Effect of pump depletion on second harmonic generation in multiple quasi-phase-matching gratings

Li-Ming Zhao; Gui-Kuan Yue; Yun-Song Zhou; Fu-He Wang

doi:10.1364/OE.21.017592

1. Introduction

In the past decades, second harmonic generation (SHG) has been extensively studied due to its advantage for obtaining a compact and high-power laser emitting in the visible or ultraviolet spectral regions. In addition, high-power laser sources for visible light are of interest in various fields, such as display technologies, biological investigations, and optical communications. It is well known that optical waves propagating in nonlinear crystals are governed by the coupled-wave equations which are based on Maxwell’s equations. However, it is very difficult to solve those complex nonlinear differential equations, and consequently several approximations should be introduced to simplify these equations. These approximation methods include the slowly varying amplitude approximation (SVAA), the infinite plane-wave approximation (PWA), and undepleted pump approximation (UPA). By adopting the above-mentioned three approximations, traditional quasi-phase-matching (QPM) technique [1], which is very attractive for realizing frequency conversion devices, can be achieved.

However, it is worth noting that these approximations are not always valid, and they should only be appropriately applied on the basis of an actual scenario. For example, recently, the QPM for SHG in a one-dimensional nonlinear crystal embedded in air has been investigated by adopting a method that does not make use of the PWA and the SVAA. The result shows that the obtained QPM was found quite different from the conventional QPM scenario, and the difference was attributed to the transmittance resonance of fundamental wave (FW) or second harmonic wave (SHW) [2]. Furthermore, QPM for SHG considering the depletion of pumping light power was amended, and the results show that the optimum domain widths deviate from the coherence length l_c = π/Δk, and the SHG conversion efficiency considering the pump depletion significantly differs from that in the case of UPA [3] when the SHG conversion efficiency is high enough.

Moreover, generally, only one wavelength can satisfy the QPM in a given nonlinear optical process. Apparently, a key issue is about how to find the optimal optical superlattice for given multiple optical parametric processes. Recently, considerable effort has been dedicated to exploring various QPM structures, including the quasi-period [4,5], aperiodic optical superlattice (AOS) [6,7], and disorder domain configuration [8,9]. However, all of these samples have been devised under UPA, and the pump depletion can not be ignored when the conversion efficiency of SHG is high enough.

Although a lot of effort has been made to study the SHG considering pump depletion by some analytical [10–12] and numerical methods [13–15], and recently, more attention has been payed to SHG of short laser pulses in the regime of pump depletion [16,17], to our knowledge, most of these are limited to exploring the feasibility of the methods themselves and research on the effects brought about by the pump depletion itself is almost non-existent. We are especially interested in the applicability of multiple-QPM gratings designed in UPA and the related conversion efficiency calculated in UPA, under the conditions when the SHG conversion efficiency is high enough and the pump depletion have to be considered, due to the practical importance. Hence, the applicability of UPA in the above-mentioned two aspects is worthy of further study, especially at higher pump intensity and SHG conversion efficiency. Our work aims to explore the possibility of addressing these problems, thereby obtaining more detailed information and providing direct guidance for practical samples, such as the design of domain configuration and the estimate for the conversion efficiency of SHG.

2. Derivation for an exact solution

We consider a one-dimensional AOS made by LiNbO₃ (LN) crystal layers, which consists of laminar ferroelectric-domains; in the following discussion, x denotes the stacked direction of the layers. In such a sample, the directions (along the z–axis) of ferroelectric polarization in the domains are alternatively changed, as are the signs of the corresponding nonlinear optical coefficient. However, the thickness of each individual domain which is dominated by specific optical parametric processes may be different. The direction of the optical wave propagation sets along the x axis.

It is assumed that a laser beam with a frequency of ω₁ = ω is perpendicularly incident onto the surface of an AOS, and the SHW with ω₂ = 2ω is generated from this AOS by nonlinear optical process. The modulation of nonlinear optical coefficient d₃₃ is described by $χ^{(2)} (x) = χ_{j}^{(2)}$ in the jth unit domain, i.e., x_j₋₁ ≤ x ≤ x_j, and $χ_{j}^{(2)}$ only takes a binary value, i.e., 2d₃₃ and −2d₃₃, corresponding to the positive and negative polarizations of the domains. In the assumption of slowly varying of field amplitude, the equations governing the propagation of the FW and the SHW are

\frac{d E_{1} (x)}{d x} = \frac{i ω_{1}^{2} χ^{(2)} (x)}{k_{1} c^{2}} E_{2} (x) E_{1}^{*} (x) e^{i Δ k x},

\frac{d E_{2} (x)}{d x} = \frac{i ω_{2}^{2} χ^{(2)} (x)}{2 k_{2} c^{2}} E_{1}^{2} (x) e^{- i Δ k x},

where k₁ = n₁ω/c (k₂ = 2n₂ω/c), c is the light speed in vacuum, n₁ (n₂) is the refractive index of the material at the FW (SHW) frequency, Δk = k₂ − 2k₁. In the AOS sample, we have

χ^{(2)} (x) = 2 | d_{33} | \tilde{d} (x),

where d̃(x) only takes a binary value of 1 or −1.

After the field is written in its real and imaginary parts E_α(x) = ρ_α(x)e^iϕ_α(x) (α = 1, 2), and variable replacements are introduced as

u_{α} (x) = {(\frac{ε_{0} c^{2} k_{α}}{2 ω_{α} I})}^{1 / 2} ρ_{α} (x), α = 1, 2,

where

u_{α}^{2} (x)

denotes the normalized intensity, therefore, the conversion efficiency is given by

η = u_{2}^{2} (L)

. Here, L is the length of the sample.

Substituting E_α(x) into Eqs. (1a) and (1b), and considering the relation of ρ_α(x) and u_α(x), the following equations can be obtained:

\frac{d u_{1}}{d x} = - ξ u_{1} u_{2} \sin θ,

\frac{d u_{2}}{d x} = ξ u_{1}^{2} \sin θ,

\frac{d ϕ_{1}}{d x} = ξ u_{2} \cos θ,

\frac{d ϕ_{2}}{d x} = \frac{ξ u_{1}^{2}}{u_{2}} \cos θ,

\frac{d θ}{d x} = Δ k + \frac{ξ u_{1}^{2}}{u_{2}} \cos θ - 2 ξ u_{2} \cos θ .

where

ξ (x) = {(\frac{8 π^{2} I}{n_{1 ω}^{2} n_{2 ω} c λ^{2} ε_{0}})}^{1 / 2} χ^{(2)} (x)

, θ(x) = Δkx + ϕ₂(x) − 2ϕ₁(x). By integrating Eq. (4e) and using Eq. (4a) and (4b), we can derive the solution in the nth domain as

\cos θ^{(n)} (x) = - \frac{Δ k}{2 ξ_{n}} \frac{{(u_{2}^{(n)} (x))}^{2} + Γ_{n}}{{(u_{1}^{(n)} (x))}^{2} u_{2}^{(n)} (x)},

where Γ_n is a constant of integration, θ⁽ⁿ⁾(x),

u_{2}^{(n)} (x)

and

u_{1}^{(n)} (x)

stand for the value of θ(x), u₂(x) and u₁(x) in the nth domain. The initial condition requires Γ₁ = 0, because the intensity of SHW at the entrance of the sample should be zero. The variations of θ(x), u₁(x) and u₂(x) along the x–axis should be continuous. Therefore, from Eq. (5) we can obtain

Γ_{n + 1} = {\begin{array}{l} Γ_{n}, & when the consecutive domains have identical polarization \\ - 2 u_{2}^{2} (x_{n}) - Γ_{n}, & otherwise \end{array}

where x_n and u₂(x_n) denote the value of x and

u_{2}^{(n)} (x)

at the exit edge of the nth domain.

It is shown that for the optimum domain widths in which the so-called QPM for a single wavelength is satisfied, the polarization direction of the domain should be changed at sinθ = 0, and correspondingly, cosθ = ±1. By substituting cosθ = ±1 into Eq. (5) and considering $u_{1}^{2} + u_{2}^{2} = 1$ , such an equation can be obtained

{(u_{2} (x_{n}))}^{3} + \frac{Δ k}{2 | ξ_{n} |} {(u_{2} (x_{n}))}^{2} - u_{2} (x_{n}) + \frac{Δ k}{2 | ξ_{n} |} Γ_{n} = 0.

Using Γ₁ = 0, the values of u₂(x_n) and Γ_n can be solved by iterative solution [3]. As to the AOS which can achieve multiple SHGs with high conversion efficiency, by using the simulated annealing method (SA), the optimum polarization direction of the domain can be arranged [6]. However, in the position where the polarization direction should be changed, sinθ = 0 is not always satisfied. Thus, a sophisticated calculation method is necessary to be employed, which will be discussed in details in the following section.

Substituting Eq. (5) into Eq. (4b), then we deduce

u_{2}^{'} (x_{n}) = s n [y, β],

where the function of sn(y, β) is defined by

y = \int_{0}^{w} \frac{d t}{{[(1 - t^{2}) (1 - β^{2} t^{2})]}^{1 / 2}}, w = s n (y, β) .

In Eq. (8),

y = \pm {(u_{3 c}^{2} - u_{3 a}^{2})}^{1 / 2} (ζ_{1} - ζ_{0}) + A_{0} .

Here, we set ζ₁ = ξ_nx_n, ζ₀ = ξ_n₋₁x_n₋₁, and

A_{0} = \int_{0}^{u_{2}^{'} (x_{n - 1})} \frac{d u_{2}^{'}}{\sqrt{[1 - u_{2}^{' 2}] [1 - (u_{3 b}^{2} - u_{3 a}^{2}) {(u_{3 c}^{2} - u_{3 a}^{2})}^{- 1} u_{2}^{' 2}]}} .

The ” ± ” in Eq. (10) is determined by the sign of sinθ; in Eq. (8), $β = (u_{3 b}^{2} - u_{3 a}^{2}) / (u_{3 c}^{2} - u_{3 a}^{2})$ , and u₃_a, u₃_b, u₃_c (u₃_a < u₃_b < u₃_c) denote the three roots of the following equation

{(u_{2}^{(n)})}^{2} (1 - {(u_{2}^{(n)})}^{2}) - \frac{Δ k^{2}}{4 ξ_{n}^{2}} {({(u_{2}^{(n)})}^{2} + Γ_{n})}^{2} = 0.

The relation of

u_{2}^{(n)}

with

u_{2}^{' (n)}

is

{(u_{2}^{' (n)})}^{2} = \frac{{(u_{2}^{(n)})}^{2} - u_{3 a}^{2}}{u_{3 b}^{2} - u_{3 a}^{2}} .

From Eqs. (8)–(11), it is found that y and β are determined by u₃_a, u₃_b, u₃_c and u′₂(x_n₋₁); according to Eq. (12), it is shown that u₃_a, u₃_b and u₃_c are determined by Γ_n. Therefore, the recursion relation of u′₂(x_n) with u′₂(x_n₋₁) could be derived from Eq. (8), considering the fact that Γ_n is completely determined by {u′₂(x_n₋₁), u′₂(x_n₋₂),..., u′₂(x₁)}. This implies that u′₂(x_n) can be obtained for a given u′₂(x₁). Furthermore, according to Eq. (13), u₂(x_n) can be solved when u′₂(x_n) is obtained.

Now, we elaborate the calculation of u′₂(x₁). Considering A₀ = 0 for the first domain and by solving Eq. (12), y and β can be obtained. After substituting y and β into Eq. (8), u′₂(x₁) can be calculated subsequently. Now, we discuss the iterations in detail. According to Eq. (8), u(x₁) can be obtained, and by substituting u(x₁) into Eq. (6), Γ₂ can be obtained. After obtaining u₃_a, u₃_b and u₃_c by Eq. (12), u(x₂) can be obtained by Eq. (8). Then, Γ₃ can be solved by Eq. (6). Using the similar process, u(x_n) can also be obtained. It is worth noting that the solution of u₂(x_n) can be obtained as long as the domain configuration is given, and consequently, Eq. (8) can be applied to any domain configuration.

3. Results and analyses

3.1. Investigation of applicability of the AOS devised in UPA for an exact solution

It was suggested that a specific design of the AOS was proposed to achieve multiple wavelength SHG with nearly identical conversion efficiency for UPA [6]. What particularly interests us may be the dependence of AOS configuration on the calculation method. That is to say, whether the AOS devised in UPA can also achieve high conversion efficiency SHG as the case when the pump depletion is considered, and how the conversion efficiency of SHG in UPA is different from that while considering the pump depletion. As an example, we assume that the pre-assigned FW wavelengths are λ₁ = 0.972μm, λ₂ = 1.064μm and λ₃ = 1.283μm, and therefore the corresponding refractive indices of LN crystal at the FW and SHW frequencies can be evaluated at T = 25°C. The number of domains is N = 600, and the thickness of a single domain is Δx = 3μm. It is assumed that the intensity of the incident light waves was set as I = 1.0 × 10¹²W/m², and d₃₃ = 27.0pm/v [18]. An AOS sample which can achieve the above-mentioned three wavelengths SHG with high and nearly identical conversion efficiency is devised by the SA method in UPA. The results show that the conversion efficiency for λ₁ is 0.938, λ₂ is 0.941 and λ₃ is 0.942. Using the same AOS configuration, the corresponding SHG conversion efficiency for an exact solution was calculated, and the conversion efficiency for λ₁, λ₂ and λ₃ is 0.558, 0.554 and 0.559, respectively. It is clearly seen that the difference of SHG conversion efficiency for the two different calculation methods is significant. It is believed that the difference maybe results mainly from the following two points: one is that the AOS devised in UPA is not applicable to the case for an exact solution; the other comes from the calculation method itself.

In order to investigate the possible factor influencing the conversion efficiency for the two different calculation methods, we now devise another AOS sample for an exact solution by the SA method [6]. The result shows that this sample can achieve high conversion efficiency SHGs, with a value of 0.607 for λ₁, 0.606 for λ₂, and 0.607 for λ₃, respectively. It is clearly suggested that there is little difference for the SHG conversion efficiencies between the two different samples. This implies that the AOS sample devised in UPA applies to a general situation of low and high conversion efficiency of SHG. In order to prove this point, we changed other parameters, such as pump intensity, nonlinear media and the preassigned wavelength, and also found that there is little difference for the calculated SHG conversion efficiency of the two AOS samples devised in UPA and by using an exact solution, typically lower than 10% in most cases. Based on the analysis, the above-noted significant difference of SHG conversion efficiency can be attributed to the fact that the UPA doesn’t work when SHG conversion efficiency is high enough.

The wavelength dependence of SHG conversion efficiency (η) calculated using an exact solution method for the pump intensity I = 1.0 × 10¹²W/m² is shown in Fig. 1 for the above-mentioned AOS sample devised in UPA. All of the three preassigned wavelengthes exhibit almost identical-height peaks after scanning a wide range of wavelengths from 0.95 to 1.30 μm. It is seen that there are also two unexpected peaks with λ = 0.984μm and λ = 1.083μm appearing in the vicinity of the preassigned wavelengthes λ₁ = 0.972μm and λ₂ = 1.064μm. This further proves the fact that the AOS sample devised in UPA, instead of an exact solution method, can also be employed to achieve high SHG conversion efficiency for the preset multiple wavelengthes.

Fig. 1 Variation of SHG conversion efficiency η with the wavelength for the pump intensity I = 1.0 × 10¹²W/m².

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3.2. Comparison of u₂(x_n) calculated by UPA and exact solution method

In addition, to investigate the effect of pump intensity on the SHG conversion efficiency, the variation of u₂(x_n) with the sequence of domain for different pumping intensities is shown in Fig. 2: solid curve denotes the intensity I = 1.0 × 10¹⁰W/m², dashed curve corresponds to I = 1.0 × 10¹¹W/m², and dotted curve is I = 1.0 × 10¹²W/m². The pre-assigned FW wavelengths are λ₁ = 0.972μm, λ₂ = 1.064μm and λ₃ = 1.283μm. The domain number is N = 1000. We devised AOS sample under the UPA, and in such an AOS sample, u₂(x_n) at different pump intensities (with λ₂ as a representation) was calculated for the two cases of UPA and pumping depletion. Red curves represent the case of adopting UPA and black curves denote the exact solution. u₂(x_n) increases nearly linearly with n for red curves. However, the increase presents nonlinear behavior for the exact solution when the value of u₂(x_n) is high enough, although u₂(x_n) increases linearly under low conversion efficiency. The difference of u₂(x_n) between UPA and exact solution increases with the u₂(x_n) enhancing. When the SHG conversion efficiency is relatively low, the red and black curves are coincident for I = 1.0 × 10¹⁰W/m², but with the conversion efficiency gradually increasing, they separate with each other, especially, for the case of I = 1.0 × 10¹²W/m². The difference of u₂(x_n) between UPA and exact solution results from the depletion of FW intensity during the conversion from FW to SHW, and the difference is larger with the increased depletion of FW intensity which is proportional to the SHG conversion efficiency. Furthermore, it is well known that the SHG conversion efficiency increases with the increase of pump intensity. Therefore, u₂(x_n) of UPA and exact solution methods deviates from each other as the pump intensity is higher. In addition, in Fig. 2, based on the result of the exact solution (curves with the black color), it is found that the value of u₂(x_n) − u₂(x_n₋₁), which represents the contribution of single domain to SHG, is related to the pumping intensity and the domain sequence (n). The contribution gradually decreases with the increase of n, and increases with the pump intensity. Obviously, the FW intensity decreases with the domain sequence owing to the conversion from FW to SHW, and thus the u₂(x_n) − u₂(x_n₋₁) decreases with the increase of n. In addition, it is worth noting that the above-mentioned AOS sample (devised in Fig. 2) will be always adopted in the following paragraphs.

Fig. 2 u₂(x_n) as a function of sequence of domain n for the three different pump intensities in the two different cases: red curve for the UPA, and black curve for an exact solution (EA). Solid curve for I = 1.0 × 10¹⁰W/m², dashed curve corresponds to I = 1.0 × 10¹¹W/m², and dotted curve is I = 1.0 × 10¹²W/m².

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3.3. Calculation of u₂(x_n) for an exact solution method by ũ₂(x_n) in UPA

We now define the relative tolerance $σ = \frac{{\tilde{u}}_{2} (x_{n}) - u_{2} (x_{n})}{u_{2} (x_{n})}$ to estimate the applicable scoped of UPA, noting that ũ₂(x_n) represents the value in the case of UPA. Figure 3 shows the variation of relative tolerance with the u₂(x_n) for the above-mentioned three pump intensities. It is found that the three curves have no relation with the pump intensity, and display nearly coincident behavior. This result implies that the relative tolerance is mainly determined by the SHG conversion efficiency, and is almost unrelated to the pump intensity. By changing other parameters such as the sample configuration, the nonlinear media and the preassigned wavelength, we found that the curves are also nearly coincident with that in Fig. 3. Especially, for the optimum domain widths in which the so-called QPM for a single wavelength is satisfied, the nearly same curves can also be obtained. This further indicates that the relative tolerance is solely determined by the SHG conversion efficiency, but unrelated to the sample configuration, the nonlinear media and the incident intensity. According to the curves in Fig. 3, two equations can be fitted as follows:

σ = 0.39 u_{2}^{2} (x_{n}) - 0.0115 u_{2} (x_{n}) (u_{2} (x_{n}) \leq 0.4),

and

σ = 0.0097 e^{u_{2} (x_{n}) / 0.217} (0.4 < u_{2} (x_{n}) < 0.9) .

This result is particularly of interest here: practical sample can be accurately and conveniently evaluated, and the applicability of UPA can also be well-estimated.

Fig. 3 Variation of relative tolerance $σ = \frac{{\tilde{u}}_{2} (x_{n}) - u_{2} (x_{n})}{u_{2} (x_{n})}$ with u₂(x_n) for the three different pump intensities, ũ₂(x_n) represents the value in the case of UPA, solid curve for I = 1.0 × 10¹⁰W/m², dashed curve corresponds to I = 1.0 × 10¹¹W/m², and dotted curve is I = 1.0 × 10¹²W/m².

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In order to prove that the Eq. (14) is useful to assess the SHG conversion efficiency for an exact solution as long as ũ₂(x_n) is obtained for a given domain configuration, we calculated the conversion efficiency as a function of the total input intensity I and the results are shown in Fig. 4. We set I₀ = 1.0 × 10⁹W/m² as a unit intensity. Solid curve is the solution solved by Eq. (8), and dotted curve is the fitted curve according to Eq. (14) provided that ũ₂(L) is known for a specific AOS sample. It is clearly shown that the two curves are nearly coincident. These results suggest that Eq. (14) can be used to assess the SHG conversion efficiency for an exact solution when SHG conversion efficiency under UPA is calculated.

Fig. 4 Variation of u₂(L) as a function of input intensity I/I₀, I₀ = 1.0 × 10⁹W/m², solid curve denotes the data for exact solution and dotted curve is the fitted curve according to Eq. (14) provided that ũ₂(L) in UPA is known for a specific sample.

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The relation between u₂(x_n) and ũ₂(x_n) is discussed in details as follows. In view of the fact that SHG conversion efficiency is mainly dependent on the intensity of FW field, and the curve u₂(x_n) verse x_n for an exact solution is nearly coincident with that in UPA when the conversion efficiency is low, such an equation could be assumed

u_{2} (x_{n}) = α u_{1}^{2} (x_{n - 1}) + u_{2} (x_{n - 1}) .

Here, α is the slope of the curve ũ₂(x_n) verse x_n in UPA (which is nearly a straight line), and α = ũ₂(x_N)/x_N. In order to verify the adaptability of the above formula, in Fig. 5, we describe the variation of u₂(x_n) as a function of x_n: solid curve denotes the data for an exact solution and dotted curve is the fitted curve according to Eq. (15). Clearly, it can be seen that the two curves are nearly coincident, and the solid curve exhibits a small fluctuation which mainly results from the imperfect QPM. The perfect QPM relates to the conventional single wavelength case, in which the domain width of the QPM is identical to the coherence length Δx = π/Δk. However, in the case of multiple wavelengthes, the sample is designed to ensure the preassigned multiple wavelengthes with high and nearly identical conversion efficiencies. In such a condition, the obtained conversion efficiency should be smaller than that in the case of the conventional QPM. As a result, the QPM is not as perfect as the case of single wavelength any more. Therefore, the SHG conversion efficiency should not always increase continuously and will show a small fluctuation with x_n. It is worth noting that for the special case of a single QPM grating,

α = \frac{2 | ξ |}{Δ k}

in Eq. (15). This result obtained here imply that Eq. (15) can be used to assess the conversion efficiency of SHG when pump depletion can not be ignored.

Fig. 5 Variation of u₂(x_n) as a function of x_n, solid curve denotes the data for exact solution and dotted curve is the fitted curve according to Eq. (15).

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4. Conclusion

In summary, we investigate the SHG in AOS with the consideration of pump depletion. The results show that the conversion efficiency of SHG when considering the pump depletion significantly differs from that in the case of UPA. The AOS sample designed in UPA, which is found sufficiently unrelated to the factors in traditional view, such as the pump intensity and the SHG conversion efficiency, could also be used to obtain high conversion efficiency SHG for an exact solution. A relative tolerance about u₂(x_n) based on both UPA and an exact solution was calculated, and it was shown that the relative tolerance is solely determined by the conversion efficiency, independent of the sample configuration, pump intensity, wavelength and nonlinear media. A model to assess u₂(x_n) was assumed, and the model proves itself easy and convenient to estimate the SHG conversion efficiency when pump depletion can not be ignored. These results will provide direct guidance for practical application, and can be used to evaluate and estimate practical samples more conveniently.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NNSFC) under grant 11274233 and 11004139, the Natural Science Foundation of Beijing under grant 1102012.

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Effect of pump depletion on second harmonic generation in multiple quasi-phase-matching gratings

Abstract

1. Introduction

2. Derivation for an exact solution

3. Results and analyses

3.1. Investigation of applicability of the AOS devised in UPA for an exact solution

3.2. Comparison of u₂(x_n) calculated by UPA and exact solution method

3.3. Calculation of u₂(x_n) for an exact solution method by ũ₂(x_n) in UPA

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (21)

Optics Express

Abstract

1. Introduction

2. Derivation for an exact solution

3. Results and analyses

3.1. Investigation of applicability of the AOS devised in UPA for an exact solution

3.2. Comparison of u2(xn) calculated by UPA and exact solution method

3.3. Calculation of u2(xn) for an exact solution method by ũ2(xn) in UPA

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (21)

Optics Express

3.2. Comparison of u₂(x_n) calculated by UPA and exact solution method

3.3. Calculation of u₂(x_n) for an exact solution method by ũ₂(x_n) in UPA