## Optical characterization of periodically-poled KTiOPO_{4}

Optics Express, Vol. 16, Issue 10, pp. 7344-7360 (2008)

http://dx.doi.org/10.1364/OE.16.007344

Acrobat PDF (562 KB)

### Abstract

We demonstrate how the Maker fringes that are observable in spontaneous parametric down-conversion (SPDC) give a direct visualization of the poling quality of a periodically-poled crystal. Identical Maker fringes are observed in the optical spectrum of collinear SPDC and the temperature dependence of second harmonic generation. We analyze these Maker fringes via a unified treatment of the tuning curve in crystals with small and slowly-varying deformations of the poling structure. Our theoretical model, based on a Fourier analysis of the poling deformations, distinguishes between duty-cycle variations and variations of the poling phase. The analysis indicates that the poling phase is approximately fixed, while the duty-cycle typically varies between 36% and 64%.

© 2008 Optical Society of America

## 1. Introduction

*χ*

^{(2)}tensor. An extensive discussion on the tuning and tolerances of QPM can be found in an article of Fejer

*et al.*[1

1. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. **28**, 2631 (1992). [CrossRef]

2. M. Houé and P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D: Appl. Phys. **28**, 1747–1763 (1995). [CrossRef]

3. C. Canalias, V. Pasiskevicius, and F. Laurell, “Periodic Poling of KTiOPO_{4}: From Micrometer to Sub-Micrometer Domain Gratings,” Ferroelectrics **340**, 27–47 (2006). [CrossRef]

4. F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO_{4} waveguides,” J. Appl. Phys. **71**, 15 (1992). [CrossRef]

5. H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, “Imaging of domain-inverted gratings in LiNbO_{3} by electrostatic force microscopy.” Appl. Phys. Lett. **71**, 146 (1997). [CrossRef]

6. G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, and M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO_{4},” Phys. Rev. B **54**, 6222 (1996). [CrossRef]

7. C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, “High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO_{4} by means of atomic force microscopy.” Appl. Phys. Lett. **83**, 734 (2003). [CrossRef]

8. S. K. Johansen and P. Baldi, “Characterization of quasi-phase-matching gratings in quadratic media through double-pass second-harmonic power measurements,” J. Opt. Soc. Am. B **21**, 1137 (2004). [CrossRef]

9. I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, “Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-*µ*m-band cascaded wavelength conversion,” Opt. Commun. **187**, 263 (2001). [CrossRef]

10. S. J. Holmgren, V. Pasiskevicius, S. Wang, and F. Laurell, “Three-dimensional characterization of the effective second-order nonlinearity in periodically poled crystals,” Opt. Lett. **28**, 1555 (2003). [CrossRef] [PubMed]

11. G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, “Mapping of periodically poled crystals via spontaneous parametric down conversion,” Appl. Phys. B **81**, 645–650 (2005). [CrossRef]

## 2. Phase matching in a periodically-poled crystal

*z*axis in domains perpendicular to the crystallographic

*x*axis. The pump beam is

*z*polarized and propagates along the

*x*axis. We consider the crystal to be infinite in the transverse directions. In the SHG configuration, we pump the crystal at radial frequency

*ω*so that the up-converted wave has radial frequency 2

*ω*. In the SPDC configuration, we pump the crystal at 2

*ω*so that the radial frequencies of the down-converted signal and idler waves can be written as

*ω*=

_{s}*ω*+Ω/2 and

*ω*=

_{i}*ω*-Ω/2.

*k*along the

*x*direction of the crystal. The tuning curve depends on the precise position-dependent nonlinear coefficient

*d*(

*x*). More specifically, the conversion efficiency as a function of Δ

*k*is proportional to the absolute value squared of the Fourier transform of

*d*(

*x*) [1

1. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. **28**, 2631 (1992). [CrossRef]

*d*(

*x*) with amplitude ±

*d*

_{eff}, where the effective nonlinearity

*d*

_{eff}is a fixed material property. The Fourier transform of a square-wave-shaped

*d*(

*x*) contains odd-

*m*harmonics of the form

*k*

^{±}

_{m}=±2

*πm*/Λ

_{0}, where Λ

_{0}is the poling period of the crystal. Hence, the conversion efficiency will become peaked around any wave vector mismatch Δ

*k*=

*k*

^{±}

_{m}. The mismatch parameter

*ϕ*, defined as the accumulated phase mismatch over half the crystal length with respect to quasi phase-matching order

_{m}*m*, is

*L*

_{0}and the poling period Λ

_{0}are specified at a certain reference temperature

*T*

_{0}. The function

*f*(

*T*) is the temperature-dependent material expansion factor which is defined to equal unity at the reference temperature. The conversion efficiency in the neighborhood of some

*m*-order quasi phase match condition now becomes [1

1. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. **28**, 2631 (1992). [CrossRef]

*m*=1, and we will drop the subscripts of

*ϕ*and

_{m}*η*. In accordance with Ref. [1

_{m}**28**, 2631 (1992). [CrossRef]

*(*η ^

*ϕ*) to its peak value via

*(*η ^

*ϕ*)=

*η*(

*ϕ*)/

*η*(

*ϕ*=0).

*ϕ*from Eq. (1), we must calculate the wave vector mismatch Δ

*k*along the

*x*direction. The wave vector mismatch is defined as the wave vector of the ‘blue’ photon minus the wave vectors of the two ‘red’ photons. For the SHG configuration, this wave vector mismatch becomes

*k*(

*ω,T*) is the temperature-dependent dispersion relation of

*z*polarized light in the crystal. Eq. (3) shows that the wave vector mismatch in SHG can be tuned in two different ways: via wavelength tuning and via temperature tuning. The wavelength and temperature dependence are often separated via

*n*(

_{z}*ω,T*)≡

*n*(

_{z}*ω,T*)-

*n*(

_{z}*ω*,

*T*

_{0}) is a shorthand notation for the change in refractive index caused by a deviation from the reference temperature

*T*

_{0}=25 °C [13

13. S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for KTiOPO_{4} and KTiOAsO_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

14. F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, “Measurement of the thermal expansion coefficients of ferroelectric crystals by a moiré interferometer,” Opt. Commun. **227**, 14 (2007). [CrossRef]

*ω*-

_{s}*ω*between the signal and idler photon is allowed. Therefore, the wave vector mismatch in SPDC can be tuned in not just two, but four different ways: via wavelength tuning, via temperature tuning, via tuning of the detection angle, and via tuning of the detection wavelength. The wave vector mismatch in the SPDC configuration can be written as

_{i}*k*and

_{s,x}*k*are the

_{i,x}*x*components of the wave vectors of the signal and idler waves in the crystal, respectively. Explicitly writing out the angle-detuning and frequency-detuning in the small angle and small frequency detuning limit, yields

*θ⃗*=(

*θ*,

_{y}*θ*) is the emission angle inside the crystal, and

_{z}*n*and

_{z}*n*are shorthand notations for the refractive index of

_{x}*z*and

*x*polarized light, respectively. The

*n*/

_{z}*n*factor in front of

_{x}*θ*originates from the fact that down-converted light with a certain transverse

_{z}*k*component carries a polarization with a small

_{z}*x*component too. The angle dependence of the refractive index is found by using the index ellipsoid for our polarization. The latter term in Eq. (6) is obtained by performing a Taylor expansion around zero frequency detuning.

20. W. Wiechmann and S. Kubota, “Refractive-index temperature derivatives of potassium titanyl phosphate,” Opt. Lett. **18**, 1208–1210 (1993). [CrossRef] [PubMed]

19. K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” Appl. Opt. **41**, 5040 (2002). [CrossRef] [PubMed]

13. S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for KTiOPO_{4} and KTiOAsO_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

13. S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for KTiOPO_{4} and KTiOAsO_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

14. F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, “Measurement of the thermal expansion coefficients of ferroelectric crystals by a moiré interferometer,” Opt. Commun. **227**, 14 (2007). [CrossRef]

*z*component of the refractive index from Ref. [18

18. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO_{4},” Appl. Phys. Lett. **74**, 914 (1999). [CrossRef]

*x*and

*y*component of the refractive index from Ref. [19

19. K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” Appl. Opt. **41**, 5040 (2002). [CrossRef] [PubMed]

_{4} and KTiOAsO_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

14. F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, “Measurement of the thermal expansion coefficients of ferroelectric crystals by a moiré interferometer,” Opt. Commun. **227**, 14 (2007). [CrossRef]

## 3. Experimental apparatuses

_{00}mode. This pump beam is mildly focussed (

*w*

_{0}=190

*µ*m is the radius at

*e*

_{-2}of maximum irradiance) into a PPKTP crystal. The crystal is manufactured by Raicol Crystals Ltd. with low temperature periodic electrical poling, based on the application of a pulsed electric switching field to a patterned electrode [25

25. G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, “Low temperature periodic electrical poling of flux-grown KTiOPO_{4} and isomorphic crystals,” Appl. Phys. Lett. **73**, 3650 (1998). [CrossRef]

*x*direction, a thickness of 1 mm in the

*z*direction and a width of 2 mm in the

*y*direction. The pump beam propagates along the crystallographic

*x*axis and is

*z*-polarized, allowing us to use the large nonlinear

*d*coefficient of KTP. The pump beam is centered on the crystal facets. The poling period of the crystal is specified to be Λ

_{33}_{0}=3.675

*µ*m which is designed for first-order quasi-phase-matching at 413.1 nm↔ 826.2 nm. The crystal is thermally contacted along the full crystal length to a bulky Aluminum mount via a 100

*µ*m thick Indium layer. The temperature is stabilized to Δ

*T*<0.1 °C using a Dale 1T1002-5 thermistor, an ILX-Lightwave LDT5910 controller and a Peltier element. The thermometer system is calibrated to an accuracy of ±(0.5%+0.2 °C) by using a commercial ATAL RTD407907 thermometer. Phase matching in forward direction is achieved at a temperature of 60.7 °C.

*f*-

*f*geometry to project the far field onto an intensified CCD camera (ICCD) of Princeton Instruments (I-MAX- 512-T,18). A narrow band color filter selects a Δ

*λ*=5 nm band around

*λ*=826 nm. This bandwidth is much smaller than the full width of the main Maker fringe in the spectral domain (see Fig. 5). The bandpass filter removes all angle-frequency correlations from the down-converted light.

^{2}in the forward direction. The selected solid angle is much smaller than the solid angle of the main central Maker fringe (≈300 mrad

^{2}), so that all angle-frequency correlations are removed. The light is then focussed onto the input slit of a spectrometer (Jobin Yvon 320) and the output is projected on an ICCD camera of Princeton Instruments (I-MAX-512-T,18). The dispersion of the spectrometer is 570

*µ*m per nanometer. The width of the ICCD chip is 12.8 mm which corresponds to a full spectral width of a single image of Δ

*λ*=22 nm. The center of the detected wavelength interval can be adjusted with a knob on the spectrum analyzer. The spectral response of the ICCD camera and spectrometer was calibrated by using a white light source and a non-intensified CCD with a well-specified spectral response. The spectrometer’s offset was calibrated by using the 826.45 nm line of Argon.

_{00}mode. The beam is mildly focussed into the center of the same PPKTP crystal. The Rayleigh range of the beam is (14±2) mm corresponding to a beam radius at

*e*

^{-2}of maximum irradiance of

*w*

_{0}=(61± 4)

*µ*m. The pump propagates along the

*x*direction of the crystal and is

*z*polarized. The PPKTP crystal, the crystal mount and the temperature control part are the same as in the SPDC setup. The intensity of the SHG light is measured with a photodiode (HUV1100BQ), a homemade amplifier, and a voltmeter.

*β*-BaB

_{2}O

_{4}crystal (BBO). The conversion efficiency for both SPDC and SHG is expected to be much higher for PPKTP than for BBO. At a cutting angle

*θ*=29.2°, the effective nonlinearity of BBO in type-I phase matching is expected to be

_{c}*d*

_{eff}=-

*d*

_{22}cos

*θ*=2.0 pm/V, based on data from Ref. [21

_{c}21. R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD-STAR-P, BaB_{2}O_{3}, LiIO_{3}, MgO-LiNbO_{3}, and KTP measured by phase-matched 2nd harmonic generation,” IEEE J. Quantum Electron. **26**, 922 (1990). [CrossRef]

*π*)

*d*

_{33}=9.8 pm/V, based on data from Ref. [22

22. M. V. Pack, D. J. Armstrong, and A. V. Smith, “Measurement of the *χ*^{(2)} tensors of KTiOPO_{4}, KTiOAsO_{4}, RbTiOPO_{4}, and RbTiOAsO_{4} crystals,” Appl. Opt. **43**, 3319 (2004). [CrossRef] [PubMed]

^{2}=24 times higher conversion efficiencies than BBO. In our case, the PPKTP crystal is 5 times longer than the BBO crystal, so we expect a conversion that is 5

^{2}×24=600 times more efficient. We observe that the SHG process is approximately 575 times more efficient than the BBO crystal. In the inverse process of SPDC we observe an enhancement factor of approximately 700. Both observations are in reasonable agreement with the expected enhancement factor of 600. We conclude that the quality of the periodically poled crystal must be very good with a duty-cycle close to 50%.

## 4. Experimental results

### 4.1. Maker fringes in angular intensity pattern of SPDC light

12. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focussing on the production of optical harmonics,” Phys. Rev. Lett. **8**, 21 (1962). [CrossRef]

*I*∝sinc

^{2}(

*A*Θ

^{2}+

*B*), where Θ is the detection angle outside the crystal. The quadratic relation between detection angle and mismatch parameter is expected from Eqs. (6) and (1). Secondly, we observe that the ring pattern in Fig. 2 is slightly elliptical. This effect is due to a difference in the refractive index of

*z*and

*x*polarized light. Down-converted light with a certain

*k*component has a polarization with a small

_{z}*x*component, which causes the fringes to be elliptical [

*n*≠

_{z}*n*in Eq. (6)]. A third aspect concerns the peculiar deviations from the ideal sinc-like fringe structure. The sixth and ninth fringe, for example, are standing out with respect to neighboring fringes. These deviations are caused by small and slowly-varying deformations of the poling structure. This experiment thus provides a simple and direct visualization of slowly-varying deviations from perfect poling.

_{x}*f*(

_{l}*r*)exp(

*ilφ*). The Fourier coefficients of this expansion depend strongly on the chosen position of the center. We have determined the optimal position of the center by minimizing the

*l*=1 coefficient. The ellipticity is now linked to the

*l*=2 coefficient of the Fourier expansion around this best center. We have accurately determined this ellipticity by stretching the image in the

*z*direction until the

*l*=2 coefficient is minimized. This procedure is applied to 25 images taken at different temperatures between 45 °C and 72 °C. The resulting ellipticity is

*n*/

_{z}*n*=1.056±0.002 where all the 25 different values are within the specified confidence region. Our measured value is in excellent agreement with existing literature giving

_{x}*n*/

_{z}*n*=1.0551 [18

_{x}18. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO_{4},” Appl. Phys. Lett. **74**, 914 (1999). [CrossRef]

19. K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” Appl. Opt. **41**, 5040 (2002). [CrossRef] [PubMed]

^{2}

_{y}for each image. A composite curve, shown in Fig. 3, is generated by cutting, shifting, and pasting 25 rotationally averaged images taken at different temperatures. The main peak is shifted such that it coincides with the null of the Θ

^{2}

_{y}axis. The resulting Θ

^{2}

_{y}axis now relates without offset to the mismatch parameter

*ϕ*in Eq. (2). Unfortunately, the node intensities in Fig. 3 have limited quantitative meaning as these intensities depend on the presence or absence of nearby bright fringes (see below).

^{2}

_{y}=(325±8) mrad

^{2}. The angular dependence of the phase-matching condition is theoretically described by Eq. (6). The measured external angle corresponds to the internal angle in Eq. (6) via Θ

_{y}≈

*n*. The expected distance between subsequent side lobs thus becomes ΔΘ

_{z}θ_{y}^{2}

_{y}=2

*πcn*/

_{z}*Lω*=299 mrad

^{2}using the refractive index of Ref. [18

18. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO_{4},” Appl. Phys. Lett. **74**, 914 (1999). [CrossRef]

^{2}

_{y}to an unintended but possibly present 5% magnification in the imaging system.

^{-2.85}≈0.15% of the peak intensity. The divergence of the pump laser of 0.7 mrad is too small to explain this observation. Additionally, we have observed that the intensity in a node is strongly reduced if we let a nearby bright fringe disappear from the image by increasing the temperature. For example, the intensity in the first minimum drops from 2.72% at

*T*=60.7 °C (collinear phase matching) to 0.56% at

*T*=63.6 °C, where the first minimum is in the forward direction. It thus seems that the intensity in a node is enhanced under the influence of nearby (bright) fringes. This effect can possibly be explained by a small degree of scattering from the crystal, predominantly in near-forward directions.

*y*direction and not in the 1mm thick

*z*direction. Two fringe patterns, corresponding to two pump positions that are 1.4 mm separated from each other along the y direction, are shown in Fig. 4. These images show the largest differences that we have observed, and even these are very small. The peak conversion efficiency (at phase-matching temperature) varies at most 2.5% over the

*y*direction and only 1% over the

*z*direction. We conclude that the observed deviations from perfect poling are present over the full cross-section, and show minor variations along the

*y*direction.

### 4.2. Maker fringes in spectrum of SPDC light

*λ*=22 nm. We have combined these images into a singe curve in Fig. 5 which thus represents the spectrum of the down-converted light. The observed spectrum contains Maker Fringes [12

12. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focussing on the production of optical harmonics,” Phys. Rev. Lett. **8**, 21 (1962). [CrossRef]

^{2}. If we convert the

*λ*axis into a (Ω/2)

^{2}axis we find, as expected, a sinc-shaped tuning curve. The central peak is neatly centered around zero detuning which corresponds to a wavelength of 826.2 nm, being twice the pump wavelength. The spacing between subsequent side lobs is Δ(Ω/2)

^{2}=(4.7±0.1)×10

^{27}rad

^{2}/s

^{2}, which is in excellent agreement with the literature value of 4.69×10

^{27}rad

^{2}/s

^{2}derived from the Sellmeier equation in Ref. [18

_{4},” Appl. Phys. Lett. **74**, 914 (1999). [CrossRef]

### 4.3. Maker fringes in temperature dependence of SHG

12. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focussing on the production of optical harmonics,” Phys. Rev. Lett. **8**, 21 (1962). [CrossRef]

_{4} and KTiOAsO_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

*ϕ*rather than temperature. In order to make this conversion, accurate knowledge of the temperature dependence of the refractive index is needed. The temperature-dependentwave vector mismatch in Eq. (4) is well described with two coefficients

*c*

_{1}and

*c*

_{2}, defined via

_{4} and KTiOAsO_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

*c*

_{1}and

*c*

_{2}from the measured tuning curve

*η*(

*T*) by using the following procedure. The peaks and nodes of the measured tuning curve serve as markers for the mismatch parameter:

*ϕ*=±(

*n*+1/2)

*π*at the peaks and

*ϕ*=±

*nπ*at the nodes. These markers yield the mismatch parameter

*ϕ*at a number of temperatures. Coefficients

*c*

_{1}and

*c*

_{2}are obtained from a parabolic fit of Eq. (1) to the obtained set of mismatch parameters, after substituting Eqs. (7) and (4) into Eq. (1). In order to obtain reliable values for

*c*

_{1}and

*c*

_{2}, we have applied this fitting method to five different PPKTP crystals. The tuning curve of one of these crystals is shown in Fig. 7. It closely resembles the ideal sinc

^{2}-shape, thus showing that the poling structure of this second 5 mm long crystal is almost without any deformation. This curve is measured at a slightly longer pump wavelength of 826.4 nm instead of 825.9 nm causing phase matching to occur at 63.7 °C instead of 54.1 °C.

*c*

_{1}=(24.0±0.2)×10

^{-6}°C

^{-1}and

*c*

_{2}=(4.8±0.3)×10

^{-8}°C

^{-2}at an approximate pump wavelength of 826 nm and a reference temperature of

*T*

_{0}=25 °C. The influence of the thermal expansion [14

**227**, 14 (2007). [CrossRef]

*ϕ*is only small (~4%) compared to the influence of the change in refractive index. The obtained coefficients are used to add a

*ϕ*axis at the top of Fig. 6 and Fig. 7. The conversion from the measured

*η*(

*T*) to the preferred form of

*η*(

*ϕ*) is now completed. We will use

*η*(

*ϕ*) for further analysis in Sec.5.

*c*

_{1}and

*c*

_{2}with existing literature. Reference [13

_{4} and KTiOAsO_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

*n*(

_{z}*ω*,

*T*) between vacuum wavelengths of 532 nm and 1585 nm. From this expression we calculate

*c*

_{1}=23.56×10

^{-6}°C

^{-1}and

*c*

_{2}=8.6×10

^{-8}°C

^{-2}. The

*c*

_{1}coefficient is in reasonable agreement with our value, but there is a distinct discrepancy between the values for

*c*

_{2}. We conclude that it is inappropriate to extrapolate the expression for Δ

*n*(

_{z}*ω*,

*T*) in Ref. [13

_{4} and KTiOAsO_{4},” Appl. Opt. **42**, 6661 (2003). [CrossRef] [PubMed]

*n*(2

_{z}*ω*,

*T*) in Eq. (7).

## 5. Interpretation of Maker fringes in terms of poling quality

### 5.1. Fourier analysis of small and slowly-varying deformations of the poling structure

^{2}-shaped tuning curve. In practice, however, small deformations of the poling structure are often present, and such deformations reveal themselves as non-sinc-like features in the tuning curve

*(*η ^

*δ*Δ

*k*). Fejer

*et al.*[1

**28**, 2631 (1992). [CrossRef]

*L*

_{0}and the number of domains

*N*relate to the design value of the poling period via Λ

_{0}=2

*L*

_{0}/

*N*. The tuning curve is peaked at any wave vector mismatch Δ

*k*

_{0}obeying |Δ

*k*

_{0}|=2

*πm*/Λ

_{0}, where

*m*can be any odd-valued quasi phase-matching order. The positions of the domain boundaries required for perfect poling are

*x*

_{n,0}=

*n*Λ

_{0}/2.

*d*(

*x*) that occupy two discrete levels ±

*d*

_{eff}. We specify the spatial deformations via the position error

*δx*

_{n}≡

*x*-

_{n}*x*

_{n,0}of the

*n*

_{th}domain boundary and introduce the phase error at the domain boundaries for fixed Δ

*k*=Δ

*k*

_{0}as

_{n}only contains the effects of the poling deformation. This definition differs from the one used by Fejer

*et al.*[1

**28**, 2631 (1992). [CrossRef]

_{n}also includes the accumulated phase due to the wave vector mismatch

*δ*Δ

*k*≡Δ

*k*-Δ

*k*

_{0}.

**28**, 2631 (1992). [CrossRef]

*x*)=0, yield

*Ê*(

*δ*Δ

*k*)=

*Ê**(-

*δ*Δ

*k*) and a symmetric tuning curve. Turning the crystal around corresponds to the operations Φ(

*x*)⇒-Φ(-

*x*),

*A*(

*x*)⇒

*A*(-

*x*), and hence

*Ê*(

*δ*Δ

*k*)⇒

*Ê**(

*δ*Δ

*k*). Therefore, a measurement of the tuning curve

*(*η ^

*δ*Δ

*k*) can not distinguish between the two possible crystal orientations. Symmetrically-deformed crystals have real-valued

*Ê*(

*δ*Δ

*k*).

*x*) and the amplitude function

*A*(

*x*) in Fourier series as

*a*,

_{n}*b*,

_{n}*c*, and

_{n}*d*are real-valued coefficients and 0≤

_{n}*A*

_{0}≤1 is the real-valued average amplitude. Any nonzero

*a*

_{0}can be interpreted as a longitudinal displacement of the crystal, which is obviously neither an internal crystal property nor a parameter that influences the detected tuning curve. We shall assume small deformations Φ(

*x*)≪1, so that a first-order Taylor expansion of exp[-

*i*Φ(

*x*)]≈1-

*i*Φ(

*x*) can be made. By inserting the Fourier decompositions in Eq. (12), and hereby neglecting second-order terms with amplitudes like

*a*, we find

_{n}c_{n}*δ*Δ

*k*is thus found to comprise a series of shifted sinc-functions with relative weights that contain essential information on the slow (=large scale) variations of the poling period. The tuning curve

*(*η ^

*ϕ*) is symmetric with respect to

*ϕ*=0 if at least all

*b*=0 in combination with either all

_{n}*d*=0 or all

_{n}*a*=0.

_{n}*,*α ˜

_{n}*,*β ˜

_{n}*, and*ζ ˜

_{s}*are linked to the deformation coefficients as*ζ ˜

_{s}*,*ζ ˜

_{s}*} coefficients are uniquely determined by the deformation coefficients {*ξ ˜

_{s}*,*α ˜

_{n}*}. The inverse transformations*β ˜

_{n}*,*α ˜

_{n}*}-set and the {*β ˜

_{n}*,*ζ ˜

_{s}*}-set individually contain all information on the poling deformations.*ξ ˜

_{s}*n*

_{th}node at each side of the main sinc-peak is determined by the

*coefficient and*α ˜

_{n}*coefficient only. This is because the maxima of the shifted sinc-functions in Eq. (16) coincide with the minima of the fundamental and all other shifted sinc-functions. The*β ˜

_{n}*coefficient affects both nodes in a symmetric way, whereas the*α ˜

*coefficient can create an unbalance between the nodes. Equation (18) shows that the*β ˜

^{s}

_{th}side maximum at each side of the main sinc-peak is determined by the

ζ ˜

_{(s+1)}coefficient and

ξ ˜

_{(s+1)}coefficient only. The reason being that

ζ ˜

_{(s+1)}and

ξ ˜

_{(s+1)}are the coefficients of sinc functions that are shifted by (

*s*+1/2)

*π*, thus exchanging the role of minima and maxima. The

*coefficient affects both side lobs in a symmetric way, whereas the*ζ ˜

*coefficient can create an unbalance between the side lobs.*ξ ˜

### 5.2. Analysis of Maker fringes in terms of poling quality

^{2}-shape as given by Eq. (2). However, small deviations from the ideal curve are clearly visible, as, for example, the sixth and ninth side lobs are standing out with respect to the neighboring peaks. These non-sinc-like features are caused by small and slowly-varying deformations in the poling structure that cover hundreds of poling periods. The reason is that the measured

*ϕ*range covers about 20 sinc-nodes only, whereas the total amount of sinc-nodes in between two phase-matching orders, like

*m*=1 and

*m*=2, equals the number of domains which is about 2

*L*

_{0}/Λ

_{0}=2770.

*c*=

_{n}*d*=0. This assumption is inspired by the fact that this category of deformations has a first-order effect on the tuning curve, whereas duty-cycle variations are only visible in second-order [see Eqs. (10) and (11)]. The symmetry with respect to

_{n}*ϕ*=0 observed in Fig. 6 indicates that

*b*≈0. The most prominent features are observed in the ±6

_{n}_{th}and ±9

_{th}side lobs (one outside figure). The tuning curve at the ±6

_{th}peaks is measured to be

*=0.54%, whereas the expected value is only*η ^

*=0.24%. From these values and the assumption of an approximate constant duty-cycle, we find*η ^

ζ ^

_{7}≈±0.11

*i*. The associated phase function

_{0}/2

*πm*=64 nm over a typical period of

_{rd}and ±8

_{th}side lobs are lower than the expected peak height at both sides of the central peak. Equation (18) shows that a pair of side lobs can be lowered symmetrically, only if ℜ(

*)<0. Any nonzero real part of*ζ ˜

*is formed by nonzero*ζ ˜

*c*coefficients and thus by variations of the poling duty-cycle. We therefore conclude that the observed non-sinc-like features in Fig. 6 can not be caused by slowly-varying deformations of the poling phase alone.

_{n}*a*=

_{n}*b*=0. This assumption is inspired by the fact that the observed tuning curve is almost perfectly symmetric, which is automatically the case when solely duty-cycle variations are assumed. Equation (18) gives the height of the side lobs, and it indicates that a real-valued

_{n}*has a first-order effect on the side lob strength, whereas an imaginary-valued*ζ ˜

_{s}*has only a second-order effect. Therefore, it is possible to make a good estimate of {*ζ ˜

_{s}*} from the measured side lob strengths alone. For example, from the measured ±6*ζ ˜

_{s}_{th}side lobs in Fig. 6, it is found that

ζ ˜

_{7}≈0.049, corresponding to the amplitude function

*A*(

*x*) corresponds to a duty-cycle variation between 36% and 64%. We thus find that the potential effect of duty-cycle variations on the tuning curve is relatively weak compared to the potential effect of poling phase deformations.

*(*η ^

*ϕ*). In general, it is not possible to do this inversion. For example, a sign-flip of all

*a*and

_{n}*d*coefficients will not change the outcome of the measurement. The large number of free parameters in the deformation model possibly limits the amount of retrievable information even further. Under some assumptions on the nature of the deformations, like the ones on duty-cycle variation discussed above, more stringent requirements on the amplitudes of the deformations apply. Whatever the assumptions, the observed tuning curve always corresponds to a Fourier analysis of the amplitude function

_{n}*A*(

*x*) and the poling phase function Φ(

*x*). Non-sinc-like features in the tuning curve

*(*η ^

*ϕ*) around the

*n*

_{th}side minimum must correspond to variations in

*A*(

*x*) and Φ(

*x*) with a typical period of

*L*/

*n*.

*b*≈0. The measurement also proves that some

_{n}*c*coefficients must be nonzero as some pairs of side lobs are lowered symmetrically at both sides of the main peak. In order to fully explain the observed symmetry in the tuning curve, it is also needed that either all

_{n}*a*≈0 or all

_{n}*d*≈0. We assume that

_{n}*a*≈0, as we suppose that any realistic duty-cycle deformation mechanism would not distinguish between the

_{n}*c*and

_{n}*d*coefficients. We thus conclude that the poling deformations in the investigated PPKTP crystal comprise duty-cycle variations only. The stronger sixth side lobs in Fig. 6 correspond to a periodic 9.8% peak-to-peak variation in the amplitude function

_{n}*A*(

*x*)=sin(

*π*×dutycycle).

23. M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase matched LiNbO_{3} waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. **62**, 435 (1993). [CrossRef]

24. G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, “Domain broadening in quasi-phase-matched nonlinear optical devices,” Appl. Phys. Lett. **73**, 865 (1998). [CrossRef]

25. G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, “Low temperature periodic electrical poling of flux-grown KTiOPO_{4} and isomorphic crystals,” Appl. Phys. Lett. **73**, 3650 (1998). [CrossRef]

*z*position of the pump beam, is consistent with the observation of Rosenman

*et al.*[24

24. G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, “Domain broadening in quasi-phase-matched nonlinear optical devices,” Appl. Phys. Lett. **73**, 865 (1998). [CrossRef]

## 6. Conclusions

_{4}crystal (PPKTP), utilizing the processes of spontaneous parametric down conversion (SPDC) and second harmonic generation (SHG). The three methods concern a measure- ment of the angular intensity pattern of collinear SPDC light, the spectrum of SPDC light, and the temperature-dependent conversion efficiency in SHG. We have shown that the three methods are fully consistent, and that the outcomes are in agreement with current knowledge of KTP’s material properties. We refer to the observed fringe pattern in the tuning curve as Maker fringes [12

**8**, 21 (1962). [CrossRef]

*y*direction of the crystal. We do not observe any variation of the fringe pattern along the

*z*direction. The ferroelectric poling structure is fabricated via low temperature electrical poling, by applying a periodic electrode pattern on one of the polar

*z*surfaces of the crystal. We conclude that the poling deformations must originate from close to the electrodes and remain uniform along the

*z*direction of the crystal.

## Acknowledgments

## References and links

1. | M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. |

2. | M. Houé and P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D: Appl. Phys. |

3. | C. Canalias, V. Pasiskevicius, and F. Laurell, “Periodic Poling of KTiOPO |

4. | F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO |

5. | H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, “Imaging of domain-inverted gratings in LiNbO |

6. | G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, and M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO |

7. | C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, “High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO |

8. | S. K. Johansen and P. Baldi, “Characterization of quasi-phase-matching gratings in quadratic media through double-pass second-harmonic power measurements,” J. Opt. Soc. Am. B |

9. | I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, “Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5- |

10. | S. J. Holmgren, V. Pasiskevicius, S. Wang, and F. Laurell, “Three-dimensional characterization of the effective second-order nonlinearity in periodically poled crystals,” Opt. Lett. |

11. | G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, “Mapping of periodically poled crystals via spontaneous parametric down conversion,” Appl. Phys. B |

12. | P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focussing on the production of optical harmonics,” Phys. Rev. Lett. |

13. | S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for KTiOPO |

14. | F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, “Measurement of the thermal expansion coefficients of ferroelectric crystals by a moiré interferometer,” Opt. Commun. |

15. | T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, “Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOP0 |

16. | V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, “Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4–4.0 |

17. | D. W. Anthon and C. D. Crowder, “Wavelength dependent phase matching in KTP,” Appl. Opt. |

18. | K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO |

19. | K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” Appl. Opt. |

20. | W. Wiechmann and S. Kubota, “Refractive-index temperature derivatives of potassium titanyl phosphate,” Opt. Lett. |

21. | R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD-STAR-P, BaB |

22. | M. V. Pack, D. J. Armstrong, and A. V. Smith, “Measurement of the |

23. | M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase matched LiNbO |

24. | G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, “Domain broadening in quasi-phase-matched nonlinear optical devices,” Appl. Phys. Lett. |

25. | G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, “Low temperature periodic electrical poling of flux-grown KTiOPO |

**OCIS Codes**

(190.4400) Nonlinear optics : Nonlinear optics, materials

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.0270) Quantum optics : Quantum optics

(130.2260) Integrated optics : Ferroelectrics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 6, 2008

Revised Manuscript: April 30, 2008

Manuscript Accepted: May 4, 2008

Published: May 6, 2008

**Citation**

W. H. Peeters and M. P. van Exter, "Optical characterization of
periodically-poled KTiOPO_{4}," Opt. Express **16**, 7344-7360 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7344

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### References

- M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631 (1992). [CrossRef]
- M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second-harmonic generation," J. Phys. D: Appl. Phys. 28, 1747-1763 (1995). [CrossRef]
- C. Canalias, V. Pasiskevicius, and F. Laurell, "Periodic poling of KTiOPO4: from micrometer to sub-micrometer domain gratings," Ferroelectrics 340, 27-47 (2006). [CrossRef]
- F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992). [CrossRef]
- H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997). [CrossRef]
- G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, and M. Roth, "Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4," Phys. Rev. B 54, 6222 (1996). [CrossRef]
- C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, "High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO4 by means of atomic force microscopy." Appl. Phys. Lett. 83, 734 (2003). [CrossRef]
- S. K. Johansen and P. Baldi, "Characterization of quasi-phase-matching gratings in quadratic media through double-pass second-harmonic power measurements," J. Opt. Soc. Am. B 21, 1137 (2004). [CrossRef]
- I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001). [CrossRef]
- S. J. Holmgren, V. Pasiskevicius, S. Wang, and F. Laurell, "Three-dimensional characterization of the effective second-order nonlinearity in periodically poled crystals," Opt. Lett. 28, 1555 (2003). [CrossRef] [PubMed]
- G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005). [CrossRef]
- P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, "Effects of dispersion and focussing on the production of optical harmonics," Phys. Rev. Lett. 8, 21 (1962). [CrossRef]
- S. Emanueli and A. Arie, "Temperature-dependent dispersion equations for KTiOPO4 and KTiOAsO4," Appl. Opt. 42, 6661 (2003). [CrossRef] [PubMed]
- F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007). [CrossRef]
- T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOP04," Appl. Opt. 26, 2390 (1987). [CrossRef] [PubMed]
- Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988). [CrossRef]
- D. W. Anthon and C. D. Crowder, "Wavelength dependent phase matching in KTP," Appl. Opt. 27, 2650 (1988). [CrossRef] [PubMed]
- K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914 (1999). [CrossRef]
- K. Kato and E. Takaoka, "Sellmeier and thermo-optic dispersion formulas for KTP," Appl. Opt. 41, 5040 (2002). [CrossRef] [PubMed]
- W. Wiechmann and S. Kubota, "Refractive-index temperature derivatives of potassium titanyl phosphate," Opt. Lett. 18, 1208-1210 (1993). [CrossRef] [PubMed]
- R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD-STAR-P, BaB2O3, LiIO3, MgO-LiNbO3, and KTP measured by phase-matched 2nd harmonic generation," IEEE J. Quantum Electron. 26, 922 (1990). [CrossRef]
- M. V. Pack, D. J. Armstrong, and A. V. Smith, "Measurement of the |(2) tensors of KTiOPO4, KTiOAsO4, RbTiOPO4, and RbTiOAsO4 crystals," Appl. Opt. 43, 3319 (2004). [CrossRef] [PubMed]
- M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435 (1993). [CrossRef]
- G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998). [CrossRef]
- G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998). [CrossRef]

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