# Optical Frequency Converters

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This is a continuation from the previous tutorial - **phase matching for nonlinear optical processes**.

A very important class of nonlinear optical devices is the optical frequency converters. Nonlinear optical frequency conversion is the only means for ** direct** conversion of optical energy from one frequency to another. Indeed, the discipline of nonlinear optics was born out of the first observation of second-harmonic generation in 1961.

There are basically two types of nonlinear optical frequency converters. The majority are based on parametric processes, particularly the parametric second-order processes, that require phase matching. Sum-frequency generators, difference-frequency generators, harmonic generators, and parametric amplifiers and oscillators belong to this type. Devices that use the nonparametric third-order processes of stimulated Raman or Brillouin scattering to shift the optical frequency are the other type. In this tutorial, we consider only those based on parametric processes. Devices based on stimulated Raman or Brillouin scattering are discussed in a later tutorial.

**Sum-frequency generators**

The basic function of a sum-frequency generator is the generation of an optical wave at a high frequency, \(\omega_3\), by mixing two optical waves at low frequencies, \(\omega_1\) and \(\omega_2\), as schematically shown in Figure 9-1(a) [refer to the nonlinear optical interactions tutorial]. The general application of a sum-frequency generator is straightforward. It is most often used to obtain, through mixing available optical waves at long wavelengths, a coherent optical beam at a desired short wavelength that is not readily available from other sources. If one of the two input waves is tunable in wavelength, a wavelength-tunable sum-frequency output wave is obtained. For example, a wavelength-tunable optical beam in the ultraviolet spectral region can be obtained with a sum-frequency generator that mixes the output of a tunable laser in the visible spectral region with that of another laser at a fixed wavelength also in the visible spectral region.

The process of sum-frequency generation is generally described by the coupled-equations in (9-60) - (9-62) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] with the condition that \(\mathcal{E}_1(0)\ne0\) and \(\mathcal{E}_2(0)\ne0\) but \(\mathcal{E}_3(0)=0\) at the input surface, \(z=0\), of a nonlinear crystal. The general solutions to these coupled equations can be found in terms of the Jacobi elliptic functions. However, simpler, and often more useful, solutions can be found for specific experimental conditions of interest.

The simplest situation is when the efficiency of a sum-frequency generator is low so that the intensities of both input waves at \(\omega_1\) and \(\omega_2\) are not depleted significantly throughout the interaction. We can then assume \(\mathcal{E}_1\) and \(\mathcal{E}_2\) to be independent of \(z\), ignore (9-61) and (9-62) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] in the coupled equations, and integrate (9-60) directly to find \(\mathcal{E}_3(z)\). Using the relation in (9-63) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] for light intensity, we find that, in the ** low-efficiency limit**, the intensity of the wave at the sum frequency as a function of the interaction length \(l\) can be expressed as

\[\tag{9-102}\begin{align}I_3(l)&=\frac{\omega_3^2|\chi_\text{eff}|^2}{2c^3\epsilon_0n_{1,z}n_{2,z}n_{3,z}}I_1I_2l^2\frac{\sin^2(\Delta{k}l/2)}{(\Delta{k}l/2)^2}\\&=\frac{8\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{1,z}n_{2,z}n_{3,z}\lambda_3^2}I_1I_2l^2\frac{\sin^2(\Delta{k}l/2)}{(\Delta{k}l/2)^2}\end{align}\]

where \(d_\text{eff}=\chi_\text{eff}/2\), and \(\lambda_3=2\pi{c}/\omega_3\) is the wavelength of the sum-frequency wave in free space. In the case of quasi-phase matching, \(\chi_\text{eff}\) and \(d_\text{eff}\) in (9-102) are replaced by \(\chi_\text{Q}\) and \(d_\text{Q}\), respectively.

The effect of phase mismatch is characterized by a function of the form

\[\tag{9-103}\frac{I_3}{I_3^\text{PM}}=\frac{\sin^2(\Delta{k}l/2)}{(\Delta{k}l/2)^2}\]

which is plotted in Figure 9-15 below.

When \(\Delta{k}\ne0\), it does not pay to have a crystal longer than the coherence length of the interaction, as discussed in the preceding section and as illustrated in Figure 9-8(c) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial]. When perfect phase matching is accomplished, the intensity of the sum-frequency wave grows quadratically with interaction length as \(I_3=I_3^\text{PM}\propto{l}^2/\lambda_3^2\).

We see from (9-102) that \(I_3\propto|d_\text{eff}|^2I_1I_2\) in the low-efficiency limit. Therefore, if the purpose of an application is to produce a significant intensity for the sum-frequency wave, the two input waves need to have high and comparable intensities.

In the above, we have assumed that the interacting waves are perfect plane waves. In reality, each optical beam has a finite cross section and a nonuniform intensity distribution. This and other spatial effects have to be carefully considered in a detailed analysis of a sum-frequency generation process, as well as in that of any other nonlinear process to be discussed later.

Without carrying out such an analysis, we point out the important, yet easily understood, fact that interaction between two or more optical beams takes place only in the area where those beams overlap spatially and, if the beams are optical pulses, also temporally. Therefore, in terms of optical power, \(I_1\), \(I_2\), and \(I_3\) in (9-102) have to be replaced by \(P_1/\mathcal{A}_1\), \(P_2/\mathcal{A}_2\), and \(P_3/\mathcal{A}_3\), respectively, where \(P_q\) is the total power of the wave at the frequency \(\omega_q\) and \(\mathcal{A}_q\) is its effective cross-sectional area. In the low-efficiency limit, we then have \(P_3\propto|d_\text{eff}|^2P_1P_2\mathcal{A}_3/\mathcal{A}_1\mathcal{A}_2\). It is important to realize that \(\mathcal{A}_3\le\min(\mathcal{A}_1,\mathcal{A}_2)\) because the sum-frequency wave is generated only in the area where the two input waves overlap.

We thus arrive at the following conclusions: (1) to maximize the efficiency of sum-frequency generation with two input waves at given power levels, it is important to collimate these two beams to the same cross-sectional area and to have them overlap uniformly so that \(\mathcal{A}_3\) is maximized; (2) it is possible to increase the conversion efficiency by focusing the input waves to reduce \(\mathcal{A}_1\) and \(\mathcal{A}_2\) simultaneously so long as the effective interaction length is not reduced due to the increased divergence of the focused beams; (3) it does not pay to just focus one input beam or to focus the two input beams unevenly because doing so results in a corresponding reduction in \(\mathcal{A}_3\).

**Difference-frequency generators**

By mixing two optical waves, taken to be at \(\omega_3\) and \(\omega_1\), respectively, a difference-frequency generator produces a third optical wave at the difference frequency \(\omega_2=\omega_3-\omega_1\), as schematically shown in Figure 9-2(a) [refer to the nonlinear optical interactions tutorial].

Difference-frequency generators are the simplest devices for the generation of coherent infrared radiation, particularly the radiation in the mid to far infrared region where efficient laser materials are rare. For this purpose, both input waves can be in the visible region, or one in the visible and another in the near infrared region, where many efficient lasers sources are available. Wavelength-tunable infrared radiation can be obtained if one of the input waves is from a wavelength-tunable source.

The equations for the description of the difference-frequency generation process are also those given in (9-60) - (9-62) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial], but the boundary conditions are \(\mathcal{E}_3(0)\ne0\), \(\mathcal{E}_1(0)\ne0\), and \(\mathcal{E}_2(0)=0\) at the input surface of a nonlinear crystal. Similarly to the case of sum-frequency generation, general solutions of the coupled equations with the boundary conditions for difference-frequency generation can be found in terms of elliptic functions. However, also similarly to the case of sum-frequency generation, simple solutions under special situations are often more useful.

In the low-efficiency limit, depletion of the intensities of the two input waves is negligible. By taking the two input fields, \(\mathcal{E}_3\) and \(\mathcal{E}_1\), to be independent of \(z\), (9-61) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] can be integrated directly for field \(\mathcal{E}_2(z)\) at the difference frequency \(\omega_2\). The following solution for the intensity of the difference-frequency wave is found:

\[\tag{9-104}\begin{align}I_2(l)&=\frac{\omega_2^2|\chi_\text{eff}|^2}{2c^3\epsilon_0n_{1,z}n_{2,z}n_{3,z}}I_3I_1l^2\frac{\sin^2(\Delta{k}l/2)}{(\Delta{k}l/2)^2}\\&=\frac{8\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{1,z}n_{2,z}n_{3,z}\lambda_2^2}I_3I_1l^2\frac{\sin^2(\Delta{k}l/2)}{(\Delta{k}l/2)^2}\end{align}\]

where \(\lambda_2=2\pi{c}/\omega_2\) is the wavelength of the difference-frequency wave in free space. In the case of quasi-phase matching, \(\chi_\text{eff}\) and \(d_\text{eff}\) in (9-104) are replaced by \(\chi_\text{Q}\) and \(d_\text{Q}\), respectively.

The relation in (9-104) has the same form as that in (9-102). The effect of phase mismatch is also that shown in Figure 9-15. With perfect phase matching, \(I_2=I_2^\text{PM}\propto{l}^2/\lambda_2^2\). To produce a significant intensity for the difference-frequency wave, it is also desirable to have two strong input waves with comparable intensities because \(I_2\propto|d_\text{eff}|^2I_3I_1\). In terms of optical power, we have \(P_2\propto|d_\text{eff}|^2P_3P_1\mathcal{A}_2/\mathcal{A}_3\mathcal{A}_1\), where \(\mathcal{A}_2\le\min(\mathcal{A}_3,\mathcal{A}_1)\). Therefore, in the low-efficiency limit, the wave produced by a difference-frequency generator has the same general characteristics as discussed for the wave produced by a sum-frequency generator.

One word of caution in the application of a difference-frequency generator goes to the generation of far infrared radiation. When the wavelength of the difference-frequency wave in the far infrared region becomes comparable to, or even larger than, one of the cross-sectional beam diameters of the input waves, the diffraction effect of the long-wavelength difference-frequency wave becomes significant. As a result, the relation in (9-104) is no longer valid. Instead, spatially nonuniform distribution of the difference-frequency wave caused by this diffraction has to be considered though the total power integrated over the entire cross section of the difference-frequency wave is not changed by the diffraction effect.

**Second-harmonic generators**

By far the most widely used nonlinear optical devices are the second-harmonic generators. An optical harmonic generator produces an optical wave at a frequency that is an integral multiple of the frequency of the input wave. A second-harmonic generator produces a wave at double the frequency of the input wave; thus it is also called an ** optical frequency doubler**. In the application of second-harmonic generator, only two optical waves are involved in the interaction: one input wave at the

**of**

*fundamental frequency**\(\omega\)*and a nonlinearly generated wave at the the

**of**

*second-harmonic frequency**\(2\omega\)*, as schematically illustrated in Figure 9-1(b) [refer to the nonlinear optical interactions tutorial].

Following a procedure similar to that leading to the coupled equations of (9-60) - (9-62) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial], we find the following two coupled equations for second-harmonic generation:

\[\tag{9-105}\frac{\text{d}\mathcal{E}_{2\omega}}{\text{d}z}=\frac{\text{i}(2\omega)^2}{2c^2k_{2\omega,z}}\chi_\text{eff}\mathcal{E}_\omega^2\text{e}^{\text{i}\Delta{k}z}=\frac{\text{i}\omega}{cn_{2\omega,z}}\chi_\text{eff}\mathcal{E}_\omega^2\text{e}^{\text{i}\Delta{k}z}\]

\[\tag{9-106}\frac{\text{d}\mathcal{E}_\omega}{\text{d}z}=\frac{\text{i}\omega^2}{c^2k_{\omega,z}}\chi_\text{eff}^*\mathcal{E}_{2\omega}\mathcal{E}_\omega^*\text{e}^{-\text{i}\Delta{k}z}=\frac{\text{i}\omega}{cn_{\omega,z}}\chi_\text{eff}^*\mathcal{E}_{2\omega}\mathcal{E}_\omega^*\text{e}^{-\text{i}\Delta{k}z}\]

where \(\chi_\text{eff}=\hat{e}^*_{2\omega}\cdot\boldsymbol{\chi}^{(2)}(2\omega=\omega+\omega):\hat{e}_\omega\hat{e}_\omega=\hat{e}_\omega\cdot\boldsymbol{\chi}^{(2)}(\omega=2\omega-\omega):\hat{e}_{2\omega}^*\hat{e}_\omega\) and \(\Delta\mathbf{k}=2\mathbf{k}_\omega-\mathbf{k}_{2\omega}=\Delta{k}\hat{z}\). Using the relation in (9-63) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial], we find that (9-105) and (9-106) lead to the following relation for the intensities of the fundamental and the second-harmonic waves:

\[\tag{9-107}\frac{\text{d}I_{2\omega}}{\text{d}z}=-\frac{\text{d}I_\omega}{\text{d}z}=-\frac{2\omega|\chi_\text{eff}|}{(2c^3\epsilon_0n_{\omega,z}^2n_{2\omega,z})^{1/2}}I_\omega{I}_{2\omega}^{1/2}\sin\varphi\]

where \(\varphi=\varphi_\chi+2\varphi_\omega-\varphi_{2\omega}+\Delta{k}z\). Therefore, we find the following Manley-Rowe relations for second-harmonic generation that involves only two optical beams:

\[\tag{9-108}\frac{\text{d}I}{\text{d}z}=\frac{\text{d}(I_\omega+I_{2\omega})}{\text{d}z}=0\]

and

\[\tag{9-109}\frac{\text{d}}{\text{d}z}\left(\frac{I_\omega}{\omega}\right)=-2\frac{\text{d}}{\text{d}z}\left(\frac{I_{2\omega}}{2\omega}\right)\]

As expected, two photons at the fundamental frequency are annihilated to create each photon at the second-harmonic frequency.

In the low-efficiency limit, depletion of the intensity of the fundamental beam can be neglected. Then, (9-105) can be integrated directly for \(\mathcal{E}_{2\omega}(z)\) by taking \(\mathcal{E}_\omega\) to be independent of \(z\). The result, expressed in terms of second-harmonic intensity as a function of interaction length, is

\[\tag{9-110}\begin{align}I_{2\omega}(l)&=\frac{\omega^2|\chi_\text{eff}|^2}{2c^3\epsilon_0n_{\omega,z}^2n_{2\omega,z}}I_\omega^2l^2\frac{\sin^2(\Delta{k}l/2)}{(\Delta{k}l/2)^2}\\&=\frac{8\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{\omega,z}^2n_{2\omega,z}\lambda^2}I_\omega^2l^2\frac{\sin^2(\Delta{k}l/2)}{(\Delta{k}l/2)^2}\end{align}\]

where \(\lambda=2\pi/\omega\) is the wavelength of the fundamental wave in free space.

In the low-efficiency limit, \(I_{2\omega}\propto|d_\text{eff}|^2I_\omega^2\), or \(P_{2\omega}\propto|d_\text{eff}|^2P_\omega^2\mathcal{A}_{2\omega}/\mathcal{A}_\omega^2\), where \(\mathcal{A}_\omega\) and \(\mathcal{A}_{2\omega}\) are the effective cross-sectional areas of the fundamental and second-harmonic beams, respectively, and \(\mathcal{A}_{2\omega}\le\mathcal{A}_\omega\) due to the nonlinear nature of the second-harmonic generation process.

Perfect phase matching is required if a high efficiency for second-harmonic generation is desired. In addition, according to the discussions in the coupled-wave analysis of nonlinear optical interactions tutorial, it is also necessary to have \(\varphi=-\pi/2\). This condition is automatically satisfied if perfect phase matching is accomplished and if the input consists of only the fundamental wave because, without any coherent second-harmonic field at the input, only the second-harmonic field that has the most favorable phase is generated and subsequently amplified. The Manley-Rowe relation in (9-108) states that the total intensity of the fundamental and second-harmonic waves remains constant throughout the interaction: \(I=I_\omega+I_{2\omega}=I_\omega(0)\) for \(I_{2\omega}(0)=0\). Under these conditions, (9-107) leads to

\[\tag{9-111}\frac{\text{d}I_{2\omega}}{\text{d}z}=\frac{2\omega|\chi_\text{eff}|}{(2c^3\epsilon_0n_{\omega,z}^3)^{1/2}}[I_\omega(0)-I_{2\omega}]I_{2\omega}^{1/2}\]

Note that with perfect phase matching, \(n_{\omega,z}=n_{2\omega,z}\). By making the change of variable \(u^2=I_{2\omega}/I_\omega(0)\) and by using the fact that \(u=\tanh{\kappa{z}}\) is the solution of the equation \(\text{d}u/\text{d}z=\kappa(1-u^2)\), we can solve (9-111) to obtain the following general results for second-harmonic generation with perfect phase matching:

\[\tag{9-112}I_{2\omega}(l)=I_\omega(0)\tanh^2\kappa{l}\]

\[\tag{9-113}I_{\omega}(l)=I_\omega(0)\text{sech}^2\kappa{l}\]

where

\[\tag{9-114}\kappa=\left[\frac{\omega^2|\chi_\text{eff}|^2}{2c^3\epsilon_0n_{\omega,z}^3}I_\omega(0)\right]^{1/2}=\left[\frac{8\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{\omega,z}^3\lambda^2}I_\omega(0)\right]^{1/2}\]

The results are plotted in Figure 9-16 below.

With perfect phase matching, it is theoretically possible to convert all of the fundamental power to the second-harmonic if the interaction length is sufficiently long. In the case of quasi-phase matching, \(\chi_\text{eff}\) and \(d_\text{eff}\) in (9-114) are replaced by \(\chi_\text{Q}\) and \(d_\text{Q}\), respectively.

The conversion efficiency of a second-harmonic generator is commonly defined as

\[\tag{9-115}\eta_\text{SH}=\frac{P_{2\omega}(l)}{P_\omega(0)}\]

In the low-efficiency limit with perfect phase matching,

\[\tag{9-116}\eta_\text{SH}=\frac{8\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{\omega,z}^3\lambda^2}\frac{\mathcal{A}_{2\omega}}{\mathcal{A}_\omega^2}P_\omega(0)l^2\]

Because \(\eta_\text{SH}\) in the low-efficiency limit is linearly proportional to the fundamental power, it is convenient to define a ** normalized second-harmonic conversion efficiency** as

\[\tag{9-117}\hat{\eta}_\text{SH}=\frac{\eta_\text{SH}}{P_\omega(0)}=\frac{8\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{\omega,z}^3\lambda^2}\frac{\mathcal{A}_{2\omega}}{\mathcal{A}_\omega^2}l^2\]

There is a relation between \(\mathcal{A}_{2\omega}\) and \(\mathcal{A}_\omega\) that depends on the cross-sectional profile of the fundamental beam. For example, \(\mathcal{A}_{2\omega}=\mathcal{A}_\omega/2\) if the beam has a Gaussian profile, but \(\mathcal{A}_{2\omega}=\mathcal{A}_\omega\) if the beam has a uniform profile.

We see from (9-117) that the conversion efficiency can be raised by focusing the fundamental beam to reduce its cross-sectional area, provided that the beam remains well collimated. Focusing the beam too tightly increases the beam divergence, thus reducing its intensity outside the Rayleigh range from the beam waist. In addition, the conversion efficiency can be reduced by any walk-off between the interacting beams.

The second-harmonic generation efficiency of a focused Gaussian beam is a function of three characteristic lengths: the crystal length \(l\), the confocal parameter \(b=2\pi{n}w_0^2/\lambda\), and the aperture length \(l_\text{a}=\pi^{1/2}w_0/\rho\) defined in (9-90) [refer to the phase matching for nonlinear optical processes tutorial].

For \(b\gg{l}\) and \(l_\text{a}\gg{l}\), the dependence of \(\hat{\eta}_\text{SH}\) on \(l^2\) seen in (9-117) is valid, and \(\hat{\eta}_\text{SH}\) can be expressed in the following form:

\[\tag{9-118}\hat{\eta}_\text{SH}=\frac{\eta_\text{SH}}{P_\omega(0)}=\frac{16\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{\omega,z}^2\lambda^3}\frac{l^2}{b}\]

For \(b\lt{l}\lt10b\) or \(l_\text{a}\lt{l}\), (9-117) and (9-118) are not valid, but the conversion efficiency can be approximated by

\[\tag{9-119}\hat{\eta}_\text{SH}=\frac{\eta_\text{SH}}{P_\omega(0)}=\frac{16\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{\omega,z}^2\lambda^3}\frac{1.068l}{1+lb/l_\text{a}^2}\]

We see that if \(lb\gg{l}_\text{a}^2\), the conversion efficiency is independent of crystal length as \(\eta_\text{SH}\propto{l}_\text{a}^2/b\) in this situation. The best efficiency that can be obtained with an optimally focused Gaussian beam is

\[\tag{9-120}\hat{\eta}_\text{SH}=\frac{\eta_\text{SH}}{P_\omega(0)}=\frac{16\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{\omega,z}^2\lambda^3}(1.068l)\]

which occurs under the conditions of no walk-off so that \(l_\text{a}=\infty\) and \(l=2.84b\).

We see that, with the fundamental beam optimally focused for the best efficiency, the conversion efficiency increases only linearly with crystal length but the focused beam waist spot area has to vary linearly with crystal length to maintain this optimum condition.

Note that in the case of quasi-phase matching, \(d_\text{eff}\) that appears in the expressions of \(\eta_\text{SH}\) and \(\hat{\eta}_\text{SH}\) given in (9-116) - (9-120) has to be replaced by \(d_\text{Q}\).

We also see that the conversion efficiency increases linearly with an increase in the input power of the fundamental beam. This statement is true as long as we stay in the low-efficiency limit so that depletion of the fundamental beam is negligible.

In the high-efficiency regime, conversion efficiency increases sublinearly with the input power of the fundamental beam. According to (9-112), it is theoretically possible to have 100% conversion efficiency for second-harmonic generation if the input power is sufficiently high and the interaction length is sufficiently large. However, the conversion efficiency of a practical device is usually limited by the damage threshold of a nonlinear crystal, as well as by many complicated spatial and temporal effects.

For many practical applications, it is often necessary to generate the third harmonic or the fourth harmonic of a fundamental wave. As discussed in the nonlinear optical interactions tutorial, the third harmonic can be generated with a parametric third-order nonlinear process characterized by \(\boldsymbol{\chi}^{(3)}(3\omega=\omega+\omega+\omega)\). However, a third-harmonic generator using a third-order nonlinear process is of little practical usefulness for two reasons: (1) the value of \(\chi^{(3)}\), though always nonvanishing, is orders of magnitude smaller than the value of \(\chi^{(2)}\) of any commonly used nonlinear crystals; (2) phase matching is very difficult for such a process. In practice, efficient third-harmonic generation is normally carried out by following second-harmonic generation with sum-frequency generation for \(\omega+2\omega\rightarrow3\omega\), as shown in Figure 9-17(a) below. Similarly, fourth-harmonic generation is accomplished by cascading two second-harmonic generators by first doubling \(\omega\) and then doubling \(2\omega\) to obtain \(4\omega\), as shown in Figure 9-17(b) below. These possibilities are already demonstrated in Example 9-4.

**Example 9-12**

In this example, we consider the second-harmonic conversion efficiency with a focused Gaussian beam at \(\lambda\) = 1.10 μm in LiNbO_{3} under different phase-matching conditions discussed in Examples 9-8 to 9-11 [refer to the phase-matching for nonlinear optical processes tutorial]. Perfect phase matching is assumed for each case, with \(\Delta{k}_\text{Q}=0\) in the case of quasi-phase matching. The fundamental beam is focused to have its beam waist located at the center of a crystal of \(l\) = 1 cm length.

**(a)** With angle phase matching as described in Example 9-8, what is the normalized efficiency \(\hat{\eta}_\text{SH}\) if the beam is focused to have a beam waist radius of \(w_0\) = 50 μm?

**(b)** With \(90^\circ\) phase matching by temperature tuning as described in Example 9-10, what is \(\hat{\eta}_\text{SH}\) for \(w_0\) = 50 μm?

**(c)** With \(90^\circ\) phase matching, the conversion efficiency can be increased by optimum focusing. What is the optimum beam waist radius for this purpose? What is the best conversion efficiency?

**(d)** With quasi-phase matching in a PPLN crystal as described in Example 9-11, what is the best attainable conversion efficiency?

**Here are the solutions:**

**(a)** With type I angle phase matching, we have \(|d_\text{eff}|=4.88\text{ pm V}^{-1}\) and \(n_{\omega,z}=n_\omega^\text{o}=2.2319\) from Example 9-8. With \(w_0\) = 50 μm and \(\lambda\) = 1.10 μm, we find that \(l_\text{a}\) = 4.54 mm from Example 9-9 and that \(b\) = 3.19 cm. Because \(l_\text{a}\lt{l}\), (9-119) has to be used to estimate the efficiency. Because \(lb\gg{l}_\text{a}^2\), we find that

\[\begin{align}\hat{\eta}_\text{SH}&\approx\frac{16\pi^2|d_\text{eff}|^2(1.068l_\text{a}^2)}{c\epsilon_0n_{\omega,z}^2\lambda^3b}\\&=\frac{16\pi^2\times(4.88\times10^{-12})^2\times1.068\times(4.54\times10^{-3})^2}{3\times10^8\times8.85\times10^{-12}\times2.2319^2\times(1.10\times10^{-6})^3\times3.19\times10^{-2}}\text{W}^{-1}\\&=0.015\%\text{ W}^{-1}\end{align}\]

**(b)** With \(90^\circ\) type I phase matching, we have \(|d_\text{eff}|=4.4\text{ pm V}^{-1}\), \(n_{\omega,z}=n_\omega^\text{o}=2.2321\) at \(T=396.7\text{ K}\), and \(l_\text{a}=\infty\) from Example 9-10. With \(w_0\) = 50 μm, \(\lambda\) = 1.10 μm, and \(n=2.2321\), we still have \(b\) = 3.19 cm. In this case, (9-118) is valid because \(b\gt3l\) and \(l_\text{a}\gg{l}\). Therefore, we find that

\[\begin{align}\hat{\eta}_\text{SH}&=\frac{16\pi^2|d_\text{eff}|^2l^2}{c\epsilon_0n_{\omega,z}^2\lambda^3b}\\&=\frac{16\pi^2\times(4.4\times10^{-12})^2\times(1\times10^{-2})^2}{3\times10^8\times8.85\times10^{-12}\times2.2321^2\times(1.10\times10^{-6})^3\times3.19\times10^{-2}}\text{ W}^{-1}\\&=0.054\%\text{ W}^{-1}\end{align}\]

We see that the conversion efficiency is 3.6 times that found in **(a)** by using \(90^\circ\) phase matching to eliminate the walk-off effect.

**(c)** In the absence of walk-off for \(90^\circ\) phase matching, the best efficiency can be obtained by making \(b=l/2.84=3.52\) mm, which can be accomplished by focusing the fundamental beam to the following beam waist radius:

\[w_0=\left(\frac{\lambda{b}}{2\pi{n}}\right)^{1/2}=\left(\frac{1.10\times10^{-6}\times3.52\times10^{-3}}{2\pi\times2.2321}\right)^{1/2}\text{ m}=16.6\text{ μm}\]

The efficiency is found by using (9-120) to be

\[\begin{align}\hat{\eta}_\text{SH}&=\frac{16\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{\omega,z}^2\lambda^3}(1.068l)\\&=\frac{16\pi^2\times(4.4\times10^{-12})^2\times1.068\times1\times10^{-2}}{3\times10^8\times8.85\times10^{-12}\times2.2321^2\times(1.10\times10^{-6})^3}\text{ W}^{-1}\\&=0.186\%\text{ W}^{-1}\end{align}\]

We see that this efficiency is more than three times that found in **(b)** by optimally focusing the beam in the absence of walk-off.

**(d)** Because there is no walk-off in the case of quasi-phase matching in a PPLN crystal described in Example 9-11, we can still take \(b=l/2.84=3.52\) mm. Because \(n_{\omega,z}=n_\omega^\text{e}=2.1536\) and \(n_{2\omega}^\text{e}=2.2260\) in this situation, we find that \(w_0=16.9\) μm, which is slightly larger than that found in **(c)**. The best efficiency is still found by using (9-120) but with \(|d_\text{eff}|\) replaced by \(|d_\text{Q}|=16.04\text{ pm V}^{-1}\) found in Example 9-11. Therefore,

\[\begin{align}\hat{\eta}_\text{SH}&=\frac{16\pi^2|d_\text{Q}|^2}{c\epsilon_0n_{\omega,z}n_{2\omega,z}\lambda^3}(1.068l)\\&=\frac{16\pi^2\times(16.04\times10^{-12})^2\times1.068\times1\times10^{-2}}{3\times10^8\times8.85\times10^{-12}\times2.1536\times2.2260\times(1.10\times10^{-6})^3}\text{ W}^{-1}\\&=2.56\%\text{ W}^{-1}\end{align}\]

This conversion efficiency is about 14 times that found in **(c)** because quasi-phase matching using a PPLN crystal allows us to take advantage of the largest nonlinear susceptibility element \(d_{33}\) of LiNbO_{3}.

This example illustrates how the efficiency of a second-harmonic generator can be substantially increased by a combination of optimization procedures. Further increase of efficiency is possible using a waveguide structure, or by using short optical pulses to increase the peak intensity at a given average power level. The same techniques can be applied generally to other nonlinear frequency converters for increasing their conversion efficiencies.

**Optical parametric frequency converters**

The function of an optical parametric frequency converter is the conversion of a signal-carrying optical wave from one carrier frequency to another through ** parametric up-conversion** or

**.**

*parametric down-conversion*Parametric up-conversion is a special case of sum-frequency generation with the objective of converting a signal-carrying optical wave at a low frequency, typically in the mid or far infrared region, where sensitive detectors are not available, to an optical wave carrying the same signal at a frequency in the visible region, where efficient detection can be easily made.

Parametric down-conversion is a special case of difference-frequency generation in which a signal-carrying optical wave at a high frequency, often in the ultraviolet region, is converted to one at a low frequency in the visible or the infrared region.

The signal-carrying input wave, which is called the ** signal**, is generally very weak in comparison to the other input wave, which is called the

*. In the following analysis, the strong pump wave is taken to be at \(\omega_2\). The signal is taken to be at \(\omega_1\) for up-conversion and is taken to be at \(\omega_3\) for down-conversion. The relation \(\omega_3=\omega_1+\omega_2\) applies to both cases.*

**pump**Because the pump is much stronger than the signal, the intensity of the pump can be considered to be constant though that of the signal is not. As a result, we have the following coupled equations for parametric conversion processes:

\[\tag{9-121}\frac{\text{d}\mathcal{E}_3}{\text{d}z}=\text{i}\left(\frac{\omega_3^2}{c^2k_{3,z}}\chi_\text{eff}\mathcal{E}_2\right)\mathcal{E}_1\text{e}^{\text{i}\Delta{k}z}=\text{i}\kappa_{31}\mathcal{E}_1\text{e}^{\text{i}\Delta{k}z}\]

\[\tag{9-122}\frac{\text{d}\mathcal{E}_1}{\text{d}z}=\text{i}\left(\frac{\omega_1^2}{c^2k_{1,z}}\chi_\text{eff}^*\mathcal{E}_2^*\right)\mathcal{E}_3\text{e}^{-\text{i}\Delta{k}z}=\text{i}\kappa_{13}\mathcal{E}_3\text{e}^{-\text{i}\Delta{k}z}\]

These two equations have the form of the coupled equations of (57) and (58) [refer to the two-mode coupling tutorial], which are solved in that tutorial. Because the signal is normally weak in the application of a parametric converter, a high conversion efficiency is most desirable. Therefore, the device is normally used under the condition of perfect phase matching.

For up-conversion, the boundary conditions are \(\mathcal{E}_1(0)\ne0\) and \(\mathcal{E}_3(0)=0\). The solutions under the condition of perfect phase matching are

\[\tag{9-123}\mathcal{E}_1(l)=\mathcal{E}_1(0)\cos\kappa{l}\]

\[\tag{9-124}\mathcal{E}_3(l)=\frac{\text{i}\kappa_{31}}{\kappa}\mathcal{E}_1(0)\sin\kappa{l}\]

where

\[\tag{9-125}\kappa=(\kappa_{31}\kappa_{13})^{1/2}=\left(\frac{\omega_1\omega_3|\chi_\text{eff}|^2}{2c^3\epsilon_0n_{1,z}n_{2,z}n_{3,z}}I_2\right)^{1/2}=\left(\frac{8\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{1,z}n_{2,z}n_{3,z}\lambda_1\lambda_3}I_2\right)^{1/2}\]

In the case of quasi-phase matching, \(\chi_\text{eff}\) and \(d_\text{eff}\) in (9-125) are replaced by \(\chi_\text{Q}\) and \(d_\text{Q}\), respectively.

The schematic diagram of an optical parametric up-converter is shown in Figure 9-18(a) below. For a parametric up-converter with perfect phase matching, the intensities of the three interacting beams vary with interaction length as

\[\tag{9-126}I_3(l)=\frac{\omega_3}{\omega_1}I_1(0)\sin^2\kappa{l}\]

\[\tag{9-127}I_1(l)=I_1(0)\cos^2\kappa{l}\]

\[\tag{9-128}I_2(l)=I_2(0)-\frac{\omega_2}{\omega_1}I_1(0)\sin^2\kappa{l}\approx{I}_2(0)\]

Figure 9-18(b) illustrates these intensity variations. Complete up-conversion of the signal occurs at an interaction length of \(l_\text{c}^\text{PM}=\pi/2\kappa\), as expected of phase-matched codirectional coupling. The value of this length can be varied by varying the pump intensity because the value of \(\kappa\) depends on that of \(I_2\). Note that when the signal intensity is completely depleted by up-conversion, the intensity of the sum-frequency wave reaches a maximum value of \(I_3^\text{max}=I_1(0)\omega_3/\omega_1\gt{I}_1(0)\) because the total number of sum-frequency photons that are created is equal to the total number of signal photons that are annihilated.

Parametric down-conversion is simply the reverse process of up-conversion, and vice versa. The same parametric converter can function as either an up-converter or a down-converter. The only difference is the initial conditions at the input. If the initial conditions are \(\mathcal{E}_1(0)=0\) and \(\mathcal{E}_3(0)\ne0\), the device functions as a down-converter. In Figure 9-18, we see clearly that when the intensity of the wave at \(\omega_1\) is completely depleted, for example, at a distance of \(l=l_\text{c}^\text{PM}\), further interaction in the parametric converter leads to down-conversion from the wave at \(\omega_3\) back to the wave at \(\omega_1\).

**Optical parametric amplifiers**

The physical process involved in an optical parametric amplifier, commonly called an OPA, is basically the same as that in a difference-frequency generator. The only difference is in the usage of the device. In either case, there are two input waves at \(\omega_1\) and \(\omega_3\). While the usage of a difference-frequency generator is for generation of a wave at the difference frequency \(\omega_2=\omega_3-\omega_1\), that of an OPA is for amplification of the input wave at \(\omega_1\). The wave at the difference frequency \(\omega_2\) is still generated in an OPA though it is not the purpose of this application. Therefore, the high-frequency input wave at \(\omega_3\) is called the ** pump wave**, the low-frequency input wave at \(\omega_1\) is called the

**, and the side product at \(\omega_2\) is called the**

*signal wave***, as shown in Figure 9-19(a) below.**

*idler wave*

Normally the pump wave of an OPA is much stronger than the signal wave and can be considered constant throughout the interaction. Therefore, only (9-61) and (9-62) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] have to be considered, and the initial conditions are \(\mathcal{E}_1(0)\ne0\) and \(\mathcal{E}_2(0)=0\). We have the following coupled equations:

\[\tag{9-129}\frac{\text{d}\mathcal{E}_1}{\text{d}z}=\text{i}\left(\frac{\omega_1^2}{c^2k_{1,z}}\chi_\text{eff}^*\mathcal{E}_3\right)\mathcal{E}_2^*\text{e}^{-\text{i}\Delta{k}z}=\text{i}\kappa_{12}\mathcal{E}_2^*\text{e}^{-\text{i}\Delta{k}z}\]

\[\tag{9-130}\frac{\text{d}\mathcal{E}_2^*}{\text{d}z}=\text{i}\left(-\frac{\omega_2^2}{c^2k_{2,z}}\chi_\text{eff}\mathcal{E}_3^*\right)\mathcal{E}_1\text{e}^{\text{i}\Delta{k}z}=\text{i}\kappa_{21}\mathcal{E}_1\text{e}^{\text{i}\Delta{k}z}\]

where (9-130) is obtained by taking the complex conjugate of (9-62) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial].

Again, these two coupled equations have the form of the coupled equations of (57) and (58) [refer to the two mode coupling tutorial], and the solutions in that tutorial can be applied directly. For efficient parametric amplification, phase matching is required. By identifying \(\beta_\text{c}=(\kappa_{12}\kappa_{21})^{1/2}=\text{i}\kappa\) in the case of perfect phase matching, we have the following solutions:

\[\tag{9-131}\mathcal{E}_1(z)=\mathcal{E}_1(0)\cos\beta_\text{c}z=\mathcal{E}_1(0)\cosh\kappa{z}\]

\[\tag{9-132}\mathcal{E}_2(z)^*=\frac{\text{i}\kappa_{21}}{\beta_\text{c}}\mathcal{E}_1(0)\sin\beta_\text{c}z=\frac{\text{i}\kappa_{21}}{\kappa}\mathcal{E}_1(0)\sinh\kappa{z}\]

where

\[\tag{9-133}\kappa=\left(\frac{\omega_1\omega_2|\chi_\text{eff}|^2}{2c^3\epsilon_0n_{1,z}n_{2,z}n_{3,z}}I_3\right)^{1/2}=\left(\frac{8\pi^2|d_\text{eff}|^2}{c\epsilon_0n_{1,z}n_{2,z}n_{3,z}\lambda_1\lambda_2}I_3\right)^{1/2}\]

In the case of quasi-phase matching, \(\chi_\text{eff}\) and \(d_\text{eff}\) in (9-133) are replaced by \(\chi_\text{Q}\) and \(d_\text{Q}\), respectively.

With perfect phase matching, the intensities of the signal, idler, and pump waves vary with interaction length as

\[\tag{9-134}I_1(l)=I_1(0)\cosh^2\kappa{l}\]

\[\tag{9-135}I_2(l)=\frac{\omega_2}{\omega_1}I_1(0)\sinh^2\kappa{l}\]

\[\tag{9-136}I_3(l)=I_3(0)-\frac{\omega_3}{\omega_1}I_1(0)\sinh^2\kappa{l}\approx{I}_3(0)\]

which are plotted in Figure 9-19(b) above.

We see that while the intensity of the signal wave grows as a result of parametric amplification, the intensity of the idler wave also increases because an idler photon is generated simultaneously with each additional signal photon generated in the parametric process.

With perfect phase matching, the ** amplification factor**, or the

**, of the signal wave for a single pass through an OPA is**

*intensity gain*\[\tag{9-137}G=\frac{I_1(l)}{I_1(0)}=\cosh^2\kappa{l}=\begin{cases}1+\kappa^2l^2,\qquad\text{in the low-gain limit}\\\frac{\text{e}^{2\kappa{l}}}{4},\qquad\qquad\text{in the high-gain limit}\end{cases}\]

Note that a large gain factor does not necessarily imply a high conversion efficiency from the pump to the signal and idler because the input signal can be extremely weak. Therefore, it is possible that the pump is not much depleted when the signal is amplified by a large gain factor but the conversion efficiency is low. When the input signal is strong, however, it is also possible that pump depletion is significant but the gain factor is small.

**Example 9-13**

An OPA for a signal wavelength at \(\lambda_1\) = 1.55 μm consists of a PPLN cystal that has a length of \(l\) = 1 cm. It is pumped with a Gaussian beam at \(\lambda_3\) = 527 nm, which is focused to a beam waist radius of \(w_0\) = 50 μm.

(a) What is the idler wavelength?

(b) What is the required first-order grating period for quasi-phase matching?

(c) What is the amplification factor for the signal if the power of the pump beam is \(P\) = 1 W?

(d) What is the required pump power for an amplification factor of \(G\) = \(10^3\)? Consider only the situation where the pump is not much depleted even when \(G\) = \(10^3\).

**Answers:**

**(a)** The wavelength of the interacting beams in a parametric amplifier have the relation \(\lambda_3^{-1}=\lambda_1^{-1}+\lambda_2^{-1}\). Therefore, the idler wavelength is

\[\lambda_2=\left(\frac{1}{\lambda_3}-\frac{1}{\lambda_1}\right)^{-1}=\left(\frac{1}{527\times10^{-9}}-\frac{1}{1.55\times10^{-6}}\right)^{-1}\text{ m}=798\text{ nm}\]

**(b)** For the most efficient interaction in a PPLN crystal, all of the interacting waves have to be extraordinary waves polarized in the \(z\) direction. Using the data given in Table 9-3 [refer to the nonlinear optical susceptibilities tutorial] for the Sellmeier equation of LiNbO_{3}, we find that \(n_3^\text{e}=2.2351\) at \(\lambda_3=527\) nm, \(n_1^\text{e}=2.1373\) at \(\lambda_1=1.55\) μm, and \(n_2^\text{e}=2.1755\) at \(\lambda_2=798\) nm. The phase mismatch is \(\Delta{k}=k_1+k_2-k_3=2\pi(n_1/\lambda_1+n_2/\lambda_2-n_3/\lambda_3)\) for collinear interaction. Therefore, according to (9-99) [refer to the phase matching for nonlinear optical processes tutorial], the required first-order grating period is

\[\begin{align}\Lambda&=\frac{2\pi}{|\Delta{k}|}=\left|\frac{n_1}{\lambda_1}+\frac{n_2}{\lambda_2}-\frac{n_3}{\lambda_3}\right|^{-1}\\&=\left|\frac{2.1373}{1.55\times10^{-6}}+\frac{2.1755}{798\times10^{-9}}-\frac{2.2351}{527\times10^{-9}}\right|^{-1}\text{ m}\\&=7.35\text{ μm}\end{align}\]

**(c)** For a Gaussian pump beam that is focused to a waist size of \(w_0=50\) μm, we find that its confocal parameter is \(b=2\pi{n}_3^\text{e}w_0^2/\lambda_3=6.66\) cm. Because \(b\gg{l}=1\) cm, we can ignore the complicated effect of focusing and take \(I_3=P_3/\mathcal{A}_3=2P/\pi{w}_0^2\) over the entire length of the PPLN crystal. For this interaction, we have \(|d_\text{Q}|=|2d_{33}/\pi|=16.04\text{ pm V}^{-1}\) from Example 9-11 [refer to the phase matching for nonlinear optical processes tutorial]. Then, using (9-133) with \(d_\text{eff}\) replaced by \(d_\text{Q}\), we can express \(\kappa^2l^2\) as a function of the pump power:

\[\kappa^2l^2=\frac{16\pi|d_\text{Q}|^2l^2}{c\epsilon_0n_1^\text{e}n_2^\text{e}n_3^\text{e}\lambda_1\lambda_2w_0^2}P_3=0.015P_3\text{ W}^{-1}\]

For \(P_3=1\) W, the single-pass amplification factor is \(G\approx1+\kappa^2l^2=1.015\) according to (9-137). The signal intensity grows only \(1.5\%\) in a single pass through the parametric amplifier.

**(d)** For an amplification factor of \(G=10^3\), we find by using the high-gain limit of (9-137) that \(\kappa{l}=4.147\) is required. From the dependence of \(\kappa^2l^2\) on \(P_3\) found in **(c)**, we find that the required pump power for \(G=10^3\) is

\[P_3=\frac{\kappa^2l^2}{0.015}\text{W}=\frac{4.147^2}{0.015}\text{W}=1.15\text{ kW}\]

This pump power looks unrealistically high. It is indeed unrealistic if we consider only the possibility of CW (continuous wave) pump beams. It is not if we consider pulse pumping. For example, by using a Q-switched laser pulse of duration \(\Delta{t}_\text{ps}=100\text{ ns}\), such a pump power requires a very common pump pulse energy of \(U_\text{ps}=P_\text{pk}\Delta{t}_\text{ps}=115\text{ μJ}\). As another example, if mode-locked pulses of pulsewidth \(\Delta{t}_\text{ps}=10\text{ ps}\) at a repetition rate of \(f_\text{ps}=100\text{ MHz}\) are used to pump the amplifier, the average power of the pulsed pump beam is again at a realistic level of \(\bar{P}=P_\text{pk}\Delta{t}_\text{ps}f_\text{ps}=1.15\text{ W}\).

**Optical parametric oscillators**

The parametric gain can be utilized to construct an optical parametric oscillator, commonly called an OPO, by placing a parametric amplifier in a resonant optical cavity that provides feedback to the parametric amplifier.

There are basically two different types of OPOs. In a ** doubly resonant** OPO, both waves at \(\omega_1\) and \(\omega_2\) are resonated because the mirrors of the optical cavity are highly reflective at both frequencies, as shown in Figure 9-20(a) below. In a

**OPO, the mirrors of the optical cavity are highly reflective at only one frequency, either \(\omega_1\) or \(\omega_2\), and only one wave is resonated, as shown in Figure 9-20(b) below. The cavity mirrors are transparent to the pump wave and, in the singly resonant case, also to the nonresonant parametric wave.**

*singly resonant*

The input to an OPO consists of only the pump wave at \(\omega_3\) to pump the nonlinear crystal for a parametric gain. When the parametric gain is high enough so that the roundtrip loss in the optical resonator is compensated by the parametric gain, the oscillator reaches its ** threshold** and parametric oscillation occurs. Because the parametric gain is a function of the pump intensity, the threshold parametric gain for an OPO translates into a threshold pump intensity required of the pump beam. Resonant oscillation builds up from the spontaneous emission noise of parametric fluorescence. No signal input is needed.

Because both low-frequency parametric waves at \(\omega_1\) and \(\omega_2\) are generated in the oscillator without a signal input, either of them can be called the signal or the idler. The designation of one particular wave to be called the signal is purely a matter of one's subjective interest. However, the choice of the resonating frequency in a singly resonant OPO is usually not arbitrary but is based on. many practical consideration, such as the availability of high-quality cavity mirrors at either of the two parametric frequencies, the spectral characteristics of the transmittance of the nonlinear crystal, and other wavelength-dependent characteristics of the optical cavity.

The frequencies and, correspondingly, the wavelengths of a parametric oscillator are subject to the following conditions:

\[\tag{9-138}\omega_3=\omega_1+\omega_2\qquad\text{and}\qquad\frac{1}{\lambda_3}=\frac{1}{\lambda_1}+\frac{1}{\lambda_2}\]

which are required by conservation of energy because one photon at \(\omega_3\) splits into a pair of photons at \(\omega_1\) and \(\omega_2\).

The exact frequencies to be generated by the oscillator are further dictated by the following two conditions: (1) the phase-matching condition

\[\tag{9-139}\mathbf{k}_3=\mathbf{k}_1+\mathbf{k}_2\]

which is determined by the properties and the physical arrangement of the nonlinear crystal, and (2) the resonance condition of the optical cavity, which depends on the physical parameters of the cavity and determines the resonance optical frequencies.

The peak parametric gain appears at frequencies that satisfy the phase-matching condition exactly. The oscillation frequencies are those, subject to the condition in (9-138), that satisfy the resonance condition of the optical resonator with the least amount of phase mismatch.

Therefore, the signal and idler frequencies of an OPO can be simultaneously tuned, though in opposite directions due to the constraint of (9-138), by varying the phase-matching condition in the crystal while the pump frequency is fixed.

This wavelength tunability is one of the most important characteristics of OPOs. Another important characteristic is that the parametric gain is not tied to any resonant transitions in the gain medium because the gain medium is a parametric nonlinear crystal.

These two characteristics make the OPOs unique devices for the generation of wavelength-tunable coherent optical waves in any spectral ranges where efficient laser materials do not exist, provided that an efficient nonlinear crystal and a commonly available laser source at a higher frequency to serve as the pump can be found.

A doubly resonant OPO generally has a lower oscillation threshold than a singly resonant one of comparable physical parameters. However, a doubly resonant OPO is difficult to operate because of its intrinsic instability.

To resonant both signal and idler waves, both frequencies \(\omega_1\) and \(\omega_2\) have to satisfy the resonance condition of the optical cavity. With the constraint of (9-138), this requirement cannot be met with an arbitrary cavity length but only with some specific values of the cavity length. This situation limits the tunability of the parametric oscillator.

In addition, any variations in the cavity length due to mechanical or thermal fluctuations can lead to instability in the oscillation frequencies and the amplitudes of the optical fields.

These problems do not exist in a singly resonant optical parametric resonator. Therefore, most OPOs designed for practical applications are of the singly resonant type.

**Example 9-14**

The PPLN parametric amplifier described in Example 9-13 is placed in a properly designed optical cavity to make a singly resonant OPO. When the OPO is sufficiently pumped above threshold with a pump beam at 527 nm of a pump power of \(P_3=2\text{ W}\), it is found that \(5\%\) of the pump power is converted to the combined output power of the signal and idler. What are the output powers of the signal and idler beams, respectively?

The total output power from this OPO is \(P_\text{out}=0.05P_3=100\text{ mW}\). In a parametric conversion process, an idler photon is simultaneously generated each time a signal photon is generated while a pump photon is annihilated because of the relation \(\omega_3=\omega_1+\omega_2\) required in (9-138). As a consequence, the total number of signal photons has to be equal to that of idler photons because there are no input signal or idler photons to an OPO. If the signal and idler photons are subject to the same fractional loss, the power ratio between the signal and the idler is

\[\tag{9-140}\frac{P_1^\text{out}}{P_2^\text{out}}=\frac{\omega_1}{\omega_2}=\frac{\lambda_2}{\lambda_1}\]

which leads to the following power split:

\[\tag{9-141}P_1^\text{out}=\frac{\lambda_2}{\lambda_1+\lambda_2}P_\text{out},\qquad{}P_2^\text{out}=\frac{\lambda_1}{\lambda_1+\lambda_2}P_\text{out}\]

With \(P_\text{out}=100\text{ mW}\), we find that \(P_1^\text{out}=34\text{ mW}\) for the signal at \(\lambda_1=1.55\text{ μm}\) and \(P_2^\text{out}=66\text{ mW}\) for the idler at \(\lambda_2=798\text{ nm}\).

For the split of output power expressed in (9-141), it is assumed that the signal and the idler suffer the same fractional loss in the OPO. In practice, this assumption may not be true, particularly when the wavelengths of the signal and the idler are far apart from each other. When the signal and the idler experience significantly disparate losses, the output power split can be very different from that described in (9-141). Even in this situation, it is still true that equal numbers of signal and idler photons are generated from converting the same number of pump photons when they interact in the nonlinear crystal.

Many lasers are available in the visible and near infrared wavelength regions for pumping second-harmonic generators to generate short-wavelength optical waves well into the deep ultraviolet region and for pumping OPOs to generate wavelength-tunable optical waves in a broad infrared region. With the advances in laser sources and crystal technology, the wavelengths that can be reached by nonlinear frequency conversion are basically only limited by the transmission windows of available nonlinear crystals. Figure 9-21 shows the transmission windows of various nonlinear crystals that can be chosen for frequency converters and the wavelengths of several lasers that can be used as pump sources. From this figure, we see that second-order nonlinear frequency conversion can cover a spectral range from about 200 nm in the deep ultraviolet to about 18 μm in the mid infrared. Depending on the pump lasers used, the coherent optical waves generated by these frequency converters in this wide spectral range cover the entire range of temporal characteristics, from CW beams through nanosecond Q-switched pulses to picosecond and femtosecond mode-locked pulses. Through these nonlinear optical devices, optical sources over a wide range of spectral and temporal characteristics are made available and flexible for many applications.

The next tutorial continues with the **nonlinear optical modulators and switches tutorial**.