## Abstract

Optical parametric mixing is a popular scheme to generate an idler wave at THz frequencies, although the THz wave is often absorbing in the nonlinear optical material. It is widely suggested that the useful material length for co-directional parametric mixing with strong THz-wave absorption is comparable to the THz-wave absorption length in the material. Here we show that, even in the limit of the absorption loss exceeding parametric gain, the THz idler wave can grows monotonically from optical parametric amplification over a much longer distance in a nonlinear optical material until pump depletion. The coherent production of the non-absorbing signal wave can assist the growth of the highly absorbing idler wave. We also show that, for the case of an equal input pump and signal in difference frequency generation, the quick saturation of the THz idler wave predicted from a much simplified and yet popular plane-wave model fails when fast diffraction of the THz wave from the co-propagating optical mixing waves is considered.

©2013 Optical Society of America

## 1. Introduction

In an ordinary optical device, optical amplification or oscillation is not possible for absorption loss exceeding amplification gain. Therefore, the operation wavelength of an optical amplifier or oscillator is usually designed in the transparent range of a gain material. This design rule is also widely adopted for an optical parametric amplifier. However, in the electromagnetic spectrum, the THz-frequency is relatively unexplored due to the scarcity of THz radiation sources. In order to generate a valuable THz wave, one often chooses to perform optical parametric amplification (OPA, input signal much weaker than the pump) or difference frequency generation (DFG, input signal comparable to the pump) in some quadratic nonlinear optical material with appreciable idler absorption at THz frequencies. One notable example is the THz-wave optical parametric generation in lithium niobate developed in the late 1960s and 1970s [1–3], which has been closely followed by Ito and his associates [4] with great improvements on output coherence and efficiency. However, the absorption length of a THz wave in lithium niobate is usually shorter than a millimeter distance [5]. Birefringence phase matching in lithium niobate requires the THz wave to propagate away from the nearly co-directional optical pump and signal beams at about a 65° angle. As soon as the THz wave propagates away from the optical beams, the THz wave is quickly absorbed in lithium niobate within a distance comparable to the THz absorption length. This walkoff-and-absorption phenomenon has prompted implementation of schemes of either coupling out the THz wave as fast as possible [6–10] or increasing the overlap of mixing waves in the THz-wave walkoff direction [11].The notion of THz OPA/DFG limited by the idler absorption was also at the same time developed across the literatures [12–15] and even emphasized in some summary papers [16–19].

By using a quasi-phase-matching (QPM) crystal, such as the periodically poled lithium niobate (PPLN) [20], it is possible to generate an idler wave at THz frequencies through OPA/DFG at an arbitrary angle with respect to the optical pump and signal beams. Despite that collinear phase matching is usually a preferred scheme to maximize the parametric gain length, the popular notion of quickly extracting the THz wave to avoid absorption was behind the motivation of several demonstrated QPM difference frequency generators with a large THz-wave emission angle [21–29]. We show in the following that, even in the regime of idler-absorption induced parametric loss exceeding parametric gain, co-propagating the idler wave with the pump and signal waves in the same direction can achieve long-range parametric amplification of the idler wave in a nonlinear crystal until pump depletion. One might point out that, for an equal input pump and signal in a DFG process, the pump depletion and thus output saturation could occur within an idler absorption length. We further show in a modified theory and an experiment that, with an equal input pump and signal, the fast diffraction of the THz wave, as compared with the optical pump and signal, slows down the pump depletion and allows the idler power to grow over a distance much longer than the idler absorption length.

In the following, we begin our discussion with the plane-wave model for parametric mixing in Sec. 2. This plane-wave model is valid when the beam size of the optical pump and signal is much larger than the THz wavelength or when the THz wave is confined in a waveguide. In Sec. 3, we modify the plane-wave model to take into account the fast diffraction of the THz wave when the optical beam size or the THz-wave’s radiation aperture is comparable to the THz wavelength. We then report a typical THz DFG experiment in Sec. 4 to show the growth of a THz wave over tens of absorption lengths in lithium niobate with an equal input intensity for the signal and pump. Sec. 5 is the conclusion.

## 2. Different regimes of OPA/DFG with idler absorption

The plane-wave formulism and solution for parametric mixing without absorption or without phase mismatch can be found in some publication and textbook [30, 31]. We first extend the theory to include both idler absorption and phase mismatch. To clearly show the influence of the idler absorption, we break down the solution into several regimes characterized by a gain-to-loss ratio. We then discuss the difference between our result and a solution widely adopted for THz OPA/DFG.

In a quadratic nonlinear optical material, assume all the three parametric mixing waves (pump, signal, and idler) are continuous plane waves with slowly varying envelopes co-propagating along the + z direction. Under the slowly varying envelope approximation, the parametric process is governed by the three well known coupled-wave equations, $\partial {\overline{E}}_{p}/\partial z=-j{\kappa}_{p}{\overline{E}}_{s}{\overline{E}}_{i}{e}^{j\Delta {k}_{f}z}$, $\partial {\overline{E}}_{s}/\partial z=-j{\kappa}_{s}{\overline{E}}_{p}{\overline{E}}_{i}^{*}{e}^{-j\Delta {k}_{f}z}$, $\partial {\overline{E}}_{i}/\partial z=-j{\kappa}_{i}{\overline{E}}_{p}{\overline{E}}_{s}^{*}{e}^{-j\Delta {k}_{f}z}-{\alpha}_{i}{\overline{E}}_{i}/2$, where the subscripts *p*, *s*, and *i* denote variables or parameters relevant to the pump, signal, and idler, respectively, $\overline{E}$ is a slowly varying envelope of the mixing field, *κ* is the nonlinear coupling coefficient, Δ*k _{f}* =

*k*−

_{p}*k*−

_{s}*k*is the wave vector mismatch for the co-directional mixing waves, and

_{i}*α*

_{i}is the intensity absorption coefficient of the idler wave. For what follows, the coupled-wave equations describing the variation of the pump, signal, and idler field envelopes are called the pump, signal, and idler equations, respectively. With strong idler absorption, it has been popular to neglect the signal and pump equations. The remaining idler equation can be solved to give the output idler photon flux density at z =

*L*[32] subject to a constant signal, a constant pump, and Δ

*k*= 0:

_{f}*ϕ*

_{s}(0) is the initial signal photon flux density, and 2Γ is the intensity parametric gain coefficient. Equation (1) indeed indicates power saturation of the idler wave in a material length comparable to the idler absorption length 1/

*α*. Equation (1) and its variants have been widely cited for THz OPA/DFG with strong idler absorption [12, 13, 25, 33–37] to form the notion of THz OPA/DFG limited to a distance comparable to the idler absorption length.

_{i}To justify the saturation predicted by Eq. (1), one has to carefully examine the assumptions made to derive Eq. (1). Removing the pump and signal equations requires a negligible change of the pump and signal amplitudes throughout the parametric mixing process. A constant-pump assumption is possible for a process without pump depletion. A constant-signal assumption is apparently invalid for an OPA process, in which the signal grows from a small value to some appreciable one. For example, the injection-seeded THz parametric generator [38] seeds a pulsed optical parametric amplifier with a weak signal in a lithium niobate crystal to generate a coherent THz wave. The constant-signal assumption *could be* valid over a very short crystal length when the input amplitude of the signal is comparable to the pump one. In this regard, the condition of an equal input pump and signal is sometimes satisfied for some DFG experiment. The validity of Eq. (1) has never been rigorously and quantitatively defined within the context of relevant physical parameters. In the following, we first show that, with a variable signal and a strong pump (typical to THz OPA), the growth of the highly absorptive idler wave is not limited to a saturation length predicted by Eq. (1). We then quantitatively show that Eq. (1) can be obtained from a joint consideration of the signal and idler equations under three conditions: (1) no pump depletion, (2) absorption loss much larger than parametric gain, and (3) material length much shorter than the absorption length. Finally we show in an experiment that the fast diffraction of the THz wave in a nonlinear optical material leaves the plane-wave model invalid even in the limit of an equal input pump and signal (typical to THz DFG).

If one allows both the signal and idler to vary in a parametric mixing process without pump depletion, only the pump equation can be abandoned from the three coupled-wave equations. The output photon flux density of the idler wave without any initial idler photon can be deduced from a general solution [39] of the same plane-wave model, given by

*k*= 0 and $g=\sqrt{{\Gamma}^{2}+{({\alpha}_{i}/4)}^{2})}$ in Eq. (2) until we come back to fit the THz spectral tuning curve from our experiment. The corresponding signal photon flux density can be found to be ${\varphi}_{s}(L)={\varphi}_{s}(0){e}^{-{\alpha}_{i}L/2}{\left|\mathrm{cos}h(gL)+{\alpha}_{i}/(4g)\times \mathrm{sin}h(gL)\right|}^{2}$. The dependence of the output idler power on the idler absorption loss is not transparent from Eq. (2) as is. In the following, we break down Eq. (2) into approximate expressions in different limits of a characteristic gain-to-loss ratio defined as

_{f}*R*= (2Γ)/(

*α*/2) = 4Γ/

_{i}*α*

_{i}, where 2Γ is the intensity parametric gain coefficient and

*α*/2 is the effective loss of the parametric mixing process (see Eq. (4) below).

_{i}#### 2.1 High-gain and short-gain-length regime

In the high-gain *R*^{2} >>1 and short-gain-length Γ*L* << 1 limit, Eq. (2) reduces to a compact expression

*L*= 2/

*α*

_{i}. However, when the crystal length approaches 2/

*α*

_{i}, the condition Γ

*L*<< 1 is violated due to the constraint

*R*

^{2}= (4Γ/

*α*

_{i})

^{2}>> 1. Therefore, it is incorrect to conclude from this regime that the maximum idler output occurs at a length comparable to the idler absorption length or

*L*= 2/

*α*

_{i}.

#### 2.2 High-gain and long-gain-length regime

In the high-gain *R ^{2}* >>1 and long-gain-length Γ

*L*>> 1 limit, the idler output at z =

*L*becomes

*z*similar to that for lossless OPA. It can be understood from Eq. (4) that 2Γ is the intensity parametric gain coefficient and

*α*

_{i}/2 is effectively the idler-absorption induced parametric loss coefficient in a high-gain OPA process. In this regime, the idler power grows exponentially until pump depletion.

#### 2.3 High-loss and short-loss-length regime

OPA or DFG with strong idler absorption is the focus of this study. In the high-loss and short-loss-length limit, *R*^{2} << 1 and *α*_{i}*L*/2 << 1, the general solution Eq. (2) reduces to Eq. (1). Therefore, quantitatively, Eq. (1) is valid under three conditions: (1) no pump depletion, (2) absorption loss much larger than parametric gain *R*^{2} << 1, (3) material length much shorter than the absorption length *α*_{i}*L*/2 << 1. Condition (3) is not typical in most THz DFG experiments. For example, the THz absorption coefficient in lithium niobate is in the range of 1-100 cm^{−1}. If one chooses *α*_{i}*L*/2 = 0.1 to satisfy the condition *α*_{i}*L*/2 << 1, Eq. (1) is valid only for a crystal length *L* between 2 and 0.02 mm. This length is unrealistically short for a practical application. On the other hand, if one chooses a crystal length *L* << 2/*α*_{i} in the first place to satisfy Condition (3), the experimental result is indeed limited to the idler absorption length as a consequence of that choice, but cannot be generalized to conclude that OPA or DFG with idler absorption is limited to a crystal length comparable to the idler absorption length.

#### 2.4 High-loss and long-loss-length regime

If one allows the crystal length to increase, in the high-loss and long-loss-length limit, *R*^{2} << 1 and *α*_{i}*L*/2 >> 1, the idler photon flux density at *L*, according to Eq. (2), becomes

*R*

^{2}<< 1, the idler intensity increases monotonically with

*L*and does not saturate until pump depletion. This result is somewhat counter-intuitive but is a consequence of the paired signal-idler photon generation in an optical parametric process. The growth of the non-absorbing signal can assist the growth of the highly absorbing idler.

Figure 1
plots Eq. (1) and Eq. (2), indicated by a dashed green line and a solid blue line, respectively, versus crystal length with idler loss 5 times the parametric gain or *R* = 0.2. The horizontal axis is the crystal length in units of the idler absorption length $\overline{L}=L{\alpha}_{i}$. It is seen that the simplified theory, Eq. (1), only overlaps with the more accurate theory, Eq. (2), for a crystal length comparable to the idler absorption length. This is a direct consequence of the short-length assumption (Condition (3)) made in deriving Eq. (1). Equation (2), the dashed curve, clearly shows monotonic growth of the idler power over the entire crystal length. Exponential growth of the idler power is also evident when $\overline{L}$ becomes large, even though in the plot the effective parametric loss is 5 times the parametric gain. The inset is the corresponding signal photon flux density (assuming no pump depletion) normalized to its initial value versus distance, in which the fast growth of the non-absorbing signal has assisted the growth of the idler.

The discussion above holds true under the plane-wave model without pump depletion. When the input signal intensity approaches the pump one, pump depletion and thus idler saturation could occur in a short crystal length. For comparison, we also plot in Fig. 1 the idler photon flux density numerically solved from the three coupled-wave equations with *r* = *ϕ _{s}*(0)/

*ϕ*

_{p}(0) = 0.01 (OPA) and 1 (DFG), where

*ϕ*

_{p}(0) is the initial pump photon flux density. As expected, pump depletion, occurring much faster for DFG (

*r*~1), results in over-estimated output power from both Eqs. (1) and (2). The saturation value of Eq. (1) is somewhat useful for estimating the maximum idler power from a DFG process with idler absorption. However, a THz wave usually diffracts away from the optical beam aperture in THz OPA/DFG. As will be shown below, given the diffraction and an equal input signal and pump, the growth distance of the idler wave is still much longer than the THz absorption length.

## 3. Diffraction-modified plane-wave model

In THz OPA/DFG, the THz wave radiates from an aperture comparable to the transverse cross section of the optical beams. The plane-wave model in the last section is only valid for an optical beam size much larger than the THz wavelength or for a THz wave confined in a waveguide. However, to obtain a strong optical intensity in a bulk nonlinear optical material, the optical beams are often focused to a size comparable to the THz wavelength, which makes the THz wave quickly diffract as soon as it is generated. Under strong THz-wave diffraction and absorption, we propose the following two modifications to the plane-wave model.

The first correction is related to the mode-area mismatch between the THz wave and the optical beams. Owing to the strong idler absorption, the field radius of the THz wave is roughly a constant equal to *w* + 2/*α*_{i} in the nonlinear optical material, where *w* is the radius of the optical beam and 2/*α*_{i} is the THz field absorption length. When the photon flux densities in Eq. (1) or (2) are converted into powers for all mixing waves in a waveguide-like medium, the parametric gain coefficient is modified by a factor *R*_{Γ} approximately equal to the square root of the mode-area ratio [40], given by

*and A*

_{THz}_{o}are the effective transverse areas of the THz and the optical waves, respectively. This gain reduction simply slows down pump depletion and thus idler saturation. However, the major conclusions drawn from Eq. (2) remain unchanged.

The second correction involves the modification of the idler absorption coefficient *α*_{i} along the crystal length. As will be shown below, the intensity attenuation coefficient *α* is not necessarily the same as the power attenuation coefficient. The THz wave in a bulk crystal is not confined to a hard boundary with a radius of *w* + 2/*α _{i}.* In fact, the THz wave continues to diffract and is nearly all absorbed outside a radius >

*w*+ 2/

*α*. For a hard-boundary waveguide with a radius of

_{i}*r*

_{0}and intensity attenuation coefficient of

*α*, the power flow of the guided wave along

*z*follows the well known expression

*P*(

*r*,

*z*) =

*P*

_{0}(

*r*) × exp(−

*αz*), where

*r*<

*r*

_{0}, and

*P*

_{0}(

*r*) is the input power at z = 0. In this case, the intensity attenuation coefficient is the same as the power attenuation coefficient. However, without the hard boundary, the power contained in the cylinder with

*r*<

*r*

_{0}becomes

*P*(

*r*,

*z*) =

*P*

_{0}(

*r*) × exp(−

*αz*) ×

*R*

_{D}(

*z*), where

*R*

_{D}(

*z*) is a diffraction-induced power reduction factor approximately equal to the ratio of the area π

*r*

_{0}

^{2}to the diffracted mode area at

*z*. For THz OPA/DFG in a bulk crystal, the THz source area is about the optical beam area or π

*w*

^{2}as

*r*

_{0}~

*w*. In most THz OPA/DFG cases,

*w*is comparable to the THz wavelength λ

_{THz}, which makes the THz wave diffract like a spherical wave with a half-spherical area ~2π

*z*

^{2}at

*z*>> λ

_{THz}in the forward direction. Therefore, in the far field

*R*

_{D}(z) ~2

*w*

^{2}/z

^{2}and in the near field

*R*

_{D}(

*z*) ~1. Given above, we approximate the THz power contained in the optical beam area as

*A*is evidently a coefficient related to how fast the THz wave diffracts, depending on the experimental conditions such as the laser focusing, THz wavelength, and refractive index of the material. With this correction, the

*power*absorption coefficient for the idler wave is a

*z*-dependent quantity given byThis expression changes the growth rate of the idler wave and will be used in the next section to explain the observed long-range idler growth in a THz DFG experiment with an equal input intensity for the signal and pump.

## 4. Experiment with equal input pump and signal

In the previous section, we have shown in theory that Eq. (1) is just a special case of Eq. (2) in the high-loss and short-loss-length limit with no pump depletion. Although pump-depletion induced idler saturation is not a subject of this paper, one might suggest that the saturation of Eq. (1) is valid for quickly pump depleted THz DFG with an equal input pump and signal as its input. In this section, we show the failure of this argument under diffraction of the THz wave in a typical THz-wave DFG experiment. In our experiment, we chose to perform co-directional optical DFG to generate an idler wave at 1.5 THz from an array of PPLN crystal strips with their lengths varying from 1 to 25 mm in 2 mm increment in a 0.5-mm thick, monolithic congruent lithium niobate substrate. The domain period of the PPLN crystals was 65 μm, which is quasi-phase matched to the generation of a THz wave at 195.7 μm at room temperature for pump and signal waves at 1.5389 and 1.5511 μm, respectively, for co-directional DFG along the crystallographic *x* direction. The two end faces of the PPLN-array crystal were optically polished and coated with anti-reflection layers at the pump and signal wavelengths.

Figure 2
shows the experimental setup. The pump and signal waves were first combined in a single-mode optical fiber from a distributed-feedback diode laser (DFBDL) with a fixed wavelength at 1.5389 μm and an external-cavity diode laser (ECDL) with a tunable wavelength covering the bandwidth of the downstream THz DFG crystal. Two independent polarization controllers were used to align the polarizations of the pump and signal waves along the crystallographic *z* direction of the PPLN crystals. A passively Q-switched Nd:YAG microchip laser pumped a pulsed optical parametric amplifier following the Erbium-doped fiber amplifier (EDFA) to produce 9.7-μJ energy in a 360-ps width for each of the pump and signal pulse. The equal-energy signal and pump pulses were then focused to a 127-μm waist radius to the center of the PPLN array crystal for THz DFG. A Ni-metal wire mesh with uniform 45 μm × 45 μm square apertures was installed between the THz-DFG PPLN crystal and the collimating off-axis parabolic mirrors to reflect 86% of the THz wave toward the 4k Si bolometer and dump 84% of the optical waves. A 3-mm thick Germanium THz filter was installed before the 4k Si bolometer to block all the residue optical waves and transmit 35% of the THz wave into the bolometer.

Figure 3
shows the measured THz-wave tuning curve from the 25 mm long PPLN strip. The plot was generated by recording the THz-wave output power in the bolometer while scanning the wavelength of the ECDL across the phase matching bandwidth of the THz DFG PPLN. It is possible to use the phase matching bandwidth of Eq. (2) to fit the experimental data to determine the absorption and parametric gain coefficients of the crystal. The continuous curve in Fig. 3 fits to the data with *α*_{i} = 40 cm^{−1}, Γ = 0.53 cm^{−1}, and *L* = 2.5 cm. The measured absorption coefficient is consistent with the reported value for THz DFG in bulk congruent lithium niobate [21, 40, 41]. With *α _{i}* = 40 cm

^{−1}and Γ = 0.53 cm

^{−1}, the gain-to-loss ratio is

*R*= 5.3 × 10

^{−2}, which sets our experiment in the high-loss regime. Given

*d*

_{eff}= 168☓2/π = 107 pm/V [42] and

*n*

_{p}=

*n*

_{s}= 2.14 [43], the parametric gain coefficient calculated from the free-space continuous plane-wave model is 2.2, which is about 4.2 times that obtained from the curve fitting in Fig. 3. The gain reduction factor

*R*

_{Γ}calculated from Eq. (6) is 4.8 with an average

*w*of 130 μm and

*α*= 40 cm

_{i}^{−1}. Given the uncertainty in the THz material parameters for lithium niobate, the experimentally deduced

*R*

_{Γ}= 4.2 is reasonably close to the theoretically estimated

*R*

_{Γ}= 4.8.

We further translated the PPLN crystal array in the transverse direction and allowed the incident optical beams to sample the PPLN gratings one by one, while keeping the input condition unchanged and measuring the generated THz-wave power in the Si bolometer. The measured THz-wave power versus the PPLN crystal length is shown in Fig. 4
(blue dots). The error bar is the range of data fluctuation about the fitted wavelength tuning curve taken for each PPLN strip. To fit the experimental data in Fig. 4, we introduced the position-dependent power absorption coefficient, Eq. (8), into the three coupled-wave equations (blue dashed curve) with A = 10 and an initial *α _{i}* = 37 cm

^{−1}. With initial signal energy of 9.7 μJ at 1539 nm and an output THz wavelength at 200 μm, the fitted curve suggests 45-pJ peak energy at 1.5 THz generated from the 25-mm long PPLN crystal. This amount of output THz-wave energy matches well to our previously reported experimental value measured by a calibrated bolometer [40]. Equation (1) (green curve) is also plot in Fig. 4 for comparison. It is seen that Eq. (1), showing fast rise and quick saturation of the THz-wave power, is clearly inconsistent with the experimental result.

## 5. Conclusion

We have studied THz-wave OPA and DFG with strong THz-wave absorption in the regime without pump depletion. This is the regime most THz-wave OPA/DFG experiments were carried out. As indicated by Eq. (2), a theory derived under a plane-wave model without pump depletion, the highly absorptive THz wave in a THz OPA process can grow monotonically over a long crystal length until pump depletion. The coherent production of the non-absorbing signal can assist the growth of the absorbing idler. The long-range parametric amplification revealed by Eq. (5), reduced from Eq. (2) in the high-loss and long-loss-length limit, is particularly useful for co-directionally phase matched THz-wave parametric amplification or oscillation built up from a small signal. For the plane-wave model to be valid, the optical beam size of the pump and signal has to be much larger than the THz wavelength or the THz wave has to be confined in a waveguide [40].

The widely adopted THz-DFG theory, Eq. (1), which shows quick saturation of an absorbing idler wave, is an approximation of a more general expression, Eq. (2), in the limit of high absorption loss *R*^{2} << 1 and short crystal length *L* << 2/*α*_{i}. The only assumptions made to derive Eq. (2) are: (1) the slowly varying envelope approximation, (2) plane-wave like fields, and (3) an undepleted pump. When the signal-to-pump ratio *r* approaches 1 for DFG, pump depletion can occur quickly in a short crystal and the valid regime of Eq. (2) merges with that of Eq. (1) in the short-crystal limit. However, in practice, the THz wave in THz DFG can diffract much faster than the optical mixing waves for an optical beam size comparable to the THz wavelength. Consequently, the pump-depletion induced idler saturation does not occur in a short crystal as predicted by Eq. (1). By using equal-amplitude pump and signal as the inputs to a co-directional THz difference frequency generator, we show the growth of an idler wave at 1.5 THz over a crystal length exceeding 40 idler absorption lengths. In this experiment, the effective parametric loss is nearly 20 times the parametric gain.

It has been widely suggested in previous publications that one should couple out the THz wave as quickly as possible from an OPA/DFG material with strong THz absorption. On the contrary, our study shows that, to maximize the THz output, the THz wave should be kept in the absorptive nonlinear optical material together with the optical waves until pump depletion. As pump depletion occurs slowly due to fast THz diffraction and strong THz absorption, the useful crystal length can be much longer than the THz absorption length even in the DFG case.

## Acknowledgments

Huang thanks Fejer and Vodopyanov of Stanford University for helpful discussion on absorptive OPA in 2009. This work was supported by National Science Council under Contract NSC 98-2923-M-007-004-MY3 and the Frontier Research Centers of National Tsinghua University.

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