Nonlinear optical coefficients and phase-matching conditions in Sn2P2S6

Phase matching conditions and second and third order nonlinear optical coefficients of Sn2P2S6 crystals are reported. The coefficients for second harmonic generation (SHG) are given at λ = 1542 nm and 1907 nm at room temperature. The largest coefficients at these wavelengths are d111 = 17± 1.5pm/V and d111 = 12± 1.5pm/V, respectively. The third-order subsceptibilities χ (3) 1111 = (17 ± 6) · 10−20 m2/V2 and χ (3) 2222 = (9 ± 3) · 10−20 m2/V2 were determined at λ = 1907 nm. All measurements were performed by the Maker-Fringe technique. Based on the recently determined refractive indices, we analyze the phase-matching conditions for second harmonic generation, sumand difference-frequency generation and parametric oscillation at room temperature. Phase-matching curves as a function of wavelength and propagation direction are given. Experimental phase-matched type I SHG at 1907 nm has been demonstrated. The results agree very well with the calculations. It is shown that phase-matched optical parametrical oscillation is possible in the whole transparency range up to 8 μm with an effective nonlinear coefficient deff ≈ 4pm/V. © 2005 Optical Society of America OCIS codes: (160.4330) Nonlinear optical materials; (190.2620) Frequency conversion References and links 1. S. G. Odoulov, A. N. Shumelyuk, U. Hellwig, R. A. Rupp, A. A. Grabar, and I. M. Stoyka, “Photorefraction in tin hypothiodiphosphate in the near infrared,” J. Opt. Soc. Am. B 13, 2352–60 (1996). 2. M. Jazbinsek, G. Montemezzani, P. Gunter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, “Fast nearinfrared self-pumped phase conjugation with photorefractive Sn2P2S6,” J. Opt. Soc. Am. B. June 20, 1241–6 (2003). 3. D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, “Electro-optical properties of Sn2P2S6,” Opt. Commun. 215, 333–43 (2003). 4. M. I. Gurzan, A. P. Buturlakin, V. S. Gerasimenko, N. F. Korde, and V. Y. Slivka, “Optical properties of Sn2P2S6 crystals,” Soviet Physics Solid State 19, 1794–5 (1977). 5. A. Anema, A. Grabar, and T. Rasing, “The nonlinear optical properties of Sn2P2S6,” Ferroelectrics 183, 181–3 (1996). 6. D. Haertle, A. Guarino, J. Hajfler, G. Montemezzani, and P. Günter, “Refractive indices of Sn2P2S6 at visible and infrared wavelengths,” Opt. Express 13, 2047–57 (2005). 7. G. Dittmar and H. Schäfer, “Die Struktur des Di-Zinn-Hexathiohypo-diphosphats Sn2P2S6,” Zeitschrift fuer Naturforschung 29B, 312–7 (1974). (C) 2005 OSA 16 May 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3765 #7179 $15.00 US Received 15 April 2005; revised 3 May 2005; accepted 5 May 2005 8. F. Brehat and B. Wyncke, “Calculation of double-refraction walk-off angle along the phase-matching directions in non-linear biaxial crystals,” J. Phys. B 22, 1891–8 (1989). 9. B. Wyncke and F. Brehat, “Calculation of the effective second-order non-linear coefficients along the phase matching directions in acentric orthorhombic biaxial crystals,” J. Phys. B 22, 363–76 (1989). 10. J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–8 (1984). 11. R. C. Miller, “Optical 2nd harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–9 (1964). 12. W. J. Alford and A. V. Smith, “Wavelength variation of the second-order nonlinear coefficients of KNbO3, KTiOPO4, KTiOAsO4, LiNbO3, LiIO3, beta -BaB2O4, KH2PO4, and LiB3O5 crystals: a test of Miller wavelength scaling,” J. Opt. Soc. Am. B 18, 524–33 (2001). 13. D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE Journal of Quantum Electronics 28, 2057–74 (1992). 14. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–81 (1970). 15. C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: Cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10,688–701 (2000). 16. M. Zgonik, M. Copic, and H. Arend, “Optical second harmonic generation in ferroand para-electric phases of PbHPO4,” J. Phys. C 20, L565–569 (1987). 17. R. S. Klein, G. E. Kugel, A. Maillard, K. Polgar, and A. Peter, “Absolute non-linear optical coefficients of LiNbO3 for near stoichiometric crystal compositions,” Opt. Mat. 22, 171–4 (2003). 18. B. Boulanger, J. P. Feve, G. Marnier, B. Menaert, X. Cabirol, P. Villeval, and C. Bonnin, “Relative sign and absolute magnitude of d(2) nonlinear coefficients of KTP from second-harmonic-generation measurements,” J. Opt. Soc. Am. B 11, 750–7 (1994).


Introduction
Tin thiohypodiphosphate (Sn 2 P 2 S 6 ) is a wide bandgap semiconductor ferroelectric with very attractive photorefractive properties [1,2] and large electro-optical coefficients [3].In addition, the wide optical transparency range extending from λ = 0.53 µm to λ = 8 µm [4] holds promise for optical parametric generation up to infrared wavelengths not accessible with standard nonlinear optical crystals.This requires the knowledge of the nonlinear optical coefficients and phase matching conditions.Up to now no coefficient had been determined; the only publication on nonlinear optics in Sn 2 P 2 S 6 reports a value for d 211 [5], but unfortunately without specifying the coordinate system being used (in Sn 2 P 2 S 6 d 211 is zero due to symmetry in the standard coordinate system).In this work we determine or estimate all 10 second-order nonlinear coefficients, as well as the third-order nonlinear optical susceptibilities χ (3) 1111 and χ (3) 2222 using the Maker-Fringe technique.
In Ref. 6 the refractive indices and the indicatrix rotation of Sn 2 P 2 S 6 are given for the wavelength range 550 − 2300 nm at room temperature.The Sellmeier coefficients determined there allow to describe the refractive indices with an accuracy of 2 • 10 −4 in the wavelength interval indicated.These data allow us to calculate phase-matching conditions for second harmonic generation (SHG), sum-and difference-frequency generation (SFG and DFG) and optical parametric oscillation (OPO).Calculated phase-matching conditions are compared with experimental data at λ = 1907 nm.A configuration for optical parametric oscillators pumped with the fundamental wavelength of a Nd:YAG laser, capable of producing radiation from 1 to 8 µm in the infrared with a high gain (d eff ≈ 4pm/V), is described.

Optical frequency conversion in Sn
For phase-matched parametric interactions among three parallel waves at the frequencies ω 1 , ω 2 , and ω 3 , where ω 3 = ω 1 + ω 2 , the vacuum wavelengths λ i of the interacting waves must satisfy where the n i are the refractive indices for the waves at frequencies ω i .We use the Cartesian coordinate system as defined in Ref. 6: unit cell of Dittmar and Schäfer [7], y b is perpendicular to the mirror plane of the crystal, z c, the positive direction of the x-axis and the z-axis so that the piezoelectric coefficients d xxx and d zzz are positive and +y so that xyz is a right-handed system.The spherical coordinates are defined in the standard way used in physics, with θ the angle between k and z, and φ the counterclockwise angle from x to the projection of k to the xy-plane.
Phase-matching conditions may be satisfied in materials with sufficient birefringence either by rotating the direction of the laser beams with respect to the main axes of the optical indicatrix (angle tuning), by adjusting the wavelengths of the interacting beams (wavelength tuning), or by temperature tuning of the birefringence of the crystal.
The advantage of wavelength and temperature tuning is that the interacting beams can travel collinearly with one of the main axes of the optical indicatrix.In this case, termed noncritical phase matching, the Poynting vectors of all the interacting waves are parallel to the wave vector: The interacting beams do not walk off from one another.
For angle tuning, there is critical phase matching: The Poynting vectors are in general not parallel to the wave vectors, and the interacting beams walk off from one another [8].The efficiency of the frequency conversion depends on the interaction length of the waves in the crystal, which in turn depends on the diameter of the beams and on the walk-off angle.Angle tuning is advantageous if the interaction length is of the order of the crystal length; the method can find application for powerful pulsed lasers when the beams do not need to be tightly focused or a long crystal length is not required.
The induced nonlinear-optical polarization P (ω 3 ) as a function of the electric fields E(ω 1,2 ) of the fundamental waves is described by where ε 0 is electric constant and d i jk are the nonlinear-optical coefficients.For second harmonic generation (ω 1 = ω 2 ), d i jk is symmetric in the last two indices, and the contracted notation can be used.
For general directions of the wave vectors and polarizations in the crystal the projection of the induced polarization at frequency ω 3 along the direction of polarization of the emitted wave with frequency ω 3 can be written as with (ω) i is the angle between the electric-field vector of the wave at frequency ω and the axis i of the Cartesian coordinate system [9].In a birefringent crystal the electric field direction in general is not perpendicular to the wave vector.The walk-off angle has to be taken into account in order to calculate the angles β i [8,10].
For type I SHG the induced nonlinear polarization is given by: where d eff can again be derived from Eq. ( 4).The difference between Eqs. ( 5) and ( 3) is consistent with a continuous transition to the degenerate case, described by (5), from the sumfrequency case, described by (3), with two distinguishable fundamental fields.
The frequency dependence of the nonlinear-optical coefficients can be approximately described with Miller's rule [11]: where δ i jk , the Miller indices, are almost independent of the frequency [12] and χ ii = n 2 i − 1 are the diagonal elements of the linear subsceptibility.

Second harmonic generation
In SHG the frequencies of the incoming beams are equal (ω 1 = ω 2 ) and the d tensor becomes symmetric in its last two indices.This allows to write it in its reduced form [13], which for the symmetry group m of Sn 2 P 2 S 6 is The nonlinear optical susceptibilities d ip were determined by a standard Maker-Fringe technique [14,15] with added suppression of laser beam intensity fluctuations.The fundamental wavelengths were λ = 1542 nm (first Stokes-line generated in a high pressure Raman cell filled with methane and pumped by a Surelite Nd:YAG laser at λ = 1064 nm, 7 ns, Q-switched at 2 Hz) and 1907 nm (same laser with the Raman cell filled with high pressure H 2 and Qswitched at 10 Hz).The samples used were an x-plate and a z-plate, from crystals grown at Uzhgorod University (Ukraine), oriented by Laue diffraction (precision ± 6'), polished to optical quality and poled by heating above T C = 66 • C and slowly cooling in an applied electric field of 1 kV/cm.The d coefficients were found by fitting the Maker-Fringe curves and comparing them to the ones of a reference crystal of α-quartz.Fig. 1 shows an example of a Maker-Fringe measurement of Sn 2 P 2 S 6 with the corresponding fitted curve.The curve is not symmetrical with respect to the angle ζ = 0 • , corresponding to beams perpendicular to the crystal, since the indicatrix is not perpendicular to the Cartesian axes, and therefore the coherence length is minimal at an angle ζ = 0 • .Nevertheless the theoretical curve describes the experiments nicely.The modified Kleinman symmetries δ 15 = δ 31 and δ 35 = δ 13 where used during fitting, while the other Kleinman symmetries where not used, since enough Maker-Fringe curves were available for those coefficients.
The resulting coefficients d ip of Sn 2 P 2 S 6 are shown in Table 1.Note that due to the contribution of several tensor elements in the Maker-Fringe experiments, some of their values could be determined only with a relatively low accuracy.The largest value is the diagonal coefficient d 111 = 17 ± 1.5pm/V at λ = 1542 nm and d 111 = 12 ± 1.5pm/V at 1907 nm, which is higher than most of the largest coefficients of standard materials for nonlinear optics.Of special interest is also that Sn 2 P 2 S 6 has very large electro-optical coefficients (r T 111 = 161 ± 8pm/V at λ = 1064 nm [3]), allowing to combine electro-optical with nonlinear-optical effects.Fig. 2 shows the temperature dependence of the largest d coefficient at λ = 1907 nm.At this wavelength the influence of temperature on the optical properties is negligible with respect to the change in nonlinear optical properties of second order.The d coefficient is proportional to the electrical polarization [16], which below T C is given mainly by the acentricity parameter, i.e. the spontaneous polarization P S .Therefore Fig. 2 represents the temperature dependence of the spontaneous polarization P S .Close to the phase transition (T C − T < 7 • C) we see the decrease proportional to the square root of T C − T (Fig. 2(b).The fact that d 111 does not vanish completely above T C is explained by thermal fluctuations in the critical region just above the second-order phase transition, induced by residual defects [16].

Third harmonic generation
The same laser source as in the case of SHG at λ = 1907 nm was used for the THG measurements, which were performed using the Maker-Fringe technique in an evacuated chamber (10 −2 bar).As reference we used an α-quartz crystal and the value χ The measurements showed that Sn 2 P 2 S 6 has very large χ (3) coefficients: corresponding to 850 and 470 times χ 2222 of KNbO 3 [15].

Second harmonic generation
In Sn 2 P 2 S 6 phase-matched SHG is possible for a fundamental wavelength in the range between 1680 nm and 8 µm.The whole range is achievable by type I phase matching (incident photons have the same polarisation), while type II phase matching (incident photons are of orthogonal polarisation) is possible for λ > 2324 nm.The upper boundary of 8 µm is given only by the end of the transparency range (see Fig. 3), which is due to phonon-phonon interactions [4].
The refractive indices used for calculating the phase-matching conditions are based on experimental data in the range of 550 − 2300 nm [6].This data is precise (∆n = 2 • 10 −4 ) and fits very well to a two-oscillator Sellmeier model, which was used to extrapolate the refractive indices at longer wavelengths.Nevertheless the precision at larger wavelengths cannot be predicted and could decrease rapidly.For the calculation of d eff , we numerically evaluated (4), taking into account the dispersion of the d i jk , given by Eq. ( 6).Some analytical expressions for d eff for a biaxial crystal can be found in Refs.9 and 10.
Fig. 4 shows the phase-matching wavelengths versus the beam direction for type I and type II SHG.The phase-matching loci at the available laser line of λ = 1907 nm are drawn by the white dashed line.Some experimental points measured at this wavelength are also plotted, demonstrating the accuracy of the calculated phase-matching curve.
In Fig. 5 the regions of internal directions not accessible in crystals cut along the Cartesian x, y, z-axes are indicated by the grey area.For those directions oblique cuts are necessary in order to access the wished internal direction from air.In the following figures this region is   either exactly equal or only slightly different than in Fig. 5.All contour plots display the data for the beam directions k with polar coordinates in the range φ ∈ [0 • , 180 • ] and θ ∈ [0 • , 90 • ] inside the crystal.In order to get the whole definition range φ ∈ [0 • , 360 • [ and θ ∈ [0 • , 180 • ] of the spherical coordinates, the following symmetries should be applied The effective nonlinear optical coefficient is shown in Fig. 6(a) for phase matching of type I.It ranges between 0 and 5 pm/V, with high values of d eff > 3pm/V for all fundamental wavelengths λ > 1980 nm. Type II phase matching has large d eff for other wave directions than type I, in particular d eff 2pm/V for all φ > 90 • .Compared to other materials, d eff of Sn 2 P 2 S 6 is similar to the ones of LiNbO 3 [17] and KTP [18], and larger than in most other standard materials.
Fig. 6(b) shows the walk-off angle inside the crystal for type I phase matching (PM).For the largest part of directions it lies between 1 • and 2 • .The corresponding data for type II PM is similar.A special point is given where the walk-off angle is zero, which corresponds to noncritical phase matching.In Sn 2 P 2 S 6 this can occur only for a beam direction parallel to the fixed main axis of the indicatrix, i. e. k y, corresponding to (φ , θ ) = (90 • , 90 • ) in the figures.The fundamental wavelength for non-critical PM is λ = 3212.5nm for type I and 4536.5nm for type II.Again, compared to other standard nonlinear optical materials, the walk-off angles are similar to the ones of LiNbO 3 , and larger than in KTP, but smaller than in BBO.
The figures in this section already give an indication of the possibilities for SFG, since the curves describing phase-matched SFG or OPO collapse in one point with λ 1 = λ 2 for the directions for phase-matched SHG with fundamental wavelength λ 1 .The positions of these points were given by the curves in Fig. 4.

Sum-frequency generation and optical parametric oscillation
In this section we discuss the phase-matching possibilities for SFG or parametric oscillation.The phase-matching condition is given by Eq. ( 1).In the case of SFG two optical waves at frequencies ω 1 and ω 2 interact to produce a wave at the sum frequency ω 3 = ω 1 + ω 2 .For parametric oscillation inside a resonant cavity a strong pump wave at frequency ω 3 can produce an idler wave at frequency ω 2 and, through difference frequency generation, a signal wave at frequency ω 1 = ω 3 − ω 2 .We consider here only configurations in which all wave vectors are collinear.There are two different possibilities for achieving phase matching: (I) The two waves with wavelengths λ 1 and λ 2 share the same polarization, and the sum frequency wave λ 3 is polarized orthogonal to λ 1 and λ 2 or (II) the two waves at the wavelengths λ 1 and λ 2 are polarized orthogonal to each other.In analogy to SHG we call these two cases type I and type II SFG.
In OPO for each beam direction a continuous range of pumping wavelengths can be phasematched.The phase-matched wavelengths for some propagation directions in the xy-plane are given in Fig. 7.With type I PM (left) the curves for λ 1 and λ 2 join smoothly at the wavelength which produces phase-matched second-harmonic radiation.For type II SFG (right) the phasematching lines intersect where phase-matched SHG is possible.In Fig. 7 the non-critical PM condition (k y) is shown by a thick line: here with wavelength tuning of λ 3 ∈ [1000, 1606] nm it is possible to get all the wavelengths between 1150 and 8000 nm.The same result can be achieved by angle-tuning with a fixed pumping wavelength: this can be seen by staying on a vertical line in Fig. 7 and choosing the corresponding value of φ for the desired wavelengths λ 1 or λ 2 .For the Nd:YAG laser wavelength λ 3 = 1064 nm this can be also seen in Fig. 8(a), looking at the bottom horizontal line, where θ = 90 • .At this wavelength λ 1,2 ∈ [1238, 7572] nm can be obtained using PM of type I.For the same configuration at the laser diode wavelength of λ 3 = 808 nm, λ 1 > 2747 nm can be accessed (Fig. 8(b)).Type II PM is also possible, with operating wavelengths similar to type I PM (see Fig. 7(b)).Generally one can access by type II PM the same signal and idler wavelengths as with type I, but the required tuning range of λ 3 is larger and the conversion efficiency lower.For the efficiency of these frequency conversions refer to Fig. 9, where d eff and the walk-off angle are displayed for k in the xy-plane (θ = 90 • ).These two parameters depend very little on the interacting wavelengths, and d eff > 3pm/V for type I PM in the range φ = 30 • ...150  , the term with d 133 becomes dominant for type I, while for type II many terms roughly balance each other.The efficiency for general beam directions is shown in Fig. 10 for λ 3 = 808 nm.It is highest for θ near 90 • , and remains large as long as the internal angle between k and the xy-plane does not exceed 20 − 30 • .The same figure at λ 3 = 1064 nm is very similar.By pumping at that wavelength, with a crystal cut perpendicularly to (θ = 90 • , φ = 45 • ), one can for example access all wavelengths λ 1,2 > 1200 nm by type I phase-matched angle-tuning and with d eff ≈ 4pm/V.

Conclusions
The nonlinear optical coefficients of ferroelectric Sn 2 P 2 S 6 were measured for λ = 1542 nm and 1907 nm at room temperature.The largest coefficients at these wavelengths are d 111 = 17 ± 1.5pm/V and d 111 = 12 ± 1.5pm/V, respectively.Third order subsceptibilities χ Based on the new refractive-index data for Sn 2 P 2 S 6 , we have analyzed various nonlinear optical second-order interactions.Phase-matching configurations for various wavelengths and beam propagation directions have been studied.The principal polarization directions, walkoff angles, acceptance angles, and effective nonlinear-optical coefficients have been calculated numerically for arbitrary beam propagation directions in this biaxial crystal.Phase matching has been found to be possible in a large variety of configurations.For example type I phasematched optical parametric generation from 1.2 to 8 µm with d eff ≈ 4pm/V is possible using a Nd:Yag pumping laser and similarly from 2.3 to 8 µm using a laser at 808 nm.
The advantages of this crystal are its large transparency range extending from 0.53 to 8 µm, the possibility for phase matching in the whole transparent range, the good nonlinear efficiency at phase matching, the very large electro-optical coefficients and the absence of hygroscopicity.The walk-off angle ranges between 0 and 2 • , similarly to LiNbO 3 but larger than in KTP.Damage threshold studies will be required to fully assess the potentiality of this crystal for high-power near infrared frequency conversion.

Fig. 1 .
Fig. 1.Maker-Fringe measurement in Sn 2 P 2 S 6 at λ = 1907 nm and fitted theoretical curve.The sample was a z-plate, which was rotated around its y-axis.The abscissa is the external angle between the fundamental beam (p-polarized) and the z-axis of the crystal.Detected was the p-polarized part of the second harmonic signal, yielding a measurement of d 11 at the angle ζ = 0 • and a combination of d 11 , d 13 , d 15 , d 31 , d 33 and d 35 for other rotation angles.

Fig. 2 .
Fig. 2. Temperature dependence of d 111 at λ = 1907 nm measured during heating until over the phase transition (a).The solid curve is according to d 111 = A(T C − T ) 1/2 with A = 4.2K −1/2 pm/V and T C = 65.7 • C. In (b) the coordinates are chosen so that the dependence of Fig. (a) is linear in the temperature range just below the phase transition.

Fig. 3 .Fig. 4 .
Fig. 3. Absorption constant of Sn 2 P 2 S 6 at room temperature for non-polarized light propagating along the z-axis.It shows the large transparency range extending from λ = 0.53 µm to λ = 8 µm.This curve is calculated from measured transmission (by a PE λ 9 spectrometer for λ < 1.6 µm and a PE Paragon FT-IR spectrometer above that wavelength) and taking into account multiple Fresnel reflections.

Fig. 5 .
Fig.5.The grey region indicates the directions that are not accessible from air in crystals cut along the Cartesian x, y, z-axes.This figure was calculated for the wavelengths in Fig.4(a), but since the dependence on the angles φ and θ is much larger than that on the wavelength, it can be assumed valid for every configuration shown in this paper.

Fig. 9 .
Fig. 9. Effective nonlinear optical coefficient (continuous line) and walk-off angle (dashed line) for phase-matched SFG or OPO with beam propagating in the xy-plane (θ = 90 • ).The dependence of d eff and the walk-off angle on the wavelengths of the interacting beams is weak.The data above is calculated for λ 1 = 2400 nm.

Fig. 10 .
Fig. 10.Effective nonlinear optical coefficient for type I phase-matched SFG or OPO with λ 3 = 808 nm, corresponding to Fig. 8(b).Some contour lines are labeled with their value in pm/V.In the inner white region one of the phase-matched wavelengths diverges.
If one neglects absorption and the dispersion of the d coefficients (Kleinman symmetry), the number of independent coefficients drops from 10 to 6, being d 15 = d 31 , d 32 = d 24 , d 26 = d 12 , and d 35 = d 13 .

Table 1 .
[13]second order nonlinear optical tensor elements d ip of Sn 2 P 2 S 6 at two wavelengths of the fundamental beam.The coefficients are given according to the standard conventions[13](e.g.d 15 = d 113 = d 131 ) and a α-quartz reference value of d Q 111 = 0.286 pm/V at λ = 1542 nm and d Q 111 = 0.277 pm/V at λ = 1907 nm[13].
13, No. 10 / OPTICS EXPRESS 3768 #7179 -$15.00US Received 15 April 2005; revised 3 May 2005; accepted 5 May 2005 OPTICS EXPRESS 3775 while for type II d eff is around 2 pm/V in the whole range of φ .The largest contributions to d eff around the non-critical PM direction (θ = φ = 90 • ) are given by d 111 and d 133 , with these two contributions having the same sign for type I and different one for type II PM.With φ moving away from 90 • , (C) 2005 OSA 16 May 2005 / Vol. 13, No. 10 /