## Abstract

We demonstrate an all-single mode structure which enables continuous phase matching of difference frequency generated THz light from the near-IR. This structure provides a long interaction length by way of well-confined collinear propagation of pumps and product without diffraction, resulting in high conversion efficiency. A LiNbO_{3} version of this structure achieved a power-normalized conversion efficiency of 1.3×10^{-7} W^{-1} - some 23 times larger than the largest previously reported results.

© 2008 Optical Society of America

## 1. Introduction

The large power densities available with fiber-coupled optical sources make far-infrared (THz) generation by way of guided wave difference frequency generation (DFG) an attractive alternative to its bulk equivalent. Incorporating optical waveguiding in *χ*
^{(2)} materials can preserve the initial optical pump power densities and provides strong pump/product overlap over long interaction lengths without concomitant diffraction. A necessary condition for an efficient nonlinear process is the maintenance of a spatial synchrony between the induced polarization wave and the electromagnetic wave product. For collinear guided wave DFG, *ω*
_{3}=*ω*
_{1}-*ω*
_{2}, this can be expressed as Δ*β*=*β*
_{1}-*β*
_{2}-*β*
_{3}=0, where *β*
_{j} are the mode propagation constants for the pumps (*j*=1,2) and the THz product (*j*=3).

While quasi-phase matching (QPM) is a powerful method for periodically re-establishing the phase matching condition [1], the conversion efficiency of the nonlinear process is reduced by [2/(*πm*)]^{2}, corresponding to a 41% reduction in conversion efficiency for first order (*m*=1) QPM. The more efficient continuously phase matched DFG has been produced by using inter-modal dispersion in waveguides [2–6]. Here, these waveguides typically support modes of mixed order at the pump and/or the product wavelengths, compromising the confinement and overlap of the optical fields. (We note that planar waveguides can be thought of as supporting many transverse modes across the large dimension.) As we demonstrate, with the proper choice of materials and dimensions, we can fabricate a pair of single mode 2-D (channel) waveguides, one for the two pumps embedded inside a second one for the THz product, and achieve continuous phase matching with good optical confinement of the pumps and low THz mode propagation loss.

## 2. Embedded waveguide structure

A LiNbO_{3}-based version of this structure is shown in Fig. 1. The pump waveguide consists of a standard Ti-diffused channel waveguide [7], single mode at 1.5 µm, in a LiNbO_{3} c-cut film.

The single mode THz waveguide core is the LiNbO_{3} film; its cladding layers are the quartz substrate and a high density polyethylene (HDPE) ridge, which also provides lateral confinement. Because of the strong confinement properties of the Ti diffused waveguide, the pump mode properties are essentially unaffected by the choice of THz waveguide parameters (i.e., the LiNbO_{3} film thickness for dimensions ~10 µm or greater, and the choice of outer cladding materials).

The phase matching condition for guided wave THz DFG from the near-IR pumps can be written as *n*
_{eff,3}≈*n*
_{eff,1,2}, where *n*
_{eff,3} is the THz mode effective index and *n*
_{eff,1,2} is the effective index of the two pumps which are assumed to be roughly equal. The THz waveguide parameters are used to control *n*
_{eff,3}, and therefore the phase matching condition, with little or no influence on *n*
_{eff,1,2}. Because the refractive index of LiNbO_{3} in the THz regime is large, *n _{bulk}*~5, low refractive index cladding layers (HDPE and quartz) are necessary to produce a THz mode effective index that is approximately equal to that of the two pumps in the near-IR (

*n*

_{eff,1,2}~2).

## 3. Analysis

The THz output power of the DFG process can be written as *P*
_{3}=*η _{p}*

*P*

_{1}

*P*

_{2}, where

*P*

_{1}and

*P*

_{2}are the input pump powers at

*ω*

_{1}and

*ω*

_{2}, respectively, and are assumed to be undepleted [8]. The pump-normalized conversion efficiency for DFG,

*η*(W

_{p}^{-1}), for single mode pumps and THz waveguides, is given by

where Δ*β*=*β*
_{1}-*β*
_{2}-*β*
_{3} is the phase mismatch, *L* is the waveguide length, and *E _{mt}* and

*H*are the transverse electric and magnetic field mode profiles at

_{mt}*ω*, for

_{m}*m*=1, 2, and 3, normalized according to $\frac{1}{2}\iint \mathrm{Re}\left[\left({\stackrel{\rightharpoonup}{E}}_{\mathrm{mt}}\times {\stackrel{\rightharpoonup}{H}}_{\mathrm{mt}}^{*}\right)\right]\xb7\hat{z}\mathrm{dxdy}=1W$ . Here,

*H⃑*=

_{mt}*i*/(

*ω*)∇×

_{m}μ_{o}*E⃑*and $\Omega =\frac{1}{2}\iint \left({\stackrel{\rightharpoonup}{E}}_{3t}\times {\stackrel{\rightharpoonup}{H}}_{3t}\right)\xb7\hat{z}\mathrm{dxdy}$ is the orthogonality constant for the lossy THz mode [9], whose amplitude absorption coefficient is

_{mt}*α*

_{3}. For the LiNbO

_{3}embedded waveguide structure the two pumps and THz modes are chosen TM polarized and as a result,

*d*=

_{eff}*d*

_{33}=175 pm/V, where

*d*

_{33}is the appropriate nonlinear coefficient of LiNbO

_{3}[8, 10]. Choosing quartz and HDPE as the cladding layers, the phase matched THz frequency (

*ω*

_{3}=

*ω*

_{1}-

*ω*

_{2}) is then determined by the LiNbO

_{3}film thickness, and to a smaller extent the HDPE ridge width.

## 4. 1-D model calculation

A one-dimensional (1-D) analysis of the LiNbO_{3} embedded waveguide structure was initially conducted to find the phase matched THz frequencies and relative output powers. A transfer matrix technique [9] was used to calculate the TM mode propagation constants and 1-D mode profiles of the two pump and THz light fields. The Ti:LiNbO_{3} pump waveguide was modeled as a step index slab waveguide, with quartz and LiNbO_{3} claddings layers and a 5 µm Ti:LiNbO_{3} core (Fig. 2(a)). The refractive index difference between the Ti:LiNbO_{3} core and LiNbO_{3} cladding was fixed at 0.005. The THz waveguide was also modeled as a slab waveguide, with a LiNbO_{3} core of thickness T surrounded by quartz and HDPE cladding layers. The frequency-dependent refractive index values in the near-IR and THz were taken from [11–15]. The material absorption of quartz, LiNbO_{3}, and HDPE were included in the THz mode calculations. See Table 1. The propagation constants and electric field mode profiles were used in a 1-D version of Eq. (1) to calculate the output power (W/m) of the LiNbO_{3} structure for a given LiNbO_{3} film thickness and THz frequency. See Fig. 2(b). The calculated maximum THz output power was found to occur at a LiNbO_{3} thickness of 14 µm, at a frequency of 1.4 THz. The lower white regions in Fig. 2(b) do not support a fundamental THz mode; the upper white regions correspond to multimode conditions for the THz waveguide.

## 5. Experiment

The LiNbO_{3}-based embedded waveguide structure [16] was fabricated as follows. Ti:LiNbO_{3} waveguides were fabricated by diffusing 100 nm thick, 9 µm wide titanium strips into the -c face of 0.5 mm LiNbO_{3}, using the diffusion procedure in [17]. The LiNbO_{3} was then bonded -c face down to 0.5 mm x-cut quartz using epoxy and mechanically lapped to a thickness of 14 µm with an optical quality polish. The end faces were blocked with quartz feet and polished to an optical finish; the exit foot was removed before measurements. The final device length was 11 mm. Figure 3 is a backlit microscope image of the polished end face showing three Ti:LiNbO_{3} waveguides embedded in the 14 µm film. Lateral confinement of the THz was produced by an inverted polyethylene ridge, 0.5×0.5×11 mm, aligned with the Ti:LiNbO_{3} waveguide (Fig. 1).

The experimental setup is shown in Fig 4. A fiber-waveguide-(apertured near field) insertion loss of -3.4 dB was measured at 1554.6 nm for the Ti:LiNbO_{3} waveguides used in the experiment. This value is in good agreement with typical insertion loss measurements of Ti-diffused waveguides in bulk LiNbO_{3} substrates [7]. An electrically modulated (on-off, 155 Hz) distributed feedback laser (DFB) at 1554.6 nm (spectral width ~2 MHz) and a continuously tunable (from 1.5450 µm to 1.5426 µm) external cavity laser diode (ECLD) (spectral width ~0.2 MHz) served as the pump sources and were coupled together through a 3dB splitter and amplified by a 2 W erbium doped fiber amplifier (EDFA). The two amplified pumps were constantly monitored to insure equal pump powers of 760 mW out of the launch fiber. Polarization rotators insured TM polarized pumps at the end of the EDFA output fiber, which was aligned directly with the Ti:LiNbO_{3} waveguide. The TM polarized THz signal was focused with a polyethylene (PE) lens (7.62 cm diameter, 20 cm focal length) into a liquid He cooled Si bolometer. Using the THz mode profile calculated from a 2-D beam propagation simulation (Fig. 5(b)), the capture efficiency of the PE lens was calculated [18] to be 20%. Unwanted radiation was filtered out with 0.4 mm thick black PE film (38% THz transmission) mounted on the bolometer entrance window and an 80 µm cut-on filter located inside the bolometer. The IR pumps were strongly scattered by the PE lens, and any residual IR at the bolometer was absorbed by the black PE film. Readings were taken from a lock-in-amplifier connected to the Si bolometer output.

The measured data was referenced to a residual background level of 1.3 nW (pumps on and difference frequency far from phase matching) and re-scaled to take into account the 20% capture efficiency of the PE lens and the 62% absorption by black PE. Figure 6 shows the measured THz output power from the LiNbO_{3} embedded waveguide structure vs. the difference frequency. A maximum output power of 75 nW was observed at 1.325 THz, corresponding to a power-normalized conversion efficiency of *η _{p}*=

*P*

_{3}/(

*P*

_{1}

*P*

_{2}=1.3×10

^{-7}W

^{-1}. The corresponding FWHM is ~0.5 THz. The THz signal could be extinguished either by rotating one of the polarization controllers (Fig. 4) or by turning off one of the pumps. It should be noted that

*P*

_{1},

*P*

_{2}, and

*P*

_{3}are instantaneous powers, i.e., peak powers in the case of an optical pulse. When considering average powers ${\overline{P}}_{m}$ and rectangular pulse sources, the power-normalized conversion efficiency becomes ${\eta}_{p}=\frac{{\overline{P}}_{3}\delta t}{\left({\overline{P}}_{1}{\overline{P}}_{2}T\right)}$ , where

*δt*and

*T*are the pulse width and period, respectively, of the pumps and THz.

#### 2-D model calculation

The output power spectrum was recalculated from Eq. (1) using mode profiles and propagation constants calculated from a 2-D beam propagation simulation including THz loss (Fig. 5). The pump mode profiles were based on a standard Ti diffused channel waveguide in a LiNbO_{3} film. Titanium diffusion constants were taken from [7]. As above, the THz waveguide consisted of the LiNbO_{3} film, clad below with quartz and above with a 500 µm×500 µm HDPE ridge to provide lateral confinement. We note that the large transverse mode profile of the THz mode (Fig. 5(b)) results in a mode propagation loss of 2*α*
_{3}=2 cm^{-1}, which is much less than bulk LiNbO_{3} (2*α _{bulk}*~32 cm

^{-1}) [4,16]. Pump powers and the device length were taken to be 760 mW and 11 mm, respectively (the values used in the experimental work). An excellent agreement between the 2-D model calculation and the experimentally measured power spectrum (Fig. 6) was obtained if a 15 µm LiNbO

_{3}film thickness was used in the calculation instead of the experimental value of 14 µm. (A 14 µm LiNbO

_{3}film thickness produced a nearly identical calculated power spectrum shifted by +57 GHz.) A maximum output power of 104 nW is calculated, corresponding to a power normalized conversion efficiency

*η*=

_{p}*P*

_{3}/(

*P*

_{1}

*P*

_{2})=1.8×10

^{-7}W

^{-1}. The two absorption troughs located at 1.312 and 1.337 THz were observed in repeated experiments, and maybe due to the thin epoxy layer or absorption features of the air path.

## Conclusions

In conclusion, a LiNbO_{3}-based embedded waveguide pair was shown to achieve 2-D confinement and single mode operation of both the near-IR pumps and the THz product, and low THz mode propagation loss. Of the previously reported THz DFG [2, 19–27], using beam optics or waveguides, we calculate that [19] has the largest normalized power conversion efficiency, estimated at *η _{p}*~5.6×10

^{-9}W

^{-1}. Our measured results reported here,

*η*=1.3×10

_{p}^{-7}W

^{-1}, are 23 times larger than this best previously reported result. Since the output power is proportional to the product of the two pump powers, the output power of our structure could be dramatically increased by replacing the very modest CW pump powers used in the experiment with the high peak output powers available from pulsed fiber lasers. In addition, the THz capture efficiency could be further improved by using ball lenses or off-axis parabolic mirrors. Other embedded waveguide configurations based on III–V materials could provide even further improvements in the power-normalized conversion efficiency due to their low THz absorption losses.

## Acknowledgments

We thank Dan van der Weide for providing the Si bolometer and Paul Voyles for use of his lapping equipment. We also thank Alan Bettermann and Charles Paulson for their assistance with the THz detection system and Lianne Streng for her work on the mechanical lapping of LiNbO_{3}. We also acknowledge the support of the NSF Division of Material Research and the facilities provided by the UW Materials Research Science and Engineering Center.

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