- Email: [email protected]
Spheres and cylinders in parametric nonlinear optics
Optical Materials 26 (2004) 459–464 www.elsevier.com/locate/optmat
Spheres and cylinders in parametric nonlinear optics B. Boulanger a
a,*
, P. Segonds a, J-P. Feve b, O. Pacaud b, B. Menaert c, J. Zaccaro
c
Laboratoire de Spectrometrie Physique, 140 avenue de la Physique BP 87, 38402 St Martin d’Heres Cedex, France b JDS Uniphase Commercial Lasers, 31 Chemin du Vieux Ch^ene, 38941 Meylan, France c Laboratoire de Cristallographie, 25 Avenue des Martyrs BP 166, 38042 Grenoble Cedex 09, France Received 24 October 2003; accepted 22 December 2003 Available online 23 April 2004
Abstract We show how crystals cut as spheres or cylinders lead to new approaches in the field of optical frequency conversion. This article is a review of our research dealing with the advantage of these curve geometries for the conception of new methods of optical characterization and new tunable parametric devices. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction and considerations of symmetry The parallelepiped is the classical shape for crystals in optics: plane and parallel surfaces are relatively easy to obtain and are suitable to a lot of situations. Nevertheless, we have shown that geometries of crystals with a higher degree of symmetry are better for the characterization and use of nonlinear optical properties. This article is a review of the main advantages provided by crystals with spherical or cylindrical shapes in parametric optics. Here we consider only uniaxial and biaxial crystals since they are the only ones to authorize phase matching. The interest of curve geometries seems logical when the symmetry of the phase-matching properties is considered. The loci of the collinear phase-matching directions are calculated from the refractive indices at the circular frequencies x1 , x2 and x3 of the three interacting waves: x1 nðx1 ; h; /Þ þ x2 nðx2 ; h; /Þ ¼ x3 nðx3 ; h; /Þ
ð1Þ
The value of each refractive index nðxi ; h; /Þ is given by one of the two possible solutions of the Fresnel equation in the considered direction of propagation, with the spherical coordinates (h, /) relative to the optical frame
*
Corresponding author. Tel.: +33-4-7651-4339; fax: +33-4-76635495. E-mail address: [email protected] (B. Boulanger). 0925-3467/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2003.12.022
[1]. The phase-matching layer, which is formed by the whole of the phase-matching directions of the considered interaction, belongs to the same symmetry group than that of the index surface, i.e. 1/mmm for the uniaxial class and mmm for the biaxial one. It is also important to consider the effective coefficient, veff ðh; /Þ, associated with each phase-matching direction (h, /). It is equal to the tensorial contraction of the second-order electric susceptibility tensor, vð2Þ , by the tensorial product of the unit electric field vectors of the interacting waves, i.e. [2]: veff ðh; /Þ ¼ vð2Þ ½eðx3 ; h; /Þ eðx1 ; h; /Þ eðx2 ; h; /Þ ð2Þ The unit electric field vectors, eðxi ; h; /Þ, are calculated from the refractive indices, nðxi ; h; /Þ, which verify the phase-matching relation (1). It appears that the effective coefficient relative to the phase-matching layer of a biaxial crystal belongs to the mmm point group, the same one than the biaxial index surface. The situation is more complicated for the uniaxial optical class because the effective coefficient layer can belong to a point group of lower symmetry than that of the index surface. For example, it is well known that for specific types of phase matching in KH2 PO4 or LiNbO3 , the effective coefficient is zero in the principal planes x–z and y–z of the optical frame, while it reaches a maximum magnitude at / ¼ 45°: so the effective coefficient layer is mmm in that case.
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B. Boulanger et al. / Optical Materials 26 (2004) 459–464
Thus it is important to have access to any phasematching direction. If we want to do that using a single crystal, it is necessary to cut it with a shape of higher degree of symmetry than that of the phase matching or effective coefficient properties. It is all the more so true if we want to have the possibility to study completely the phase-matching properties, i.e. as a function of the wavelength. Then, the spherical geometry is the best one because it belongs to the group of highest symmetry, i.e. 111; but the cylinder, which is 1/mmm, can also be interesting for devices considerations. With a spherical shape, it is possible to propagate a beam successively in any direction of a single crystal by a simple rotation on it-self. In this case, the incident beam has to be focused by a spherical lens in order to be parallel inside the crystal as it is shown in Fig. 1. A cylindrical lens is obviously sufficient when a cylindrical crystal is considered. The suitable focusing conditions can be easily defined using the ABCD matrix transformation formalism when the curve radius, the wavelength and the refractive index of the lens and of the crystal are known [3,4]. The idea of using spheres and cylinders in nonlinear optics relies on the ability to cut and polish spheres and oriented cylinders in crystals. For that, we have con-
ceived specific apparatus which are based on the rotation of the sample during its cutting and polishing [4]. The orientation of the cylinders has to be chosen before the cutting, and controlled during the process. For the spheres, the orientation is only done after the polishing. The ‘‘asphericity’’ or ‘‘acylindricity’’ are better than 1% and the accuracy of orientation is of about 0.1°. The spheres are polished over all the surface while it is only done on the edge for the cylinders, as it is shown in Fig. 2. We cut spheres with a radius ranging between 2 and 7 mm, and between 10 and 40 mm for cylinders.
2. Crystal characterization Our challenge was to be able to measure directly any phase-matching direction and the associated conversion efficiency in only one sample of the studied crystal. It was motivated by the weakness of the classical methods. Generally the phase-matching angles are calculated from the dispersion equation of the refractive indices determined by the prism method. The calculation is often wrong because the accuracy of measurement of the refractive indices is not sufficient: indeed, the refractive indices at x1 , x2 and x3 have to be measured with a
Fig. 1. Schematic focusing conditions of the incident beam. Incident (w0 , d) and refracted (w00 , d 0 ) Gaussian beam parameters. D is the sphere diameter.
Fig. 2. Ca4 YO(BO3 )3 crystal sphere with a diameter of 5.54 mm (a) and KTiOPO4 crystal cylinder with a diameter of 21.2 mm and a thickness of 5 mm (b).
B. Boulanger et al. / Optical Materials 26 (2004) 459–464
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YCOB x-z plane 32
30
(a, z)angle (˚)
precision of about 10 4 in order to calculate the phasematching angles to a precision of about 1°. Such an accuracy can be reached in the visible spectrum, but it is more difficult in the infrared. Moreover the classical methods require the cutting of several oriented plates and prisms of a few mm3 , such volumes being usually not compatible with the early stages of the growth of a new material. At the opposite, only one crystal cut as a sphere allows us to perform direct measurements of the parametric optical properties, the point group of the studied crystal being the only previous data which is mandatory to know. The polished sphere is oriented using an automatic Xrays diffractometer and then it is stuck on a goniometric head. From this point, nonlinear measurements can be performed for crystals belonging to the trigonal, quadratic or orthorhombic crystal systems because the crystallographic and optical frames have the same orientation. It is not the case for monoclinic or triclinic crystals, for which it is thus necessary to use optics as complementary technique for the determination of the relative orientation of the two frames. It is classically done by using a X-ray technique associated with the conoscopic method [5]. Recently we have shown that it is also possible to determine this relative orientation with a crystal cut as a sphere, by coupling X-ray diffraction with internal conical refraction of a laser. By this way, we were also able to check the possible wavelength dispersion of the orientation of the optical frame. This first demonstration has been done for the study of the monoclinic crystal Ca4 YO(BO3 )3 (YCOB) [6]. The sphere was stuck on the goniometric head in the binary b-axis which is also the y-axis of the optical frame. The two other axes of the optical frame, i.e. x and z, are tilted around b from the a- and c-crystallographic b and cx b as axes. The goal is then to measure the angles az a function of the wavelength. In a first step, the sphere was mounted on the X-rays automatic diffractometer and was simultaneously illuminated by a HeNe beam at k ¼ 0:6328 lm. The laser beam was properly focused through the sphere as described above. From X-rays orientation, we marked out the goniometric positions of the a- and c-axes. Then by rotating the sphere through 360° in that plane, we can observe the four hollow cones which correspond by pairs to the two optical axis of internal conical refraction. This specific phenomenon of biaxial crystals is enhanced by the spherical shape of the crystal, as we had shown previously [7]. Since the two optical axes of internal conical refraction are symmetrical in comparison with x- and z-axes, we can mark out their goniometric position with a very good accuracy of about ±0.1°. In a second step, the YCOB sphere was placed at the center of an Euler circle and coupled to a tunable laser, emitting between the visible and the infrared. We took the same goniometric head as the one mounted previously in order to keep the goniometric
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20 0.4
0.8
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Wavelength (µm) Fig. 3. Angle between the a-axis of the crystallographic frame and the z-axis of the optical frame of Ca4 YO(BO3 )3 as a function of wavelength.
positions of the four axes a, c, z and x. The incoming tunable laser beam was allowed to propagate in any direction of the x–z plane since the sphere can be rotated around the y-axis using an Euler Circle. We still used the observation of the four hallow cones. The observation was made directly on a screen in the visible, and with a visual card in the infrared. The corresponding results for b azðkÞ are reported in Fig. 3 with an accuracy which reaches ±0.5°. The error was mainly due to the difficulty of observation in the infrared and to the asphericity of the sample. It clearly appears from Fig. 3 that the angle between the a- and z-axes can be considered as wavelength independent within the accuracy of our measurements. The determination of the orientation of the optical axes of a triclinic crystal will be more complicated because no axes of the crystallographical and optical axes are joint. We can also use a method based on the coupling of a diffractometer and a laser beam. But in that case, it will be necessary to scan the whole space of the sphere because the plane where lie the optical axes of internal conical refraction is not known a priori. Nonlinear measurements can start when the optical frame orientation is known. The sphere placed at the center of an Euler circle is coupled with two laser beams at different wavelengths k1 and k2 which can be tuned or not in order to use two configurations [8]: second-harmonic generation from the fundamental k1 or k2 , and sum- or difference-frequency generation at k3 such as 1 1 k 1 3 ¼ k1 k2 . The polarization of the incident beams is controlled in order to realize any of the three possible
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phase-matching types referred as types I, II and III. The phase-matching measurement method consists in rotating the sphere on it-self in order to propagate the incident beams successively in different directions of the crystal. A phase-matching direction is located when the conversion efficiency is maximum. The generated and incident beams are detected by diodes after being separated and filtered by a prism and several spectral filters. The typical accuracy for a detection out of the principal planes, i.e. x–y, x–z and y–z, is of about ±1°. Nevertheless, this precision can be notably improved for the directions contained in the principal planes where it is then possible to use the detection of the four symmetrical directions. In the principal planes the accuracy is generally better than ±0.3°. The measured phase-matching directions can be directly used for the cutting of the crystal at the right angle. We can also consider the phase-matching curves measured for different wavelengths and types: their simultaneous fit can lead to the determination of dispersion equations of the refractive indices. The reliability of these equations will be all the more so better since the three refractive indices are implicated over a wavelength range as large as possible. This goal can be reached by using tunable incident beams and all the possible types of phase matching, that can provide hundreds experimental phase-matching data. We did that for the study of KTiOPO4 [9], RbTiOPO4 [10], the Incoming wavelength λi (µm) 1.76
1.72
1.68
1.65
1.62
1.59
1.57
1.55
90
arsenates KTiOAsO4 , RbTiOAsO4 and CsTiOAsO4 [11] and YCOB [6]. Fig. 4 shows the example of type I difference frequency generation in RbTiOPO4 [10], where it clearly appears that the calculation from the measured refractive indices do not agree with the experiment, contrarily to our fit. Note that absolute values of the three principal refractive indices are fixed by adding to our experimental data the absolute value of one refractive index at one wavelength. If we are only interested in the phase-matching properties, any value can be taken because the phase-matching properties do not depend on the absolute values of the refractive indices, but only on the ratii nx =ny , nx =nz and ny =nz . In other words, the same phase-matching curve will be obtained by multiplying the dispersion equations of the three principal refractive indices by any factor. The sphere allows us also to measure the different properties which are associated to each phase-matching direction: the spectral and angular acceptances, and the conversion efficiency with an accuracy of ±10%. From all these data we can choose the best phase-matching direction for the aimed application, but we can also determine the relative sign and the absolute values of the second order electric susceptibility tensor vð2Þ [9]. In order to measure the maximum number of independent coefficients, it is necessary to carefully consider the expression of the effective coefficient (2) as a function of the phase-matching direction (h, /), and that for the three types of phase matching. An appropriate analysis of the field tensor, i.e. F ð2Þ ðh; /Þ ¼ eðx3 ; h; /Þ eðx1 ; h; /Þ eðx2 ; h; /Þ, allows us to know which nonlinear coefficients are involved in given direction of propagation and configuration of polarization [12].
Phase-matching direction θPM (˚)
85
3. Tunable parametric devices 80
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55 2.6
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Generated wavelength λg (µm) Fig. 4. Measured and calculated phase-matching curves in the x–z plane (squares) and y–z plane (circles) for type I difference frequency generation in RbTiOPO4 : kop koi ! keg . ki is generated from a tunable OPO, and kp ¼ 1:064 lm. The solid lines are calculated with the dispersion equations resulting from the fit of sphere data, while the dotted lines are deduced from Sellmeier equations established from refractive index measurement using the prism technique.
For tunable optical parametric generators (OPG) or oscillators (OPO), it is not necessary to consider all the phase-matched parametric interactions and directions of a given crystal, contrarily to the characterization problematic: only one interaction is chosen, and the phasematching directions contained in one plane, usually a principal plane, are sufficient to exploit at best the tunability of the crystal. In that case, a cylinder whose the revolution axis is orthogonal to the useful plane is required: the propagation is then ensured in this plane, and the tunability is obtained by rotation of the crystal cylinder around its revolution axis. There are two advantages of such a geometry compared with parallelepiped crystals [13]: the angular tunability is infinite and the beams can propagate at normal incidence for each phase-matching direction. Due to the last point, there are less noncollinear interactions and Fresnel losses than in the case of parallelepiped crystals, so better spatial and spectral properties can be expected. But in
B. Boulanger et al. / Optical Materials 26 (2004) 459–464
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Fig. 5. Schematic focusing conditions of an OPO using a cylindrical nonlinear crystal.
The cylinder is also a satisfactory alternative for continuous tuning of quasi-phase-matching. Fig. 6 shows that the rotation of a periodically polled cylinder with a single periodicity leads to the variation of the effective periodicity as seen by the beams: it varies from K to K= cos a when the cylinder is rotated by an angle a [13]. We have demonstrated that a rotation over an angle a ¼ 26° of a 12.1 mm diameter periodically poled KTiOPO4 cylinder with a thickness of 0.5 mm and a periodicity of 35 lm leads to a broad tenability: the OPO pumped at 1.064 lm emits a signal tunable between 1.515 and 2.040 lm, while the corresponding idler is tuned between 2.220 and 3.560 lm as shown in Fig. 7 [15]. The main advantage of cylindrical quasi-phasematched OPOs compared with the multi-grating devices is that the tunability is perfectly continuous, and the wavelength over the transverse section of the beam is more homogeneous than in the case of fan-shaped gratings. We have recently generalized this concept to OPG. We cut a cylinder with a diameter of 40 mm and a thickness of 0.5 mm in a periodically poled LiNbO3 crystal with a periodicity of 27.5 lm [16]. The experiments were performed at 140 °C in order to prevent photo-refractive damage. The pump was a microchip
3500
Signal and idler wavelengths λs,i(nm)
return, it is necessary to focus the pump beam in the plane of cylindricity at least, that can limit the input energy according to the damage threshold intensity. The emitted signal and idler beams have also to be collected by a lens. Then the OPO cavity, with its axis orthogonal to the revolution axis of the cylinder, is placed between the nonlinear crystal and the lenses, as shown in Fig. 5. We have performed a complete theoretical study of the different possible cavity geometries which are compatible with a cylindrical dioptre [14]. Our Gaussian model was based on the coupling of the pump and resonating signal beams in the cavity. We have investigated the effects of different parameters on the output beam quality: radius of curvature of the mirrors, cavity length, crystal cylinder radius, pump-beam waist size, and spectral-tuning range of the signal beam. These calculations lead to the definition of the optimum configurations associated with a low M2 beam quality factor. On this theoretical basis, we have conceived a phasematched singly resonant OPO using a KTiOPO4 cylinder with a diameter of 21.2 mm and a thickness of 5 mm [14]. The revolution axis is the y-axis and so the used phase-matching directions lie in the x–z plane where the effective coefficient has the strongest value. The OPO was pumped at 1.064 lm with a repetition rate of 10 Hz and a pulse duration of 10 ns. We have demonstrated an angular tunability of 31.5° corresponding to the emission of the resonating signal beam between 1.570 and 1.830 lm. The associated M2 was less than 1.5, that is remarkable for an OPO without any restricting element inside the cavity.
3000
2500
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1500 0
Fig. 6. Schematic diagram of a tunable quasi-phase-matched OPO using a periodically poled nonlinear crystal. KðaÞ is the effective periodicity corresponding to an angle of rotation a. The wavelengths kp , ks ðaÞ and ki ðaÞ are relative to the pump, signal and idler beams respectively.
5
10 15 Rotation angle α (˚)
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Fig. 7. OPO wavelengths as a function of the revolution angle a of a periodically poled KTiOPO4 cylinder pumped at 1.064 lm. The squares are the experimental points. The solid lines are calculated with two different models of quasi-phase-matching coupling.
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Idler wavelength λi (µm)
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References
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same crystal. In particular, we are thinking of performing direct measurements of the laser properties by using a sphere.
1. 65 1. 60 1. 55 1. 50 1. 45 1. 40
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Propagation angle α (˚)
Fig. 8. Angular tuning curve of an optical parametric generator based on a periodically poled LiNbO3 cylinder pumped at 1.064 lm. The solid line is calculated from Sellmeier equations.
laser at 1.064 lm with a repetition rate of 1 kHz and a pulse duration of 430 ps. We realized a continuous tuning OPG over a broad spectral range, as shown in Fig. 8. This device is more simple and stable than an OPO because there is no cavity.
4. Conclusion and perspectives We have shown that spheres and cylinder allows us a new approach in nonlinear optics. We are going to apply these concepts to the study and the use of bi-functional materials, in which the laser effect and the nonlinear frequency conversion occur simultaneously inside the
[1] J.P. Feve, B. Boulanger, G. Marnier, Opt. Commun. (99) (1993) 284. [2] B. Boulanger, J. Zyss, Non-linear optical properties, in: A. Authier (Ed.), International Tables for Crystallography, Physical Properties of Crystals, vol. D, International Union of Crystallography, Kluwer Academic Publisher, Dordrecht, Netherlands, 2003, p. 177 (Chapter 1.8). [3] B. Boulanger, J.P. Feve, G. Marnier, B. Menaert, Pure Appl. Opt. (7) (1998) 239. [4] O. Pacaud, Oscillateurs parametriques optiques bases sur des cristaux de geometrie cylindrique, PhD thesis, Joseph Fourier University, 13 June 2001. [5] See for example: Precis d’optique cristalline appliquee a l’identification des mineraux, Masson & Cie, Paris, 1968. [6] P. Segonds, B. Boulanger, J.P. Feve, B. Menaert, J. Zaccaro, G. Aka, D. Pelenc, J. Opt. Soc. Am. B 21 (4) (2004). [7] J.P. Feve, B. Boulanger, G. Marnier, Opt. Commun. (105) (1994) 243. [8] I. Rousseau, B. Boulanger, J.P. Feve, O. Pacaud, Appl. Opt. (36) (1999) 7406. [9] B. Boulanger, J.P. Feve, G. Marnier, C. Bonnin, P. Villeval, J.J. Zondy, J. Opt. Soc. Am. B 14 (1997) 1380. [10] Y. Guillien, B. Menaert, J.P. Feve, P. Segonds, J. Douady, B. Boulanger, O. Pacaud, Opt. Mat. (22) (2002) 155. [11] J.P. Feve, B. Boulanger, O. Pacaud, I. Rousseau, B. Menaert, G. Marnier, P. Villeval, C. Bonnin, G.M. Loiacono, D.N. Loiacono, J. Opt. Soc. Am. B (17) (2000) 775. [12] B. Boulanger, G. Marnier, J. Phys.: Condens. Matter (3) (1991) 8327. [13] B. Boulanger, J.P. Feve, B. Menaert, G. Marnier, French patent 2771519 (28 May 1999), International PCT WO99/28785 (6 October 1999). [14] J.P. Feve, O. Pacaud, B. Boulanger, B. Menaert, M. Renard, J. Opt. Soc. Am. B (19) (2002) 222. [15] J.P. Feve, O. Pacaud, B. Boulanger, B. Menaert, J. Hellstr€ om, F. Laurell, Opt. Lett. (26) (2001) 1882. [16] J.P. Feve, B. Boulanger, B. Menaert, O. Pacaud, Opt. Lett. (28) (2003) 1028.
Spheres and cylinders in parametric nonlinear optics B. Boulanger a
a,*
, P. Segonds a, J-P. Feve b, O. Pacaud b, B. Menaert c, J. Zaccaro
c
Laboratoire de Spectrometrie Physique, 140 avenue de la Physique BP 87, 38402 St Martin d’Heres Cedex, France b JDS Uniphase Commercial Lasers, 31 Chemin du Vieux Ch^ene, 38941 Meylan, France c Laboratoire de Cristallographie, 25 Avenue des Martyrs BP 166, 38042 Grenoble Cedex 09, France Received 24 October 2003; accepted 22 December 2003 Available online 23 April 2004
Abstract We show how crystals cut as spheres or cylinders lead to new approaches in the field of optical frequency conversion. This article is a review of our research dealing with the advantage of these curve geometries for the conception of new methods of optical characterization and new tunable parametric devices. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction and considerations of symmetry The parallelepiped is the classical shape for crystals in optics: plane and parallel surfaces are relatively easy to obtain and are suitable to a lot of situations. Nevertheless, we have shown that geometries of crystals with a higher degree of symmetry are better for the characterization and use of nonlinear optical properties. This article is a review of the main advantages provided by crystals with spherical or cylindrical shapes in parametric optics. Here we consider only uniaxial and biaxial crystals since they are the only ones to authorize phase matching. The interest of curve geometries seems logical when the symmetry of the phase-matching properties is considered. The loci of the collinear phase-matching directions are calculated from the refractive indices at the circular frequencies x1 , x2 and x3 of the three interacting waves: x1 nðx1 ; h; /Þ þ x2 nðx2 ; h; /Þ ¼ x3 nðx3 ; h; /Þ
ð1Þ
The value of each refractive index nðxi ; h; /Þ is given by one of the two possible solutions of the Fresnel equation in the considered direction of propagation, with the spherical coordinates (h, /) relative to the optical frame
*
Corresponding author. Tel.: +33-4-7651-4339; fax: +33-4-76635495. E-mail address: [email protected] (B. Boulanger). 0925-3467/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2003.12.022
[1]. The phase-matching layer, which is formed by the whole of the phase-matching directions of the considered interaction, belongs to the same symmetry group than that of the index surface, i.e. 1/mmm for the uniaxial class and mmm for the biaxial one. It is also important to consider the effective coefficient, veff ðh; /Þ, associated with each phase-matching direction (h, /). It is equal to the tensorial contraction of the second-order electric susceptibility tensor, vð2Þ , by the tensorial product of the unit electric field vectors of the interacting waves, i.e. [2]: veff ðh; /Þ ¼ vð2Þ ½eðx3 ; h; /Þ eðx1 ; h; /Þ eðx2 ; h; /Þ ð2Þ The unit electric field vectors, eðxi ; h; /Þ, are calculated from the refractive indices, nðxi ; h; /Þ, which verify the phase-matching relation (1). It appears that the effective coefficient relative to the phase-matching layer of a biaxial crystal belongs to the mmm point group, the same one than the biaxial index surface. The situation is more complicated for the uniaxial optical class because the effective coefficient layer can belong to a point group of lower symmetry than that of the index surface. For example, it is well known that for specific types of phase matching in KH2 PO4 or LiNbO3 , the effective coefficient is zero in the principal planes x–z and y–z of the optical frame, while it reaches a maximum magnitude at / ¼ 45°: so the effective coefficient layer is mmm in that case.
460
B. Boulanger et al. / Optical Materials 26 (2004) 459–464
Thus it is important to have access to any phasematching direction. If we want to do that using a single crystal, it is necessary to cut it with a shape of higher degree of symmetry than that of the phase matching or effective coefficient properties. It is all the more so true if we want to have the possibility to study completely the phase-matching properties, i.e. as a function of the wavelength. Then, the spherical geometry is the best one because it belongs to the group of highest symmetry, i.e. 111; but the cylinder, which is 1/mmm, can also be interesting for devices considerations. With a spherical shape, it is possible to propagate a beam successively in any direction of a single crystal by a simple rotation on it-self. In this case, the incident beam has to be focused by a spherical lens in order to be parallel inside the crystal as it is shown in Fig. 1. A cylindrical lens is obviously sufficient when a cylindrical crystal is considered. The suitable focusing conditions can be easily defined using the ABCD matrix transformation formalism when the curve radius, the wavelength and the refractive index of the lens and of the crystal are known [3,4]. The idea of using spheres and cylinders in nonlinear optics relies on the ability to cut and polish spheres and oriented cylinders in crystals. For that, we have con-
ceived specific apparatus which are based on the rotation of the sample during its cutting and polishing [4]. The orientation of the cylinders has to be chosen before the cutting, and controlled during the process. For the spheres, the orientation is only done after the polishing. The ‘‘asphericity’’ or ‘‘acylindricity’’ are better than 1% and the accuracy of orientation is of about 0.1°. The spheres are polished over all the surface while it is only done on the edge for the cylinders, as it is shown in Fig. 2. We cut spheres with a radius ranging between 2 and 7 mm, and between 10 and 40 mm for cylinders.
2. Crystal characterization Our challenge was to be able to measure directly any phase-matching direction and the associated conversion efficiency in only one sample of the studied crystal. It was motivated by the weakness of the classical methods. Generally the phase-matching angles are calculated from the dispersion equation of the refractive indices determined by the prism method. The calculation is often wrong because the accuracy of measurement of the refractive indices is not sufficient: indeed, the refractive indices at x1 , x2 and x3 have to be measured with a
Fig. 1. Schematic focusing conditions of the incident beam. Incident (w0 , d) and refracted (w00 , d 0 ) Gaussian beam parameters. D is the sphere diameter.
Fig. 2. Ca4 YO(BO3 )3 crystal sphere with a diameter of 5.54 mm (a) and KTiOPO4 crystal cylinder with a diameter of 21.2 mm and a thickness of 5 mm (b).
B. Boulanger et al. / Optical Materials 26 (2004) 459–464
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YCOB x-z plane 32
30
(a, z)angle (˚)
precision of about 10 4 in order to calculate the phasematching angles to a precision of about 1°. Such an accuracy can be reached in the visible spectrum, but it is more difficult in the infrared. Moreover the classical methods require the cutting of several oriented plates and prisms of a few mm3 , such volumes being usually not compatible with the early stages of the growth of a new material. At the opposite, only one crystal cut as a sphere allows us to perform direct measurements of the parametric optical properties, the point group of the studied crystal being the only previous data which is mandatory to know. The polished sphere is oriented using an automatic Xrays diffractometer and then it is stuck on a goniometric head. From this point, nonlinear measurements can be performed for crystals belonging to the trigonal, quadratic or orthorhombic crystal systems because the crystallographic and optical frames have the same orientation. It is not the case for monoclinic or triclinic crystals, for which it is thus necessary to use optics as complementary technique for the determination of the relative orientation of the two frames. It is classically done by using a X-ray technique associated with the conoscopic method [5]. Recently we have shown that it is also possible to determine this relative orientation with a crystal cut as a sphere, by coupling X-ray diffraction with internal conical refraction of a laser. By this way, we were also able to check the possible wavelength dispersion of the orientation of the optical frame. This first demonstration has been done for the study of the monoclinic crystal Ca4 YO(BO3 )3 (YCOB) [6]. The sphere was stuck on the goniometric head in the binary b-axis which is also the y-axis of the optical frame. The two other axes of the optical frame, i.e. x and z, are tilted around b from the a- and c-crystallographic b and cx b as axes. The goal is then to measure the angles az a function of the wavelength. In a first step, the sphere was mounted on the X-rays automatic diffractometer and was simultaneously illuminated by a HeNe beam at k ¼ 0:6328 lm. The laser beam was properly focused through the sphere as described above. From X-rays orientation, we marked out the goniometric positions of the a- and c-axes. Then by rotating the sphere through 360° in that plane, we can observe the four hollow cones which correspond by pairs to the two optical axis of internal conical refraction. This specific phenomenon of biaxial crystals is enhanced by the spherical shape of the crystal, as we had shown previously [7]. Since the two optical axes of internal conical refraction are symmetrical in comparison with x- and z-axes, we can mark out their goniometric position with a very good accuracy of about ±0.1°. In a second step, the YCOB sphere was placed at the center of an Euler circle and coupled to a tunable laser, emitting between the visible and the infrared. We took the same goniometric head as the one mounted previously in order to keep the goniometric
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Wavelength (µm) Fig. 3. Angle between the a-axis of the crystallographic frame and the z-axis of the optical frame of Ca4 YO(BO3 )3 as a function of wavelength.
positions of the four axes a, c, z and x. The incoming tunable laser beam was allowed to propagate in any direction of the x–z plane since the sphere can be rotated around the y-axis using an Euler Circle. We still used the observation of the four hallow cones. The observation was made directly on a screen in the visible, and with a visual card in the infrared. The corresponding results for b azðkÞ are reported in Fig. 3 with an accuracy which reaches ±0.5°. The error was mainly due to the difficulty of observation in the infrared and to the asphericity of the sample. It clearly appears from Fig. 3 that the angle between the a- and z-axes can be considered as wavelength independent within the accuracy of our measurements. The determination of the orientation of the optical axes of a triclinic crystal will be more complicated because no axes of the crystallographical and optical axes are joint. We can also use a method based on the coupling of a diffractometer and a laser beam. But in that case, it will be necessary to scan the whole space of the sphere because the plane where lie the optical axes of internal conical refraction is not known a priori. Nonlinear measurements can start when the optical frame orientation is known. The sphere placed at the center of an Euler circle is coupled with two laser beams at different wavelengths k1 and k2 which can be tuned or not in order to use two configurations [8]: second-harmonic generation from the fundamental k1 or k2 , and sum- or difference-frequency generation at k3 such as 1 1 k 1 3 ¼ k1 k2 . The polarization of the incident beams is controlled in order to realize any of the three possible
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phase-matching types referred as types I, II and III. The phase-matching measurement method consists in rotating the sphere on it-self in order to propagate the incident beams successively in different directions of the crystal. A phase-matching direction is located when the conversion efficiency is maximum. The generated and incident beams are detected by diodes after being separated and filtered by a prism and several spectral filters. The typical accuracy for a detection out of the principal planes, i.e. x–y, x–z and y–z, is of about ±1°. Nevertheless, this precision can be notably improved for the directions contained in the principal planes where it is then possible to use the detection of the four symmetrical directions. In the principal planes the accuracy is generally better than ±0.3°. The measured phase-matching directions can be directly used for the cutting of the crystal at the right angle. We can also consider the phase-matching curves measured for different wavelengths and types: their simultaneous fit can lead to the determination of dispersion equations of the refractive indices. The reliability of these equations will be all the more so better since the three refractive indices are implicated over a wavelength range as large as possible. This goal can be reached by using tunable incident beams and all the possible types of phase matching, that can provide hundreds experimental phase-matching data. We did that for the study of KTiOPO4 [9], RbTiOPO4 [10], the Incoming wavelength λi (µm) 1.76
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arsenates KTiOAsO4 , RbTiOAsO4 and CsTiOAsO4 [11] and YCOB [6]. Fig. 4 shows the example of type I difference frequency generation in RbTiOPO4 [10], where it clearly appears that the calculation from the measured refractive indices do not agree with the experiment, contrarily to our fit. Note that absolute values of the three principal refractive indices are fixed by adding to our experimental data the absolute value of one refractive index at one wavelength. If we are only interested in the phase-matching properties, any value can be taken because the phase-matching properties do not depend on the absolute values of the refractive indices, but only on the ratii nx =ny , nx =nz and ny =nz . In other words, the same phase-matching curve will be obtained by multiplying the dispersion equations of the three principal refractive indices by any factor. The sphere allows us also to measure the different properties which are associated to each phase-matching direction: the spectral and angular acceptances, and the conversion efficiency with an accuracy of ±10%. From all these data we can choose the best phase-matching direction for the aimed application, but we can also determine the relative sign and the absolute values of the second order electric susceptibility tensor vð2Þ [9]. In order to measure the maximum number of independent coefficients, it is necessary to carefully consider the expression of the effective coefficient (2) as a function of the phase-matching direction (h, /), and that for the three types of phase matching. An appropriate analysis of the field tensor, i.e. F ð2Þ ðh; /Þ ¼ eðx3 ; h; /Þ eðx1 ; h; /Þ eðx2 ; h; /Þ, allows us to know which nonlinear coefficients are involved in given direction of propagation and configuration of polarization [12].
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Generated wavelength λg (µm) Fig. 4. Measured and calculated phase-matching curves in the x–z plane (squares) and y–z plane (circles) for type I difference frequency generation in RbTiOPO4 : kop koi ! keg . ki is generated from a tunable OPO, and kp ¼ 1:064 lm. The solid lines are calculated with the dispersion equations resulting from the fit of sphere data, while the dotted lines are deduced from Sellmeier equations established from refractive index measurement using the prism technique.
For tunable optical parametric generators (OPG) or oscillators (OPO), it is not necessary to consider all the phase-matched parametric interactions and directions of a given crystal, contrarily to the characterization problematic: only one interaction is chosen, and the phasematching directions contained in one plane, usually a principal plane, are sufficient to exploit at best the tunability of the crystal. In that case, a cylinder whose the revolution axis is orthogonal to the useful plane is required: the propagation is then ensured in this plane, and the tunability is obtained by rotation of the crystal cylinder around its revolution axis. There are two advantages of such a geometry compared with parallelepiped crystals [13]: the angular tunability is infinite and the beams can propagate at normal incidence for each phase-matching direction. Due to the last point, there are less noncollinear interactions and Fresnel losses than in the case of parallelepiped crystals, so better spatial and spectral properties can be expected. But in
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Fig. 5. Schematic focusing conditions of an OPO using a cylindrical nonlinear crystal.
The cylinder is also a satisfactory alternative for continuous tuning of quasi-phase-matching. Fig. 6 shows that the rotation of a periodically polled cylinder with a single periodicity leads to the variation of the effective periodicity as seen by the beams: it varies from K to K= cos a when the cylinder is rotated by an angle a [13]. We have demonstrated that a rotation over an angle a ¼ 26° of a 12.1 mm diameter periodically poled KTiOPO4 cylinder with a thickness of 0.5 mm and a periodicity of 35 lm leads to a broad tenability: the OPO pumped at 1.064 lm emits a signal tunable between 1.515 and 2.040 lm, while the corresponding idler is tuned between 2.220 and 3.560 lm as shown in Fig. 7 [15]. The main advantage of cylindrical quasi-phasematched OPOs compared with the multi-grating devices is that the tunability is perfectly continuous, and the wavelength over the transverse section of the beam is more homogeneous than in the case of fan-shaped gratings. We have recently generalized this concept to OPG. We cut a cylinder with a diameter of 40 mm and a thickness of 0.5 mm in a periodically poled LiNbO3 crystal with a periodicity of 27.5 lm [16]. The experiments were performed at 140 °C in order to prevent photo-refractive damage. The pump was a microchip
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Signal and idler wavelengths λs,i(nm)
return, it is necessary to focus the pump beam in the plane of cylindricity at least, that can limit the input energy according to the damage threshold intensity. The emitted signal and idler beams have also to be collected by a lens. Then the OPO cavity, with its axis orthogonal to the revolution axis of the cylinder, is placed between the nonlinear crystal and the lenses, as shown in Fig. 5. We have performed a complete theoretical study of the different possible cavity geometries which are compatible with a cylindrical dioptre [14]. Our Gaussian model was based on the coupling of the pump and resonating signal beams in the cavity. We have investigated the effects of different parameters on the output beam quality: radius of curvature of the mirrors, cavity length, crystal cylinder radius, pump-beam waist size, and spectral-tuning range of the signal beam. These calculations lead to the definition of the optimum configurations associated with a low M2 beam quality factor. On this theoretical basis, we have conceived a phasematched singly resonant OPO using a KTiOPO4 cylinder with a diameter of 21.2 mm and a thickness of 5 mm [14]. The revolution axis is the y-axis and so the used phase-matching directions lie in the x–z plane where the effective coefficient has the strongest value. The OPO was pumped at 1.064 lm with a repetition rate of 10 Hz and a pulse duration of 10 ns. We have demonstrated an angular tunability of 31.5° corresponding to the emission of the resonating signal beam between 1.570 and 1.830 lm. The associated M2 was less than 1.5, that is remarkable for an OPO without any restricting element inside the cavity.
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Fig. 6. Schematic diagram of a tunable quasi-phase-matched OPO using a periodically poled nonlinear crystal. KðaÞ is the effective periodicity corresponding to an angle of rotation a. The wavelengths kp , ks ðaÞ and ki ðaÞ are relative to the pump, signal and idler beams respectively.
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Fig. 7. OPO wavelengths as a function of the revolution angle a of a periodically poled KTiOPO4 cylinder pumped at 1.064 lm. The squares are the experimental points. The solid lines are calculated with two different models of quasi-phase-matching coupling.
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same crystal. In particular, we are thinking of performing direct measurements of the laser properties by using a sphere.
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Fig. 8. Angular tuning curve of an optical parametric generator based on a periodically poled LiNbO3 cylinder pumped at 1.064 lm. The solid line is calculated from Sellmeier equations.
laser at 1.064 lm with a repetition rate of 1 kHz and a pulse duration of 430 ps. We realized a continuous tuning OPG over a broad spectral range, as shown in Fig. 8. This device is more simple and stable than an OPO because there is no cavity.
4. Conclusion and perspectives We have shown that spheres and cylinder allows us a new approach in nonlinear optics. We are going to apply these concepts to the study and the use of bi-functional materials, in which the laser effect and the nonlinear frequency conversion occur simultaneously inside the
[1] J.P. Feve, B. Boulanger, G. Marnier, Opt. Commun. (99) (1993) 284. [2] B. Boulanger, J. Zyss, Non-linear optical properties, in: A. Authier (Ed.), International Tables for Crystallography, Physical Properties of Crystals, vol. D, International Union of Crystallography, Kluwer Academic Publisher, Dordrecht, Netherlands, 2003, p. 177 (Chapter 1.8). [3] B. Boulanger, J.P. Feve, G. Marnier, B. Menaert, Pure Appl. Opt. (7) (1998) 239. [4] O. Pacaud, Oscillateurs parametriques optiques bases sur des cristaux de geometrie cylindrique, PhD thesis, Joseph Fourier University, 13 June 2001. [5] See for example: Precis d’optique cristalline appliquee a l’identification des mineraux, Masson & Cie, Paris, 1968. [6] P. Segonds, B. Boulanger, J.P. Feve, B. Menaert, J. Zaccaro, G. Aka, D. Pelenc, J. Opt. Soc. Am. B 21 (4) (2004). [7] J.P. Feve, B. Boulanger, G. Marnier, Opt. Commun. (105) (1994) 243. [8] I. Rousseau, B. Boulanger, J.P. Feve, O. Pacaud, Appl. Opt. (36) (1999) 7406. [9] B. Boulanger, J.P. Feve, G. Marnier, C. Bonnin, P. Villeval, J.J. Zondy, J. Opt. Soc. Am. B 14 (1997) 1380. [10] Y. Guillien, B. Menaert, J.P. Feve, P. Segonds, J. Douady, B. Boulanger, O. Pacaud, Opt. Mat. (22) (2002) 155. [11] J.P. Feve, B. Boulanger, O. Pacaud, I. Rousseau, B. Menaert, G. Marnier, P. Villeval, C. Bonnin, G.M. Loiacono, D.N. Loiacono, J. Opt. Soc. Am. B (17) (2000) 775. [12] B. Boulanger, G. Marnier, J. Phys.: Condens. Matter (3) (1991) 8327. [13] B. Boulanger, J.P. Feve, B. Menaert, G. Marnier, French patent 2771519 (28 May 1999), International PCT WO99/28785 (6 October 1999). [14] J.P. Feve, O. Pacaud, B. Boulanger, B. Menaert, M. Renard, J. Opt. Soc. Am. B (19) (2002) 222. [15] J.P. Feve, O. Pacaud, B. Boulanger, B. Menaert, J. Hellstr€ om, F. Laurell, Opt. Lett. (26) (2001) 1882. [16] J.P. Feve, B. Boulanger, B. Menaert, O. Pacaud, Opt. Lett. (28) (2003) 1028.