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Experiments and numerical simulations on beam bending and transverse beam encoding by spatial self- and cross-phase modulation in ruby
Optics Communications 103 (1993) 285-296 North-HoUand
OPTICS COMMUNICATIONS
Full length article
Experiments and numerical simulations on beam bending and transverse beam encoding by spatial self- and cross-phase modulation in ruby J.P. B e r n a r d i n , A . S . L . G o m e s 1, J.L. C o h e n a n d N . M . L a w a n d y 2 Division of Engineering, Brown University, Providence, RI 02912, USA Received 12 April 1993; revised manuscript received 10 June 1993
Experiments and numerical simulations are performed on the nonlinear beam evolution of simultaneously propagating cw argon-ion and HeNe laser beams through a ruby rod. The HeNe probe beam exhibits a deflection of nearly 12 mrad when a 1.5 W, 514.5 nm laser beam is injected into the ruby rod at an angle to the probe. The copropagating geometry is also studied where the azimuthally symmetric transverse beam encoding leads to complex ring structures in both the HeNe and argon-ion laser beams. The numerical simulations are compared with the experimental data to obtain the cross-phase modulation term. Using previously reported values for the saturable nonlinear index of ruby and a nonlinear index for cross-phase modulation no= 1.35Re (n 2), the nonlinear wave equation in two transverse dimensions is numerically solved to predict the bending angles and emerging beam profiles. These findings are discussed in light of previous experiments in ruby as well as the possibility of using these effects for producing nanosecond optical switching systems.
1. Introduction Transverse effects in n o n l i n e a r b e a m p r o p a g a t i o n have been studied in a wide range o f n o n l i n e a r optical materials such as a t o m i c vapors, liquid crystals, thermal materials, semiconductors, a n d photorefractive materials [ 1 ]. Recently, e x p e r i m e n t a l a n d theoretical work has been p e r f o r m e d on transverse optical propagation effects in Cr3+-doped crystals [2,3 ]. Theoretical work has also been done on b e a m propagation in ruby subject to a thin sample approxim a t i o n [4]. In this regime, diffraction effects such as self-focusing or defocusing do not take place during the nonlinear encoding. The p r o b l e m is therefore reduced to a n u m e r i c a l calculation o f a diffraction integral. Analytic work in this a p p r o x i m a t i o n has been done in various regimes o f encoding by Le Berre et al. and is referred to as diffraction-free encoding [ 5 ]. To be within the e x p e r i m e n t a l limit o f this app r o x i m a t i o n requires that the crystal length does not 1 Permanent address: Departamento de Fisica, Universidade Federal de Pernambuco, 50739 Recife, Brazil. 2 Also Department of Physics.
exceed the Rayleigh length or self-focusing length. F o r cw laser sources in the 1.0 W power range nonlinear b e a m encoding must take place b e y o n d these length scales in order to see significant transverse b e a m evolution a n d m a x i m u m b e a m bending. Adequate theoretical t r e a t m e n t o f b e a m b e n d i n g and encoding in ruby requires a full c o m p u t a t i o n o f the nonlinear wave equation. The study o f nonlinear p r o p a g a t i o n in laser materials is o f critical i m p o r t a n c e because the complex index of refraction in a l o a d e d laser cavity is strongly coupled to the laser field by way o f propagation. Inclusion o f transverse effects in lasers has been quite simple thus far, due to the e n o r m o u s complexity already i m p o s e d by the time and longitudinal space variables. Nonetheless, nonlinear phase shifts o f over 2n can be achieved at the center o f the b e a m propagating through a few centimeters o f ruby at low powers. F u r t h e r m o r e , a rich variety o f d y n a m i c s has been seen in ruby lasers that could originate in nonlinear p r o p a g a t i o n effects [ 6 ]. The nonlinear index o f ruby has been measured using m a n y techniques including two-wave mixing [ 7 - 9 ], nonlinear interferometry [ 10,11 ], and four-
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
285
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wave mixing [ 12-15 ]. Although a consensus for the approximate value of the nonlinear index has been reached (n2 ~ 10- 8 cm2/W ), the microscopic theory for the nonlinearity is not complete. The n2 of ruby (and other C r 3+-doped crystals) has been found to saturate at an intensity which clearly suggests that the value of n2 is proportional to the polarizability difference between the ground and long lived metastable 2E state. The saturating behavior has been shown by various groups with concurring saturation intensities and n2 values [7,15 ]. The nonlinear index of ruby, however, has not been measured at intensities considerably higher than the saturation intensity. It is possible that at very high intensities, levels in between the 2E state and charge transfer state can become partially occupied. This would change the polarizability since these levels are closer to the charge transfer state. Recent theories have been proposed for changes in the imaginary part of n2 at high laser intensities based on photon-assisted energy transfer [ 16 ]. Optical self-bending is an effect that takes place if an asymmetric laser beam propagates through a quadratic index medium. The laser beam thus propagates through its own gradient index prism. This effect was first proposed by Kaplan [17] and experimentally observed by Brodin and Kamuz [ 18]. Maximum bending angles to date have exceeded 2 mrad [ 19 ] with a cw beam. Possible applications include optical switching and interconnecting, resonatorless optical bistability [ 20 ], and power limiting [21]. Using a strong pump beam, it is possible to encode phase information onto a weak probe beam. Spatial pattern formation by transverse cross-phase modulation has been observed in ruby [2,3 ] and stable guiding of weak probe beam was induced by spatial solitons in CS2 [22]. When asymmetric encoding takes place, it is possible to bend or steer one beam with another. Recent work in this area includes a report of beam steering by way of a nonlinear prism created in mesotetraphenyl-porphine by a strong beam propagating perpendicular to the weak beam [23]. In that study beam deflection of about 0.1 mrad was seen. More recently Golub demonstrated a 3.0 mrad beam deflection experiment in CS2 [24] using a pulsed laser. Instead of inducing a prism transverse to the probe beam propagation, the intense asymmetric beam at 1.06 gm was aligned with 286
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a weak probe at 532 nm and copropagated through a C S 2 cell. The weak beam at 532 nm was steered along with the self-deflecting 1.06 gm beam.
2. Theoretical model
The general mechanism for the nonlinear index of ruby is known to be caused by the polarizability difference between the ground state and long lived metastable 2E state. An energy level diagram depicting the relevant states is shown in fig. 1. To obtain an estimate for the nonlinear index we briefly outline the argument made by Catunda [4], and assume that all fields are perpendicular to the c-axis, thus avoiding birefringent effects. The population of the 2E state as a function of pump intensity is given by
I/I~
( 1)
N(2E) =No 1 +Ilia'
where No is the C r 3+ concentration, Is= h v/ar, and the relaxation time from the 4T state is assumed to be short compared with r, the relaxation time from the 2E state. Based on a variety of experimental data, it is well established that Is= 1500 W / c m 2 for the 514.5 nm argon-ion line. Using the Clausius-Mosoti equation we can obtain the index of refraction as a function of pump power: x 10 4 cm 1
Charge transfer state 4 TI
4F
2 11
I,, 41- 2
21!
Fig. 1. Energylevel diagram depicting the relevant states of ruby from ref. [25].
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3 n2-1 1 [N(2E) z(2E) +N(4A2) ~((4A2) ] , 4n n 2 + 2 - No
(2) where Z is the linear susceptibility at the wavelength of interest. Assuming that n = no + An and An << no, we can express eq. (2) in the form
n2I n=no+ 1 + I / ~
(3)
where 2~ (noZ+2~ 2 n2= ~ \ no / [ z ( 2 E ) - z ( a A 2 ) ] "
(4)
An estimate of the value of n2 can be made from the excited state spectrum of ruby found in ref. [25]. Assuming that Z (2E) >> x(4A2 ), one can estimate n2 by assuming a two-level model for z(2E) where the domain contribution to the real part ofz(2E) stems from the charge transfer state at aproximately 40 000 cm-~ above the 2E state. The temperature dependence as well as spectral dependence of n2 has been studied and it was shown that the polarizability at the 2E state is in fact dominated by the charge transfer state [9]. From a two-level model, we can estimate the upper limit of n2. The susceptibility of an unsaturated two-level system is Re(z)-
nZaNo A n~aNo k I+A 2~ kA
(5)
Using the following values from ref. [25 ], g ~ 10-~7 cm 2, A~ 10 (normalizing detuning), and No~ 1019 cm-% we obtain n2,~ 5 × 10 -7 cm2/W. As previously stated many groups have experimentally measured n2 and it is generally accepted that n2~ 1.5× 10 -a cm2/W for a Cr 3÷ concentration of 1019 c m - L The imaginary part of n2 is dominated by the transition from the 2E state to one of the 2T~ states below the charge transfer band. Regardless of the exact state, the imaginary part of n2 can be determined by the excited state spectra found in ref. [25 ] and the following equation: Im(n2) = a ( 2 E ) -- °L(4A2) 2knoIs '
(6)
where a is the linear absorption. At 514.5 nm, the
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absorption can be bleached by as much as 30%. In two-beam coupling experiments, the Im(n2) gives rise to an observed asymmetry in the gain curve. McMichael et al. determined the relative values of Im (n2) for various wavelengths in their two-beam coupling experiments and the results are in reasonable agreement with the I m (n2) values predicted by the excited state spectra of ruby [7]. Since the I m ( n 2 ) / R e ( n 2 ) is less than 10% in ruby, the nonlinear absorption only gives rise to slight asymmetries in gain curves for two-beam coupling. This Im(n2) effect however, has very important ramifications in beam propagation experiments, as we will show. An important quantity that has not been modeled or experimentally measured is the cross-phase modulation term. For a weak HeNe probe beam the index of refraction has the form r/el n=no + 1 +I/-----~'
(7)
where I is the argon-ion laser intensity and nc is the cross phase modulation term given by 2zc (no2 + 2 ~ 2 n c = ~ - s \ no / [ x ( 2 E ) - • ( 4 A 2 ) ] '
(8)
where the susceptibilities and no are at the 633 nm probe wavelength. Although nc has not been experimentally determined, it should be similar in value to n2, assuming that the dominant contribution comes from the charge transfer state. Two-beam nonlinear propagation experiments have shown that the onset of spatial ring formation of both the argon-ion and HeNe laser beams occur at approximately the same argon-ion pump intensities [2,3]. This gives strong support to the assumption that the polarizability in the metastable state is dominated by the influence of the charge transfer state. For this study we begin by assuming that nc is equal to n2, and then adjust it to fit the experimental data. To model the propagation of this two-beam system, we begin with the nonlinear wave equation:
VZE + k2n 2E + k2n2no cE IEI 2E= 0 ,
(9)
where E = E ( x , y , z) exp(io~t) and nE<
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r/2----r/2(let2). A spatial phase term is assumed and the variables are normalized in a typical manner: Ej( X, y, z) =Es wj( ax', ay',
2kl
noa2z ')
X exp(i2k2noa2z ') ,
- a (~Jl)2~Z2 +i ~kj- ~a~llj 7z +v2vj ~9=0,
(11)
where 6=22/16rr2a 2, V~ is the transverse laplacian, flj=a:a2k:no, ql =2nok~Isnza 2, q:=2nok21snca 2. In the paraxial approximation the first term ofeq. ( 11 ) would be dropped. However, we account for the dominant contribution from that term in the following way. By defining H: as ]~l 12 Hy=V~ + (ifly + t/j 1 ~--i~,(] 5]
(12)
the paraxial approximation gives
kja
H : ~ - i k l Oz'
(13)
and therefore the next term in the binomial expansion of the propagator is obtained in a recursive manner,
i kJ O~uj -~ O~z + (Hj+6H]) ~,j~O.
(14)
Since the beams are never less than 40 gm in diameter (as we will show later), the dominant contribution to H~ is the diagonal term. In some experimental scenarios, 6t-I]/Hy can be close to 1% due to the large n2 of ruby. The final equation then reads:
i kj a~,j
[
288
(
3. Numerical analysis
We numerically solve eq. ( 15 ) in both cylindrical and cartesian coordinates. For the radially symmetric transverse beam encoding problem, where various symmetric ring patterns form, cylindrical coordinates are used. The following second order finite difference operator, D r, is used for the transverse Laplacian in cylindrical symmetry: 1
Dr~k=
rk Ar 2
(rk+l/2 ~+l--2r~k+rk_l/2~ ~-l)
.
(16) The numerical technique used to solve the parabolic system of equations is the well-known Crank-Nicholson method. It has been shown that this unconditionally stable technique is actually slightly faster than operator splitting methods based on the FFT [26]. Additionally, in cylindrical coordinates, the FFT would have to be replaced by an appropriate algorithm for computing fast Fourier-Bessel transforms. The nonlinearity was first treated by iteration, but we found little advantage in speed over using the field values of the previous step for the nonlinear matrix elements. Modeling the asymmetric beam bending problem requires two independent transverse dimensions. Standard second order difference operators (D x, Dy) were used and the alternating direction implicit method was employed to solve the problem. Making the following definition:
[
' ~/'/i [ 2
(
[ ~'l [2 "~2l
V(I#j)= iflj+~/)l+lWll2+6 qJl+l~ul~]_]
(17)
the splitting method is expressed as {1 + ½iaz[DX+ ½V(¢")]} ¢:+1/:
I~l 12
~O~-z + v2~9+ _i[3j+tb 1 + I~, 12 +6
After solving eq. ( 15 ) in the crystal region, the farfield propagation patterns outside the crystal is obtained by using a Kirchhoff-Huygens diffraction integral.
(10)
where E~ is the field saturation amplitude, a is the beam radius and kl is the argon-ion laser wavenumber. The primed coordinate system is dropped for convenience, and the subscripts j = 1, 2 refer to the argon-ion and HeNe laser fields, respectively. In these normalized variables, the equation is
+ iflg+tbl+[~,[2
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= { 1 - ½iAz[IY+ ½V(~:) ]} ~u",
(18)
{ 1 + ½iAz[DY+ ½V(q/") 1} q : + ' . ~/j=0.
(15)
= {1
-
½iAz[DX+ ½V(~")1} ~/]n+1/2
(19)
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All computations modeled a two-beam system propagating through a 7.0 cm ruby rod with an initial beam radius of 40 ~tm and 57 ~tm for the argon-ion and HeNe laser beam, respectively. For the radially symmetric computations the spatial domain extended over 12 beam radii (0.48 ram), with a mesh spacing Ar=0.0117 (0.47 ~tm), and Az= 1.96× 10 -3 (0.1 mm). In the asymmetric problems the domain extended 24 beam radii (0.96 mm), Ax, Ay=0.086 (3.4 ~tm), and Az= 1.96× l0 -3 (0.1 mm). Dirichlet conditions were used for the boundaries, and the domain was increased until changes in the generated light patterns were less than 10-3. The step sizes and mesh spacings were tested to ensure that the total computational error also remained less than 10 -3. The classic results of Boshier and Sandle were repeated as a test of the numerical procedure [27 ]. In the beam bending simulations, the light fields are propagated at different angles through the ruby crystal. If both equations were treated in the same coordinate system, one of the equations would become hyperbolic and the numerical procedure would have to be significantly modified to account for the convective nature of the problem. To avoid these difficulties, the light fields are solved in independent coordinate systems and the nonlinear coupling is imposed on the HeNe beam by a space transformation at each longitudinal mesh point. The far-field propagation outside the crystal was performed by a simple Riemann sum of the discretized diffraction integral.
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camera, place at a distance 20 cm to 30 cm away from the exit face of the rod coupled to a Spiricon LBA100A laser beam profiler. Figure 2 shows a schematic of the experimental set-up. For the initial alignment the beams are made to coincide at BS and M2, with fine adjustments made to ensure symmetric ring patterns result in the encoded HeNe beam. Since calculations proved to be quite sensitive to the initial beam radii we carefully measured the spot sizes of the beams at the focus using variable slits on an x - y translation stage. The beam radii were computed to be 40 ~tm for the 514.5 nm beam and 57 ~tm for the 633 nm probe beam. For the beam bending experiments the mirror M1 is placed on a translation state. By adjusting the distance between the parallel HeNe and argon-ion laser beams, the input angle of argon-ion beam could be adjusted without effecting the initial overlap at the ruby rod face. The lens has a focal length of 10 cm, giving a 1.0 mrad angle for every 0.1 mm displacement of the translation stage. Using the camera mounted on a translation stage at the focus of the light beam, we verified that the initial overlap was maintained while the relative angle between the two beams coincided with the angle predicted by ray optics. Figure 3 displays the technique used to set-up the initial conditions for the beam bending experiments.
T1
4. Experimental set-up All experiments were performed using a Coherent Innova argon-ion laser in single line operation at the 514.5 nm wavelength. The maximum available power at this line is approximately 1.6 W. The ruby rod (R) is placed on a translation stage for initial spot size and phase front curvature adjustments. The ruby rod has Cr 3+ density of about 1.1 X 1019 c m -3 and is antireflection coated on one end. The c-axis of the ruby rod is at 60 degrees to the rod axis. All experiments were performed with the c-axis perpendicular to the argon-ion light field to eliminate birefringent effects and afford the maximum n2. The beam profile emerging from the crystal is targeted at a silicon
Argon-ion
He-Ne
% Fig. 2. Experimental set-up. M 1 and M2 represent high reflecting
mirrors, BS is a Fresnel beam sampler, T1 and T2 are translation stages. The lens has a focal length of 10 cm and the ruby rod is 7.0 cm long. The beam profiles are targeted at a silicon camera (C), whichis fed into the laser beam profiler (LBP). 289
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L Rub'), Rod
~ [ ~ 0
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Argon-ion d / 2 He-Ne
~
= d/f
Fig. 3. Beam bending set-up. The mirror M 1 in fig. 2 is placed on a translation stage to adjust the distance between the parallel HeNe and argon-ion laser beams. The input angl: of the argon-ion laser beam can be adjusted without effecting the initial overlap of the two beams at the crystal face.
5. Experimental and numerical results
(a) ~.,
Recent results on two-bean[ p r o p a g a t i o n experiments in ruby have revealed interesting pattern form a t i o n including multiple ring structures and on axis nulls [ 2,3 ]. Experiments revealed only small changes in the b e a m width when a 1.0 W, 514.5 nm, argonion laser b e a m passed through a 1 cm ruby crystal. However, u p o n passing through a longer crystal, a rich variety o f p a t t e r n formations can occur in b o t h a strong b e a m a n d simultaneously propagating weak probe. This length r e q u i r e m e m makes it necessary to model the propagation by solving the nonlinear wave equation. The curves in fig. 4 show the calculated b e a m waist for b e a m s o f different initial intensity u p o n traversal through the ruby crystal. The incident b e a m is 40 ~tm in radius. At a p p r o x i m a t e l y 200 m W a significant ring structure begins to develop. At nearly 400 m W , the on-axis i~tensity begins to d r o p to a m i n i m u m . Figure 5a shows the far-field profile o f the argon-ion b e a m incident u p o n the crystal with
2
~
6
distance in ruby rod (cm) Fig. 4. The argon-ion beam is shown to undergo significant changes in its waist size upon propagation through the crystal, (a) 1.0 mW; (b) 100 mW; (c) 500 roW. 290
1o o
>-,
05
.4
O0 -8
4
4
o
x axis (mrad)
(b)
j
.6
-8.0
i -4.0
i 0.0
4.0
8.0
x axis (mrad) Fig. 5. The far field profile of the argon-ion laser beam incident upon the crystal with power of 500 mW (a), and the corresponding numerical calculation with Re(n2) = 1.6 × 10-8 cm2/W (b). power o f 500 mW. The corresponding numerical calculation with t/z-- 1.6( 1 - 0.06i) × 10 -8 c m Z / W a n d o q = 1.3 cm -~ is shown in fig. 5b. Increasing the power to 1.1 W produces a far-field profile shown in fig. 6a and the numerical calculation in fig. 6b. Calculations predict that new ring structures and patterns develop b e y o n d 1.1 W. Changing the value of n2 by 10% drastically changes the far field patterns to the point where there is little or no resemblance to the experimental data. However, using n2 values that are nearly multiples o f the
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(a) ~
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There has been no experimental measurement of the cross-phase m o d u l a t i o n in term in eq. (7). Past experiments on two-beam nonlinear propagation indicate that the spatial evolution of both argon-ion and HeNe beams occur at the same argon-ion p u m p intensities [2,3]. In this study, we found that n¢= 1.35Re(nz) best fit the experimental data. Figure 7a shows the experimental result for the far-field
10
4 ~0.5&)
00 -8
-4
0
4
x axis (mrad)
(a)
(b)
1.0
Zo ~2
~
0.5
00 8 -6.0
-4.0
0.0
4.0
4
8.0
x axis (mrad) Fig. 6. The far field profile of the argon-ion laser beam incident upon the crystal with power of 1.1 W (a), and the corresponding numerical calculation (b). measured value yield similar pattern formations for at least one initial i n p u t power, i.e. one profile at a given power may match the experimental data, but not a series of profiles at different powers. We conclude that using an F/2= 1 . 6 ( 1 - - 0 . 0 6 i ) X 10 -8 c m 2 / W gives a good fit to the experimental data. Previous theoretical work on n o n l i n e a r pattern formation in thin G d A 1 0 3 : C r +3 samples indicates distinctly different pattern formation for input beam having either negative or positive phase front curvature [4]. In our experiments it was found that there was an initial phase front curvature dependence on the resulting beam profile, but strong pattern form a t i o n was also seen when the ruby rod face was at the center of the focal region. For a thin sample significant ring formation is not predicted in this region. However, since the beam is significantly encoded during self-focusing in our experiment, it does not seem likely that any clear distinction can be made about the initial phase front curvature dependence on the resulting profile. For this reason, all computations and experiments are carried out with the ruby rod face positioned at the center of the focal region.
0
4
x axis (mrad)
(b) ~2 7,
-8.0
-4.0
0.0
4.0
8.0
x axis (mrad)
(c)
.:2
-8.0
-4.0
0.0
4.0
8.0
x axis (mrad) Fig. 7. (a) The experimental result for the far field HeNe laser beam profile subject to the nonlinear phase encoding of a copropagating argon-ion laser beam. The initial 514.5 nm radiation power is 1.1 W upon entering the ruby crystal. (b) The computational result for these initial conditions with no=Re(n2). (c) The computational results with no= 1.35Re(n2). In the calculations for the nonlinear beam steering we use a value n2 = 1.35Re(nz) for the cross phase modulation term. 291
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HeNe beam profile subject to the nonlinear phase encoding of a copropagating argon-ion beam. The initial 514.5 nm radiation power is 1.1 W upon entering the ruby crystal. Figure 7b shows the computational result for these imtial conditions with no = Re (n2) and fig. 7c shows the computational results with nc= 1.35Re(n2). In the calculations for the nonlinear beam steering, we use a value of 1.35Re(n2) for the cross phase modulation term. Experiments were performed Io determine the imaginary part of the cross phase modulation term; however, no change in the absorption of the HeNe laser beam could be detected in the presence of the 514.5 nm pump beam. This null result is supported by the excited state spectrum reported in ref. [25], which suggests that the difference in the absorption coefficient at 633 nm between the ground and 2E state is very small. Therefore, the 633 nm beam undergoes only linear absorption in the ruby crystal, with O/2=0.1 cm -x. Earlier studies on this two-beam system showed that the ring patterns of the HeNe and argon-ion beams evolved with increasing argon-ion intensity in distinctly different manners. In light of the computations, we can attribute this interesting feature to the difference in the beam sizes, the difference in the phase modulation terms n2 and no, and the fact that nc is assumed to be purely real. Experiments in transverse self-phase modulation in ruby at rhodamine 6G wavelength indicated a similar pattern formation compared with the HeNe probe experiments. At these wavelength Im(n2) is small but slightly positive (limiting action) [7], while at 514.5 nm Im(n2) is small but negative [ 15 ]. The beam bending experiments were performed by injecting the argon-ion and HeNe beams into the ruby crystal at relative angles with respect to each other. In all experiments, the beams initially overlapped at the face of the crystal, and the cw argonion beam was measured to be 1.5 W. Figure 8a shows the experimental beam pattern for the symmetric coproprapagating situation where the relative angle between the beams is zero. Figure 8b is the numerical calculation of the beam pattern obtained by solving eqs. (18) and ( 19 ) at every mesh point, followed by numerical integration of the diffraction integral at the end of the 7.0 cm crystal. Adjusting the translation stage under mirror M 1 in fig. 2 causes the ar292
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-5.0
5.0
~5.0
x axis (mrad)
-5.0
5.0
~.5.(~
x axis (mrad) rig. 8. (a) The experimental beam pattern for the symmetric copropagating situation where the relative angle between the beams is zero and the initial argon-ion power is 1.5 W. (b) The numerical calculation of the beam pattern.
gon-ion beam to propagate at an angle to the HeNe beam. Figure 9a demonstrates the bending of the HeNe beam when the argon-ion is injected into the crystal at a relative angle of 3 mrad. Figure 9b shows the concomitant numerical calculation for this initial condition. Adjusting the translation stage to produce an input angle of 6 mrad produces the pattern shown in fig. fig. 10a, and the numerical calculation in fig. 10b. When the argon-ion injection angle reaches approximately 7.0 mrad, the steering angle
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5.0
15.~
x axis (mrad)
-5.0
5.0
x axis (mrad)
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I -
.0
5,0
15.0
x axis (mrad)
-5.0
5.0
15.0
Fig. 9. (a) The deflected HeNe laser beam when the argon-ion is injected into the crystal at a relative angle of 3 mrad. (b) The concomitant numerical calculation for this initial condition.
x axis (mrad) Fig. 10. Adjusting the translation stage to produce an input angle of 6 mrad produces the pattern (a) and the numerical calculation (b).
begins to saturate, until at a p p r o x i m a t e l y 9.0 mrad, the m a j o r i t y o f the power in the H e N e b e a m returns to the center o f the field. Figure 11 shows experimental and numerical plots o f the angle o f the peak H e N e intensity as a function o f the the argon-ion injection angle. The calculations were also p e r f o r m e d in one cartesian transverse d i m e n s i o n for c o m p a r i son. Keeping the argon-ion injection angle fixed at 6.0 m r a d and adjusting the power from zero to 1.5 W, the deflected H e N e b e a m was seen to m o n o t o n ically bend from its centered position to nearly 10 mrad. Figure 12 shows a plot o f the deflected peak
as a function o f p u m p power for this experimental scenario. Again we include a calculation with one cartesian transverse dimension for comparison. It was found that the calculations did not predict the correct a m p l i t u d e for the deflected spot for this power d e p e n d e n c e study. Figure 13a shows the experimentally observed beam profile obtained with a argonion p u m p power o f 0.9 W and an 6.0 m r a d injection angle and fig. 13b is the calculated b e a m profile. Although the steering angle is found to be approximately the same for the experimental and numerical b e a m profiles, the a m p l i t u d e s are shown to be different. 293
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15 November 1993
2O
E ¢)
~D
g 0 2
4
6
8
10
pump angle (mrad) Fig. 11. Experimental and numerical curves of the angle of the peak HeNe intensity as a function of the the argon-ion injection angle. The calculations were also performed in one cartesian transverse dimension for comparison. Experiment, black diamonds; one transverse dimensional calculation, hollow squares; two transverse dimensional calculation, black squares.
os
lo
g
p
-5.0
~
5.0
,
1.~.~3
x axis (mrad)
~5
pump power (watts) Fig. 12. Keeping the argon-ion injection angle fixed at 6.0 mrad and adjusting the power from zero to 1.5 W, the deflected HeNe beam is seen to monotonically bend from its centered position to nearly 10 mrad. The calculations were also performed in one cartesian transverse dimension for comparison. Experiment, black diamonds; one transverse dimensional calculation, black squares; two transverse dimensional calculation, hollow squares. F o r a p p l i c a t i o n s i n v o l v i n g optical switching, it is i m p o r t a n t to k n o w the speed at w h i c h the steering can be p e r f o r m e d and the f r a c t i o n o f the p u m p and signal fields w h i c h exit the ruby crystal. Since the 514.5 n m light is strongly a b s o r b e d ( f c q ( I ) d L ~ 8 ), the p u m p b e a m c a n n o t be effectively used after exiting the crystal. T h e a b s o r p t i o n o f the signal b e a m , h o w e v e r , is v e r y slight ( c ~ 2 L ~ 0 . 7 ) , and a p p r o x i m a t e l y 50% o f the signal b e a m exits the crystal. T h e switch-on t i m e o f the system is g o v e r n e d by the p u m p a b s o r p t i o n rate, and the s w i t c h - o f f t i m e is g o v e r n e d by the lifetime o f the long lived m e t a s t a b l e state ( r ~ 3 m s ) . T h e p r o b l e m i m p o s e d by this long s p o n t a n e o u s 294
-5.0
5.0
t533
x a x i s (inrad) Fig. 13. (a) The experimentally observed beam profile obtained with an argon-ion pump power of 0.9 W and a 6.0 mrad injection angle; (b) the calculated beam profile. Although the steering anNe is found to be approximately the same for the experimental and numerical beam profiles, the amplitudes are shown to be different. lifetime could be c i r c u m v e n t e d by using a pulsed laser at 693 n m to stimulate transitions to the ground state. T h i s c o u l d lead to a 50% m o d u l a t i o n on the nonlinearity, which is certainly significant in light o f this study. T h e pulse energy r e q u i r e d to m o d u l a t e the n o n l i n e a r i t y by a p p r o x i m a t e l y 50% is AtP~ 0.5Ah v~ or, w h e r e AtP is the pulse energy, A is the area, and is the cross section o f the ruby line. T h i s predicts
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a pulse energy o f 400 ~tJ, which is easily o b t a i n e d in Q-switched n a n o s e c o n d dye laser systems. The same analysis can be m a d e for the switch-on time; however, the cross section for the absorption o f 514.5 n m r a d i a t i o n is a p p r o x i m a t e l y tenfold that o f the ruby wavelength. This predicts that a pulse energy o f 30 pJ could m o d u l a t e the index o f refraction by nearly 50% o f the m a x i m u m (An ~ 0.5Isn2 ). The rate o f this process is limited by the relaxation time from the 4T 2 state to the 2E state. Although there has not been any recent m e a s u r e m e n t o f this lifetime using m o d e r n picosecond p u m p - p r o b e techniques, there appears to be some consensus that the lifetime is in the few nanoseconds regime [28,29 ].
6. Conclusions We p e r f o r m e d experiments a n d numerical simulations on the nonlinear b e a m evolution o f simultaneously propagating cw argon-ion a n d H e N e laser b e a m s through a 7.0 cm ruby rod. A H e N e probe b e a m is deflected by nearly 12 m r a d by a 1.5 W, 514.5 n m laser b e a m injected into the ruby rod at an angle to the probe. Previous nonlinear b e n d i n g experiments have p r o d u c e d m a x i m u m b e n d i n g angles o f nearly 3 m r a d in pulsed o p e r a t i o n in CS2. We predict that switching times in the few nanosecond time scale would be possible using a Q-switched dye laser tuned to the ruby line. The copropagating geometry is also studied where the s y m m e t r i c transverse b e a m encoding leads to complex ring structures in both the H e N e and argon-ion beams. Interesting differences in the p a t t e r n f o r m a t i o n s o f the H o N e and argon-ion laser b e a m s are found to have their origins in slight differences in p a r a m e t e r values such at the crossphase m o d u l a t i o n term a n d the different b e a m widths. The n u m e r i c a l simulations are c o m p a r e d with the e x p e r i m e n t a l d a t a to obtain the cross-phase m o d u l a t i o n term. Using previously reported values for the saturable n o n l i n e a r index o f ruby and a crossphase m o d u l a t i o n t e r m nc = 1.3 5Re (/,12), the nonlinear wave equation in two transverse d i m e n s i o n s is numerically solved to predict the bending angles a n d emerging b e a m profiles.
15 November 1993
Acknowledgements The authors wish to thank Dr. J. Abshire o f NASA G o d d a r d and the Air Force Office o f Scientific Research for supporting this research. We also thank George Loriot for p r o v i d i n g us with u n l i m i t e d use o f an IBM RS6000 c o m p u t e r a n d his invaluable system support. A.S.L. G o m e s thanks C N P q for financial support. J.P. Bernardin would like to thank Sari Gilm a n for p r o o f reading the manuscript.
References [1] For recent examples, see the special issue entitled: Transverse effects in nonlinear-optical systems, J. Opt. Soc. Am. B June 1991, and July 1991, and references therein. [2] R.S. Afzal and N.M. Lawandy, Optics Lett. 14 (1989) 794. [3] R.S. Afzal and N.M. Lawandy, Optics Comm. 86 (1991) 307. [4IT. Catunda and L.A. Cuff, J. Opt. Soc. Am. B 7 (1990) 1445. [ 5 ] M. Le Berre, E. Ressayre, A. Tallet, K. Tai, H.M. Gibbs, M.C. Rushford and N. Peyghambarian, J. Opt. Soc. Am. B 1 (1984) 591. [6] R.S. Afzal, W.P. Lin and N.M. Lawandy, J. Opt. Soc. Am. B6 (1989) 2348. [7] I. McMichael, P. Yeh and P. Beckwith, Optics Lett. 13 (1988) 500. [ 8 ] S.A. Boothroyd, J. Chrostowski and M.S. O'Sullivan, J. Opt. Soc. Am. B 6 (1989) 766. [9] C.L. Adler and N.M. Lawandy, Optics Comm. 81 (1991) 33. [ 10] T.N.C. Venkatesan and S.L. McCall, Appl. Phys. Lett. 30 (1977) 282. [ 11 ] T. Catunda, J.P. Andreeta and J.C. Castro, Appl. Optics 25 (1986) 2391. [ 12] P.F. Liao and D.M. Bloom, Optics Lett. 3 (1978) 4. [13]D.S. Hamilton, D. Heiman, Jack Feinberg and R.W. Hellwarth, Optics Lett. 4 (1979) 124. [ 14] S.C. Weaver and S.A. Payne, Phys. Rev. B 40 (1989) 10727. [ 15 ] T. Catunda, A.M. Cansian and J.C. Castro, J. Opt. Soc. Am. B8 (1991) 820. [ 16] E.A. Gouveia, I. Guedes, J.C. Castro and S.C. Zilio, Phys. Rev. B 46 (1992) 14387. [ 17] A.E. Kaplan, JETP Lett. 9 (1969) 33. [ 18] M.S. Brodin and A.M. Kamuz, JETP Lett. 9 (1969) 351. [ 19 ] G.A. Swartzlander Jr., H. Yin and A.E. Kaplan, Optics Lett. 13 (1988) 1011. [20] A.E. Kaplan, Optics Lett. 6 ( 1981 ) 360. [21 ] J.A. Hermann, Optics Comm. 62 (1987) 367. [22 ] R. de la Fuente and A. Barthelemy, J. Quantum Electron. QE-28 (1992) 547. [23] M. Miguel Cervantes, Optics Comm. 90 (1992) 144. [24] I. Golub, Optics Comm. 94 (1992) 143. 295
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[25] T. Kushida, J. Quantum Electr:m. QE-2 (1966) 524. [26] Y. Chung and N. Dagli, J. Quantum Electron. QE-26 ( 1990 ) 1335. [27] M.G. Boshier and W.J. Saddle, Optics Comm. 42 (1982) 371.
296
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[ 28 ] S.A. Pollack, J. Appl. Phys. 38 (1967) 5083. [29] M. Anson and R.C. Smith, J. Quantum Electron. QE-6 (1970) 268.
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Full length article
Experiments and numerical simulations on beam bending and transverse beam encoding by spatial self- and cross-phase modulation in ruby J.P. B e r n a r d i n , A . S . L . G o m e s 1, J.L. C o h e n a n d N . M . L a w a n d y 2 Division of Engineering, Brown University, Providence, RI 02912, USA Received 12 April 1993; revised manuscript received 10 June 1993
Experiments and numerical simulations are performed on the nonlinear beam evolution of simultaneously propagating cw argon-ion and HeNe laser beams through a ruby rod. The HeNe probe beam exhibits a deflection of nearly 12 mrad when a 1.5 W, 514.5 nm laser beam is injected into the ruby rod at an angle to the probe. The copropagating geometry is also studied where the azimuthally symmetric transverse beam encoding leads to complex ring structures in both the HeNe and argon-ion laser beams. The numerical simulations are compared with the experimental data to obtain the cross-phase modulation term. Using previously reported values for the saturable nonlinear index of ruby and a nonlinear index for cross-phase modulation no= 1.35Re (n 2), the nonlinear wave equation in two transverse dimensions is numerically solved to predict the bending angles and emerging beam profiles. These findings are discussed in light of previous experiments in ruby as well as the possibility of using these effects for producing nanosecond optical switching systems.
1. Introduction Transverse effects in n o n l i n e a r b e a m p r o p a g a t i o n have been studied in a wide range o f n o n l i n e a r optical materials such as a t o m i c vapors, liquid crystals, thermal materials, semiconductors, a n d photorefractive materials [ 1 ]. Recently, e x p e r i m e n t a l a n d theoretical work has been p e r f o r m e d on transverse optical propagation effects in Cr3+-doped crystals [2,3 ]. Theoretical work has also been done on b e a m propagation in ruby subject to a thin sample approxim a t i o n [4]. In this regime, diffraction effects such as self-focusing or defocusing do not take place during the nonlinear encoding. The p r o b l e m is therefore reduced to a n u m e r i c a l calculation o f a diffraction integral. Analytic work in this a p p r o x i m a t i o n has been done in various regimes o f encoding by Le Berre et al. and is referred to as diffraction-free encoding [ 5 ]. To be within the e x p e r i m e n t a l limit o f this app r o x i m a t i o n requires that the crystal length does not 1 Permanent address: Departamento de Fisica, Universidade Federal de Pernambuco, 50739 Recife, Brazil. 2 Also Department of Physics.
exceed the Rayleigh length or self-focusing length. F o r cw laser sources in the 1.0 W power range nonlinear b e a m encoding must take place b e y o n d these length scales in order to see significant transverse b e a m evolution a n d m a x i m u m b e a m bending. Adequate theoretical t r e a t m e n t o f b e a m b e n d i n g and encoding in ruby requires a full c o m p u t a t i o n o f the nonlinear wave equation. The study o f nonlinear p r o p a g a t i o n in laser materials is o f critical i m p o r t a n c e because the complex index of refraction in a l o a d e d laser cavity is strongly coupled to the laser field by way o f propagation. Inclusion o f transverse effects in lasers has been quite simple thus far, due to the e n o r m o u s complexity already i m p o s e d by the time and longitudinal space variables. Nonetheless, nonlinear phase shifts o f over 2n can be achieved at the center o f the b e a m propagating through a few centimeters o f ruby at low powers. F u r t h e r m o r e , a rich variety o f d y n a m i c s has been seen in ruby lasers that could originate in nonlinear p r o p a g a t i o n effects [ 6 ]. The nonlinear index o f ruby has been measured using m a n y techniques including two-wave mixing [ 7 - 9 ], nonlinear interferometry [ 10,11 ], and four-
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
285
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wave mixing [ 12-15 ]. Although a consensus for the approximate value of the nonlinear index has been reached (n2 ~ 10- 8 cm2/W ), the microscopic theory for the nonlinearity is not complete. The n2 of ruby (and other C r 3+-doped crystals) has been found to saturate at an intensity which clearly suggests that the value of n2 is proportional to the polarizability difference between the ground and long lived metastable 2E state. The saturating behavior has been shown by various groups with concurring saturation intensities and n2 values [7,15 ]. The nonlinear index of ruby, however, has not been measured at intensities considerably higher than the saturation intensity. It is possible that at very high intensities, levels in between the 2E state and charge transfer state can become partially occupied. This would change the polarizability since these levels are closer to the charge transfer state. Recent theories have been proposed for changes in the imaginary part of n2 at high laser intensities based on photon-assisted energy transfer [ 16 ]. Optical self-bending is an effect that takes place if an asymmetric laser beam propagates through a quadratic index medium. The laser beam thus propagates through its own gradient index prism. This effect was first proposed by Kaplan [17] and experimentally observed by Brodin and Kamuz [ 18]. Maximum bending angles to date have exceeded 2 mrad [ 19 ] with a cw beam. Possible applications include optical switching and interconnecting, resonatorless optical bistability [ 20 ], and power limiting [21]. Using a strong pump beam, it is possible to encode phase information onto a weak probe beam. Spatial pattern formation by transverse cross-phase modulation has been observed in ruby [2,3 ] and stable guiding of weak probe beam was induced by spatial solitons in CS2 [22]. When asymmetric encoding takes place, it is possible to bend or steer one beam with another. Recent work in this area includes a report of beam steering by way of a nonlinear prism created in mesotetraphenyl-porphine by a strong beam propagating perpendicular to the weak beam [23]. In that study beam deflection of about 0.1 mrad was seen. More recently Golub demonstrated a 3.0 mrad beam deflection experiment in CS2 [24] using a pulsed laser. Instead of inducing a prism transverse to the probe beam propagation, the intense asymmetric beam at 1.06 gm was aligned with 286
15 November 1993
a weak probe at 532 nm and copropagated through a C S 2 cell. The weak beam at 532 nm was steered along with the self-deflecting 1.06 gm beam.
2. Theoretical model
The general mechanism for the nonlinear index of ruby is known to be caused by the polarizability difference between the ground state and long lived metastable 2E state. An energy level diagram depicting the relevant states is shown in fig. 1. To obtain an estimate for the nonlinear index we briefly outline the argument made by Catunda [4], and assume that all fields are perpendicular to the c-axis, thus avoiding birefringent effects. The population of the 2E state as a function of pump intensity is given by
I/I~
( 1)
N(2E) =No 1 +Ilia'
where No is the C r 3+ concentration, Is= h v/ar, and the relaxation time from the 4T state is assumed to be short compared with r, the relaxation time from the 2E state. Based on a variety of experimental data, it is well established that Is= 1500 W / c m 2 for the 514.5 nm argon-ion line. Using the Clausius-Mosoti equation we can obtain the index of refraction as a function of pump power: x 10 4 cm 1
Charge transfer state 4 TI
4F
2 11
I,, 41- 2
21!
Fig. 1. Energylevel diagram depicting the relevant states of ruby from ref. [25].
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3 n2-1 1 [N(2E) z(2E) +N(4A2) ~((4A2) ] , 4n n 2 + 2 - No
(2) where Z is the linear susceptibility at the wavelength of interest. Assuming that n = no + An and An << no, we can express eq. (2) in the form
n2I n=no+ 1 + I / ~
(3)
where 2~ (noZ+2~ 2 n2= ~ \ no / [ z ( 2 E ) - z ( a A 2 ) ] "
(4)
An estimate of the value of n2 can be made from the excited state spectrum of ruby found in ref. [25]. Assuming that Z (2E) >> x(4A2 ), one can estimate n2 by assuming a two-level model for z(2E) where the domain contribution to the real part ofz(2E) stems from the charge transfer state at aproximately 40 000 cm-~ above the 2E state. The temperature dependence as well as spectral dependence of n2 has been studied and it was shown that the polarizability at the 2E state is in fact dominated by the charge transfer state [9]. From a two-level model, we can estimate the upper limit of n2. The susceptibility of an unsaturated two-level system is Re(z)-
nZaNo A n~aNo k I+A 2~ kA
(5)
Using the following values from ref. [25 ], g ~ 10-~7 cm 2, A~ 10 (normalizing detuning), and No~ 1019 cm-% we obtain n2,~ 5 × 10 -7 cm2/W. As previously stated many groups have experimentally measured n2 and it is generally accepted that n2~ 1.5× 10 -a cm2/W for a Cr 3÷ concentration of 1019 c m - L The imaginary part of n2 is dominated by the transition from the 2E state to one of the 2T~ states below the charge transfer band. Regardless of the exact state, the imaginary part of n2 can be determined by the excited state spectra found in ref. [25 ] and the following equation: Im(n2) = a ( 2 E ) -- °L(4A2) 2knoIs '
(6)
where a is the linear absorption. At 514.5 nm, the
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absorption can be bleached by as much as 30%. In two-beam coupling experiments, the Im(n2) gives rise to an observed asymmetry in the gain curve. McMichael et al. determined the relative values of Im (n2) for various wavelengths in their two-beam coupling experiments and the results are in reasonable agreement with the I m (n2) values predicted by the excited state spectra of ruby [7]. Since the I m ( n 2 ) / R e ( n 2 ) is less than 10% in ruby, the nonlinear absorption only gives rise to slight asymmetries in gain curves for two-beam coupling. This Im(n2) effect however, has very important ramifications in beam propagation experiments, as we will show. An important quantity that has not been modeled or experimentally measured is the cross-phase modulation term. For a weak HeNe probe beam the index of refraction has the form r/el n=no + 1 +I/-----~'
(7)
where I is the argon-ion laser intensity and nc is the cross phase modulation term given by 2zc (no2 + 2 ~ 2 n c = ~ - s \ no / [ x ( 2 E ) - • ( 4 A 2 ) ] '
(8)
where the susceptibilities and no are at the 633 nm probe wavelength. Although nc has not been experimentally determined, it should be similar in value to n2, assuming that the dominant contribution comes from the charge transfer state. Two-beam nonlinear propagation experiments have shown that the onset of spatial ring formation of both the argon-ion and HeNe laser beams occur at approximately the same argon-ion pump intensities [2,3]. This gives strong support to the assumption that the polarizability in the metastable state is dominated by the influence of the charge transfer state. For this study we begin by assuming that nc is equal to n2, and then adjust it to fit the experimental data. To model the propagation of this two-beam system, we begin with the nonlinear wave equation:
VZE + k2n 2E + k2n2no cE IEI 2E= 0 ,
(9)
where E = E ( x , y , z) exp(io~t) and nE<
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r/2----r/2(let2). A spatial phase term is assumed and the variables are normalized in a typical manner: Ej( X, y, z) =Es wj( ax', ay',
2kl
noa2z ')
X exp(i2k2noa2z ') ,
- a (~Jl)2~Z2 +i ~kj- ~a~llj 7z +v2vj ~9=0,
(11)
where 6=22/16rr2a 2, V~ is the transverse laplacian, flj=a:a2k:no, ql =2nok~Isnza 2, q:=2nok21snca 2. In the paraxial approximation the first term ofeq. ( 11 ) would be dropped. However, we account for the dominant contribution from that term in the following way. By defining H: as ]~l 12 Hy=V~ + (ifly + t/j 1 ~--i~,(] 5]
(12)
the paraxial approximation gives
kja
H : ~ - i k l Oz'
(13)
and therefore the next term in the binomial expansion of the propagator is obtained in a recursive manner,
i kJ O~uj -~ O~z + (Hj+6H]) ~,j~O.
(14)
Since the beams are never less than 40 gm in diameter (as we will show later), the dominant contribution to H~ is the diagonal term. In some experimental scenarios, 6t-I]/Hy can be close to 1% due to the large n2 of ruby. The final equation then reads:
i kj a~,j
[
288
(
3. Numerical analysis
We numerically solve eq. ( 15 ) in both cylindrical and cartesian coordinates. For the radially symmetric transverse beam encoding problem, where various symmetric ring patterns form, cylindrical coordinates are used. The following second order finite difference operator, D r, is used for the transverse Laplacian in cylindrical symmetry: 1
Dr~k=
rk Ar 2
(rk+l/2 ~+l--2r~k+rk_l/2~ ~-l)
.
(16) The numerical technique used to solve the parabolic system of equations is the well-known Crank-Nicholson method. It has been shown that this unconditionally stable technique is actually slightly faster than operator splitting methods based on the FFT [26]. Additionally, in cylindrical coordinates, the FFT would have to be replaced by an appropriate algorithm for computing fast Fourier-Bessel transforms. The nonlinearity was first treated by iteration, but we found little advantage in speed over using the field values of the previous step for the nonlinear matrix elements. Modeling the asymmetric beam bending problem requires two independent transverse dimensions. Standard second order difference operators (D x, Dy) were used and the alternating direction implicit method was employed to solve the problem. Making the following definition:
[
' ~/'/i [ 2
(
[ ~'l [2 "~2l
V(I#j)= iflj+~/)l+lWll2+6 qJl+l~ul~]_]
(17)
the splitting method is expressed as {1 + ½iaz[DX+ ½V(¢")]} ¢:+1/:
I~l 12
~O~-z + v2~9+ _i[3j+tb 1 + I~, 12 +6
After solving eq. ( 15 ) in the crystal region, the farfield propagation patterns outside the crystal is obtained by using a Kirchhoff-Huygens diffraction integral.
(10)
where E~ is the field saturation amplitude, a is the beam radius and kl is the argon-ion laser wavenumber. The primed coordinate system is dropped for convenience, and the subscripts j = 1, 2 refer to the argon-ion and HeNe laser fields, respectively. In these normalized variables, the equation is
+ iflg+tbl+[~,[2
15 November 1993
= { 1 - ½iAz[IY+ ½V(~:) ]} ~u",
(18)
{ 1 + ½iAz[DY+ ½V(q/") 1} q : + ' . ~/j=0.
(15)
= {1
-
½iAz[DX+ ½V(~")1} ~/]n+1/2
(19)
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All computations modeled a two-beam system propagating through a 7.0 cm ruby rod with an initial beam radius of 40 ~tm and 57 ~tm for the argon-ion and HeNe laser beam, respectively. For the radially symmetric computations the spatial domain extended over 12 beam radii (0.48 ram), with a mesh spacing Ar=0.0117 (0.47 ~tm), and Az= 1.96× 10 -3 (0.1 mm). In the asymmetric problems the domain extended 24 beam radii (0.96 mm), Ax, Ay=0.086 (3.4 ~tm), and Az= 1.96× l0 -3 (0.1 mm). Dirichlet conditions were used for the boundaries, and the domain was increased until changes in the generated light patterns were less than 10-3. The step sizes and mesh spacings were tested to ensure that the total computational error also remained less than 10 -3. The classic results of Boshier and Sandle were repeated as a test of the numerical procedure [27 ]. In the beam bending simulations, the light fields are propagated at different angles through the ruby crystal. If both equations were treated in the same coordinate system, one of the equations would become hyperbolic and the numerical procedure would have to be significantly modified to account for the convective nature of the problem. To avoid these difficulties, the light fields are solved in independent coordinate systems and the nonlinear coupling is imposed on the HeNe beam by a space transformation at each longitudinal mesh point. The far-field propagation outside the crystal was performed by a simple Riemann sum of the discretized diffraction integral.
15 November 1993
camera, place at a distance 20 cm to 30 cm away from the exit face of the rod coupled to a Spiricon LBA100A laser beam profiler. Figure 2 shows a schematic of the experimental set-up. For the initial alignment the beams are made to coincide at BS and M2, with fine adjustments made to ensure symmetric ring patterns result in the encoded HeNe beam. Since calculations proved to be quite sensitive to the initial beam radii we carefully measured the spot sizes of the beams at the focus using variable slits on an x - y translation stage. The beam radii were computed to be 40 ~tm for the 514.5 nm beam and 57 ~tm for the 633 nm probe beam. For the beam bending experiments the mirror M1 is placed on a translation state. By adjusting the distance between the parallel HeNe and argon-ion laser beams, the input angle of argon-ion beam could be adjusted without effecting the initial overlap at the ruby rod face. The lens has a focal length of 10 cm, giving a 1.0 mrad angle for every 0.1 mm displacement of the translation stage. Using the camera mounted on a translation stage at the focus of the light beam, we verified that the initial overlap was maintained while the relative angle between the two beams coincided with the angle predicted by ray optics. Figure 3 displays the technique used to set-up the initial conditions for the beam bending experiments.
T1
4. Experimental set-up All experiments were performed using a Coherent Innova argon-ion laser in single line operation at the 514.5 nm wavelength. The maximum available power at this line is approximately 1.6 W. The ruby rod (R) is placed on a translation stage for initial spot size and phase front curvature adjustments. The ruby rod has Cr 3+ density of about 1.1 X 1019 c m -3 and is antireflection coated on one end. The c-axis of the ruby rod is at 60 degrees to the rod axis. All experiments were performed with the c-axis perpendicular to the argon-ion light field to eliminate birefringent effects and afford the maximum n2. The beam profile emerging from the crystal is targeted at a silicon
Argon-ion
He-Ne
% Fig. 2. Experimental set-up. M 1 and M2 represent high reflecting
mirrors, BS is a Fresnel beam sampler, T1 and T2 are translation stages. The lens has a focal length of 10 cm and the ruby rod is 7.0 cm long. The beam profiles are targeted at a silicon camera (C), whichis fed into the laser beam profiler (LBP). 289
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L Rub'), Rod
~ [ ~ 0
15 November 1993
Argon-ion d / 2 He-Ne
~
= d/f
Fig. 3. Beam bending set-up. The mirror M 1 in fig. 2 is placed on a translation stage to adjust the distance between the parallel HeNe and argon-ion laser beams. The input angl: of the argon-ion laser beam can be adjusted without effecting the initial overlap of the two beams at the crystal face.
5. Experimental and numerical results
(a) ~.,
Recent results on two-bean[ p r o p a g a t i o n experiments in ruby have revealed interesting pattern form a t i o n including multiple ring structures and on axis nulls [ 2,3 ]. Experiments revealed only small changes in the b e a m width when a 1.0 W, 514.5 nm, argonion laser b e a m passed through a 1 cm ruby crystal. However, u p o n passing through a longer crystal, a rich variety o f p a t t e r n formations can occur in b o t h a strong b e a m a n d simultaneously propagating weak probe. This length r e q u i r e m e m makes it necessary to model the propagation by solving the nonlinear wave equation. The curves in fig. 4 show the calculated b e a m waist for b e a m s o f different initial intensity u p o n traversal through the ruby crystal. The incident b e a m is 40 ~tm in radius. At a p p r o x i m a t e l y 200 m W a significant ring structure begins to develop. At nearly 400 m W , the on-axis i~tensity begins to d r o p to a m i n i m u m . Figure 5a shows the far-field profile o f the argon-ion b e a m incident u p o n the crystal with
2
~
6
distance in ruby rod (cm) Fig. 4. The argon-ion beam is shown to undergo significant changes in its waist size upon propagation through the crystal, (a) 1.0 mW; (b) 100 mW; (c) 500 roW. 290
1o o
>-,
05
.4
O0 -8
4
4
o
x axis (mrad)
(b)
j
.6
-8.0
i -4.0
i 0.0
4.0
8.0
x axis (mrad) Fig. 5. The far field profile of the argon-ion laser beam incident upon the crystal with power of 500 mW (a), and the corresponding numerical calculation with Re(n2) = 1.6 × 10-8 cm2/W (b). power o f 500 mW. The corresponding numerical calculation with t/z-- 1.6( 1 - 0.06i) × 10 -8 c m Z / W a n d o q = 1.3 cm -~ is shown in fig. 5b. Increasing the power to 1.1 W produces a far-field profile shown in fig. 6a and the numerical calculation in fig. 6b. Calculations predict that new ring structures and patterns develop b e y o n d 1.1 W. Changing the value of n2 by 10% drastically changes the far field patterns to the point where there is little or no resemblance to the experimental data. However, using n2 values that are nearly multiples o f the
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(a) ~
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There has been no experimental measurement of the cross-phase m o d u l a t i o n in term in eq. (7). Past experiments on two-beam nonlinear propagation indicate that the spatial evolution of both argon-ion and HeNe beams occur at the same argon-ion p u m p intensities [2,3]. In this study, we found that n¢= 1.35Re(nz) best fit the experimental data. Figure 7a shows the experimental result for the far-field
10
4 ~0.5&)
00 -8
-4
0
4
x axis (mrad)
(a)
(b)
1.0
Zo ~2
~
0.5
00 8 -6.0
-4.0
0.0
4.0
4
8.0
x axis (mrad) Fig. 6. The far field profile of the argon-ion laser beam incident upon the crystal with power of 1.1 W (a), and the corresponding numerical calculation (b). measured value yield similar pattern formations for at least one initial i n p u t power, i.e. one profile at a given power may match the experimental data, but not a series of profiles at different powers. We conclude that using an F/2= 1 . 6 ( 1 - - 0 . 0 6 i ) X 10 -8 c m 2 / W gives a good fit to the experimental data. Previous theoretical work on n o n l i n e a r pattern formation in thin G d A 1 0 3 : C r +3 samples indicates distinctly different pattern formation for input beam having either negative or positive phase front curvature [4]. In our experiments it was found that there was an initial phase front curvature dependence on the resulting beam profile, but strong pattern form a t i o n was also seen when the ruby rod face was at the center of the focal region. For a thin sample significant ring formation is not predicted in this region. However, since the beam is significantly encoded during self-focusing in our experiment, it does not seem likely that any clear distinction can be made about the initial phase front curvature dependence on the resulting profile. For this reason, all computations and experiments are carried out with the ruby rod face positioned at the center of the focal region.
0
4
x axis (mrad)
(b) ~2 7,
-8.0
-4.0
0.0
4.0
8.0
x axis (mrad)
(c)
.:2
-8.0
-4.0
0.0
4.0
8.0
x axis (mrad) Fig. 7. (a) The experimental result for the far field HeNe laser beam profile subject to the nonlinear phase encoding of a copropagating argon-ion laser beam. The initial 514.5 nm radiation power is 1.1 W upon entering the ruby crystal. (b) The computational result for these initial conditions with no=Re(n2). (c) The computational results with no= 1.35Re(n2). In the calculations for the nonlinear beam steering we use a value n2 = 1.35Re(nz) for the cross phase modulation term. 291
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HeNe beam profile subject to the nonlinear phase encoding of a copropagating argon-ion beam. The initial 514.5 nm radiation power is 1.1 W upon entering the ruby crystal. Figure 7b shows the computational result for these imtial conditions with no = Re (n2) and fig. 7c shows the computational results with nc= 1.35Re(n2). In the calculations for the nonlinear beam steering, we use a value of 1.35Re(n2) for the cross phase modulation term. Experiments were performed Io determine the imaginary part of the cross phase modulation term; however, no change in the absorption of the HeNe laser beam could be detected in the presence of the 514.5 nm pump beam. This null result is supported by the excited state spectrum reported in ref. [25], which suggests that the difference in the absorption coefficient at 633 nm between the ground and 2E state is very small. Therefore, the 633 nm beam undergoes only linear absorption in the ruby crystal, with O/2=0.1 cm -x. Earlier studies on this two-beam system showed that the ring patterns of the HeNe and argon-ion beams evolved with increasing argon-ion intensity in distinctly different manners. In light of the computations, we can attribute this interesting feature to the difference in the beam sizes, the difference in the phase modulation terms n2 and no, and the fact that nc is assumed to be purely real. Experiments in transverse self-phase modulation in ruby at rhodamine 6G wavelength indicated a similar pattern formation compared with the HeNe probe experiments. At these wavelength Im(n2) is small but slightly positive (limiting action) [7], while at 514.5 nm Im(n2) is small but negative [ 15 ]. The beam bending experiments were performed by injecting the argon-ion and HeNe beams into the ruby crystal at relative angles with respect to each other. In all experiments, the beams initially overlapped at the face of the crystal, and the cw argonion beam was measured to be 1.5 W. Figure 8a shows the experimental beam pattern for the symmetric coproprapagating situation where the relative angle between the beams is zero. Figure 8b is the numerical calculation of the beam pattern obtained by solving eqs. (18) and ( 19 ) at every mesh point, followed by numerical integration of the diffraction integral at the end of the 7.0 cm crystal. Adjusting the translation stage under mirror M 1 in fig. 2 causes the ar292
15 November 1993
-5.0
5.0
~5.0
x axis (mrad)
-5.0
5.0
~.5.(~
x axis (mrad) rig. 8. (a) The experimental beam pattern for the symmetric copropagating situation where the relative angle between the beams is zero and the initial argon-ion power is 1.5 W. (b) The numerical calculation of the beam pattern.
gon-ion beam to propagate at an angle to the HeNe beam. Figure 9a demonstrates the bending of the HeNe beam when the argon-ion is injected into the crystal at a relative angle of 3 mrad. Figure 9b shows the concomitant numerical calculation for this initial condition. Adjusting the translation stage to produce an input angle of 6 mrad produces the pattern shown in fig. fig. 10a, and the numerical calculation in fig. 10b. When the argon-ion injection angle reaches approximately 7.0 mrad, the steering angle
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5.0
15.~
x axis (mrad)
-5.0
5.0
x axis (mrad)
15 November 1993
I -
.0
5,0
15.0
x axis (mrad)
-5.0
5.0
15.0
Fig. 9. (a) The deflected HeNe laser beam when the argon-ion is injected into the crystal at a relative angle of 3 mrad. (b) The concomitant numerical calculation for this initial condition.
x axis (mrad) Fig. 10. Adjusting the translation stage to produce an input angle of 6 mrad produces the pattern (a) and the numerical calculation (b).
begins to saturate, until at a p p r o x i m a t e l y 9.0 mrad, the m a j o r i t y o f the power in the H e N e b e a m returns to the center o f the field. Figure 11 shows experimental and numerical plots o f the angle o f the peak H e N e intensity as a function o f the the argon-ion injection angle. The calculations were also p e r f o r m e d in one cartesian transverse d i m e n s i o n for c o m p a r i son. Keeping the argon-ion injection angle fixed at 6.0 m r a d and adjusting the power from zero to 1.5 W, the deflected H e N e b e a m was seen to m o n o t o n ically bend from its centered position to nearly 10 mrad. Figure 12 shows a plot o f the deflected peak
as a function o f p u m p power for this experimental scenario. Again we include a calculation with one cartesian transverse dimension for comparison. It was found that the calculations did not predict the correct a m p l i t u d e for the deflected spot for this power d e p e n d e n c e study. Figure 13a shows the experimentally observed beam profile obtained with a argonion p u m p power o f 0.9 W and an 6.0 m r a d injection angle and fig. 13b is the calculated b e a m profile. Although the steering angle is found to be approximately the same for the experimental and numerical b e a m profiles, the a m p l i t u d e s are shown to be different. 293
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2O
E ¢)
~D
g 0 2
4
6
8
10
pump angle (mrad) Fig. 11. Experimental and numerical curves of the angle of the peak HeNe intensity as a function of the the argon-ion injection angle. The calculations were also performed in one cartesian transverse dimension for comparison. Experiment, black diamonds; one transverse dimensional calculation, hollow squares; two transverse dimensional calculation, black squares.
os
lo
g
p
-5.0
~
5.0
,
1.~.~3
x axis (mrad)
~5
pump power (watts) Fig. 12. Keeping the argon-ion injection angle fixed at 6.0 mrad and adjusting the power from zero to 1.5 W, the deflected HeNe beam is seen to monotonically bend from its centered position to nearly 10 mrad. The calculations were also performed in one cartesian transverse dimension for comparison. Experiment, black diamonds; one transverse dimensional calculation, black squares; two transverse dimensional calculation, hollow squares. F o r a p p l i c a t i o n s i n v o l v i n g optical switching, it is i m p o r t a n t to k n o w the speed at w h i c h the steering can be p e r f o r m e d and the f r a c t i o n o f the p u m p and signal fields w h i c h exit the ruby crystal. Since the 514.5 n m light is strongly a b s o r b e d ( f c q ( I ) d L ~ 8 ), the p u m p b e a m c a n n o t be effectively used after exiting the crystal. T h e a b s o r p t i o n o f the signal b e a m , h o w e v e r , is v e r y slight ( c ~ 2 L ~ 0 . 7 ) , and a p p r o x i m a t e l y 50% o f the signal b e a m exits the crystal. T h e switch-on t i m e o f the system is g o v e r n e d by the p u m p a b s o r p t i o n rate, and the s w i t c h - o f f t i m e is g o v e r n e d by the lifetime o f the long lived m e t a s t a b l e state ( r ~ 3 m s ) . T h e p r o b l e m i m p o s e d by this long s p o n t a n e o u s 294
-5.0
5.0
t533
x a x i s (inrad) Fig. 13. (a) The experimentally observed beam profile obtained with an argon-ion pump power of 0.9 W and a 6.0 mrad injection angle; (b) the calculated beam profile. Although the steering anNe is found to be approximately the same for the experimental and numerical beam profiles, the amplitudes are shown to be different. lifetime could be c i r c u m v e n t e d by using a pulsed laser at 693 n m to stimulate transitions to the ground state. T h i s c o u l d lead to a 50% m o d u l a t i o n on the nonlinearity, which is certainly significant in light o f this study. T h e pulse energy r e q u i r e d to m o d u l a t e the n o n l i n e a r i t y by a p p r o x i m a t e l y 50% is AtP~ 0.5Ah v~ or, w h e r e AtP is the pulse energy, A is the area, and is the cross section o f the ruby line. T h i s predicts
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a pulse energy o f 400 ~tJ, which is easily o b t a i n e d in Q-switched n a n o s e c o n d dye laser systems. The same analysis can be m a d e for the switch-on time; however, the cross section for the absorption o f 514.5 n m r a d i a t i o n is a p p r o x i m a t e l y tenfold that o f the ruby wavelength. This predicts that a pulse energy o f 30 pJ could m o d u l a t e the index o f refraction by nearly 50% o f the m a x i m u m (An ~ 0.5Isn2 ). The rate o f this process is limited by the relaxation time from the 4T 2 state to the 2E state. Although there has not been any recent m e a s u r e m e n t o f this lifetime using m o d e r n picosecond p u m p - p r o b e techniques, there appears to be some consensus that the lifetime is in the few nanoseconds regime [28,29 ].
6. Conclusions We p e r f o r m e d experiments a n d numerical simulations on the nonlinear b e a m evolution o f simultaneously propagating cw argon-ion a n d H e N e laser b e a m s through a 7.0 cm ruby rod. A H e N e probe b e a m is deflected by nearly 12 m r a d by a 1.5 W, 514.5 n m laser b e a m injected into the ruby rod at an angle to the probe. Previous nonlinear b e n d i n g experiments have p r o d u c e d m a x i m u m b e n d i n g angles o f nearly 3 m r a d in pulsed o p e r a t i o n in CS2. We predict that switching times in the few nanosecond time scale would be possible using a Q-switched dye laser tuned to the ruby line. The copropagating geometry is also studied where the s y m m e t r i c transverse b e a m encoding leads to complex ring structures in both the H e N e and argon-ion beams. Interesting differences in the p a t t e r n f o r m a t i o n s o f the H o N e and argon-ion laser b e a m s are found to have their origins in slight differences in p a r a m e t e r values such at the crossphase m o d u l a t i o n term a n d the different b e a m widths. The n u m e r i c a l simulations are c o m p a r e d with the e x p e r i m e n t a l d a t a to obtain the cross-phase m o d u l a t i o n term. Using previously reported values for the saturable n o n l i n e a r index o f ruby and a crossphase m o d u l a t i o n t e r m nc = 1.3 5Re (/,12), the nonlinear wave equation in two transverse d i m e n s i o n s is numerically solved to predict the bending angles a n d emerging b e a m profiles.
15 November 1993
Acknowledgements The authors wish to thank Dr. J. Abshire o f NASA G o d d a r d and the Air Force Office o f Scientific Research for supporting this research. We also thank George Loriot for p r o v i d i n g us with u n l i m i t e d use o f an IBM RS6000 c o m p u t e r a n d his invaluable system support. A.S.L. G o m e s thanks C N P q for financial support. J.P. Bernardin would like to thank Sari Gilm a n for p r o o f reading the manuscript.
References [1] For recent examples, see the special issue entitled: Transverse effects in nonlinear-optical systems, J. Opt. Soc. Am. B June 1991, and July 1991, and references therein. [2] R.S. Afzal and N.M. Lawandy, Optics Lett. 14 (1989) 794. [3] R.S. Afzal and N.M. Lawandy, Optics Comm. 86 (1991) 307. [4IT. Catunda and L.A. Cuff, J. Opt. Soc. Am. B 7 (1990) 1445. [ 5 ] M. Le Berre, E. Ressayre, A. Tallet, K. Tai, H.M. Gibbs, M.C. Rushford and N. Peyghambarian, J. Opt. Soc. Am. B 1 (1984) 591. [6] R.S. Afzal, W.P. Lin and N.M. Lawandy, J. Opt. Soc. Am. B6 (1989) 2348. [7] I. McMichael, P. Yeh and P. Beckwith, Optics Lett. 13 (1988) 500. [ 8 ] S.A. Boothroyd, J. Chrostowski and M.S. O'Sullivan, J. Opt. Soc. Am. B 6 (1989) 766. [9] C.L. Adler and N.M. Lawandy, Optics Comm. 81 (1991) 33. [ 10] T.N.C. Venkatesan and S.L. McCall, Appl. Phys. Lett. 30 (1977) 282. [ 11 ] T. Catunda, J.P. Andreeta and J.C. Castro, Appl. Optics 25 (1986) 2391. [ 12] P.F. Liao and D.M. Bloom, Optics Lett. 3 (1978) 4. [13]D.S. Hamilton, D. Heiman, Jack Feinberg and R.W. Hellwarth, Optics Lett. 4 (1979) 124. [ 14] S.C. Weaver and S.A. Payne, Phys. Rev. B 40 (1989) 10727. [ 15 ] T. Catunda, A.M. Cansian and J.C. Castro, J. Opt. Soc. Am. B8 (1991) 820. [ 16] E.A. Gouveia, I. Guedes, J.C. Castro and S.C. Zilio, Phys. Rev. B 46 (1992) 14387. [ 17] A.E. Kaplan, JETP Lett. 9 (1969) 33. [ 18] M.S. Brodin and A.M. Kamuz, JETP Lett. 9 (1969) 351. [ 19 ] G.A. Swartzlander Jr., H. Yin and A.E. Kaplan, Optics Lett. 13 (1988) 1011. [20] A.E. Kaplan, Optics Lett. 6 ( 1981 ) 360. [21 ] J.A. Hermann, Optics Comm. 62 (1987) 367. [22 ] R. de la Fuente and A. Barthelemy, J. Quantum Electron. QE-28 (1992) 547. [23] M. Miguel Cervantes, Optics Comm. 90 (1992) 144. [24] I. Golub, Optics Comm. 94 (1992) 143. 295
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[25] T. Kushida, J. Quantum Electr:m. QE-2 (1966) 524. [26] Y. Chung and N. Dagli, J. Quantum Electron. QE-26 ( 1990 ) 1335. [27] M.G. Boshier and W.J. Saddle, Optics Comm. 42 (1982) 371.
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[ 28 ] S.A. Pollack, J. Appl. Phys. 38 (1967) 5083. [29] M. Anson and R.C. Smith, J. Quantum Electron. QE-6 (1970) 268.