Poling Quality Evaluation of Optical Superlattice Using 2D Fourier Transform Method

X. J. Lv; L. N. Zhao; J. Lu; G. Zhao; H. Liu; Y. Q. Qin; S. N. Zhu

doi:10.1364/OE.17.018241

1. Introduction

The development of domain-engineered crystals has gained significant importance not only in nonlinear optical frequency conversion process [1], but also in linear polarization control [2] or modulation [3] devices. Quasi-phase-matching (QPM) devices have been successfully realized in ferroelectric crystals such as lithium niobate (LiNbO₃), lithium tantalate (LiTaO₃) and potassium titanyl phosphate (KTiOPO₄) by the process of electric field poling. Utilizing this approach, QPM pattern with desired reciprocal vectors is able to be fabricated. Due to substrate inhomogeneity and imperfect photolithography process, two kinds of error are inherent: period error and duty cycle error [4]. There are several methods to visualize the resulting domain structures such as second harmonic generation microscopy [5], confocal luminescence microscopy [6], optical near field microscopy [7], etc. However, in these techniques only a small area of the poled pattern can be visualized. An alternative approach to investigate the poling quality of QPM device using diffraction method has been described by Krishnamoorthy et al, in which the duty cycle error is quantified [8].

Effective nonlinear coefficient (d_eff) is proportional to the magnitude of the reciprocal vector. Period and duty cycle error influence the magnitude of the reciprocal vector and finally reduce the nonlinear conversion efficiency. In this article, we present a technique to quantify the magnitude of the reciprocal vectors using 2D Fourier transform method. By comparing the measured result with the ideal or designed magnitude of the reciprocal vector, the poling quality of an OSL can be evaluated.

2. Theory

The second-order nonlinear polarization is given by:

(\begin{array}{c} P_{1} \\ P_{2} \\ P_{3} \end{array}) = (\begin{array}{c} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\ d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{array}) = (\begin{array}{c} E_{1}^{2} \\ E_{2}^{2} \\ E_{3}^{2} \\ {2 E}_{2} E_{3} \\ {2 E}_{3} E_{1} \\ {2 E}_{1} E_{2} \end{array}) .

For QPM materials, e.g. periodically poled LiTaO₃ (PPLT), the maximum second-order nonlinear optical susceptibility is d ₃₃=13.8pm/v@1064nm[9]. Define rect function as

rect (t) = {\begin{array}{c} \begin{array}{c} 0 & , & if & ∣ t ∣ & > \frac{1}{2} \\ \frac{1}{2} & , & if & ∣ t ∣ & = \frac{1}{2} . \\ 1 & , & if & ∣ t ∣ & < \frac{1}{2} \end{array} \end{array}

We assume the propagation to be along the x-axis. In order to compensate phase mismatch, the modulation function of d ₃₃ should be:

f (x) = Σ_{j = - m}^{m} [rect (\frac{j + \frac{x}{Λ}}{D}) - rect (\frac{j - D + \frac{x}{Λ}}{1 - D}],

where Λ is the period and D is the duty cycle, satisfying $\frac{1}{Λ} = \frac{n (λ_{1})}{λ_{1}} - \frac{n (λ_{2})}{λ_{2}} - \frac{n (λ_{3})}{λ_{3}} (λ_{1} < λ_{2}, λ_{1} < λ_{3})$ . The length of the superlattice is a=(2m+1)Λ. Effective nonlinear coefficient is given by

d_{eff} = d_{33} . ∣ g (f) ∣,

g (f) = \frac{1}{a} \int_{- \frac{a}{2}}^{\frac{a}{2}} f (x) \exp (2 π i f x) d x .

At the phase-matching points, $f_{n} = \frac{n}{Λ} (n = 1,2,3, . . .),$ ,

g (f_{n}) = \frac{1}{a} \int_{- \frac{a}{2}}^{\frac{a}{2}} f (x) e x p (2 π i f_{n} x) d x = \frac{e x p (2 π i n D) - 1}{i n π} .

If the superlattice is poled perfectly $(D = \frac{1}{2}), | g (f_{n}) | = 0$ when n is even, and $∣ g (f_{n}) ∣ = \frac{2}{πn}$ when n is odd. The most commonly used one is the first order reciprocal vector: when n=1, |g ₁|=0.6366.

The ferroelectric domain boundaries of a perfectly poled one-dimensional (1D) OSL parallel to each other. The substrate inhomogeneity and imperfect photolithography result in the distorted domain boundaries. In order to evaluate these error, two-dimensional (2D) evaluation method should be introduced. Similarly, we extend the g(f_x) to the 2D case,

g (f_{x}, f_{y}) = \frac{1}{a \cdot b} \int_{- \frac{b}{2}}^{\frac{b}{2}} \int_{- \frac{a}{2}}^{\frac{a}{2}} f (x, y) \exp [2 πi (x f_{x} + y f_{y})] dxdy,

and

d_{eff} = d_{33} . ∣ g (f_{x}, f_{y}) ∣,

where a is the length and b is the width of the sampling area. Generally the sampling area should contain enough periods, so the condition of a,b≫Λ should be satisfied. To explore the domain walls, the -Z and +Z surfaces of the same OSL are etched with hydro-fluoric solution. It etches smoothly at -Z face whereas the inverted domain etching rate is negligible. Using polarizing microscope or phase contrast microscope, we can distinguish the +Z domain from the -Z domain. The domain wall in the micrograph usually has a certain width. In order to find the accurate position of the domain walls, image processing operations [10] including sharpening and thinning methods are preformed. After image processing operations, the +Z domains are assigned +1 and -Z domains are assigned -1, by which we can calculate the reciprocal vectors of the superlattice.

However, the view-field of the microscope is limited to a small region. Now we consider how to extrapolate the g(f_x, f_y) of the OSL from the region we observed. A limited region $f (x, y) = u (x, y) \cdot rect (\frac{x}{a}) \cdot rect (\frac{y}{b})$ is intercepted from the whole region u(x,y). Periodic prolongation of f(x,y) will rebuild U(x,y) as:

U (x, y) = f (x, y) * \sum_{j = - m}^{m} \sum_{k = - n}^{n} [δ (ja + x) . δ (kb + y)] .

m, n are integers and δ(x) is Dirac function. If the sampling area is representative, we have U(x,y)≈u(x,y). Then the reciprocal vector of U(x,y) is

G (f_{x}, f_{y}) = \frac{1}{S} ℱ (U (x, y))

This means that at $(f_{x} = \frac{M}{a}, f_{y} = \frac{N}{b}),$ , the Fourier transform result of a limited region is equal to that of the expanded region. If the observed region is representative, by calculating the reciprocal vector at these points, we can obtain the $g (\frac{M}{a}, \frac{N}{b})$ of the whole OSL. The fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT). In 1D case, the separated frequency components of FFT are at $f N = \frac{N}{a}$ (assume the sample length in the real space to be a and N=0,1,2, …). The first order reciprocal vector (g ₁) of the superlattice with a period of Λ is at $f = \frac{1}{Λ}$ . In order to ensure the peak of the first order reciprocal vector coincides with the discrete frequency components of FFT, the condition of $\frac{N}{a} = \frac{1}{Λ}$ should be satisfied, which means that $\frac{a}{Λ}$ should be integer and the sample region should contain integer periods.

An alternative method is high resolution Fourier transform (HRFT) [11], which is capable of improving the frequency resolution. But the sample window will impact on the reciprocal vectors. If the periodic structure is along x-axis,

g (f_{x}, f_{y}) = \frac{1}{a \cdot b} ℱ f (x, y)

Theoretically, the magnitude of the even order reciprocal vectors are zero, and the maximum influence of the odd order vectors(3th order) to the first order vector is $\frac{Λ}{3 πa} .$ . If the sample region contains enough period $(\frac{Λ}{3 πa} ≪ 1),$ , this influence is negligible (e.g., when $\frac{a}{Λ} = 50,$ , $\frac{Λ}{3 πa} = 0.002$ . So it is capable to obtain the reciprocal vector by HRFT with very small error. But HRFT algorithm is much slower than FFT. In practice, we use FFT to obtain the approximate position of the reciprocal vector and then use HRFT to calculate the peak value of it. The calculation speed and precision can both be satisfied.

In addition, the sampling interval will also cause calculation error. Assume the sampling interval to be d. The sign of the whole interval is decided by the value of the point at the center. So the sampled duty cycle is $D_{s} = \frac{p}{Λ} + D,$ , where D is the actual duty cycle and p is random number uniformly distributed over the interval $(- \frac{d}{2}, \frac{d}{2}) .$ . Mathematical expectation of random variable X is noted as E(X). Considering E(X ₁+X ₂+⋯+X_n)=E(X ₁)+E(X ₂)+⋯+E(X_n), the Mathematical expectation of Eq. (5) is given by

E (g (f) D_{s}) = \frac{1}{a} E {\int_{- \frac{a}{2}}^{\frac{a}{2}} \sum_{j = - m}^{m} ⌊ rect (\frac{j + \frac{x}{Λ}}{D_{j}}) - rect (\frac{j - D_{j} + \frac{x}{Λ}}{1 - D_{j}}) ⌋ \exp (2 πifx) dx}

where $D_{j} = \frac{p_{j}}{Λ} + D .$ . Considering that {p_j} are independent with each other and under the same distribution,

E (g (f) D_{s}) = \frac{1}{ad} \int_{- \frac{d}{2}}^{\frac{d}{2}} \int_{- \frac{a}{2}}^{\frac{a}{2}} \sum_{j = - m}^{m} [rect (\frac{j + \frac{x}{Λ}}{D_{j}}) - rect (\frac{j - D_{j} + \frac{x}{Λ}}{1 - D_{j}})] . \exp (2 πifx) dxdp

When $\frac{a}{Λ} = 50$ (it means that the sampling area contains 50 periods) and the sampling number is 1024,

∣ E (g_{1}) ∣ - ∣ E (g_{1} D_{s}) ∣ = {\begin{cases} 0.0013, D = 0.5 \\ 0.0011, D = 0.3 \end{cases} .

The measured result is always smaller than the actual value. This is easy to understand because the sampling error always decreases the vector magnitude. If the sampling number of each domain is large enough, this error is negligible.

3. Experiment and discussion

The periodic poled LiTaO₃ (PPLT) and LiNbO₃ (PPLN) samples are fabricated by electric field poling technique [12]. To explore the superlattice, the -Z and +Z surfaces of the same OSL are etched with hydro-fluoric solution. The original micrography of the +Z surface of sample 1 (PPLT, Λ=28.6µm) is shown in Fig. 1(a). Figure 1(b) shows the image processing result of Fig. 1(a). Then fill the positive domains with black(+1) and the negative domains with white(-1) as shown in Fig. 1(c). The 1024×1024 pixels image is sampled point-by-point and FFT result is shown in Fig. 2(a) and 2(b). Due to the limited resolution of FFT, the peak amplitude of the reciprocal vectors in Fig. 2(b) are not accurate. But we can find the approximate positions of these peaks. Figure 2(c) shows the HRFT result of Fig. 1(c) at the first order reciprocal vector position. The peak value is |g ₁|=0.625. This means that the poling quality of sample1’s +Z surface is very high (the perfect value is 2/π=0.6366). If the d33 is given as d ₃₃=13.8pm/v, d_eff=d ₃₃ · |g ₁|=8.63pm/v.

Fig. 1. Sample 1: +Z surface (Λ=28.6µm). (a) The original micrograph; (b) The processed image; (c) The final image before analysis.

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Figure 3 shows the -Z surface of sample 1 and the FFT result at f_y=0. The HRFT result of the first order reciprocal vector is |g ₁|=0.592. From Fig. 3(b) we can see that due to the decreased poling quality, the second-order reciprocal vector is higher than the third one.

Figure 4 shows the -Z surface of sample 2 (PPLT, Λ=7µm) and it’s FFT result. The HRFT result of the first order reciprocal vector is |g ₁|=0.187. From Fig. 4(a) we can see that due to the poor poling quality, the one-dimensional superlattice nearly turns in to a two-dimensional one. The nonuniformity of the 2D pattern introduced by the poling process greatly decreases the quality of the superlattice.

Fig. 2. Fourier transform result of Fig. 1(c). (a) 2D FFT result; (b) 2D FFT result at f_y=0; (c) HRFT result is |g ₁|=0.625.

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Fig. 3. Sample 1: -Z surface. (a) The processed micrograph; (b) 2D FFT result at f_y=0. The HRFT result is |g ₁|=0.592.

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Fig. 4. Sample 2: -Z surface (Λ=7µm). (a) The original micrograph; (b) 2D FFT result. The HRFT result is |g ₁|=0.187.

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Figure 5(a) shows the +Z surface of a square poled lithium tantalate (Squ-PLT, sample 3, square lattice of circular patterns) with structure parameter a=9.05µm. The FFT result is shown in Fig. 5(b). The HRFT result of the maximal reciprocal vector is |g ₀₁|=0.293, which is designed to be 0.4. Now we derive the analytical expression of the reciprocal vector of the 2D OSL with rectangle lattice. The OSL can be expressed as

f (x, y) = {2 P (x, y) * [comb (\frac{x}{Λ_{x}}) comb (\frac{y}{Λ_{y}})] - 1} \cdot rect (\frac{x}{a}) \cdot rect (\frac{y}{b}),

where P(x,y) is the function of the positive domain in a period, Λ_x and Λ_y are the period of lattice along x-axis and y-axis, a and b are the width and length of the OSL, respectively. The reciprocal vector of 2D OSL is given as:

g (f_{x}, f_{y}) = {2 Λ_{x} Λ_{y} ℱ [P (x, y)] \cdot comb (f_{x} Λ_{y}) comb (f_{x} Λ_{y}) - δ (f_{x}) δ (f_{y})} * [\sin c (a f_{x}) \sin c (b f_{y})] .

For square poled OSL, if we assume the positive domain to be circular and Λ_x=Λ_y=a, the maximum g ₀₁ is about 0.4 (which is the reciprocal vector we desire) when the radius of the circle is about 0.39a. From the evaluation we can see that due to poling error, the g ₀₁ of the fabricated 2D SQL-PLT is smaller than the perfect value. If the pattern is irregular, it is difficult to measure the duty cycle and period error. In this case, our method is still valid, which indicates that it is quite fit for the evaluation of 2D OSL.

Fig. 5. Sample 3: +Z surface (Squ-PLT, a=9.05µm). (a) The original micrograph; (b) 2D FFT result. The HRFT result is |g01|=0.293.

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In addition, the process of 1024×1024 FFT and HRFT is very fast (several seconds using a computer with 2GHz CUP clock speed). The necessities of this method are only microscope with CCD camera and computer. The simplicity and rapidity of this method make it promising in poling quality evaluation.

In order to validate the evaluation, SHG experiment is performed to inspect the poling quality. Under small-signal condition, the generated second harmonic intensity is

I_{2} = \frac{2 ω_{1}^{2} d_{eff}^{2} L^{2} I_{1}^{2}}{c^{3} n_{1}^{2} n_{2} ε_{ο}} \frac{\sin^{2} (△ k L / 2)}{{(△ k L / 2)}^{2}} = C {∣ g ∣}^{2},

where I ₁ and I ₂ are the optical power density of fundamental and second-harmonic wave, respectively. The direct measurement of |g| is difficult because the above equation is derived using the approximation of plane wave, single-longitudinal mode and continuous wave conditions. What’s more, the d ₃₃ of LiNbO₃ varies from 25pm/v to 42pm/v at different references [13], which will have great impact on the calculation of |g|. The relative |g|² is easier to be achieved. Assuming that the fundamental beam propagates along Y-axis, when scan the fundamental beam along X-axis, we will achieve I ₂(x)=C|g(x)²|, where C is a constant if we keep the other parameters (I ₁, oven temperature, etc.) constant. In our experiment, the fundamental wave is outputted from a Q-switched 1064nm Nd:YAG laser with pulse duration of 50ns and repetition rate of 0.2KHz. The average power is 560mw. Then it is focused to about 0.1mm diameter spot with a f=200mm lens. The fluctuation of fundamental power is less than 2% within 10 minutes. The PPLN oven is fixed at 146.1°C±0.1 to ensure negligible photorefractive effects. The beam is propagating along Y-axis and moving along X-axis. The measured second harmonic power is shown in Fig. 6(b). The maximum output power is about 50mW and the small-signal approximation is still valid. We divide the +Z surface of the 6.14mm long (Y-axis, periodically poled), 5mm wide (X-axis) PPLN (sample 4) into 5×10 areas. The poling quality of the -Z surface approximately equals to that of the +Z surface. The period of the OSL is about 6.6µm and the first reciprocal vector is used for SHG process. We calculate the |g ₁| of each area separately (Fig. 6(a)). Each g ₁ represents the average intensity of the reciprocal vector at the single area. The average |g ₁| of the whole OSL (50 areas) is 0.181 and the standard deviation (SD) is 0.052. In general, the higher of |g1| and the smaller of SD, the higher quality of the whole OSL. Due to 𝓕(f ₁+f ₂)=𝓕(f ₁)+𝓕(f ₂), the |g ₁(x)| equals to the average |g ₁| of each area along Y-axis at x position. Fig. 6(b) shows the comparison of the calculated |g ₁(x)|² and the measured output SHG power. The fundamental beam is influenced by the edge of the OSL, so the output power is smaller than it should be at the edge of the OSL. At other points, the proportional relation between output power and the calculated |g ₁|² is good. If we define C(x)=I ₂(x)/|g ₁(x)|², the relative deviation can be expressed as $∣ C (x) - \overset{ˉ}{C (x)} ∣ / \overset{ˉ}{C (x)}$ which is smaller than 5% except for the two points at the edge of the OSL. Considering the fluctuation of fundamental power and temperature, we think this error is acceptable.

Fig. 6. (a) Evaluated |g ₁| at the +Z surface of sample 4 (PPLN, Λ=6.6µm); (b) SHG power and evaluated |g ₁|² at different X-axis position.

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4. Conclusion

In conclusion, we fabricate PPLT and PPLN samples using standard electric field poling technique. A method is proposed to evaluate the poling quality by analyzing the processed micro-graph of the etched surface. The error caused by sampling is derived and estimated. Utilizing this method, the amplitude of the reciprocal vectors are able to be calculated directly. Poling quality of the whole superlattice is evaluated and average quality and deviation are given. SHG experiment is performed to validate the evaluated result, which demonstrates that this method is reliable. In principle, this technique is appropriate for evaluating the poling quality of the periodically poled 1D or 2D superlattice.

Acknowledgments

This work is supported by the National Natural Science Foundation of China Grant (Nos.10776011, Nos.10534020) and by the National Key Projects for Basic Research of China (No.2006CB921804).

References and links

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Poling Quality Evaluation of Optical Superlattice Using 2D Fourier Transform Method

Abstract

1. Introduction

2. Theory

3. Experiment and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (26)

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