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OPTICS LETTERS / Vol. 32, No. 1 / January 1, 2007
Parametric excitation of X-waves by downconversion of Bessel beams in nonlinear crystals S. Orlov, A. Stabinis, V. Smilgevicius, G. Valiulis, and A. Piskarskas Department of Quantum Electronics, Faculty of Physics, Vilnius University, Saule˙tekio 9, Building 3, 10222 Vilnius, Lithuania Received July 6, 2006; revised October 3, 2006; accepted October 6, 2006; posted October 10, 2006 (Doc. ID 72725); published December 13, 2006 We predict that in traveling-wave degenerate parametric downconversion the Bessel beam pump stimulates the appearance of a nondiffracting X-wave from quantum noise amplification. Numerical simulation results of downconversion in ADP crystal are presented, along with preliminary experimental data. © 2006 Optical Society of America OCIS codes: 050.0050, 190.4410, 190.4970.
The diffraction-free and dispersion-free propagation of pulsed beams (usually called X-waves) can be achieved in linear1,2 as well as in nonlinear media. In a quadratic medium, X-waves can be formed spontaneously through conical emission via mismatched second-harmonic generation.3–8 It was shown that angular dispersion of the waves excited in a traveling-wave parametric amplifier by a quasimonochromatic plane pump wave is quite similar to the angular dispersion of X-waves, especially at degenerate parametric interaction.9,10 In this case an excitation of X-waves is feasible if the components of its spatial–temporal spectrum are phased.10 In what follows, we show that the phasing becomes possible under downconversion of a quasi-monochromatic Bessel beam pump in nonlinear crystal. First, we discuss the appearance of an azimuthal correlation in an incoherent conical beam under parametric amplification by a Bessel beam. The conical beam is a superposition of plane waves with wave vectors that lie upon the surface of a cone. In this case the amplitude A共x , y兲 of a light field is given by 1 A共x,y兲 =
2
冕
2
S共⌿兲exp关− i共x cos ⌿ + y sin ⌿兲兴d⌿,
0
共1兲 where x and y are transverse coordinates, ⌿ is an azimuthal angle, and  is the transverse wave vector of the constituent plane waves. When amplitudes S共⌿兲 are uncorrelated, the conical beam is incoherent. In the polar system of coordinates x = r cos , y = r sin , Eq. (1) can be written as A共r, 兲 =
1 2
冕
2
S共⌿兲exp关− ir cos共⌿ − 兲兴d⌿. 共2兲
0
By use of the expansion exp关−ir cos共⌿ − 兲兴 ⬁ 共−i兲nJn共r兲exp关−in共⌿ − 兲兴, we obtain = 兺n=−⬁ ⬁
A共r, 兲 =
共− i兲nanJn共r兲exp共− in兲, 兺 n=−⬁
共3兲
where an = 1 / 2兰02S共⌿兲exp共in⌿兲d⌿ is the amplitude 0146-9592/07/010068-3/$15.00
of the nth azimuthal harmonic. So an incoherent signal with a ringlike angular spectrum can be represented as a superposition of Bessel modes with random amplitudes. Further, we discuss the phasematched parametric amplification of an incoherent conical beam with amplitude A0共r , 兲 by a J0 pump beam. In the case of an undepleted pump beam, for large parametric gain we have A1共r, 兲 ⬇ A0 exp关G兩J0共pr兲兩兴/2,
共4兲
⬁ where A0 = 兺n=−⬁ 共−i兲nan0Jn共r兲exp共−in兲, G is a parametric gain factor, and p is a transverse wave vector of the Bessel beam. Large parametric gain, G Ⰷ 1, is possible for pr Ⰶ 1. In this case J0共pr兲 ⬇ 1 − p2r2 / 4 and
A1共r, 兲 ⬇
exp共G兲 2
⬁
共− i兲nan0fn共r兲exp共− in兲, 兺 n=−⬁
共5兲
where fn共r兲 = Jn共r兲exp共−r2 / d2兲 and d = 2 / 共p冑G兲. The aperture function exp共−r2 / d2兲 is strictly determined by the Bessel beam central peak. As a result, the parametric amplification of various Bessel modes is different. For d 艋 2, only an amplification of J0 beam is effective. This means that for incoherent conical beams with transverse wave vectors  艋 2 / d the appearance of an azimuthal correlation is feasible at large parametric gain, G Ⰷ 1. We note that the selection of Bessel modes under parametric amplification is also possible by use of a narrow Gaussian pump beam if the diffraction of this beam can be neglected. An appearance of J0 Bessellike beams from the quantum noise in the parametric amplifier with ring-shaped gain profile pumped by narrow Gaussian beam was demonstrated by Di Trapani et al.11 The probability of a J0-like beam increased with the decrease of the pump beam diameter. The radial nonhomogeneity of parametric gain caused by the Bessel beam central peak can also stimulate the appearance of radial coherence of the optical field excited from the quantum noise level. The case of a narrow Gaussian pump beam was demonstrated in Ref. 12. We predict that the coherence © 2006 Optical Society of America
January 1, 2007 / Vol. 32, No. 1 / OPTICS LETTERS
A1 z
=−
1 A1 u1 t
+i
g 1 2A 1 2 t2
−i
⌬⬜ 2k1
69
A1 + iApJ0共pr兲A1* , 共6兲
Fig. 1. Intensity distribution of the spatial–temporal spectrum of the signal wave in the ADP crystal at different propagation distances z (in cm): (a) 1, (b) 2, (c) 4. (d) Contour plot of the intensity distribution of the spatial– temporal spectrum of the signal wave at the output plane of the ADP crystal.
Fig. 2. Phase distribution of the spatial–temporal spectrum of the signal wave in the ADP crystal at different propagation distances z (in cm): (a) 1, (b) 2, (c) 4. (d) Contour plot of the phase distribution of the spatial–temporal spectrum of the signal wave at the output plane of the ADP crystal. The phase step between two adjacent lines is / 4.
arising in the light field parametrically amplified from the noise level by the Bessel beam stimulates the phasing of temporal spectral components because of the strong angular dispersion that is typical for parametric processes. In this case an excitation of X-waves becomes feasible. Further, we present results of numerical simulation of degenerate parametric downconversion in an ADP crystal for type I temperature-tuned noncritical phase matching. All results refer to pump and central signal wavelengths p = 266 nm and 1 = 532 nm. The calculations were made by use of the refractive index data on ADP crystal presented in Ref. 13. For simplicity, we assume that all optical fields are axially symmetric. Then, an evolution of optical field from the noise level can be described by the differential equation
where t is the time, z is the longitudinal coordinate, and ⌬⬜ = / 共rr兲 + 2 / r2. Here k1, u1, and g1 are the wave vector, group velocity, and group velocity dispersion coefficient of the signal wave at degeneracy 1 = p / 2, and is the coefficient of nonlinear coupling of the signal and pump waves. Ap is the amplitude of the Bessel pump beam, and p = kp sin p is the transverse wave vector of the pump wave, where p is a half-cone angle. We include in our consideration the normalized variables = t / 0, = r / r0, = z / l and the normalized amplitude of signal wave B = A1 / a0, where l is the crystal length. It was assumed that 0 = 100 ps and r0 = 120 m. Then, Eq. (6) can be written as
B
= − D1
B
+ iD2
2B 2
− iD3⌬B + iGJ0共0兲B* ,
共7兲
where ⌬ = 1 / 共兲 + 2 / 2, D1 = l / 0u1, D2 = g1l / 202, D3 = l / 2k1r02, G = Apl, and 0 ⬇ kppr0. The following values of the parameters of Eq. (6) were used: u1 = 1.92⫻ 108 m / s, g1 = 7.8⫻ 10−26 s2 / m, k1 = 2n1 / 1 = 1.8⫻ 107 m−1, and kp = 2np / p = 3.6⫻ 107 m−1, where n1 = np = 1.526, p = 35 mrad, p = 1.26 ⫻ 106 m−1, l = 4 cm. As a result, for the parameters of Eq. (7) we have D1 ⬇ 2.08, D2 ⬇ 1.55⫻ 10−9, D3 ⬇ 0.075, and 0 = 153. For the parametric gain factor it was assumed that G = 8. Equation (7) was solved numerically for 0 艋 , , 艋 1 with initial random amplitude B共 , 兲 and the following boundary conditions: B共0 , , 兲 = B共1 , , 兲 = B共 , 1 , 兲 = B / 共 , 0 , 兲 = 0. After finding signal amplitude B共 , , 兲, we obtained the spatial–temporal spectrum of signal wave S共 , , 兲 for a given propagation distance. The simulation results for three propagation distances inside the ADP crystal are shown in Figs. 1 and 2. Two pairs of straight lines intersecting at points A and B situated on the pump cone [Fig. 1(c)] correspond to the phase-matching curve of degenerate parametric interaction. The angular dispersion of the obtained spectrum can also be calculated by use of 1 = ± p ± 冑g1 / k112共 / 1 − 1兲,10 and the numerical results are in good agreement with the theoretical predictions. The evolution of the spectral phase is
Fig. 3. Intensity profile of the signal beam at the output plane of the ADP crystal 共z = 4 cm兲 for two values of pump beam half-cone angle p (in mrad): (a) 35, (b) 70. The contour plots correspond to the half-intensity level.
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OPTICS LETTERS / Vol. 32, No. 1 / January 1, 2007
Fig. 4. Experimental setup.
Fig. 5. Angular wavelength distribution of the output radiation excited in the traveling-wave parametric amplifier under downconversion of the Bessel beam pump in the 4 cm long ADP crystal.
presented in Fig. 2. The spatial–temporal profile of the signal wave at the output of the ADP crystal is shown in Fig. 3. The structure of the signal beam clearly resembles the X-shape known from classical studies of nondiffractive and nondispersive optical fields. The localization of the excited light field increases with an increase of J0 pump beam cone angle (Fig. 3). The beam width of the central peak of pump J0 beam (in m兲 is 3.2 [Fig. 3(a)] and 1.6 [Fig. 3(b)]. The experimental setup for measurement of the angular wavelength distribution of the output radiation excited in the traveling-wave parametric amplifier from the noise level under downconversion of quasi-monochromatic Bessel beam pump is shown in Fig. 4. The radiation of the Nd:YAG amplifier was frequency doubled in a 3 cm long type I KDP crystal. The fourth harmonic 共266 nm兲 was produced in a 5 mm long type I BBO crystal, and 2 mJ pulses 共35 ps兲 in Gaussian mode were obtained. The Bessel beam was formed by a quartz axicon with an apex angle of 174°. The parametric downconversion of the Bessel beam pump was performed in 4 cm long type I ADP crystal, which was placed in an oven to achieve noncritical phase matching. The energy conversion efficiency was ⬍5%. The pump radiation after the parametric amplifier was cut off by the glass plate. The spectrograph entrance slit was placed in the focal plane of the lens. The CCD camera was placed at the output plane of the spectrograph, and the angular frequency distributions of the output radiation at different temperatures of the ADP crystal were obtained. An angular wavelength distribution measured under degenerate parametric downconversion
at a crystal temperature of about 55° C is presented in Fig. 5. Good qualitative agreement with theoretical predictions is obtained. We note that the conditions of the experiment are somehow different from the conditions of numerical simulation, and for this reason discrepancies are seen in the intensity distribution in Fig. 5 in comparison with theoretical predictions [Fig. 1(c)]. In conclusion, it is shown that a noise burst (duration 100 ps, transverse size 120 m) under parametric amplification by a quasi-monochromatic Bessel beam is transformed into a nondiffracting X-like pulsed beam with a pulse duration and a beam width of three and one order less, respectively, than those of an initial random signal. This is the result of azimuthal and radial correlations that arise in a random signal as a result of the action of the Bessel beam central peak, as well as the appearance of angular dispersion caused by the phase-matching condition. Time-domain measurements of the signal beam at the output plane of parametric amplifier are in progress. We acknowledge technical assistance from S. Ališauskas, R. Butkus, and L. Dabkevičius; fruitful discussions with A. Dubietis; and partial financial support from EC FW-6 project LASERLAB-Europe and the Lithuanian Science and Studies Foundation. A. Stabinis’s e-mail address is
[email protected]. References 1. K. Reivelt and P. Saari, Phys. Rev. E 65, 046622 (2002). 2. E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez, IEEE J. Sel. Top. Quantum Electron. 9, 59 (2003). 3. S. Trillo, C. Conti, P. Di Trapani, O. Jedrkiewicz, G. Valiulis, and G. Bellanca, Opt. Lett. 27, 1451 (2002). 4. C. Conti and S. Trillo, Opt. Lett. 28, 1251 (2003). 5. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003). 6. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and D. Trillo, Phys. Rev. Lett. 91, 093905 (2003). 7. O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, Phys. Rev. E 68, 026610 (2003). 8. J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijosius, A. Varanavicius, G. Valiulis, R. Danielius, E. Kucinskas, and A. Piskarskas, Phys. Rev. E 69, 026607 (2004). 9. S. Orlov, A. Piskarskas, and A. Stabinis, Opt. Lett. 27, 2103 (2002). 10. R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevicius, and A. Stabinis, Opt. Commun. 244, 411 (2005). 11. P. Di Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, Phys. Rev. Lett. 81, 5133 (1998). 12. P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, Phys. Rev. Lett. 80, 265 (1998). 13. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook Of Nonlinear Optical Crystals (Springer, 1991).
