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Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal
Optik 115, No. 7 (2004) 317–321 http://www.elsevier.de/ijleo
International Journal for Light and Electron Optics
Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal Zhancheng Guo, Zhaoqi Wang, Hongli Liu Institute of Modern Optics, Nankai University, the Key Laboratory of Opt-electronic Information Science and Technology, M. E. C., Tianjin 300071, P.R. China
Abstract: We have observed two-dimensional bright photorefractive spatial soliton for the first time in a Ce : KNSBN fiber-like crystal at wave-length of 514.5 nm with the intensity range between 2.83 103 W/cm2 and 4.71 103 W/cm2, then compared it with the one formed in a bulk sample. By comparison we have found that the intensity of forming a soliton in a fiber-like crystal is 7 9 times that in a bulk crystal due to stronger scattering light and complicated multiwave interaction. By comparison we have also found that the transverse dimensions of a soliton formed in a fiber-like crystal are about 4% smaller than the ones in a bulk crystal when the experimental geometry remains the same, then we have exploited the theory including the contribution of scattering field to interpret the experimental phenomenon. Key words: Photorefractive spatial soliton – bright photovoltaic photorefractive spatial soliton – fiber-like crystal
1. Introduction In 1992 Segev et al. theoretically predicted a new type of optical spatial soliton (SS) later named photorefractive spatial soliton [1]. The formation of this soliton results from photorefractive effect that at very low input power (of milliwatts or microwatts) the nonlinearity of photorefractive crystal can completely compensate for the diffraction of laser beam and so trap the beam. Shortly after that in 1993 Galen et al. observed photorefractive spatial soliton in their experiment for the first time [2]. Since these two fundamental work researchers have done much theoretical and experimental investigation in recent decades, which has greatly enriched the knowledge in this field. So far, three types of SS’s known are quasi-steady-state SS [3–5], screening SS [6–9], and photovoltaic (PV)SS [10–13]. Of these three types of solitons, the PV SS’s formation, distinguished from the other types of soli-
tons, doesn’t necessarily have the external electric field involved but requires the photorefractive crystal with strong enough PV effect. Taya et al. observed one dimensional dark PV SS in LiNO3 crystal [12], She et al. reported the observation of two dimentional bright PV SS in Cu:KNSBN [13]. However, almost all the investigation with respect to photorefractive SS has confined to bulk crystal. Fiber-like photorefractive crystals have many advantages compared to the bulk materials. They are easier to grow, they permit high energy densitities because of channeling radiation over much longer distances, and they can be effectively controlled by applying much lower transverse voltage [14, 15]. In this paper, to our knowledge for the first time, we have observed two dimentional bright PV SS in a fiber-like photorefractive crystal. And then we compared the difference of bright PV SSs between in fiber-like photorefractive crystal and in bulk photorefractive crystal theoretically and experimentally.
2. Theory According to reference [10], the Kukhtarev equations set describing photorefractive dynamics in steady state can be rewritten: 8 ð1Þ > ðSI þ bÞ ðNd Ndi Þ gnc Ndi ¼ 0 ; > > > > > Jc ¼ qmnc Esc þ SKeff IðNd Ndi Þ þ kB Tmrn > > ð2Þ > > ¼ const: ; > > > < q ð3Þ r Esc þ ðnc þ NA Ndi Þ ¼ 0 ; e s > > > > > @ i @2 ik > ð4Þ > Dnb Aðx; zÞ ; Aðx; zÞ ¼ > > 2 > @z 2k n @x b > > > : Dmb ¼ 12 n3b reff Esc ; ð5Þ
Received 22 April 2004; accepted 25 June 2004. Correspondence to: Z. Wang Fax: ++86-22-2350 8332 E-mail: [email protected]
where we assume the propagating direction of the beam is along with z coordinate, and the beam’s polarization is parallel to x coordinate. The directions of x 0030-4026/04/115/07-317 $ 30.00/0
318
Z. Guo et al., Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal
G A1
L1
Lfl2
C axis
R
F
L3
Ar+ laser y x
CCD
z
mkB TtR es k e and e2 ¼ 2 eff . Under the e keff d q mdtR Nd condition of open circuit and bright soliton, J is equal to zero and only ‘+’ out of ‘’ in eq. (9) is taken. Substituting the expression of “n” from eq. (6) into eq. (7), we get the expression below: where e1 ¼
A2 S
Fig. 2. The experimental setup for observation of the formation of bright PV SS.
E¼
ðF u2 Þ ; ð1 þ u2 Þ
ð10Þ
where and z are shown in fig. 2. The five dependent variables are nc, the electronic number density; Ndi , the number ! density of ionized donors; Jc , the current density; Esc , the space-charge field inside the crystal, of which the magnitude is Esc ; and A, the slowly varying amplitude of the optical field, defined by Eopt ðx; z; tÞ 2pnb , and w is ¼ Aðx; zÞ exp ðikz iw tÞ þ c:c: k ¼ l the frequency . Relevant parameters of the crystal are Nd , the total donor number density; NA , the number density of acceptors; b, the dark generation rate; S, the photoionization cross section; g, the recombination coefficient; m, the electron mobility; es , the low-frequency dielectric constant; nb , the background index of the crystal; reff , the effective electro-optic coefficient; and keff , the effective photovoltaic constant. q, the charge on the electron, kB is Boltzmann’s constant, and T is the absolute temperature. Finally the optical and the dark irradiances are defined as I ¼ jAj2 and b Id ¼ , respectively. Usually stationary soliton’s soluS pffiffiffiffi tion can be written as: Aðx; zÞ ¼ uðxÞ exp ðiGzÞ Id , where G is the soliton propagation constant. We denote: a¼
SId ; gNA
E¼
Esc ; Ep
n¼
J¼
x ; d
N
r¼
Nd ; NA
N¼
Ndi ; Nd
where the double prime stands for the derivative with respect to the variable x. For bright soliton one requires three boundary conditions: (i) uð1Þ ¼ u0 ð1Þ ¼ u00 ð1Þ ¼ 0 ; (ii) u0 ð0Þ ¼ 0 ; u00 ð0Þ < 0; uð0Þ after the initial integral of eq. (12), we can get u2 u02 ¼ ðF þ 1Þ ln ð1 þ u2 Þ 2 ln ð1 þ u20 Þ : ð13Þ u0 (iii)
Let i ¼ u2 , i0 ¼ u20 , then substitute them into eq. (13) and we can get: 1=2 di i ¼ 2 ðF þ 1Þ i ln ð1 þ iÞ ln ð1 þ i0 Þ : dx i0 ð14Þ
keeff gNA ; qm k 1 3 e n r Ep ; b¼ nb 2 b eff
Jc ; qmaNd Ep
d ¼ ð2kbÞ1=2 ; x¼
nc ; aNd
e1
dn ; ð11Þ 1 dx 1 r because of 0 < F < u20 ¼ uð0Þ2, F can be roughly treated as a constant. Substituting the eq. (10) into (9), we can get: G F u2 00 u ¼ þ u; ð12Þ b 1 þ u2 F ¼
Ep ¼
The numerical simulation result from the equation above is shown in fig. 1. From fig. 1, we can see that
1 1: r
Then we can have the dimensionless equations set corresponding to the eq. set (1)–(4) above: 8 1N > > ð1 þ u2 Þ ¼ 0 ; n > > Nr > > > > dn > > > ¼ const ; J ¼ nE þ ð1 NÞ u2 þ e1 > < dx 1 dE > > N an e2 ¼ 0; > > > r dx > > > > > d2 u G > > : 2 ¼ þ E uðxÞ ; b dx
ð6Þ ð7Þ ð8Þ ð9Þ
Fig. 1. Bright PV SS’s intensity distribution curve.
Z. Guo et al., Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal
the curve becomes narrower and narrower when the value of F rises from 0 to 1.2, that is, stronger scattering effect narrows the soliton.
3. Experiment and discussion We conduct our experiment with two different Ce : KNSBN samples measuring 1.4 mm 1.0 mm 8.0 mm and 5.0 mm 5.0 mm 5.5 mm. Rectangular coordinates system selected is shown in fig. 2, x coordinate parallel to the crystallographic axis direction, z coordinate orthogonal to x coordinate and along the beam’s propagating direction, and y coordinate orthogonally penetrating the paper inside. At room temperature the crystallographic parameters concerned are listed as: r13 ¼ 30 pm/V, r33 ¼ 200 pm/V, r42 ¼ 820 pm/V, no ¼ 2:35, ne ¼ 2:37, e11 ¼ 588, e33 ¼ 500. The experimental setup is shown in fig. 2. The beam from the argon ion laser at wave-length of 514.5 nm passes through a polarization rotator R, a GlanThompson prism G, an attenuator A1, two convex lenses L1 (focal length: 17.2 cm) and L2 (focal length: 10.3 cm), then focuses on a point F, three millimeters away from the crystallographic front face, finally orthogonally enters the crystal. G is used to make the beam’s polarization horizontal (e ray), R is used to adjust the intensity incident on the crystal,
as well as A1. The convex lens L3 is used to image various cross sections of the crystal onto the CCD camera. Attenuator A2 is used to adjust the intensity onto the CCD camera, and diaphragm S is used to block stray light. The magnification of the imaging system is determined by placing a thin film on the crystal exit face and imaging the film onto the CCD camera. Using the known size of the reference film, the magnification is determined to be about 8 and the positions of the exit and entrance faces are located. Various cross sections throughout the crystal are imaged by moving L3 and CCD camera synchronally with the same step length of 1 mm. P0 stands for the entrance face, Pk stands for the cross section inside the crystal along with the beam’s propagating direction, k mm away from the entrance face. P8 stands for the extit face in a fiber-like crystal, and P5 stands for the extit face in a bulk crystal. The outpower of Argon ion laser at wave-length of 514.5 nm is 250 mW. We record the spots images of various cross sections throughout the crystal along the beam’s propagating direction. Fig. 3a shows spots images when the crystal is removed with the intensity of 60 W/cm2; fig. 3b shows the ones in a fiber-like crystal with the intensity of 3.77 103 W/cm2, and fig. 3c shows the ones in a bulk crystal with the intensity of 361 W/cm2. Moreover the data of the spots’ dimensions (FWHM) are listed in table 1. We can see clearly that
P0
P0
P1
P2
P2
P4
P6
P3
P4
P8
Fig. 3. Beam spots images and x directional intensity distribution profiles of various cross sections. In a) the intensity I = 60 W/cm2 when the crystal is removed; In b) the intensity I = 3.77 103 W/cm2 in a fiber-like crystal; In c) the intensity I = 361 W/cm2 in a bulk crystal.
319
P5
a.
b.
c.
320
Z. Guo et al., Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal
Table. 1. Beam spots data corresponding to fig. 2. Unit: mm. Without crystal
P0
Fiber-like crystal
x
y
x
y
x
y
46
47
45
44
46
46
47
46
47
46
47
47
47
47
47
47
P1 P2
49
52
44
45
P3 P4
55
58
45
44
P5 P6
Bulk crystal
62
66
45
45
69
73
44
44
P7 P8
the dimensions of the spots in group a gradually become bigger, while the ones in group b and group c remain almost the same. According to fig. 3 along with table 1, we can see that the dimensions of the spot on the exit face of the crystal in group a are about 20 mm larger than the one on the entrance face, while in group b and in group c the variation of the spots’ dimensions between on the entrance face and on the exit face of the crystal are both less than 2 mm. Evidently bright PV SSs have both formed in group b and in group c. Furthermore, there is a noticeable phenomenon that generally the soliton’s dimensions in group b are about 4% smaller than the ones in group c. Through a further set of experiment and analyses we can draw the following conclusion: 1) In the fiber-like photorefractive crystal the intensity range to support a bright PV SS is between 2.83 103 W/cm2 and 4.71 103 W/cm2; 2) In the bulk crystal the intensity range to support a bright PV SS is between 3.78 102 W/cm2 and 6.39 102 W/cm2. 3) By comparison of 1) and 2), we can find that the intensity range for the soliton formation in the fiber-like crystal is about 7 9 times that in the bulk crystal and the soliton’s dimensions in the fiber-like crystal are found to be a little smaller than the ones in the bulk crystal.
4. Discussion In the crystal Ce:KNSBN, the bright PV SS is formed when the self focusing effect completely compensates for the beam’s diffraction and fanning effects. From eq. (2), we can get [16]: ESC ¼
1 ½ke ðNd Ndi Þ I þ kB Tm rnc qm nc eff
¼ EPV þ ED :
ð15Þ
1 keff ðNd Ndi Þ I, qmnc standing for the contribution of photovoltaic fields, and 1 kB Tm rnc , standing for the contribution ED ¼ qmnc of scattering fields. In former theoretical analysis for bulk photorefractive samples [16], ED can be neglected, but it is not the case for fiber-like crystal. In fiber-like crystal: rnc is larger due to smaller transverse dimensions. The incident light, the scattering light and their multireflected light form complicated multi-wave couple progress due to larger longitudinal dimension. So ED is larger and can not be neglected. From expression (15), the item EPV which results in forming a soliton is negative, while the item ED which counteracts the role of EPV is positive. Therefore, on one hand, the soliton formation requires larger intensity in fiber-like crystal and this qualitative analysis coincides with the conclusion drawn from the experiment above. On the other hand, the larger ED narrows the soliton as observed in the experiment above, which can be also explained through fig. 1 obtained by the numerical simulation of differential eq. (14). In fig. 1, it is evident that the soliton’s intensity profiles become narrower and narrower by the order of F ¼ 0 ( negligible scattering effect), F ¼ 0:6 (strong scattered effect), F ¼ 1:2 (stronger scattering effect). ESC consists of two parts: EPV ¼
5. Conclusion In conclusion we have observed two-dimensional bright PV photorefractive spatial soliton in a Ce : KNSBN fiber-like crystal and compared it with the one formed in a bulk sample for the first time. By comparison we can see the intensity of forming a soliton for fiber-like crystal is 7 9 times that for bulk crystal, which originates from stronger scattering field and complicated multi-wave interaction. The theoretical analysis including the contribution of scattering field explains the experimental phenomenon that the transverse dimensions of a soliton is smaller in a fiber-like crystal than the ones in a bulk crystal when the experimental geometry remains the same. Our research lays a foundation upon which people can probably make further investigation such as: adjust of the intensity of forming soliton, control of soliton’s transverse dimensions, etc. This Research is supported by the National Science Foundation of China (Grant No. 60177006).
References [1] Segev M, Crosignani B, Yariv A, et al.: Spatial solitons in phtorefractive media. Phys. Rev. Lett. 68 (1992) 923–926 [2] Duree GC, Shultz JL, Salamo G, Segev M, Yariv A, et al.: Observation of self-trapping of an optical beam due to the photorefrective effect. Phys. Rev. Lett. 71 (1993) 533 [3] Duree GC Jr., Shultz JL, Salamo GJ, Segev M, Yariv A, Crosignani B, Porto PD, Sharp EJ, Neurgaonker RR: Phys. Rev. Lett. 71 (1993) 533
Z. Guo et al., Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal [4] Segev M, Crosignani B, Yariv A, Fischer B: Phys. Rev. Lett. 68 (1992) 923 Maufoy J, Fressengeas N, Wolfersberger D, Kugel G: Phys. Rev. E 59 (1999) 6116 [5] Duree G, Morin M, Salama G, Segev M, Crosignani B, Porto PD, Sharp E, Yariv A: Phys. Rev. Lett. 74 (1995) 1978 [6] Csstillo MDI, Aguilar PAM, Sanchez Mondragon JJ, Stepanov S, Vysloukh V: Appl. Phys. Lett. 64 (1994) 408 [7] Segev M, Valley GC, Crosignani B, Porto PD, Yariv A: Phys. Rev. Lett. 73 (1994) 3211 [8] Segev M, Shih M, Valley GC: J. Opt. Soc. Am. B 12 (1996) 1628 [9] Kos K, Meng H, Salamo G, Shih M, Segev M, Valley GC: Phys. Rev. E 53 (1996) R4330
[10] [11] [12] [13] [14] [15] [16]
321
Ryf R, Wiki M, Monte Mezzani G, Guter P, Zozulya AA: Opt. Commun. 159 (1999) 339 Segev M, Valley GC, Bashaw MC, Taya M, Fejer MM: Photovoltaic spatial solitons. J. Opt. Soc. Am. B 14 (1997) 1772 Valley GC, Segev M, Crosignani B, Yariv A, Fejer MM, Bashaw MC: Phys. Rev. A 50 (1994) R4457 Taya M, Bashaw MC, Fejer MM, Segev M, Valley GC: Phys. Rev. A 52 (1995) 3095 She WL, Lee K, Lee WK: Phys. Rev. Lett. 78 (1999) 646 Kamshilin AA, Ravattinen R, et al.: Optics Commun. 103 (1993) 221 Raita E, Kamshilin AA, Jaaskelainen T: Opt. Lett. 21 (1996) Valley GC, Segev M, Crosignani B, Yariv A, Fejer MM, Bashaw MC: Phys. Rev. A 50 (1994) R4458
International Journal for Light and Electron Optics
Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal Zhancheng Guo, Zhaoqi Wang, Hongli Liu Institute of Modern Optics, Nankai University, the Key Laboratory of Opt-electronic Information Science and Technology, M. E. C., Tianjin 300071, P.R. China
Abstract: We have observed two-dimensional bright photorefractive spatial soliton for the first time in a Ce : KNSBN fiber-like crystal at wave-length of 514.5 nm with the intensity range between 2.83 103 W/cm2 and 4.71 103 W/cm2, then compared it with the one formed in a bulk sample. By comparison we have found that the intensity of forming a soliton in a fiber-like crystal is 7 9 times that in a bulk crystal due to stronger scattering light and complicated multiwave interaction. By comparison we have also found that the transverse dimensions of a soliton formed in a fiber-like crystal are about 4% smaller than the ones in a bulk crystal when the experimental geometry remains the same, then we have exploited the theory including the contribution of scattering field to interpret the experimental phenomenon. Key words: Photorefractive spatial soliton – bright photovoltaic photorefractive spatial soliton – fiber-like crystal
1. Introduction In 1992 Segev et al. theoretically predicted a new type of optical spatial soliton (SS) later named photorefractive spatial soliton [1]. The formation of this soliton results from photorefractive effect that at very low input power (of milliwatts or microwatts) the nonlinearity of photorefractive crystal can completely compensate for the diffraction of laser beam and so trap the beam. Shortly after that in 1993 Galen et al. observed photorefractive spatial soliton in their experiment for the first time [2]. Since these two fundamental work researchers have done much theoretical and experimental investigation in recent decades, which has greatly enriched the knowledge in this field. So far, three types of SS’s known are quasi-steady-state SS [3–5], screening SS [6–9], and photovoltaic (PV)SS [10–13]. Of these three types of solitons, the PV SS’s formation, distinguished from the other types of soli-
tons, doesn’t necessarily have the external electric field involved but requires the photorefractive crystal with strong enough PV effect. Taya et al. observed one dimensional dark PV SS in LiNO3 crystal [12], She et al. reported the observation of two dimentional bright PV SS in Cu:KNSBN [13]. However, almost all the investigation with respect to photorefractive SS has confined to bulk crystal. Fiber-like photorefractive crystals have many advantages compared to the bulk materials. They are easier to grow, they permit high energy densitities because of channeling radiation over much longer distances, and they can be effectively controlled by applying much lower transverse voltage [14, 15]. In this paper, to our knowledge for the first time, we have observed two dimentional bright PV SS in a fiber-like photorefractive crystal. And then we compared the difference of bright PV SSs between in fiber-like photorefractive crystal and in bulk photorefractive crystal theoretically and experimentally.
2. Theory According to reference [10], the Kukhtarev equations set describing photorefractive dynamics in steady state can be rewritten: 8 ð1Þ > ðSI þ bÞ ðNd Ndi Þ gnc Ndi ¼ 0 ; > > > > > Jc ¼ qmnc Esc þ SKeff IðNd Ndi Þ þ kB Tmrn > > ð2Þ > > ¼ const: ; > > > < q ð3Þ r Esc þ ðnc þ NA Ndi Þ ¼ 0 ; e s > > > > > @ i @2 ik > ð4Þ > Dnb Aðx; zÞ ; Aðx; zÞ ¼ > > 2 > @z 2k n @x b > > > : Dmb ¼ 12 n3b reff Esc ; ð5Þ
Received 22 April 2004; accepted 25 June 2004. Correspondence to: Z. Wang Fax: ++86-22-2350 8332 E-mail: [email protected]
where we assume the propagating direction of the beam is along with z coordinate, and the beam’s polarization is parallel to x coordinate. The directions of x 0030-4026/04/115/07-317 $ 30.00/0
318
Z. Guo et al., Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal
G A1
L1
Lfl2
C axis
R
F
L3
Ar+ laser y x
CCD
z
mkB TtR es k e and e2 ¼ 2 eff . Under the e keff d q mdtR Nd condition of open circuit and bright soliton, J is equal to zero and only ‘+’ out of ‘’ in eq. (9) is taken. Substituting the expression of “n” from eq. (6) into eq. (7), we get the expression below: where e1 ¼
A2 S
Fig. 2. The experimental setup for observation of the formation of bright PV SS.
E¼
ðF u2 Þ ; ð1 þ u2 Þ
ð10Þ
where and z are shown in fig. 2. The five dependent variables are nc, the electronic number density; Ndi , the number ! density of ionized donors; Jc , the current density; Esc , the space-charge field inside the crystal, of which the magnitude is Esc ; and A, the slowly varying amplitude of the optical field, defined by Eopt ðx; z; tÞ 2pnb , and w is ¼ Aðx; zÞ exp ðikz iw tÞ þ c:c: k ¼ l the frequency . Relevant parameters of the crystal are Nd , the total donor number density; NA , the number density of acceptors; b, the dark generation rate; S, the photoionization cross section; g, the recombination coefficient; m, the electron mobility; es , the low-frequency dielectric constant; nb , the background index of the crystal; reff , the effective electro-optic coefficient; and keff , the effective photovoltaic constant. q, the charge on the electron, kB is Boltzmann’s constant, and T is the absolute temperature. Finally the optical and the dark irradiances are defined as I ¼ jAj2 and b Id ¼ , respectively. Usually stationary soliton’s soluS pffiffiffiffi tion can be written as: Aðx; zÞ ¼ uðxÞ exp ðiGzÞ Id , where G is the soliton propagation constant. We denote: a¼
SId ; gNA
E¼
Esc ; Ep
n¼
J¼
x ; d
N
r¼
Nd ; NA
N¼
Ndi ; Nd
where the double prime stands for the derivative with respect to the variable x. For bright soliton one requires three boundary conditions: (i) uð1Þ ¼ u0 ð1Þ ¼ u00 ð1Þ ¼ 0 ; (ii) u0 ð0Þ ¼ 0 ; u00 ð0Þ < 0; uð0Þ after the initial integral of eq. (12), we can get u2 u02 ¼ ðF þ 1Þ ln ð1 þ u2 Þ 2 ln ð1 þ u20 Þ : ð13Þ u0 (iii)
Let i ¼ u2 , i0 ¼ u20 , then substitute them into eq. (13) and we can get: 1=2 di i ¼ 2 ðF þ 1Þ i ln ð1 þ iÞ ln ð1 þ i0 Þ : dx i0 ð14Þ
keeff gNA ; qm k 1 3 e n r Ep ; b¼ nb 2 b eff
Jc ; qmaNd Ep
d ¼ ð2kbÞ1=2 ; x¼
nc ; aNd
e1
dn ; ð11Þ 1 dx 1 r because of 0 < F < u20 ¼ uð0Þ2, F can be roughly treated as a constant. Substituting the eq. (10) into (9), we can get: G F u2 00 u ¼ þ u; ð12Þ b 1 þ u2 F ¼
Ep ¼
The numerical simulation result from the equation above is shown in fig. 1. From fig. 1, we can see that
1 1: r
Then we can have the dimensionless equations set corresponding to the eq. set (1)–(4) above: 8 1N > > ð1 þ u2 Þ ¼ 0 ; n > > Nr > > > > dn > > > ¼ const ; J ¼ nE þ ð1 NÞ u2 þ e1 > < dx 1 dE > > N an e2 ¼ 0; > > > r dx > > > > > d2 u G > > : 2 ¼ þ E uðxÞ ; b dx
ð6Þ ð7Þ ð8Þ ð9Þ
Fig. 1. Bright PV SS’s intensity distribution curve.
Z. Guo et al., Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal
the curve becomes narrower and narrower when the value of F rises from 0 to 1.2, that is, stronger scattering effect narrows the soliton.
3. Experiment and discussion We conduct our experiment with two different Ce : KNSBN samples measuring 1.4 mm 1.0 mm 8.0 mm and 5.0 mm 5.0 mm 5.5 mm. Rectangular coordinates system selected is shown in fig. 2, x coordinate parallel to the crystallographic axis direction, z coordinate orthogonal to x coordinate and along the beam’s propagating direction, and y coordinate orthogonally penetrating the paper inside. At room temperature the crystallographic parameters concerned are listed as: r13 ¼ 30 pm/V, r33 ¼ 200 pm/V, r42 ¼ 820 pm/V, no ¼ 2:35, ne ¼ 2:37, e11 ¼ 588, e33 ¼ 500. The experimental setup is shown in fig. 2. The beam from the argon ion laser at wave-length of 514.5 nm passes through a polarization rotator R, a GlanThompson prism G, an attenuator A1, two convex lenses L1 (focal length: 17.2 cm) and L2 (focal length: 10.3 cm), then focuses on a point F, three millimeters away from the crystallographic front face, finally orthogonally enters the crystal. G is used to make the beam’s polarization horizontal (e ray), R is used to adjust the intensity incident on the crystal,
as well as A1. The convex lens L3 is used to image various cross sections of the crystal onto the CCD camera. Attenuator A2 is used to adjust the intensity onto the CCD camera, and diaphragm S is used to block stray light. The magnification of the imaging system is determined by placing a thin film on the crystal exit face and imaging the film onto the CCD camera. Using the known size of the reference film, the magnification is determined to be about 8 and the positions of the exit and entrance faces are located. Various cross sections throughout the crystal are imaged by moving L3 and CCD camera synchronally with the same step length of 1 mm. P0 stands for the entrance face, Pk stands for the cross section inside the crystal along with the beam’s propagating direction, k mm away from the entrance face. P8 stands for the extit face in a fiber-like crystal, and P5 stands for the extit face in a bulk crystal. The outpower of Argon ion laser at wave-length of 514.5 nm is 250 mW. We record the spots images of various cross sections throughout the crystal along the beam’s propagating direction. Fig. 3a shows spots images when the crystal is removed with the intensity of 60 W/cm2; fig. 3b shows the ones in a fiber-like crystal with the intensity of 3.77 103 W/cm2, and fig. 3c shows the ones in a bulk crystal with the intensity of 361 W/cm2. Moreover the data of the spots’ dimensions (FWHM) are listed in table 1. We can see clearly that
P0
P0
P1
P2
P2
P4
P6
P3
P4
P8
Fig. 3. Beam spots images and x directional intensity distribution profiles of various cross sections. In a) the intensity I = 60 W/cm2 when the crystal is removed; In b) the intensity I = 3.77 103 W/cm2 in a fiber-like crystal; In c) the intensity I = 361 W/cm2 in a bulk crystal.
319
P5
a.
b.
c.
320
Z. Guo et al., Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal
Table. 1. Beam spots data corresponding to fig. 2. Unit: mm. Without crystal
P0
Fiber-like crystal
x
y
x
y
x
y
46
47
45
44
46
46
47
46
47
46
47
47
47
47
47
47
P1 P2
49
52
44
45
P3 P4
55
58
45
44
P5 P6
Bulk crystal
62
66
45
45
69
73
44
44
P7 P8
the dimensions of the spots in group a gradually become bigger, while the ones in group b and group c remain almost the same. According to fig. 3 along with table 1, we can see that the dimensions of the spot on the exit face of the crystal in group a are about 20 mm larger than the one on the entrance face, while in group b and in group c the variation of the spots’ dimensions between on the entrance face and on the exit face of the crystal are both less than 2 mm. Evidently bright PV SSs have both formed in group b and in group c. Furthermore, there is a noticeable phenomenon that generally the soliton’s dimensions in group b are about 4% smaller than the ones in group c. Through a further set of experiment and analyses we can draw the following conclusion: 1) In the fiber-like photorefractive crystal the intensity range to support a bright PV SS is between 2.83 103 W/cm2 and 4.71 103 W/cm2; 2) In the bulk crystal the intensity range to support a bright PV SS is between 3.78 102 W/cm2 and 6.39 102 W/cm2. 3) By comparison of 1) and 2), we can find that the intensity range for the soliton formation in the fiber-like crystal is about 7 9 times that in the bulk crystal and the soliton’s dimensions in the fiber-like crystal are found to be a little smaller than the ones in the bulk crystal.
4. Discussion In the crystal Ce:KNSBN, the bright PV SS is formed when the self focusing effect completely compensates for the beam’s diffraction and fanning effects. From eq. (2), we can get [16]: ESC ¼
1 ½ke ðNd Ndi Þ I þ kB Tm rnc qm nc eff
¼ EPV þ ED :
ð15Þ
1 keff ðNd Ndi Þ I, qmnc standing for the contribution of photovoltaic fields, and 1 kB Tm rnc , standing for the contribution ED ¼ qmnc of scattering fields. In former theoretical analysis for bulk photorefractive samples [16], ED can be neglected, but it is not the case for fiber-like crystal. In fiber-like crystal: rnc is larger due to smaller transverse dimensions. The incident light, the scattering light and their multireflected light form complicated multi-wave couple progress due to larger longitudinal dimension. So ED is larger and can not be neglected. From expression (15), the item EPV which results in forming a soliton is negative, while the item ED which counteracts the role of EPV is positive. Therefore, on one hand, the soliton formation requires larger intensity in fiber-like crystal and this qualitative analysis coincides with the conclusion drawn from the experiment above. On the other hand, the larger ED narrows the soliton as observed in the experiment above, which can be also explained through fig. 1 obtained by the numerical simulation of differential eq. (14). In fig. 1, it is evident that the soliton’s intensity profiles become narrower and narrower by the order of F ¼ 0 ( negligible scattering effect), F ¼ 0:6 (strong scattered effect), F ¼ 1:2 (stronger scattering effect). ESC consists of two parts: EPV ¼
5. Conclusion In conclusion we have observed two-dimensional bright PV photorefractive spatial soliton in a Ce : KNSBN fiber-like crystal and compared it with the one formed in a bulk sample for the first time. By comparison we can see the intensity of forming a soliton for fiber-like crystal is 7 9 times that for bulk crystal, which originates from stronger scattering field and complicated multi-wave interaction. The theoretical analysis including the contribution of scattering field explains the experimental phenomenon that the transverse dimensions of a soliton is smaller in a fiber-like crystal than the ones in a bulk crystal when the experimental geometry remains the same. Our research lays a foundation upon which people can probably make further investigation such as: adjust of the intensity of forming soliton, control of soliton’s transverse dimensions, etc. This Research is supported by the National Science Foundation of China (Grant No. 60177006).
References [1] Segev M, Crosignani B, Yariv A, et al.: Spatial solitons in phtorefractive media. Phys. Rev. Lett. 68 (1992) 923–926 [2] Duree GC, Shultz JL, Salamo G, Segev M, Yariv A, et al.: Observation of self-trapping of an optical beam due to the photorefrective effect. Phys. Rev. Lett. 71 (1993) 533 [3] Duree GC Jr., Shultz JL, Salamo GJ, Segev M, Yariv A, Crosignani B, Porto PD, Sharp EJ, Neurgaonker RR: Phys. Rev. Lett. 71 (1993) 533
Z. Guo et al., Observation of the bright photovoltaic spatial soliton in fiber-like photorefractive crystal [4] Segev M, Crosignani B, Yariv A, Fischer B: Phys. Rev. Lett. 68 (1992) 923 Maufoy J, Fressengeas N, Wolfersberger D, Kugel G: Phys. Rev. E 59 (1999) 6116 [5] Duree G, Morin M, Salama G, Segev M, Crosignani B, Porto PD, Sharp E, Yariv A: Phys. Rev. Lett. 74 (1995) 1978 [6] Csstillo MDI, Aguilar PAM, Sanchez Mondragon JJ, Stepanov S, Vysloukh V: Appl. Phys. Lett. 64 (1994) 408 [7] Segev M, Valley GC, Crosignani B, Porto PD, Yariv A: Phys. Rev. Lett. 73 (1994) 3211 [8] Segev M, Shih M, Valley GC: J. Opt. Soc. Am. B 12 (1996) 1628 [9] Kos K, Meng H, Salamo G, Shih M, Segev M, Valley GC: Phys. Rev. E 53 (1996) R4330
[10] [11] [12] [13] [14] [15] [16]
321
Ryf R, Wiki M, Monte Mezzani G, Guter P, Zozulya AA: Opt. Commun. 159 (1999) 339 Segev M, Valley GC, Bashaw MC, Taya M, Fejer MM: Photovoltaic spatial solitons. J. Opt. Soc. Am. B 14 (1997) 1772 Valley GC, Segev M, Crosignani B, Yariv A, Fejer MM, Bashaw MC: Phys. Rev. A 50 (1994) R4457 Taya M, Bashaw MC, Fejer MM, Segev M, Valley GC: Phys. Rev. A 52 (1995) 3095 She WL, Lee K, Lee WK: Phys. Rev. Lett. 78 (1999) 646 Kamshilin AA, Ravattinen R, et al.: Optics Commun. 103 (1993) 221 Raita E, Kamshilin AA, Jaaskelainen T: Opt. Lett. 21 (1996) Valley GC, Segev M, Crosignani B, Yariv A, Fejer MM, Bashaw MC: Phys. Rev. A 50 (1994) R4458