## Abstract

We present a photonic band gap (PBG) structure (or nonlinear photonic crystal) design for terahertz (THz) wave parametric generation, whose component materials have a small refractive index difference in the near infrared and a large index difference for THz waves. The structural dispersion of such a PBG structure is strong in the THz range but negligible in the optical range. The former allows the phase-matched pump wavelength to be placed in the near infrared to eliminate two-photon absorption of the pump and signal beams. The latter leads to a crystal layer fabrication tolerances of a few micrometers and traditional polishing methods are suitable for device fabrication. The added design flexibility also allows the use of the most efficient crystal orientations.

©2006 Optical Society of America

## 1. Introduction

Many future earth science remote sensing programs and missions require microwave to submillimeter wavelength high performance components, in particular sources, operating in the 1-cm to 100-μm wavelength range (or a frequency range of 30 GHz to 3 THz) because the atmospheric attenuation of this range is relatively low compared with infrared and optical wavelengths. The components used in this range are relatively small and light compared with those required for microwaves, which is an important consideration for military and space-borne applications.[1, 2] In addition, millimeter and submillimeter wave have enormous potential for imaging and sensing applications in biology, medicine and homeland security.[3, 4]. Hence, many researchers have been attracted to the science and technology of this mostly unexplored spectrum range.

Several techniques have been applied in developing THz sources [5–14], Among them, optical parametric conversion, as well as free electron lasers, have the advantage of the widest tunable range over all other technologies. The potential of being combined with fiber optical communication systems allows possibly easier modulation and demodulation. However, high efficiency optical parametric conversion cannot be readily obtained in THz parametric generation because the parametric conversion efficiency decreases with decreasing idler frequency. One common way to increase the efficiency is to use pulsed pump sources. Unfortunately, the efficiency improvement based on increased pump intensity is limited by two-photon absorption.[15]

Another effective way to improve conversion efficiency is to use artificial periodic structures. By periodically alternating ferroelectric domain structures or materials to generate photonic band gap structures it is possible to use the most efficient nonlinear optical coefficients. Both these methods also have the advantage that the walk-off effect is eliminated so that a long effective interaction length can be achieved. Quasi-phase-matched (QPM) crystals based on the former structure have been well studied with the fabrication methods of periodic poling, optical contact and diffusion-bonded stacking.[16–18]. They have been successfully used in near IR for second harmonic and parametric generation, as well as in THz for difference frequency generation.[19] Nonlinear photonic crystals formed by the latter structure have also been investigated, in particular, for second harmonic generation.[20
21]. Compared with QPM, the construction of nonlinear photonic crystals is more flexible in material selection and has the new features of Fabry-Perot resonance and structural dispersion.[22] However, due to requirements imposed by the design of the band gaps for the near infrared pumping wavelengths, the fabrication tolerance of a nonlinear photonic crystal is much smaller than that of a QPM crystal, otherwise the beam propagation loss will be very high. To our best knowledge, so far few practical nonlinear photonic crystals for three wave mixing have been reported. Recently, the possibility of using a photonic band gap (PBG) structures (or photonic crystals) to enhance the conversion efficiency of THz difference frequency generation has been investigated.[23, 24]. Theory shows that the Fabry-Perot resonance effect is strong on the band-edge and can improve the efficiency of nonlinear optical parametric processes by several orders of magnitude through the use of one-dimensional photonic bandgap (PBG) structures.[23–24] By using this effect, structures with layer thickness smaller than the pump wavelength for THz generation have been investigated by J. W. Haus *et al*. [Ref.24]. Such a structure has an advantage that the phase mismatch of each crystal layer is negligible.[24] By contrast, we focus on the use of structural dispersion for phase matched THz wave generation in structures with layer thicknesses in the submillimeter range, which can be made by traditional crystal cutting and polishing techniques.[23] Normally, the need to control the bandgaps for the near infrared pumping wavelengths would require fabrication tolerances in the sub-micrometer range or even smaller that would be prohibitive for manufacture by stacking of layers fabricated by cutting and polishing.

In this paper, we present the design of a THz generation PBG device with the component materials that have a small refractive index difference at near infrared pump wavelengths and a large refractive index difference for THz waves. This results in substantial PBG features in the THz range but not in the optical or near infrared range. Negligible PBG features in the optical range lead to micrometer scale fabrication tolerances. Large structural dispersion at THz frequencies allows the phase matched pump wavelength to be placed in the near infrared to eliminate two-photon absorption of the pump and signal beams. Also broad tuning range can be achieved by tuning the pump wavelength. The added design flexibility also allows the use of the most efficient crystal orientations.

## 2. Analysis

It is well known that the size of the band gap of a PBG device depends on the refractive index difference between the two component materials.[25] If it is desired to reduce the band gap, the refractive index difference between the two materials should be small. If such a design were used for optical parametric conversion, in which all the three beams had wavelengths of the same order of magnitude, we would lose all the features of the PBG structure. However, for THz difference frequency generation, the wavelengths of pump and signal beams are far smaller than that of THz wave. It is not difficult to find two electro-optic (EO) crystals with refractive indexes that are almost the same in the optical frequency range but very different in the THz frequency range. By using this design, we will lose the PBG features in the optical range, but still have large structural dispersion in the THz range, which can be used for phase matching and to efficiently utilize the nonlinear coefficients of the EO crystals. Note that for most non-birefringent EO crystals, the refractive indexes in the THz range are about 3~4, but for some birefringent EO crystals, it could be as large as 5~7. So we can use both types of crystals to construct the PBG device.

To efficiently utilize the nonlinear coefficient of the nonlinear crystals, the coherence length of each component crystal should be larger than the layer thickness. The coherence length of a bulk crystal, as well as a PBG device, can be calculated as *π*/∣Δ*k*∣ with phase mismatch:

where *n*(*λ*) is the refractive index of a bulk crystal or the effective index of a PBG device^{22} as a function of the wavelengths of the pump (*λ _{p}*), signal (

*λ*) and THz (

_{s}*λ*) beams. It is worth mentioning that Eq. (1) can be simplified to Δ

_{T}*k*= 2

*π*(

*n*(

_{g}*λ*)-

_{p}*n*(

*λ*))/

_{T}*λ*with the group velocity index

_{T}*n*(

_{g}*λ*) =

*n*(

*λ*)[1-

*λ*(

*d*(

*n*(

*λ*))/

*dλ*)/

*n*(

*λ*)] for a non-birefringent crystal or for cases in which the pump and signal have parallel polarization and the phase matching condition becomes

*n*(

_{g}*λ*) =

_{p}*n*(

*λ*), which is often used for THz generation with ultrafast lasers. However, the group velocity approximation is not available for the case of a birefringent crystal using orthogonally polarized pump and signal. In general, the refractive index of a crystal in the THz range is smaller than the group velocity index in the UV and larger than that in the near IR range so that one can get phase matching (Δ

_{T}*k*= 0) for certain ranges of THz waves with a special pump wavelength. For most EO crystals, this special wavelength is much smaller than 1 μm, and two-photon absorption (or even the band to band transition absorption) of the pump and signal beams seriously reduce the parametric conversion efficiency. To eliminate these losses, the pump wavelength should be not shorter than 1 μm. With a near IR pump wavelength, most non-birefringent crystals can have a coherence length of a few hundred micrometers or more, regardless of whether the polarizations of the pump and signal beams are parallel or perpendicular. For birefringent crystals, the case of pump and signal beams with parallel polarization is similar to that of a non-birefringent crystal, but if the pump and signal beams have perpendicular polarizations, the coherence length is smaller than 100μm for most birefringent crystals at the orientation with most efficient nonlinear optical coefficient, due to the large refractive index difference between two polarizations.

For a PBG device constructed with *m* layers each of two kinds of crystal with thickness of *L*
_{1} and *L*
_{2} respectively, if structural dispersion is neglected for the moment (i.e. without considering boundary conditions or the Bloch condition), from Eq. (1), we know that the total mismatch phase is

where Δ*k _{j}* = 2

*π*⌊

*n*(

_{j}*λ*)/

_{p}*λ*-

_{p}*n*(

_{j}*λ*)/

_{s}*λ*-

_{s}*n*(

_{j}*λ*)/

_{T}*λ*⌋ for each crystal type, j=1,2 and the average phase mismatch of the device is:

_{T}with *n*¯(*λ*) = (*L*
_{1}
*n*
_{1}(*λ*) + *L*
_{2}
*n*
_{2}(*λ*))/(*L*
_{1} + *L*
_{2}), average refractive index of PBG device. If we take structural dispersion into account, the phase mismatch of the PBG device can be expressed as:

where *n _{eff}*(

*λ*) is the effective index. We choose component materials so that the difference between

*n*

_{1}(

*λ*)and

*n*

_{2}(

*λ*) is small enough in the near IR for structural dispersion to be negligible there and

*n*(

_{eff}*λ*) =

*n*(

*λ*), which is demonstrated by our simulation results [see Fig. 1(b)]. To achieve phase matching for the PBG device (Δ

*k*= 0), we can show from Eqs. (3) and (4) that

_{eff}should be satisfied. With a pump wavelength sufficiently far from the intrinsic band edge, say longer than 1 μm, Δ*k* is generally negative for non-birefringent bulk EO crystals (except for GaAs) and for birefringent crystals if the pump and signal beams’ polarizations are parallel. In order to minimize the demands on structural dispersion for phase matching, we should design our structure to minimize Δ*k*¯. This is easier to do if we choose the signs of Δ*k*
_{1} and Δ*k*
_{2} to be opposite. That is, one of the Δ*k _{j}* should be positive, a circumstance that can be achieved with orthogonally polarized near IR pump and signal beams in a birefringent crystal such as LiTaO

_{3}. In this paper, we consider the case of generating the THz signal at the atmospheric transmission window of ~0.2 THz with pump wavelength near 1.55μm.

## 3. Simulation

Among conventional birefringent EO crystals, LiTaO_{3} has a very small birefringence (*n*
_{e}-*n*
_{o} ≈ 0.004 @ 1.55 μm at room temperature), with a reasonable EO coefficient (γ_{13} ≈ 8.9pm/V) and a small enough absorption coefficient (< 5 cm^{-1}) for o-beam generation at frequencies below 0.5THz. The refractive index of LiTaO_{3} is ~2.2 at near IR wavelengths and ~ 6.4 in the THz range.[26,27] The phase mismatch is calculated as +0.0012 μm^{-1}, corresponding to a coherence length of ~2.6mm at ~0.2 THz. We chose ZnSe as the non-birefringent material. It has a refractive index in near infrared close to that of LiTaO_{3} (~2.47) and a refractive index (~2.93) much smaller than that of LiTaO3 in the THz range.[28] It also has a fair EO coefficient (*γ*
_{41} = 1.8pm/V), and a very small absorption coefficient at THz range.[29] The phase mismatch is calculated as -0.0016 μm^{-1}, corresponding to a coherence length of ~1.9mm at 0.2 THz. So we use LiTaO_{3} and ZnSe to design the PBG device with the polarization of the pump beam perpendicular to that of signal beam and THz wave. To get good transmission at 0.2THz, we design the device as a half-wave/half-wave stack at that frequency. Accordingly, the layer thickness of LiTaO_{3} and ZnSe crystals are 116-μm and 256-μm, respectively. Below we consider a PBG device of 20 periods identically.

The transmission spectrum of a PBG structure will show interference effects due to multiple reflections in the PBG device layers and on the two end surfaces of the device if the device is surrounded by air. Eliminating surface effects will help us understand the dependence of the field enhancement effect on the PBG structure. Therefore, below we will consider the case of a PBG device is sandwiched by two thick ZnSe crystals, and calculate the transmission from the middle of one end ZnSe crystal to the middle of the other one to model a device with antireflection coatings.

The transmission of the PBG device was calculated using the well-known matrix transfer method.[30] Figure 2 shows the intensity transmission of ZnSe sandwiched PBG device (a) below 0.5 THz and (b) at 1.55-μm range of o-beam and (c) at 1.55μm range of e-beam. From Fig. 2 we know that the device has a transmission window between 0.13 THz and 0.28 THz and the band gaps of the device at the pump and signal wavelengths of o-beam and e-beam are very narrow. The interference signals on the spectra (Fig. 2) reveal that the field enhancement caused by the PBG structure is strong at the THz frequencies, in particular at the band-edges, but very weak at the optical range except in narrow resonances and can be neglected for most pump wavelengths. The variations in the depth of the narrow forbidden bands might be due to the difference of the dispersion of the two materials in the PBG device, causing a perturbation of the phase variation with wavelength.

To calculate the coherence length of the PBG device, first we need to know its effective index. The effective refractive index of the PBG device in the THz range and optical ranges are calculated with the method used by Centini et al. [Ref. 22], which derives the effective index from transmission of the PBG devices and takes the Bloch effect into account. It includes the multi-reflection forward- and backward-field overlap and phase modulation, which can be seen from the fluctuations on the effective index curves in Fig. 1. We know, from Fig. 1(a), that the structural dispersion in the working transmission window of 0.124THz to 0.274THz is much stronger than that of a bulk crystal. On contrary, it is shown in Fig. 1(b) that the effective refractive index (solid line) is close to the average refractive index (open circle or square points) in the optical range.

Figure 3 shows the coherence length of PBG device with pump wavelength of 1 μm, 1.064 μm, 1.31 μm, 1.55 μm or 2.00μm as labeled. From Fig. 3 we know by tuning the pump wavelength from 1 μm to 2 μm we tune the THz frequency from ~0.14THz to ~0.26 THz. To demonstrate this result, we also calculated the parametric conversion efficiency following the matrix formalism of Y. Jeong and B. Lee, which takes the boundary condition (or multiple reflection effects) at each interface and the phase mismatch at each layer into account.[31]. Assuming the PBG device is pumped by two near infrared laser pulses of 10-ns and 1 mJ/pulse, the THz photon conversion efficiencies in the range from 0.1 THz to 0.3 THz with pump frequencies from 1 μm to 2 μm (corresponding to frequency from 300THz to 150THz) are calculated and plotted in Fig. 4. For each pump frequency, there is one THz frequency with maximum conversion efficiency due to phase-matching, which is about three orders of magnitude larger than a bulk LiTaO_{3} or ZnSe crystal of same length and same orientation pumped by above same conditions. The phase-matching THz frequencies in Fig. 4 are totally consistent with the numerical results of maximum coherent lengths shown in Fig. 3. Place ment of the phase-matched pump wavelength in the near infrared will eliminate the two-photon absorption losses.

Because the transmission and the refractive index of the PBG device at the optical range is very similar to a bulk crystal, the fabrication tolerance will not be limited to the order of pump wavelength. The only thing we need to take care with is the effective index in the THz range. The simulation results tell us that 10-μm fabrication error only causes the transmission window at the THz range shift 0.004THz. Because a THz wave with a long coherence length is not so close to the edge of the transmission window (0.127THz), such a small shift will not cause a large change of the effective index in the THz range and cause obvious coherence length change. So the fabrication tolerance could be as large as a few micrometers.

## 4. Conclusion

In conclusion, we introduce a practicable PBG structure for ~0.2 THz wave difference frequency generation, which has a fabrication tolerance as large as a few micrometers and can be fabricated by traditional polishing. This PBG structure efficiently utilizes the nonlinear coefficient of the construction crystals, and has lower two-photon absorption at phase-matched pump wavelength than the two bulk crystals and a lower absorption coefficient at THz range than LiTaO_{3} by itself.

## Acknowledgments

This work was supported by MDA grant.

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