asymmetrical anisotropic film waveguides,” Appl. Opt. 13(6), 1363–1371 (1974). [CrossRef] [PubMed]
  28. S. Nemoto and T. Makimoto, “Further discussion of the relationship between phase and group indices in anisotropic inhomogeneous guiding media,” J. Opt. Soc. Am. 67(9), 1281–1283 (1977). [CrossRef]
  29. D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media-Part I: Optical axis not in plane of slab,” IEEE J. Quantum Electron. 14(10), 736–741 (1978). [CrossRef]
  30. D. Marcuse and I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. 15(2), 92–101 (1979). [CrossRef]

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