1. Introduction

After the successful development of a periodical poling technique for lithium niobate (LiNbO₃) second-order nonlinear frequency mixing can be quasi-phase-matched (QPM) in a wide wavelength range. Many applications in laser technology and optics communications, as well as pure scientific investigations benefit from nowadays available bulk and waveguide LiNbO₃ devices with QPM gratings for efficient frequency mixing [1, 2]. Because of a wide range of applied frequencies the magnitude and the frequency-dependence of the second-order nonlinear optical susceptibility tensor χ⁽²⁾ of LiNbO₃ are interesting to know. Since 1964 measurements of second-order nonlinear tensor elements of LiNbO₃ have been reported. However, the reported values in different publications deviate by up to 100% [3]. Only since the late 90s, more consistent values for all three independent tensor elements at a few wavelengths have become available. In this article we use the definition of the susceptibility from classical nonlinear optics [4, 5]. The quadratic nonlinear susceptibility tensor χ⁽²⁾(2ω;ω,ω) for second-harmonic generation (SHG) is equal to the half of the nonlinear dielectric polarization response tensor κ⁽²⁾(2ω;ω,ω). The tensor coefficients of the quadratic susceptibility for SHG are often simply called nonlinear coefficients d. Values for ${\chi }_{zzz}^{\left(2\right)}=d_{33}$ and ${\chi }_{zxx}^{\left(2\right)}=d_{31}$ for SHG in LiNbO₃ with different compositions are reported in Refs. [3, 6] which agree well to the old measurements from Miller at a wavelength of 1.064 μm [7]. The d₃₁ value at 1.064 μm was later confirmed in good agreement in Ref. [8] where additionally a value for ${\chi }_{yyy}^{\left(2\right)}=d_{22}$ is presented. However, the discussion continues with deviating experimental results for d₃₁ at 1.32 μm in Ref. [9]. The measured values for d₃₃ in congruent LiNbO₃ from Shoji are plotted in Fig. 1 for all the measured wavelengths [3].

 

Fig. 1 d₃₃ in LiNbO₃ at room temperature from Ref. [3] and from calculations in Ref. [10] compared to the measurement at 1.52 μm from this work. The error for the older measurement was estimated to be better than 10%. The most probable 5% error bars are shown.

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Calculations based on the Chemical Bound Model of the dielectric properties of LiNbO₃ including also second-order susceptibilities were reported for a wavelength of 1.064 μm [10]. The calculated nonlinear coefficients compare well with Shoji’s measurements as shown in Fig. 1. The small theoretically expected composition-dependence [10] could not experimentally be verified within the measurement accuracy in congruent and stoichiometric material [6].

We are not aware of a publication of directly measured values for the nonlinear coefficients of LiNbO₃ for SHG in the communications wavelength range near 1.55μm, neither in the pure bulk material nor in waveguide structures. In order to obtain reliable information on the of-ten applied largest nonlinear coefficient d₃₃ in waveguides, calibrated SHG efficiency measurements in the low-fundamental-depletion regime were performed in titanium-indiffused LiNbO₃ waveguides with a QPM grating. It showed that the most direct measurement of the nonlinearity in a frequency mixing experiment needs a detailed system analysis. All important experimental parameters and their tolerances were measured to analyze their influence on the accuracy of the measured susceptibility. A careful setup calibration, measurements of the waveguide modes, losses and the quality of the QPM grating, and a strict analysis of the interference fringes due to multiple reflections at the end facets of the waveguide yield a value for the second-order susceptibility element $d_{33}=\left(-20.6\pm 2.1\right)\frac{\text{pm}}{\text{V}}$ for SHG in periodically poled, titanium-indiffused waveguides in congruent composition lithium niobate at room temperature for a fundamental wavelength of 1.52 μm. Despite the complexity of the waveguide experiment due to the large number of system parameters a measurement accuracy of 10% could be quantitatively determined. The nonlinear coefficient presented in this work fits well to the previously measured nonlinearity of the bulk material at lower wavelengths as shown in Fig. 1. Therefore, the titanium indiffusion has only limited influence on the nonlinear coefficient, and the over the waveguide mode area averaged measured “effective” nonlinear coefficient is close to the d₃₃ of LiNbO₃.

2. Linear waveguide characteristics

2.1. Samples and experimental technique

Optical waveguides were fabricated by indiffusion of 5-, 6- and 7-μm-wide and 100-nm-thick titanium stripes in a Z-cut LiNbO₃ crystal in the Department of Physics at the University of Paderborn. The diffusion time was 8.5 hours at a temperature of 1060°C. With electric-field poling a QPM grating with a periodicity of Λ = 16.4 μm was prepared. To measure d₃₃ both, the fundamental wave (FW) and the second-harmonic wave (SH) have to propagate as quasi-linear polarized TM waveguide modes. With the chosen QPM period a FW TM₀₀ mode with a wavelength close to 1.52 μm is quasi-phase-matched to a SH TM₀₀ mode at room temperature. The waveguides are monomode guides for the FW and guide 7 to 8 modes at the SH wavelength. The sample and the waveguides are 2 cm long with polished end facets for end-fire coupling. The end facets are not anti-reflection coated. The sample was cut out of a fabricated 7-cm-long sample to obtain a very uniform waveguide.

The FW mode was excited by focusing a continuous-wave beam from a wavelength-tunable external-cavity laser diode with a linewidth of 150 kHz with a microscope objective onto the input surface of the waveguide. To prevent photorefraction at room-temperature the FW input power was kept below 0.25 mW. The wavelength was tuned around phase-matching and the output power of the FW and the SH were measured separately with scaled Ophir PD300 detectors. The generated SH at the waveguide output depends in the low-FW-depletion regime on the square of the nonlinear coefficient d₃₃, on the wavelength-dependent phase-mismatch and on waveguide parameters like the mode-overlap integral, losses and end-facet reflectivities. After measuring all these waveguide characteristics d₃₃ could be determined from the tuning curve of the measured SH versus the FW input wavelength. All the presented data stem from 6-μm-wide waveguides. Measurements with 7-μm-wide waveguides gave the same results. The waveguide width is simply defined as the titanium-stripe width before diffusion.

2.2. TM-mode profiles, mode-overlap integral for SHG

Exciting the FW TM₀₀ mode at phase-matching the waveguide end facets were imaged with a 100X microscope objective directly on the detectors of both, an infrared and a silicon digital camera. The accuracy of the mode width measurement is in the order of 1%. The focusing of the modes was relatively easy with the strong intensity gradient close to the waveguide-air interface. The width error was estimated by comparing repeated measurements after refocusing and measurements from different waveguides and different end facets. Fig. 2 compares FW and SH mode-intensity scans. In a perfect waveguide the intensity spot of a mode does not change along the propagation. Indeed FW mode patterns in our waveguides at the output and input facet are identical. However, the modes at smaller wavelengths are more sensitive to small modifications of the waveguide index profile, and in some waveguides differences in the SH mode-intensity patterns are observable. As shown in Fig. 2 the width of the SH TM₀₀ mode may vary by up to 10% from the input to the output side, indicating end facet imperfections or index-profile nonuniformities which are too small for detection in the FW mode pattern. The resulting change of the mode-overlap or coupling integral $K^{\left(2\right)}=\iint \text{d}x\text{d}y{e}_{\text{FW}}\left(x,y\right)e_{\text{FW}}\left(x,y\right)e_{\text{SH}}^{*}\left(x,y\right)$ along the waveguide adds directly to the error of the susceptibility measurement. e_FW/SH are transverse electric fields of the interacting modes normalized to 1 W at the FW and SH wavelength. In order to find the value of K⁽²⁾ the mode fields and the propagation constants were calculated using the model presented in Refs. [11,12]. With a diffusion width of D_x = 3.5μm and a diffusion depth of D_y = 4.7μm in the refractive index model for the mode solver a good agreement between calculated and measured mode profiles was obtained (see Fig. 2). Dependent on sample preparation and fabrication parameters the diffusion lengths had to be adjusted for different samples in a range of ±20% for a good fit between measured and calculated mode intensity patterns. The refractive indices are calculated with the temperature- and wavelength-dependent Sellmeier equation from Edwards and Lawrence [13].

 

Fig. 2 Measured FW and SH TM₀₀ mode-intensity scans of a 6-μm-wide waveguide at the output and input facet (for forward and backward propagation) in comparison with calculated mode profiles, λ = 1.52μm and λ = 0.76μm.

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Based on the mode patterns a coupling integral $K^{\left(2\right)}=\left(9.17\times {10}^8\pm 4\%\right)\frac{{\text{V}}^3}{\text{m}}$ was calculated. The tolerance was estimated by calculating K⁽²⁾ for narrower and wider mode spots, corresponding to the measured mode widths. The width uncertainty of the FW mode in the order of ±1% introduces an uncertainty of ∓1.5% for K⁽²⁾. Fortunately the relative influence of the dominant SH mode uncertainty is smaller. A deviation of the SH mode width by ±1% would change K⁽²⁾ by only ±0.5%.

2.3. Effective mode indices

The mode solver provides also the effective mode indices of the FW and SH TM₀₀ modes. Close to the experimental wavelengths and at room temperature they are expanded in Taylor series with expansion coefficients given in Table 1. Mixed second-order and higher-order derivatives are neglected.

Table 1. Taylor Coefficients for FW and SH TM₀₀ Mode-Index Series^*

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2.4. Losses and reflectivity

Fig. 3 shows the FW throughput of the waveguide when the wavelength is tuned. The fast oscillations in the throughput are due to external cavity resonances in the incoupling setup. Theoretical considerations show that a simple averaging yields the fringes in the throughput of the Fabry-Perot resonator formed by the end facets of the waveguide, which are shown in red in Fig. 3. From the fringe contrast K_FW = 1.605 ± 0.015 the loss coefficient of the FW mode a_FW = (0.057 ± 0.01)cm⁻¹ is determined. This corresponds to $a_{\text{FW}\hspace{0.17em}\text{dB}}=\left(0.25\pm 0.04\right)\frac{\text{dB}}{\text{cm}}$ and a FW waveguide transmittance of T_FW = (89.2 ± 1.8)%. A calculated FW reflectivity for an air-LiNbO₃ interface of R_FW = 13.2% was used for the waveguide end facet reflectivities.

 

Fig. 3 FW waveguide throughput including the end-facet transmittances, the input-mode overlap and the waveguide loss itself; red line: throughput after Fourier filtering.

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Absolute power measurements confirm the determined waveguide transmittance. A transmittance of 88.8% is found by comparing the power of calculated and measured FW output fringes for the measured input power. The same waveguide transmittance is calculated from the ratio between output and input power at a fringe maximum taking the field enhancement in the resonator into account. For our specific operation parameters the reflections from the resonator are negligible when the output fringes are maximal. The overlap between a gaussian input beam with spot size ω_IN = 4.3μm and the FW mode was calculated to be 90.4%. All measurements, also in forward and backward propagation, had a maximum transmittance deviation of ±2%. Therefore the uncertainty in the input overlap and the transmittance of the waveguide end facets add up to a maximum of only 2%.

Losses of the SH TM₀₀ mode of a_SH = 0.17cm⁻¹ are calculated by fitting the fringe contrast of the measured normalized SH tuning curve to a model assuming Fresnel reflections R_SH = 13.8% (see Section 4.1). This corresponds to $a_{\text{SH}\hspace{0.17em}\text{dB}}=0.74\frac{\text{dB}}{\text{cm}}$ and a SH waveguide transmittance of T_SH = 71.2%. This transmittance has not yet been compared to a direct SH throughput measurement because the multi-mode character of the waveguide at the SH wavelength had prevented an unmistakable external SH TM₀₀ mode excitation.

3. Nonlinear waveguide characteristics

3.1. Measured tuning curves

The FW wavelength was tuned and the SH at the output was monitored. With the measured transmittance of the outcoupling optics and with the SH transmittance of the output facet from Fresnel’s formula we can determine the SH power P_SH just before the waveguide output. The main resonance at phase-matching between the FW and SH TM₀₀ modes of such measured SH tuning curves are presented in Fig. 4. Tuning curves with a forward and a backward oriented crystal – rotating the sample by 180° to exchange input and output side – were measured because a congruency of both curves is a proof for good in- and outcoupling and sample symmetry. Measurements were performed at room temperature to prevent nonuniformities due to a temperature profile. The temperature in the laboratory was between 22 and 24°C. An aluminum holder of the sample ensures a homogeneous sample temperature.

 

Fig. 4 (a) SH output just before the waveguide end; black: backward orientation, red: forward orientation, room temperature. (b) Detail of the backward measurement; black: before smoothing, orange: after smoothing. The FW input power was ≈ 0.23 mW. The waveguide length is L = 1180.44Λ = 19.3592 mm.

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The SH tuning curves show a superposition of three sets of fringes. Dominant are those SH fringes which are conform to the Fabry-Perot fringes in the FW (which were shown in Fig. 3). The smaller fast oscillations due to the external cavity resonance of the input FW are also visible. Because they disappear after the normalization during the data processing (see Section 3.2) they can be neglected in the following discussion. After an averaging we observe the remaining superposition of two fringe sets in Fig. 4(b). The dominant fringes which follow the FW Fabry-Perot fringes are superimposed by a fringe set with approximately half of the main periodicity and with a smaller amplitude. These are the Fabry-Perot fringes of the SH. The fringe periodicities are approximated by the Fabry-Perot formula Δλ = λ²/(2Ln) with L as waveguide length and n as effective mode index. In Fig. 4 with a FW wavelength scale the periodicity of the FW cavity fringes is not exactly equal to twice the periodicity of the SH fringes because in quasi-phase-matched samples the effective indices of FW and SH are not equal at phase-matching. The superposition of these two fringe sets yields the formation of three “lobes” within the main SH resonance in Fig. 4(a). The width of each of these lobes w_lobe = 2Δλ_SHΔλ_FW/(4Δλ_SH – Δλ_FW) is determined by the difference of the fringe periodicities of FW and SH. Because all the FW and the SH fringe period (and therefore the lobe width) are inverse proportional to the sample length, as it is also the width of the main resonance of the SH tuning curve, the general appearance of the main resonance of the tuning curve in LiNbO₃ waveguides with reflecting end facets and with QPM gratings for phase-matching near 1.52 μm should be similar to our measured tuning curves, with double fringes and three lobes, independent of the sample length. For our measurement the calculation of the lobe width w_lobe = 0.274 nm using accurate numeric solutions for Δλ_SH and Δλ_FW reproduces the measured w_lobe very well. For the backward propagation curve in Fig. 4b we find at 1516.798 nm that both the FW and the SH fringes have maxima yielding also a lobe maximum. At 1516.931 nm the FW fringes have a maximum and simultaneously the SH fringes have a minimum. Both wavelengths differ correctly by half the calculated lobe width w_lobe/2.

The relative position or phase of the FW and the SH fringes determine the position of the three lobes in the tuning curve maximum. Because the maxima of the FW fringes and the SH fringes in the tuning curve shift with slightly different rates to new wavelengths when changing the sample length also the lobe positions move with a length change. Assuming both FW and SH fringe maxima coincide at a certain wavelength for a given sample length. In order to observe the FW and SH maxima again after a length change (at the same wavelength), the sample length needs to be changed by multiples of half the FW or SH mode wavelength λ_FWm/2 or λ_SHm/2. After a length change of ΔL = νλ_FWm/2 = (2ν + 1)λ_SHm/2 FW and SH maxima coincide again and the tuning curve with an identical lobe form repeats (ν is an integer number). The period ΔL is easily shown to be half of the QPM period ΔL = 0.5λ_FWmλ_SHm/(λ_FWm – 2λ_SHm) = Λ/2. In Fig. 5 the length-dependence of the position of the lobes in tuning curves from different-long waveguides is illustrated. The small differences in the SH output power of the different waveguides are caused by quality deficits like loss and nonuniformity. The position of the lobes in the tuning curve in Fig. 5(c) is very similar to the one in the tuning curve in Fig. 4. Indeed the length difference of both waveguides is 0.54Λ ≈ Λ/2.

 

Fig. 5 SH tuning curves from 6-μm-wide waveguides with different lengths: (a) L = 1179.5Λ, (b) L = 1179.75Λ and (c) L = 1179.9Λ (no backward measurement available); black: backward orientation, red: forward orientation, room temperature, the FW input power was ≈ 0.23 mW.

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The overall appearence of the fringe superposition picture looks very similar for forward and backward propagation for the same waveguide. Equal power levels are an indication for perfect in- and outcoupling. The observed fine shift between fringes in the forward and backward tuning curve for some waveguides is only due to small temperature differences at the moments of each measurement. A shift of half a fringe period corresponds to a temperature change of ≈ 0.3K. Tuning curves measured with years in between overlap exactly when the crystal temperature is controlled to the same value.

Summarizing, we find that the 13 to 14% reflectivity of the end facets strongly affects SHG tuning curves by introducing very dominating sets of interference fringes with interesting and QPM-specific details. These details can be resolved only when the tuning curve is measured with high enough phase-mismatch resolution. In SHG waveguides without QPM the effective mode indices of FW and SH are equal at phase-matching and the lobe structure in the double fringes is not present. In addition to the double-fringe structure of the SHG tuning curves we shall discuss in the next section another feature unique to QPM waveguides that becomes clearly visible only after eliminating the FW fringes by a normalization.

3.2. Normalized tuning curves

The influence of FW power fluctuations in SHG is usually cancelled in a normalization process by the definition of the SH efficiency $\eta =P_{\text{SH}}/P_{\text{FW}}^2$ with P_FW and P_SH as the input FW power and the output SH power. The FW input power (inside the waveguide) is determined by the FW output power divided by the transmittance of the outcoupling optics, the end facet and the waveguide. A corresponding calculation of η from the data in Fig. 4 yields the efficiency tuning curves for forward and backward crystal orientation shown in Fig. 6. The normalized tuning curves show as a main feature SH Fabry-Perot fringes. However, the asymmetry of the curves and even a dependence on the sample orientation indicate that the SH Fabry-Perot fringes are disturbed. The resulting asymmetry does not allow a straightforward nonlinear coefficient evaluation. The external induced fast oscillations of the FW are completely compensated by the normalization because the SHG process, even in the resonator, depends in the low-FW-depletion regime purely on the square of the FW input. The FW Fabry-Perot fringes introduced by the waveguide resonator can not be cancelled completely by the normalization. Because both FW and SH are reflected at both waveguide end facets we have seeded SHG in multiple-reflected forward and backward propagating waves. This seeded SHG is a phase-sensitive process. The generated SH is no longer purely dependent on the square of the FW power but also on the relative phase of reflected FW and SH. Different phase conditions in the seeded SHG in the reflected signals determine amplification or back-conversion which introduce an asymmetry in the tuning curve by enlarged or reduced SH fringes for different wavelengths.

 

Fig. 6 Measured and simulated SH efficiency: (a) backward, 23.08°C, and (b) forward orientation, 22.88°C.

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Despite the envelope similarity of the forward and backward tuning curves in Fig. 4 the differences between Figs. 6(a) and 6(b) indicate that the details of the beating pattern of FW and SH fringes in Fig. 4 must have been very different for both orientations. This can only be explained with different phase conditions in the SHG seeding process for reflected waves in forward and backward orientation. Indeed it is known from SHG in QPM gratings that the length of the first domain of the QPM grating determines the phase of the SH at the sample end. In QPM samples the phase conditions for SHG of reflected waves for forward and backward orientation can therefore be different when the boundaries of the poled regions do not coincide with the end facets of the crystal, i.e. the first and/or the last poled region is only a fraction of Λ/2 long. Slightly different temperatures at the measurements for Figs. 6(a) and 6(b) are not responsible for the different efficiency curves in both orientations. A small room temperature change can only shift the whole fringe beating pattern by the above given rate, but not change the pattern form.

3.3. QPM grating

In order to verify this theory we made the QPM gratings visible by cleaning the sample in a solution of water, ammoniac and hydrogen peroxide for a few minutes. This process etches the surface slightly with different rates at the +z (slow etching) and the −z face (fast etching) and the QPM grating becomes visible in a phase-contrast microscope. Fig. 7 shows the different domains of the QPM grating close to the end facets. At the beginning of the waveguide the first domain in Fig. 7(a) is only (0.35±0.1)Λ long. Because of deviations from a perfect 50 : 50 duty cycle the measuring tolerance is not better than 25%. At the waveguide end Fig. 7(b) shows a domain border that almost coincides with the crystal end. We checked the QPM grating along the whole waveguide. For 99% of all periods the quality of the grating was comparable to the quality of the parts shown in Fig. 7. Only in two approximately six-period-long areas the duty cycle reduces down to 10 : 90. In one of these areas two or three poled regions are completely missing. All of the domain lengths were finally measured with the statistical distribution shown in Fig. 8. Both the inverted and uninverted domains follow very well a normal distribution with standard deviations of σ = 0.853μm and σ = 0.815μm. The shorter domains are in average 7.6μm long and the longer domains have an average length of 8.8μm giving a duty cycle of the grating of 46:54.

 

Fig. 7 QPM grating in the investigated waveguide: (a) beginning of the crystal, (b) end of the crystal.

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Fig. 8 Histogram of 1100 inverted and uninverted domain widths (bars), and fit to Gaussian normal distributions (red curves). The standard deviations are 0.853 and 0.815μm.

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Counting the domains the waveguide length was found to be (1180.35 ± 0.1)Λ. The crystal was exposed in the lithographic grating definition with a mask with Λ = 16.4μm at room temperature. In agreement with a microscopic measurement the waveguide length at this temperature is (19.3577 ± 0.0016) mm.

The investigated waveguide itself was found under the microscope to be very good along its whole length.

4. The nonlinear susceptibility

4.1. Measured value for d₃₃

In order to find the second-order nonlinear susceptibility we calculate the tuning curves of SHG with the standard coupled-mode theory for SHG taking the multiple reflections of both FW and SH into account. A simple iteration for the seeded nonlinear mixing is used and converges within the first 4 to 8 iterations. All necessary parameters for the simulation except d₃₃ had been measured and presented with tolerances in the preceding text. An excellent fit between calculated and measured tuning curves in Fig. 6 is obtained with $|d_{33}|=20.29\frac{\text{pm}}{\text{V}}$ for the backward orientation and with $|d_{33}|=19.37\frac{\text{pm}}{\text{V}}$ for the forward orientation. Fig. 9 shows in more detail how good all of the measured data, also the scaled absolute data, are described by the theoretical fit. Because the SH efficiency is proportional to the square of d₃₃ we can determine only the absolute value |d₃₃| and the sign is taken from literature.

 

Fig. 9 Comparison of measured and calculated data: (a) Absolute power tuning curve (SH power just before the waveguide output); forward orientation, 22.88°C, 0.228 mW FW input power. (b) Detail from the absolute power tuning curve in (a). (c) Detail of the normalized SH tuning curve with SH fringes from Fig. 6(a). (d) Absolute FW output power just after the waveguide output; backward orientation, 23.08°C, 0.228 mW FW input power. The input power is the FW power that is focused onto the waveguide input facet reduced by the 90.4% overlap between the gaussian beam and the waveguide mode.

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For a good fit three parameter groups needed a fine tuning, however, without influence on the determined d₃₃ value. First, a fine adjustment of the crystal length within its tolerances to L = 1180.44Λ = 19.3592 mm enables a correct position of the lobes in the not normalized tuning curve in Figs. 9(a) and 9(b). Together with the crystal length the correct domain boundary positions relative to the end facets determine the asymmetries of the normalized forward and backward tuning curves in Figs. 6 and 9(c). The first domain boundary has a distance of 5.0μm from the input facet [domain width of 5.0μm according to Fig. 7(a)]. The last domain boundary has a distance of only 0.3μm from the output facet [see Fig. 7(b)]. Second, the Taylor coefficients for the SH effective index in Table 1 were slightly modified to shift the phase-matching wavelength together with the temperature to the measured value and to adjust the width of the tuning curve resonance. The original derivatives $\frac{\partial n}{\partial \lambda }$ from the Sellmeier equation predicted a 12% narrower resonance of the tuning curve than it was measured. Finally the temperature was fine adjusted to shift the fringe positions to the experimental finding as shown in Fig. 9(d). The SH loss coefficient, which was previously introduced in Section 2.4, was obtained by fitting the SH fringe contrast in the normalized tuning curves to the measured one [see Figs. 6 and 9(c)].

For the simulations a uniform phase-mismatch in a perfect QPM grating with a duty cycle of 50:50 and without any defects was assumed. However, from the microscope pictures of the QPM grating it is known that three QPM periods were missing and a few have a very small duty cycle. This has an effect on the SH efficiency similar to a reduction of the sample length by 9Λ. In the 2-cm-long sample this would reduce the SH efficiency by ≈ 1.5%. The non-perfect duty cycle further reduces the SH efficiency. According to the amplitude of the first harmonic of the Fourier series of a square function with a 46 : 54 duty cycle η is reduced by 1.5% compared to the SH efficiency in a perfect grating. The reduction of the SH efficiency due to random fluctuations of the domain lengths is discussed in Refs. [14, 15]. With a standard deviation of σ = 0.83μm we calculate a reduced η = exp(−2π²σ²/Λ²) = 0.95 or an efficiency-reduction of 5%.

A nonuniform phase-mismatch along the waveguide due to nonuniform QPM gratings, together or in addition to crystal and waveguide inhomogeneities causes asymmetric tuning curves with deviations from the ideally expected sinx/x-shape (sinc-function) [14]. Long-range tuning curves from our waveguides in Fig. 10 show only small asymmetries in the height of the side lobes near the main resonance. A slightly larger wavevector-mismatch at the ends of the waveguide would explain the measured asymmetry [16]. Introducing such a nonuniformity in a simulation a reduction of the SH efficiency by only 0.5% could be estimated. The observation of a perfect waveguide under the microscope and the good congruence of results for forward and backward propagation leads us to simply neglect any kind of nonuniformity in the sample, nonuniformities of the phase-mismatch as well as possible localized losses along the waveguides yielding nonuniform loss distributions.

 

Fig. 10 Normalized measured SH tuning curve (in black, averaged in yellow) compared with the sinc-form of a simulated tuning curve of a reflection-free waveguide (in red); forward orientation, 22.88°C.

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Combined, the imperfections of the QPM grating reduce the SH efficiency by approximately 8%. In our fit procedure we assumed a perfect QPM grating. Because in theory the SH efficiency decreases when QPM imperfections are considered, a larger d₃₃ value needs to be assumed to find again agreement between theory and experiment. Because the SH efficiency depends on the square of d₃₃ the found nonlinear coefficient needs to be corrected by an increase of 0.5×8% = 4%. Therefore the measured value of the nonlinear susceptibility element is in the range of $|d_{33}|=\left(20.6\pm 0.5\right)\frac{\text{pm}}{\text{V}}$.

After the correct data analysis it is worth to point out that a strongly simplified analysis with a fit of the averaged normalized tuning curve to a sinc-curve from a reflection-free waveguide in Fig. 10 yields a good guess for the nonlinear coefficient, too. The result $|d_{33}|=20.2\frac{\text{pm}}{\text{V}}$ falls in the above mentioned range.

4.2. Accuracy of measurement

The accuracy of the d₃₃ measurement depends on two parameter sets. First, there are the tolerances of the equipment yielding a limited accuracy for the measured SH efficiency. The FW and the SH power detection is limited by the accuracy of the detector reading, the errors in the values of the throughput of the beamsplitter for FW and SH separation, and the errors in the throughput of the outcoupling microscope objective and the waveguide end facet. Because the FW input power was determined by the measured FW output also errors in the FW waveguide transmittance influence the value of the calculated SH efficiency. After a careful setup calibration the relative errors for transmittance coefficients shown in Table 2 were found. ΔT/T consists for the FW of the uncertainty of the transmittance ΔT_FW–OE/T_FW–OE = 1% of beam-splitter, microscope objective and crystal end facet (Optical Elements) plus the uncertainty of the waveguide transmittance ΔT_FW–WG/T_FW–WG = 2%. Comparing different newly calibrated detector heads from the Ophir PD300 series we found that the accuracy ΔD_FW/SH/D_FW/SH with a reading deviation of a maximum of only 1.5% among them seemed better than the 3 and 4% that are given in the specifications. With these values we calculate straightforward a maximum relative error for the measured SH efficiency

Table 2. Relative Error of the Transmittance ΔT/T of a Combination of Diverse Optical Elements and Surfaces for FW and SH^*

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Comparing simulated and measured SH efficiencies a resulting maximum relative error for the extracted nonlinear susceptibility is found:

Comparable results within the given tolerance range were measured in other waveguides, with and without anti-reflection coatings, and with waveguide lengths between 1.5 and 7 cm. Also for the FW TM₀₀ to SH TM₁₀ SHG resonance the measured d₃₃ value was confirmed.

5. Conclusion

We presented the first direct measurement of the second-order nonlinear susceptibility d₃₃ in titanium indiffused waveguides for a wavelength of 1.52μm in the communication band. It has been shown that the absolute scaling of a simple SHG experiment in a waveguide becomes relatively complex due to the many system parameters. However, all of these parameters and their tolerances could be measured and a rigorous error analysis ensures an accuracy of 10% for the measurement. The here reported susceptibility determines the nonlinearity in many frequency mixing optical waveguide devices. For example, the description of most of our experiments on the cascaded quadratic nonlinearity, like soliton propagation, all-optical switching in single and coupled waveguides and waveguide arrays during the last ten years, rely on the here measured parameter.

The result is of interest even in a more general context. As shown in Fig. 1 the here reported nonlinear coefficient fits well to the formerly published data at lower wavelengths for pure congruent LiNbO₃ from Ref. [3] when using the upper limit of Shoji’s tolerance specification of “5 to 10%” (in Fig. 1 error bars of only ±5% are shown). Within the measuring tolerance an influence of the indiffused titanium on the nonlinear coefficient can not be quantified. Therefore, the usually in waveguides measured “effective” nonlinear coefficient (spatial average of the mode-field-weighted nonlinearity) equals in our system the d₃₃ value of the pure LiNbO₃ crystal.

We would also like to point out still unsolved problems for a future improved accuracy. Till now no direct measurement of the influence of titanium to the nonlinearity has been available. Neither a quantification of the increased nonlinear coefficients at the domain borders [17] and their influence on SHG in waveguides is known. The most unassured parameter in our evaluation is the QPM quality in the waveguide. Even when we observed a good poling at the surface, confocal SHG and Raman microscopy measurements [17] indicate a possible degradation of the domain boarders just a few μm below the surface. In case of a really reduced quality of the QPM grating in the waveguide volume, our measurement would provide only a lower limit for the nonlinear coefficient.