Forward and backward THz-wave difference frequency generations from a rectangular nonlinear waveguide

Yen-Chieh Huang; Tsong-Dong Wang; Yen-Hou Lin; Ching-Han Lee; Ming-Yun Chuang; Yen-Yin Lin; Fan-Yi Lin

doi:10.1364/OE.19.024577

1. Introduction

Difference frequency generation (DFG) of two lasers beating at THz frequencies in a nonlinear optical material is a popular optical technique to generate coherent THz wave radiation. However, owing to the vast difference of the mixing wavelengths, the THz wave is usually more absorptive in the material than the optical mixing waves. For example, lithium niobate, while being transparent in the optical spectrum, has a typical absorption coefficient of a few tens of cm⁻¹ at THz frequencies [1]. The THz wave experiences a pure absorption loss as soon as it leaves the gain region of the optical pump and signal beams. To overcome this problem, several schemes have been successfully implemented to couple out the THz wave as soon as possible for non-collinearly phase matched THz-wave DFG in lithium niobate [2]. For parametric frequency mixing, however, collinear phase matching is a preferred configuration to increase the parametric gain length and thus the wavelength conversion efficiency. With quasi-phase-matching (QPM), we have previously demonstrated collinearly phase matched THz-wave DFG in bulk periodically poled lithium niobate (PPLN) [3]. Unfortunately, the much longer wavelength of the THz wave could still make the generated THz wave quickly diffracted and absorbed outside the gain region of the optical pump and signal beams. GaAs is also a demonstrated QPM material for THz-wave DFG with much less absorption in the THz spectrum [4]. However, LN has a three-time larger nonlinear coefficient. If the diffraction induced absorption in LN can be alleviated, LN is still a promising material for high-efficiency THz DFG.

To reduce the diffraction-induced absorption, one could in principle design a nonlinear optical waveguide that guides the THz wave as well as the optical waves [5]. However, guiding an optical wave requires a waveguide aperture comparable to the optical wavelengths, which severely restricts the input and output powers of the mixing waves. Previously we have shown an experimental evidence of enhanced, non-collinearly phase matched THz-wave DFG from a 0.5-mm thick crystal slab of lithium niobate [6]. This crystal slab guides the THz wave in one transverse direction but behaves like a bulk crystal to the optical mixing waves. For what follows, we call such a waveguide a one-dimensional (1-D) nonlinear optical semi-waveguide (NOSW). In this paper, we compare collinearly phase matched THz-wave DFG in rectangular (2-D) and 1-D NOSWs of the same length made of PPLN. As will be shown below, the additional confinement of the THz wave in the other transverse dimension of the crystal indeed enhances the THz-wave output power under the same pump condition.

2. Theory

To be consistent with our following experiment using type-0 phase matched PPLN crystals, we choose z as the direction of polarization for all mixing waves and + x as the propagation direction for the pump and signal waves. Forward or backward THz-wave DFG refers to the propagation of the THz wave in the +x or −x direction, respectively. In a NOSW, the unguided optical component has a varying beam size along the propagation direction x. However, in our case, the optical mixing waves have a depth of focus significantly longer than the crystal length and the mode-radius variation is less than 2% over the whole crystal length. Therefore, it is a good approximation to write the electric fields of the collinearly propagating optical pump and signal waves with a constant Gaussian field profile in the transverse direction, given by $E_{i} (t, y, z, x) = Re [\sqrt{2 η_{i}} e_{i} (y, z) A_{i} (x) e^{j (ω_{i} t - β_{i} x)}]$ , where the subscript i = p, s denote the pump and signal waves, respectively, β is the propagation constant in the x direction, η = η₀/n is the intrinsic wave impedance in a material of refractive index n, and $e (y, z)$ is the transverse field profile with the normalization $\int \int_{- \infty}^{\infty} {| e (x, y) |}^{2} d x d y = 1$ , so that ${| A (x) |}^{2} = P (x)$ is the power of the wave at x. For the THz-wave DFG, the optical pump and signal wavelengths are nearly the same and $e_{p} (y, z) \approx e_{s} (y, z)$ for a fundamental Gaussian beam can be written as

e_{p, s} (y, z) = \frac{1}{w} \sqrt{\frac{2}{π}} e^{- (y^{2} + z^{2}) / w_{}^{2}},

where w is the average mode radius of the pump and signal waves in the nonlinear optical material. Likewise, the electric field of the guided THz wave can be expressed as

E_{T H z} (t, y, z, x) = Re [\sqrt{2 η_{T H z}} e_{T H z} (y, z) A_{T H z} (x) e^{j (ω_{T H z} t \mp β_{T H z} x)}],

where the sign

\mp

preceding the propagation constant β_THz denotes a THz wave propagating along ± x.

In our experiment, pump depletion was negligible. Without pump depletion and signal absorption, the coupled-mode equations for continuous-wave, collinear DFG are given by [7]

\frac{\partial A_{s}}{\partial x} = - j κ_{s} A_{p} A_{T H z}^{*} e^{- j Δ β x},

\frac{\partial A_{T H z}}{\partial x} = \mp j κ_{T H z} A_{p} A_{s}^{*} e^{- j Δ β x} \mp \frac{α_{T H z}}{2} A_{T H z},

where α_THz is the absorption coefficient of the THz wave, κ_i is the nonlinear coupling coefficient, and

Δ β = β_{p} - β_{s} \mp β_{T H z} - k_{Q P M}

is the wave-number mismatch among the collinear pump, signal, THz waves, and the quasi-phase-matching (QPM) grating k_QPM. Without any initial THz-wave power at x = 0, the forward THz-wave output power at x = L normalized to the input signal power P_s(0) is given by

\frac{P_{T H z} (L)}{P_{s} (0)} = \frac{λ_{s}}{λ_{T H z}} e^{- α_{T H z} L / 2} \frac{Γ^{2}}{{| g_{f} |}^{2}} {| \sinh (g_{f} L) |}^{2},

where λ is the wavelength in vacuum, Γ is the parametric gain coefficient, and

g_{f} \equiv \sqrt{{(α_{T H z} / 4 - j Δ β_{f} / 2)}^{2} + Γ^{2}}

with

Δ β_{f} = β_{p} - β_{s} - β_{T H z} - k_{Q P M} .

The specific expression of the parametric gain coefficient Γ is given by

Γ^{2} = κ_{s} κ_{T H z} P_{p} (0) = Γ_{0}^{2} A_{p} ϑ^{2} = \frac{8 π^{2} d_{e f f}^{2} η_{0}}{n_{p} n_{s} n_{T H z} λ_{s} λ_{T H z}} I_{p} (0) A_{p} ϑ^{2},

where P_p(0) and I_p(0) are the initial pump power and intensity at x = 0, respectively, Γ₀ is the free-space plane-wave parametric gain coefficient [8], d_eff is the effective nonlinear coefficient,

A_{p}

is the pump-mode area, and

\sqrt{A_{p}} ϑ

is a modification factor to the free-space plane-wave parametric gain coefficient Γ₀ with the mode-overlapping integral

ϑ

defined as

ϑ = \int \int_{- \infty}^{\infty} e_{p} (y, z) e_{s} (y, z) e_{T H z} (y, z) d y d z .

It is straightforward to show that in the limit of a much larger THz-wave mode size than the optical one, $ϑ^{2}$ approaches the inverse of the THz-wave mode area or $ϑ^{2} \to 1 / A_{T H z}$ and $Γ^{2}$ is reduced from its free-space plane-wave value by a factor of $A_{p} / A_{T H z} .$ Since the THz wavelength is much longer than that of the optical mixing waves, this parametric gain reduction can be very significant due to fast diffraction of the THz wave in a bulk nonlinear optical material.

For the case of a backward THz wave propagating in the −x direction, we derive the THz-wave output power at x = 0 with zero initial THz-wave power at x = L, given by

\frac{P_{T H z} (0)}{P_{s} (0)} = \frac{λ_{s}}{λ_{T H z}} Γ^{2} {| \frac{\sin (g_{b} L)}{\frac{α_{T H z}}{4} \sin (g_{b} L) + g_{b} \cos (g_{b} L)} |}^{2},

where

g_{b} \equiv \sqrt{Γ^{2} - {(α_{T H z} / 4 - j Δ β_{b} / 2)}^{2}}

with

Δ β_{b} = β_{p} - β_{s} + β_{T H z} - k_{Q P M} .

It can be seen from Eq. (7) that the THz absorption increases the oscillation threshold and broadens the parametric gain bandwidth.

3. Experiment

We fabricated three 2-D PPLN NOSWs from congruent lithium niobate, as shown in Fig. 1(a) , but unfortunately broke the two shorter ones during polishing. For comparison, we also fabricated a 1-D PPLN NOSW with no wave confinement in the y direction. Both NOSWs are 25 mm long and 0.5 mm thick in the crystallographic x and z directions, respectively. The 2-D NOSW has a width of 0.6 mm along the y direction. All the guiding surfaces of the two crystals were optically polished. The ±x faces of the crystals were coated with anti-reflection layers at the pump and signal wavelengths. The QPM domain period of the two PPLN NOSWs is 65 μm, which permits phase matching for the generation of forward and backward THz waves at λ_THz = 197 and 469 μm, respectively, at room temperature with a pump wavelength at λ_p = 1538.9 nm.

Fig. 1 (a) Photograph of the three 2-D NOSWs fabricated from PPLN for our experiment. Experimental data in this paper were taken from the longest one. (b) Schematic of the forward and backward THz-wave DFG in PPLN NOSW. The pump and signal are initially combined from a distributed feed-back diode laser (DFBDL) and a tunable external cavity diode laser (ECDL), and then boosted up in power by an Erbium-doped fiber amplifier (EDFA) and a pulsed optical parametric amplifier (OPA). A 4K silicon bolometer detects the backward and forward THz waves before and after the PPLN NOSW, respectively.

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In our experiment, the waist radius of the optical pump and signal waves was 127-μm at the center of the PPLN crystals. Since the aperture of the PPLN crystals is 4-5 times the waist radius of the signal and pump waves, the PPLN crystals appear as a bulk material to the optical mixing waves. However, the very same crystals act as a waveguide for the THz waves, because the diffraction angles of the forward and backward THz waves are 0.1 and 0.23 rad, respectively, for an initial beam radius comparable to that of the optical waves. The crystal apertures are relatively large and capable of accommodating many waveguide modes for the THz waves. However, the fundamental THz-wave mode overlaps well with the Gaussian optical mode and is the dominant mode to build up in such a highly absorptive NOSW [6].

Figure 1(b) shows the schematic of the THz-wave DFG experiment. The pump and signal waves are first combined from a fixed-frequency distributed-feedback diode laser (DFBDL) at 1538.9 nm and an external-cavity diode laser (ECDL) with its wavelength tuned to the phase matching one for the downstream THz-wave DFG. An amplifier system, consisting of an Erbium-doped fiber amplifier (EDFA) followed by a pulsed two-color optical parameter amplifier (OPA), boosts each of the pump and signal energy to 9.7 μJ/pulse in a 360-ps pulse width. The forward propagating signal and pump pulses were then focused to the center of the PPLN NOSW. A silicon bolometer was installed before and after the PPLN NOSW to detect the backward and forward THz waves, respectively [3].

The largest possible THz mode area is the aperture area of the 2-D PPLN NOSW, which can be described by $e_{T H z} (y, z) = rect (y / l_{y}) \times rect (z / l_{z}) / \sqrt{l_{y} l_{z}},$ where $rect (r / l)$ is a rectangular function with a unit amplitude inside and zero amplitude outside the range of −l/2 < r < l/2. The maximum gain reduction factor thus estimated from Eq. (6) is $\sqrt{A_{p}} ϑ \approx 0.29$ , which is about the square root of the area ratio $A_{p} / A_{T H z}$ , as expected for a large mode-area mismatch. Therefore, the actual parametric gain coefficient $Γ$ for the 2-D NOSW could be reduced to about 1/3 of its free-space plane-wave value. Given d_eff = 168 × 2/π = 107 pm/V [9] and n_p = n_s = 2.14 [10], the reduced parametric gain is estimated to be $Γ = Γ_{0} \sqrt{A_{p}} ϑ = 0.65, 0.44$ cm⁻¹ for the forward and backward THz waves with refractive indices of 5.22 and 5.05 [3], respectively. For the 1-D NOSW, the aforementioned theory is not valid due to the fast variation of the THz wave beam size along the propagation direction. However, one would expect a smaller effective parametric gain coefficient Γ and thus a smaller growth rate for the THz wave resulting from worsened mode mismatch and diffraction-induced absorption in the 1-D NOSW.

Figure 2(a) shows the measured DFG tuning curves for the forward THz waves at 197 μm generated from the 1-D (crosses) and 2-D (dots) PPLN NOSWs with a pump intensity of 102 MW/cm². In the plot, the dashed and continuous lines are fitting curves using Eq. (4) with Γ = 0.53 and 0.65 cm⁻¹ for the 1-D and 2-D NOSWs, respectively, given α_THz = 40 cm⁻¹ at 1.5 THz for congruent lithium niobate [1]. As expected, the effective parametric gain coefficient for the 1-D NOSW is smaller due to diffraction of the THz wave in the y direction. The parametric gain of Γ = 0.65 cm⁻¹ for the 2-D NOSW agrees well with the theory for a large mode-area mismatch. The measured tuning curves clearly show an enhancement factor of 1.6 at the phase matching wavelength for the THz-wave output power from the 2-D NOSW. Figure 2(b) shows the THz-wave output power versus pump intensity from the 1-D (squares) and 2-D (circles) NOSWs at the phase matching wavelength. During the measurement, the ratio of the pump to signal intensity remained one. The power enhancement factor of the THz wave is nonlinearly increased with the pump intensity, which indicates some exponential gain for the THz wave as predicted by Eq. (4). The estimated forward THz-wave pulse energies generated inside the 1-D and 2-D NOSWs are 41 and 63 pJ, respectively.

Fig. 2 (a) Measured forward THz-wave tuning curves from the 1-D (dots) and 2-D (crosses) PPLN NOSWs. The dashed and continuous lines are fitting curves of Eq. (4) with Γ = 0.53 and 0.65 cm⁻¹ for the 1-D and 2-D NOSW, respectively, given an attenuation coefficient of 40 cm⁻¹. (b) The Measured THz-wave output power versus pump intensity from the 1-D (squares) and 2-D (circles) NOSWs, indicating some nonlinear THz-wave power enhancement in the 2-D NOSW.

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Figure 3(a) shows the measured DFG tuning curves for the backward THz wave at 469 μm generated from the 1-D (cross) and 2-D (dot) PPLN NOSWs with a pump intensity of 104 MW/cm². The dashed and continuous lines are fitting curves using Eq. (7) with Γ = 0.34 and 0.44 cm⁻¹ for the 1-D and 2-D NOSWs, respectively, and with an assumed attenuation coefficient of 6 cm⁻¹. The reduced parametric gain coefficient for the 1-D NOSW also indicates a poorer beam overlap between the backward THz and forward optical waves due to diffraction of the THz wave in the y direction. The parametric gain of Γ = 0.44 cm⁻¹ for the 2-D NOSW agrees well with the theory for a large mode-area mismatch. The measured tuning curves clearly show a power enhancement factor of 1.8 at the phase matching wavelength for the 2-D NOSW. Figure 3(b) shows the THz-wave output power versus pump intensity from the 1-D (squares) and 2-D (circles) NOSWs at the phase matching wavelength, indicating enhanced output power from the 2-D NOSW over the whole range of measurement. The estimated backward THz-wave pulse energies generated inside the 1-D and 2-D NOSWs are 0.3 and 0.5 nJ, respectively.

Fig. 3 (a) Measured backward THz-wave tuning curves from the 1-D (dots) and 2-D (crosses) PPLN NOSWs. The dashed and continuous lines are fitting curves of Eq. (7) with Γ = 0.34 and 0.44 cm⁻¹ for the 1-D and 2-D NOSWs, respectively, and with an assumed attenuation coefficient of 6 cm⁻¹. (b) The Measured backward THz-wave output power versus pump intensity from the 1-D (squares) and 2-D (circles) NOSWs, indicating enhanced THz-wave power from the 2-D NOSW over the whole range of measurement.

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It should be pointed out that the THz wave in the 1-D NOSW could have a smaller diffraction angle in the unconfined y direction. With a fixed detection path for the THz wave from the 1-D and 2-D NOSW, the bolometer could have collected less THz-wave power from the 2-D NOSW. If so, the power enhancement factor reported above is conservative. However, the high absorption of the THz wave in lithium niobate would have concentrated the THz-wave power to an emitting aperture comparable to that of the optical beam for both NOSWs. Under this situation, the comparison curves reported above do not require further correction.

4. Summary

In summary, we have reported enhanced THz-wave DFG from a rectangular PPLN crystal rod with an aperture of 0.5 × 0.6 mm². This crystal appears as a bulk material to the optical mixing waves near 1.54 μm but behaves like a 2-D waveguide to the forward and backward THz waves at 197 and 469 μm, respectively. The THz-wave output power from this 2-D NOSW is increased by a factor of 1.6 and 1.8 for the forward and backward THz waves, respectively, at pump and signal intensities of ~100 MW/cm², when compared with that from a 1-D PPLN NOSW of the same thickness and length under the same pump condition. The 2-D waveguide confinement reduces the diffraction-induced absorption for the THz wave during the DFG process. In addition, we pointed out that, for most low-efficiency THz-wave DFG in bulk nonlinear optical materials, the major gain reduction mechanism could be the large mode-area mismatch between the THz and optical waves. This proof-of-principle experiment indeed points out a direction of potentially high-efficiency THz-wave DFG in a PPLN NOSW with further improved mode-area overlap for the THz and optical waves.

Acknowledgments

Shayeganrad has helped to correct some minor mistakes in this paper. This work is supported by National Science Council under Contract NSC99-2622-M-007-001-CC1.

References and links

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Forward and backward THz-wave difference frequency generations from a rectangular nonlinear waveguide

Abstract

1. Introduction

2. Theory

3. Experiment

4. Summary

Acknowledgments

References and links

Cited By

Figures (3)

Equations (7)

Optics Express