- Email: [email protected]
Phase modulation spectroscopy of space-charge wave resonances in Bi12SiO20
1.5April 1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications 137 (1997) 181- 191
Full length article
Phase modulation spectroscopy of space-charge wave resonances in Bi,,SiO, Mikhail Vasnetsov ‘, Preben Bud-have *, Sergei Lyuksyutov Depurtment
ofPhysics,
’
Optics Group, Technical University of Denmark, Bid. 309, DK-2800 Lyngby, Denmark
Received 2 July 1996; revised 24 October 1996; accepted 26 November 1996
Abstract A new experimental method for the study of resonance effects and space-charge wave excitation in photorefractive Bi I2SiO,, crystals by using a combination of frequency detuning and phase modulation technique has been developed. The accuracy of the method allows a detection of resonance peaks of diffraction efficiency within 0.5 Hz. Numerical simulations of the nonlinear differential equations describing the behaviour of the space-charge waves in photorefractive crystals have been performed and found to be in a good agreement with experiment. We have measured the photocurrent through the crystal and revealed its resonance dependence. A minimum of electric current through the sample corresponds to the main resonance detected by phase modulation technique. PACS: 42.65
1. Introduction The photorefractive effect has been intensively investigated since the first experiments were reported demonstrating the refractive index change in a LiNbO, crystal under influence of light illumination [l]. Free carrier generation due to the excitation from traps in illuminated areas and charge migration into dark regions lead to the formation of the internal space-charge field which modulates the crystal refractive index via the electro-optic effect. The specific nonlinear material response to the spatially inhomogeneous illumination in photorefractive crystals permits a recording of phase volume holograms which may possess high diffraction efficiency. One of the most prominent features of the photorefractive crystals is the possibility to record and read the holograms in real time (dynamic holography) [2,3]. The light-induced gratings which are the result of
* Corresponding author. Fax: t(45) 45 93 16 69; E-mail: [email protected]. ’ On leave of absence from the Institute of Physics, National Ukrainian Academy of Sciences, 252 650, Prospekt Nauki 46, Kiev, Ukraine.
small variations of the material refractive index in bulk crystals can follow the moving light pattern within the limitations defined by the response time of the material. High speed materials such as Bi ,2Si02,, (BSO) crystals are very promising for dynamic holography applications. The running grating technique in combination with the application of an external dc electric field significantly increases the amplitude of the space-charge field and therefore the grating strength [4,5]. The resonance bchaviour of the running grating strength measured by its diffraction efficiency with respect to the velocity of the grating in a crystal was the reason for the introduction of the idea of a space-charge field in the form of a running space-charge wave [6]. Our goal is to investigate both experimentally and by numerical simulations the main features of the resonance space-charge wave excitation in photorefractive BSO crystals. In this paper we introduce a new experimental technique for the study of space-charge wave excitation and resonance phenomena based on the combination of a frequency shift between the two coherent beams writing a grating and phase modulation of one of the beams. The phase modulation technique [7] permits the production of
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running light fringes with variable contrast, which is important in order to study the dependence of the space-charge wave excitation on the light modulation. We detected a shift of about 60% of the resonance frequency as a result of variations of the modulation. The resonance behaviour of the photocurrent through the crystal was also established experimentally. In the numerical calculations a new technique that gives us the possibility to determine the resonance features and spatial structure of a space-charge wave was used. The structure of this paper is as follows: Section 2 describes the experimental arrangement used for the investigation of the running gratings in the BSO crystal. Section 3 deals with the detailed description of the phase modulation technique. Section 4 gives the theoretical description of the running resonant photorefractive grating excitation by a moving light interference pattern. In Section 5 the experimental results are discussed, and conclusions are summarised in Section 6.
2. Experimental
setup
The experimental setup is shown in Fig. 1. The light source is a single mode Ar-ion laser operating at a wavelength of 514.5 run. The light is brought from the laser to the optical table through a single mode optical fibre. A radial holographic grating serves as a combined beamsplitter and frequency shifter. When the grating rotates, the first and minus first order diffracted beams are frequency shifted by equal amounts of So, but with opposite sign. This arrangement allows precise control of the frequency shift over a wide range without phase jumps. The direct transmitted beam with unshifted frequency w is blocked, and the two diffracted beams of equal intensities and with frequency detuning 60= f(2?r/a)~, where v is the
powt?r meter / or photodetector
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angular velocity of the grating and (Yis the angular grating spacing, are collimated, expanded and directed to intersect in the BSO crystal. The BSO crystal with dimensions 10 X 10 X 3 mm was antireflection coated on the input and output surfaces. Gold electrodes were plated onto the side faces of the crystal. A dc electric field E, = 6 kV/cm was applied perpendicular to the (1 IO) surface. The intersection angle between the writing beams 28 was 1.47” resulting in a fringe spacing A = 20.1 pm, The total Ar-laser power incident on the crystal was 3 mW/cm’. Both incident waves were polarised perpendicularly to the plane of incidence. Phase modulation was applied to one of the writing beams via a piezoelectric mirror driven by a sinusoidal voltage from a frequency generator. The accuracy of the frequency control was about 0.1 Hz. Low power He-Ne laser radiation was used as a probe beam to test the diffraction efficiency of the running gratings. The angle of incidence of the probe beam was Bragg-matched to the spatial frequency of the grating recorded in the crystal. The intensity of the diffracted wave which serves as a relative measure of the diffraction efficiency of the grating was picked up by a photodetector and digitised by an oscilloscope. An electric current through the crystal in the direction of the wave vector of the running grating has been measured. With an applied electric voltage of 6 kV and for an incident light intensity of 3 mW/cm2 typical values of the current were of the order of 30 pA.
3. The phase modulation technique The general scheme for analysing the running grating excitation in the BSO crystal is as follows. Two coherent light waves of electric field strength amplitudes E, and E, intersect in the crystal to record a grating in a symmetrical scheme with the intersection angle 20. The frequency shift 260 is created between the waves by the rotating diffraction grating (Fig. 1). the first wave has positive shift &IJ,
(3.1) where k is the wavevector modulus 27r/h where A is the wavelength of the incident light. The other wave has negative shift - 60, and may be phase modulated by a piezoelectric mirror with the amplitude of modulation A and frequency R: E,(x,z,t)=E,exp(i[k(xsin0-zcos0) He-Ne
laser
mlrmrs
Fig. 1. Experimental setup for phase modulation spectroscopy. Inset: rotating diffraction grating producing frequency shift 260 betweenwriting beams.
-6wt+Asin
at]},
(3.2)
where E, and E, are real amplitudes and the phase factor describing the light frequency w has been omitted. The distribution of the resulting light illumination I(x,t) on the
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crystal input face ( ;: = 0) has, apart from a constant of proportionality, the form: I( XJ) = lE,( XJ) + E,(XJ)12.
(3.3)
After evaluation we have I( x,?) = Ef + Ez + [E, E, exp[ i(2kx sin 0 -2Sor+Asin
fir)] +c.c.].
(3.4)
Further we shall use the following notations: m = 2E,E,/(Ef + Ez) (modulation); 1, = E: + Ez (total light illumination); K = 2k sine (grating wavenumber); and decomposition exp[i(A sin fir)] = Cy= _m J,(A) exp(ilRt), where J,(A) are Bessel functions of order 1 with argument A. Inserting into (3.4) results in I(x,f)=I,
1 +T i
i
J,(A)(exp[i(Kx
,= --Ix
.
-260t+lRt)]+c.c.}
I
(3.5)
Light intensity distribution in the form (3.5) is a decomposition into an infinite set of running light fringes with the same spatial frequency K but different velocities V, = (260 - lfi)/K. For a negative sign of the term 2 SW - I R, the fringes move backwards with respect to the x axis. The contrast of the light fringe pattern running with the velocity V, is a product of m and the Bessel function of the corresponding order 1 with argument A. Below we analyse three combinations of frequency shift 2 60 and the frequency 0 produced by phase modulation: (i) no phase modulation, A = 0; (ii) both frequency shift and phase modulation are applied to the writing beams; (iii) frequency shift 26~ is zero, phase modulation is active. 3.1. Space-charge wave excitation without phase modulation
= 1 + m cos( Kx - 260 t) = 1 +mcos[K(x-V,,t)]
(3.6)
and only one cosinusoidal light fringe pattern exists running with the velocity V, = 2&0/K. This case is thoroughly investigated theoretically and experimentally [6,8lo]. Periodical running light fringes write a photorefractive grating, which is supposed to have the same fundamental spatial frequency. To describe the grating diffraction efficiency n, we introduce the response function f(m,V,,>, which is the amplitude of the fundamental spatial harmonic of the internal space-charge field as a function of the interference
183
pattern contrast and the velocity of the light fringe motion. The amplitude of the probe wave diffracted in the Bragg diffraction order Ed will be proportional to the grating strength, which is in turn proportional through the electrooptic coefficient to the space-charge field ES, (more precisely to the fundamental component of the space-charge field flm,V,) cos[ K(x - V,t)]: Ed aflm,V,)). The measured diffraction efficiency n of the running grating is thus proportional to the square of the modulus of the response function, rj a If(m.VJl*. As follows from the dependence of diffraction efficiency on the frequency detuning [6,8], the response function has a maximum (resonance) for a certain frequency w,, or the fringe velocity V, = w,/K, which depends on the light intensity, applied voltage, material parameters and the fringe contrast. Far from the resonance the diffraction efficiency is negligible and flm,V,) = 0. The analytical approach to the determination of the response function is discussed in Section 4. In order to determine the main features of the response function experimentally, we may scan the frequency shift 280 from negative values to positive ones and measure the corresponding diffraction efficiency n of the running grating. In this way we can detect the position of the resonance and determine the shape of the resonance curve $2 6~) for the given value of the modulation m and light intensity 1,. 3.2. Combination offrequency
shift and phase modulation
In the case when both frequency shift and phase modulation are applied to the writing beams, a variety of running light patterns with the same spatial period but different velocities and contrasts are present. The resonance will occur for the following combinations: 2s030r
(f=O);
2&w-R=o, 2&0+n=
In the case when the phase modulation is not applied to the writing beam, (3.5) turns into I( x,r)/l,
137 (1997) 181-191
(I-l); 0,
(I-
(3.7) -1);
and analogously for the following I values. To simplify the analysis, we shall take the frequency of phase modulation 0 higher than the resonance frequency o, so that only one frequency combination 260 - In can appear within the region of the resonance. At least three variables may be used to test the diffraction efficiency 7: 0, 2Sw and A. If we take a constant value of fi > w, and scan the frequency shift 2Sw, we shall observe the main resonance corresponding to I= 0 for the grating recorded by fringes running with a contrast ml,(A) and velocity V, = 2&0/K = V, corresponding to the condition 2 So - w,: 1,[1 +m&(A)cos(Kx-26wt)] =I,,{1 +mJ,(A)cos[K(x-V,,r)]}
(3 *8)
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and additional peaks for combinations of the frequency shift and phase modulation frequency producing the running fringes with the same velocity (2 SW- 10 )/K = V,.: I,[1 +mJ,(A)cos(Kx-26ut+lRr)] = IO{ 1 + mJ,( A) cos[ K( x - V/t)]}.
(3.9)
As seen from (3.8), the shape of the main peak will correspond to the modulation depth of the light distribution or fringe contrast, which is equal to ml& A). Varying the amplitude of phase modulation A, we can easily detect the dependence of the resonance curve on the light fringe contrast without any change in total light intensity incident on the crystal. With zero phase modulation (A = 0) we shall have the same resonance curve as discussed in the previous subsection, for J,(O) = 1. For A f 0 additional peaks appear, which are shifted from the resonance frequency 0, by +lR. These new resonances display a reduction in fringe contrast according to the dependence mJ,( A> in relation (3.9). Another way to measure the resonance curve is to keep the frequency shift 260 constant (larger than the resonance frequency mr) and scan the frequency of the phase modulation R. The fringe velocity will be close to the resonance value when the condition 26~ - I.0 = w, is satisfied. Again, a set of peaks occurs with different 1. The first peak will appear for 1= 1 corresponding to the moving fringe pattern: I,[1 +mJ,(A)cos(Kx-26wt+Rr)] =I,{1
+m.J,(A)cos[K(x-V,t)]}.
(3.10)
Changing A from zero to - 1.9, we have the possibility of varying the contrast &,(A) from zero to = 0.58, which is the maximum of the function J,(A) (m = 1 for equal writing beams intensities). From the point of view of experimental convenience, this combination is preferable, because we can control the frequency of the sinusoidal voltage driving the piezoelectric mirror with higher accuracy than the angular velocity of the rotating grating. By measuring the resonance curves of the diffraction efficiency as a function of 0 for different values of the amplitude A, we are able to obtain a de ndence of the p” square of the response function IjfV,)l on the fringe contrast.
137 (1997) 181-191
the stationary grating will be detected. When R is smaller than the resonant frequency of the space-charge excitation w,, several frequency components may be located within the region of the resonance (V, = V,). For the next analysis, two assumptions are made. First, we suppose that the electric space-charge field inside a crystal has the same spatial structure as the light distribution: only running waves of the form exp(iKx + ilot) are involved. (In the general case this is not valid, because higher spatial harmonics with grating vectors 2K, 3K, ... may be excited, and also subharmonic components with grating vectors K/2 and others may occur.> To obey this condition, we need to restrict ourselves to weak contrast of running light fringes, i.e. A +C 1. Secondly, each running component of light distribution is responsible for the corresponding component of the space-charge field: Es,=
f(m,A,l~){exp[i(Kx+l~t)]+c.c.},
i
t= -zz (3.11) where the amplitude of the space-charge field component flm,A,la) is a response function for frequency 10 with parameters m ,and A. This function describes as before the relative values of space-charge field components created by light fringes running with velocities V,. Further, we suppose proportionality between the running refractive-index grating strength g, and the corresponding component of the space-charge field: g,(m,AJn)
(3.12)
af(m,A,lfi),
and the whole time-dependent grating recorded by the phase-modulated wave and the ordinary wave has the form of a composition of running gratings: G(x,t)a
i f(m,A,lR){exp[i(Kx+lflt)]+c.c.}, ,= -r (3.13)
(the permanent variation of dielectric permittivity due to the constant applied field E, is omitted). In the approximation of small diffraction efficiency, the amplitude of the Bragg-matched diffraction order will be proportional to the grating strength. For the incident probe wave the amplitude of the Bragg diffracted order will have time dependence
3.3. Phase modulation spectroscopy
Ed(t) a
In the case when the frequency shift 280 is zero, the set of the light fringes running with velocities V, = -lo/K and contrast ml,(A), corresponding to the frequency of the phase modulation 0 and integer multiples of it, are present in the light distribution. Fringes running towards the x-axis have positive velocities, and vice versa. A stationary light fringe pattern corresponding to I = 0 also exists. When the frequency of phase modulation R is much larger than the resonant frequency, only the response from
and the diffraction efficiency
2 f(m,A,lfI)e-““‘, I= --P
(3.14)
2
Tot
i f(m,A,lLf)e-““’ ,= --r
.
(3.15)
The analysis of the response function flm,A,l.fI) the time dependence of the diffraction efficiency r] general case is rather complicated. To simplify the lem, we neglect all terms but flm,A,O) and flm,A,
from in the prob- 0)
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X e’“’ due to the fact that running light fringes with negative velocities (I> 0) cannot be in resonance, and negligible contrast rnJ,( A) for 1II > 1. For two remaining terms flm,A,O) =flO, and flm,A,C2)ein’=fe’“’ we can write (3.15) as 7~a If(O) +feinr12.
(3.16)
The multiplication will give the (asterisk means complex conjugation):
following
r] a ]f(O)l’ + IfI* +f(O)
(0) fe”“.
f” eCir” +f*
no a If(
and a time-dependent qfjaf(0)f*
efficiency
con-
(3.18)
+ Ml’. term oscillating
at frequency
eelR’ +f’(O)fe’“‘.
0: (3.19)
The complex response functions fl0) and f may be determined from the measurement of Q and no. The relation (3.19) can be rewritten as nf)aIf(0)flcos(nr+
cp).
4. Space-charge wave excitation
dN,+ -=d(N,-N&) at
- yRN,fn.
we need to by the light the x-axis differential [2,3]:
(4.1)
aj dP x++=o,
(4.2)
JE E&O- = p,
(4.3)
dX
p=e(N,+-N,-n),
(4.4)
an j = plc,Tz
+ penE,
(4.5)
where e is the absolute charge of the electron, p is the electron mobility, k, is Boltzmann’s constant, T is the absolute temperature and NA is the density of compensating negatively charged centres. To simplify the analysis, we neglect the contribution of free electrons to the spacecharge distribution p and rewrite (4.4) as p=e(Nz
-NA),
orNz
-N*+p/e.
(4.6)
Following the method used in Ref. [6] to obtain the equation for the space-charge field amplitude in a stationary regime of running light fringes, we may rewrite Eqs. (4.1)-(4.6) in a coordinate system moving with the light fringes. Using the variable .$ = x - Vat and neglecting the term responsible for the diffusion, Eqs. (4.1)-(4.6) may now be combined into one differential equation for E( 5 ):
(3.20)
where cp is the phase shift between the running interference fringes and the space-charge field. The amplitude of oscillations of the diffracted wave intensity with the frequency 0 is found to be proportional to IflO)fl, that may be used to find the position of the resonance and its dependence on running fringe contrast (within the small contrast limitation).
To calculate the response function fim,V,), determine the space-charge field E,,( x,t) built interference pattern I = I(x,t) running along (3.6). We start from the well-known set of equations describing the photorefractive effect
section for the active center photoexcitation, ~a is the recombination constant and p and j are the charge density along the x axis and the electric current defined as
terms
(3.17) As follows from (3.17). the diffraction tains a steady-state term
18.5
where E is the internal electric field, which is a sum of the field E, produced by the applied voltage and the spacecharge field ES, created by charge redistribution, No, Ng, are the total density of active centres and density of ionised centres (positively charged) respectively, n is the concentration of free photoexcited electrons, s is the cross
Volvo d2E/a.g 2 + sl( eN, - eN, - &co aE/&g ) PeE
x(
aE + Vo.sxO-- = j,, 86
eN, +
880
aE/%
>
(4.7)
where jo is a constant of integration, having a physical sense of electric current density across the crystal. Eq. (4.7) may be further simplified if the relatively small term in parentheses EEL i)E/i)S could be omitted:
(4.8) where No., = N, - NA_ When the light fringes do not move (V, = 0) the standard solution appears E = (joy, NA)/( peslkdN,_,); for moving fringes Eq. (4.8) becomes a nonlinear second-order differential oscillatory equation. The light distribution I( 5 ) = I,[ 1 + mcos( K 5 )] stands for the forcing term in Eq. (4.8) in the case of moving fringes. For a particular value of V, the amplitude of E( 5) may exhibit resonant behaviour, i.e. a running space-charge wave is excited. We note that the current j(, will depend on the value of the free electron concentration and hence may also show resonance features. To analyse the resonance conditions, we let the modulation m be small, which results in a small value of I E,( 5 )I compared to the applied field E,. First, j, = ~eiiE,, where ii = sle ND_J~a NA is the average free electrons concentration. Without the small nonlinear terms proportional to E,,Er,
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linear approximation for m (< 1 is the amplitude of the space-charge wave (4.10). To analyse the behavior of the space-charge wave I?,,( 5) for the case of arbitrary m the nonlinear secondorder differential equation (4.8) could be calculated numerically. As we seek a stationary solution in the form of a running wave and suppose that this solution does exist for the nonlinear second-order differential equation (4.8). the existence of a stable attractor (limiting cycle) is further postulated. Eq. (4.8) may be rewritten for more convenience as
and mE,, we obtain an equation for a linear oscillator driven by an external force:
~w,Eo”o YRNA +
tL+%A
EzC+ V, .wOE;, +
&eND., YRN.
ES,
YRNA
EOmcos( Kt)
= 0,
where a prime denotes derivative with respect to 5. The solution has a form of a running wave E,,( 5) mEOsIOeND_,cos( Kt+
=
137 (1997) 181-191
(4.14)
yy”+yy’+Hy(l+mcosKl)=s”J,,
‘p)
where
y( 5) = HO/Eat E,, and
Y = ynNA/pEo+
B =
sI,~N,.,/EE,V~ J, = j,-
where the phase shift cp between the light fringe pattern and the space-charge field E,, is determined as cp= arctan
YRNAEEO”OK
PC EE~E~V,,K~ - sI,eN,.,)
The resonant condition is easily derived from (4.10) as the space-charge field amplitude has a maximum when the fringes move with a velocity
V,., (4.12)
The same condition turns phase shift cp to 7r/2. In the analysis used in Ref. [8] the calculated optimum velocity
has a negative sign due to the definition of the charge e; we use the absolute value of the electron charge instead of the negative in Ref. [8]. In a physical sense the drift of electrons in the applied field occurs in the opposite direction with respect to the fringe motion. The resonance frequency may also be determined as or-KV,----
do eN,_
A
.q,EOK
(4.13)
’
The resulting response function flm,V,,> derived in the
0.0 0.0
0.5
1.0
1.5
2 0
) 0
sI, eEOND. A
, 10
1 20
30
40 s
EWE, (a)
El.
is the normalised current, far from the resonance or for very small modulations J, = 1. We used the following numerical values for the coefficients: m = 0.4, K = 2, y = 0.3 (the choice of the value y/K = 0.15 is explained in Section 6), B ranged from 2 to 200, J, was usually less than 1 and was determined by a fitting procedure described below. In numerical simulations the periodical function y( 5) will come closer and closer to the attractor, independent of the initial values of y(O) and y’(0). Starting from arbitrary (with some reasonable choice) initial values, we integrate Eq. (4.14) along the 8 axis and wait until the periodic structure of the space-charge field will not change more with the growth of 5: this is evidence that the steady-state solution is reached. Then Fourier transform is used to calculate the power spectrum of the oscillations with respect to the spatial frequency. The response to the fundamental spatial frequency K and higher harmonics 2K, 3 K, ... (if necessary) is determined, and their dependence on fringe velocity V, and modulation m may be found. The average value of the calculated electric field E( 5 ) must be equal to the applied field E,. To reach the average value ( y( 6 )) = 1 we need to vary the parame-
(4.11)
’
1
YRNA
(b)
0
K 2K 3K 4K 5K 6K
Spatial frequency CC)
Fig. 2. Numerical calculations. (a) Limiting cycle corresponding to the periodic solution of Eq. (4.14) with parameters m = 0.4, y = 0.3, K = 2. (b) Electric field distribution along the t-axis. (c) Power spectrum of the space-charge field (E(c) - &J/E,. Normalised fringe velocity a- ’ = 0.08. Calculated current J, equals 0.65. Excitation of high spatial harmonics of the space-charge field is clearly seen as peaks corresponding to spatial frequencies 2 K, 3 K, ... (the peak width is determined by numerical FFT procedure).
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o.oj,. 15
2.0
:, 0
10
I, 20
ter J, until the component in the oscillation spectrum corresponding to zero spatial frequency becomes unity. In our calculations the Fourier transform was applied to the function y([ ) - 1 = E,,(~)/E,, and we reach zero for the average ( E,,( 5 1) value by a proper choice of ./a. Fig. 2 demonstrates the limiting cycle in coordinates E( 5 j/E, = y( 51, E’( 5 I/E, = y’( 5) and the shape of the normalised field E( 5 )/E, combined with a power spectrum of the space-charge field E,,( .$I/E, obtained in the calculations for the nonnalised fringe velocity Es”-’ = 0.08. The calculated diffraction efficiency of the running grating is thus proportional to the magnitude of the response on the fundamental spatial frequency K. Second and higher spatial harmonics are excited in addition to the fundamental component of E,,(t). Similar calculations for the normalised fringe velocity E- ’ = 0.3 1 are shown in Fig. 3. Mostly oscillations of the space-charge field in the fundamental spatial frequency are excited in this case. Further results of the calculations are plotted as a dependence of the diffraction efficiency on the normalised fringe
10
1.0
/-------
A--
1
09 0.8
0.8
m z-. 0.7 E” : $ ” P g
0.6 0.5 0.4
00
01
0.2
03
04
0.5
06
0.7
0.8
0.9
10
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0.0 40
187
11 0
K
2K
3K
4K
5K
6K
Spatial frequency Cc)
W
Fig. 3. The same as in Fig. 2 but for normalised fringe velocity 8-l spatial harmonic of the space-charge field is excited.
$f
I,
5
(a)
3
(1997)
30
EWE,
09.
137
= 0.31. Calculated current J, equals 0.57. Mostly the fundamental
velocity in Fig. 4. The resonance behaviour of the diffraction efficiency around E:“- ’ = 0.4 is evident, the excitation of the running space-charge wave is optimal. Despite this resonance, the diffraction efficiency at low fringe velocities also demonstrates the resonance peaks. According to the theoretical model proposed in Ref. [6] the excitation of the fundamental harmonic of the space-charge field wave may occur also via excitation of its higher spatial harmonics. As they have spatial frequencies twice, triple and larger (2 K, 3K, ...) than the fundamental spatial frequency, the resonance fringe velocity for them will be four, nine etc. times lower that for the main resonance. It is seen from Fig. 4, that the excitation through the second spatial harmonic has the maximum around ? ’ = 0.1, what is four times lower than that for the main resonance. We detected in numerical calculations a hysteresis around the value s-’ = 0.1, which may be the consequence of a strong interaction between the fundamental and the second spatial harmonic of the space-charge field. The calculated dependence of the normalised current through the crystal Jo is also shown in Fig. 4. The normalised current tends to unity outside the resonance region and decreases at the resonance, both around the peaks of the main resonance and at higher harmonic resonances. Using the method of numerical simulation of the steady-state space-charge field, we calculated the normalised diffraction efficiency for a small value of modulation (m = 0.01) and for a high value (m = 0.99). In the small-modulation regime the effects connected with the resonance wave excitation through high spatial harmonics disappear and only the main resonance exists. High modulation results in the appearance of nonlinear effects: the excitation of the space charge wave through higher harmonics becomes even more efficient than the direct excitation, also the shift of the main resonance position was detected.
1.1 1.2
Normalisedfringevelchty5'
Fig. 4. Numerical calculations. Dependence of the diffraction efficiency (solid line) and current J, (dashed line) on the normalised velocity of interference fringes 5- ’ .
5. Experimental
results and discussion
In our experiments we keep equal intensities of the writing beams to avoid change of the total light intensity
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0
Frequency
10
20
detuning
I37 (1997) 181-191
30
40
26~ (Hz)
Fig. 5. Experiment. Dependence of diffraction efficiency on frequency detuning 260. Total intensity of light is 3 mW/cm’.
when changing the beam intensity ratio. The first set of experiments was performed without application of the phase modulation to the writing beam. The frequency shift 2Sw produced by the rotating grating was measured from the beatings between writing beams. By changing the direction of the grating rotation, we were able to scan the frequency shift 260~ from negative to positive values with accuracy within 0.5 Hz. Fig. 5 represents the dependence of the diffracted signal with a maximum located around 1 Hz. This peak is known as a low-frequency maximum [9]. The shift of its position from exact zero detuning was specially checked. The measured diffraction efficiency shows a decrease at the region of negative frequency shift, while there is another maximum for positive values around 9 Hz, which we shall denote as the main resonance. No other maxima were detected. The second set of measurements included the phase modulation of one of the writing beams by a piezoelectric mirror. We controlled the amplitude of the phase modulation A in the following manner. Keeping the frequency of the phase modulation far from the resonance position, we detected the maximum of the diffraction efficiency at the resonance and changed the amplitude of the driving voltage I/ of the frequency generator. The contrast of the running fringes equals mJ,(A) and becomes zero when J,(A) = 0: no grating is recorded. As the amplitude of the piezoelectric mirror displacement is proportional to the applied voltage, we define the proportionality coefficient between A and Cr. The value of U = U. when diffraction efficiency drops to zero corresponds to the first zero of the J,(A) Bessel function: A = 2.4. Thus the driving voltage U was calibrated in terms of the amplitude of phase modulation: A = 2.4U/Uo. Fig. 6 demonstrates the resonances obtained. The applied frequency of the piezoelectric mirror vibration 0 was 35 Hz and the frequency shift produced by rotating grating 2Sw was varied. The used value of A was 1,
which gives J,(A)-0.77, J,(A) -0.44 and J,(A) = 0.11. Some changes in the central resonance curve in comparison with the one plotted in Fig. 5 are observed: The decrease of the modulation from 1 to 0.77 leads to the relative suppression of the low-frequency maximum. The position of the main resonance is now around 7 Hz which is somewhat less than in Fig. 5. Side resonances corresponding to I = &-1 are shifted at the value of the frequency D = 35 Hz, also resonances for 1= + 2 are resolved. The shifted peaks with the depths of modulation 0.44 and 0.11 do not exhibit low-frequency maxima at all. To resolve the small changes in the position of the main resonance, we varied the amplitude of the phase modulation A from zero to the value A = 2.4 corresponding to the first zero of the .I,( A) Bessel function. For each value of A the position of the main resonance maximum was detected. The data obtained are plotted in Fig. 7 as the resonance frequency w, as a function of the value of
Frequency detuning 280 (Hz)
Fig. 6. Experiment. Dependence of diffraction efficiency on frequency dehming 260. Phase modulation is imparted on one of the beams at frequency D = 35 Hz. Side resonance peaks are shifted with respect to the main resonance by 35 Hz.
M.
”
Vasnetsov
et al./Optics
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137 (1997) 181-191
7-400 2 d E
300
s 'G E .E
200
5 J! cl 100
. 0 0.0
0.1
0.2
0.3
0.4
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0.8
0.9
1.0
-6 -4 -2 0
Fringe contrast
2
4
6
8 1012141618202224262830
Frequency difference 260 - C?(Hz)
Fig. 7. Experiment. Dependence of resonance frequency on the contrast of the interference pattern (light modulation depth).
modulation of the running light fringes. The observed shift exceeds the experimental error and amounts to about 3.6 Hz from the lowest to highest values of fringe contrast. We note that the position of the resonance is very sensitive to the light power incident on the crystal. Therefore the absolute values of the resonant frequency may differ in different sets of measurements due to the laser output power variations, but the total shift remains the same. In addition to the measurement of the maximum of the diffraction efficiency position we also detected its absolute value, the corresponding dependence is plotted in Fig. 8. The origin of the low-frequency peak was observed in the
Fig. 9. Experiment. Dependence of diffraction efficiency on the frequency difference between constant frequency shift 280 and variable frequency of phase modulation R. Open circles correspond to contrast 0.13; full circles correspond to contrast 0.58.
set of measurements with constant frequency shift 260~ and varied frequency of phase modulation 0. Two resonance curves are shown in Fig. 9 for the modulations 0.58 (maximum for J,(A) Bessel function) and 0.13 as the dependence of the diffraction efficiency on the difference between frequency shift 280 and the frequency of phase modulation 0. The curve for small contrast (0.13) does not contain the low-frequency peak, whereas the appearance of the peak is resolved for a contrast of 0.58.
120 110 100
. 400
l
. .
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.
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.
.Fj 200
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/
50
02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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Fringe contrast
Fig. 8. Experiment. Dependence of diffraction efficiency on the contrast of interference pattern.
”
,A”
/I’
’
0.04
/”
/’
0
,,,I’,,,,,‘,,
I,
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/’
. 0.0
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. 0
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,’
.
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/
.
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5
10
15
20
25
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Frequency of phase modulation Q
35
40
(Hz)
Fig. IO. Experiment. Amplitude of Fourier component of the diffracted signal for a particular value of phase modulation fl as function of 0 for three values of light fringes contrast.
190
M. Vasnetsov et al./Optics 1.0
Communications
. l
. .
0.9
$ g
0.7-
a 0 5 E
0.6 -*
.
05-
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. l
. 0.4
,
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,
,
/
,
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0 1.1 Velocity
of fringes
(mm/s)
Fig. 1I. Experiment. Dependence of photocurrent through the crystal j normalised to its maximum value j,,,,, on velocity of interference fringes. Applied electric voltage is 6 kV and fringe contrast is 0.4. The electric current minimum corresponds to the main resonance of space-charge wave excitation, 260~ = 8.5 Hz.
To complete the measurements of the dependence of the main resonance maximum on the value of modulation, we used phase modulation without frequency shift (2 60 = 0) between the writing beams. In accordance with the description of the phase modulation technique given in Subsection 3.3, an amplitude of phase modulation was chosen to be small. The resonance curves shown in Fig. 10 correspond to the modulation depths 0.04, 0.08 and 0.15. The value of the response was measured as the Fourier component of the time-dependent signal from the photodetector corresponding to the frequency of phase modulation 0. No frequency shift of the resonance discussed above was detected for these small modulation values. There is no contradiction with the measurements of the resonance frequency shift as plotted in Fig. 7: resonance frequency w, seems to be independent of the fringe contrast in the region of small modulations. In this experimental arrangement we also measured an electric current, which is induced in the BSO crystal by the applied voltage and the light illumination. Photoexcited carriers in the crystal move in the applied electric field thus creating a photocurrent (parameter j, in Eqs. (4.7), (4.8)). The value of the current depends on the carrier modulation. The higher the modulation, the lower the current. We have experimentally detected that the minimum of the current exactly corresponds to the resonance of the space-charge wave (Fig. 11). This is in good agreement with our numerical calculations.
6. Conclusions Phase modulation technique was used to study the main features of the resonant excitation of photorefractive running gratings in BSO induced by moving light fringes. The
137 (1997) IN-191
merits of this technique are the possibility to vary the contrast of the running light fringes without changing the total light intensity and easily control the velocity of the fringe motion. When the velocity of the running fringes is close to the resonance velocity of the space charge wave propagation, strong enhancement of the running grating diffraction efficiency is detected. We observed the development of a low-frequency peak with the growth of light fringe contrast. This low frequency peak is related to the nonlinear effect of space charge wave excitation through higher spatial harmonics. The appearance of additional peaks due to space-charge wave excitation by a combination of the frequency shift produced by the rotating grating and a phase modulation generated by vibration of a piezoelectric mirror was detected. This effect explains the periodical diffraction efficiency peaks observed by us earlier when the crystal vibrated [I 11. The high resolution of the phase modulation spectroscopy technique enables us to measure the resonance frequency dependence on the contrast of the light fringes. We detected a shift of the resonance frequency from 6.2 Hz to 9.8 Hz, which amounts to about 60% of the resonance frequency. The results obtained in the experiments are compared to numerical simulations based on the model of a running grating recorded by a moving light interference pattern. We used the following material constants: ya = 1.65 X IO-” m3/s, No = 1O26 rne3, NA = 2 X lO22 rnm3, s = lo-’ m2/J, CL= 10m5 m*/(V s). The calculated resonance velocity, from relation (4.12), corresponding to our experimental parameters K = 3.1 X lo5 m-l, E. = 6 X lo5 V/m and I, = 30 W/m* attains the value V, = 0.17 mm/s, and the resonance frequency is 8.3 Hz, or 52 s-‘. This is in good agreement with our experimental result o, = 6.2 Hz, regardless of some uncertainty in the sNo value. The direct measurements of the fringe velocity for the resonance condition at contrast m = 0.4 give the value V, = 0.18 mm/s (Fig. 1 I>. In the computer simulation of the periodic electric field distribution in the crystal we applied a new technique of calculation based on the assumption of the convergence of the solution to a stable limiting cycle. The choice of the used y/K value can again be verified by substituting the material parameters, which gives y/K = 0.18. The main features of the resonance space-charge wave excitation observed experimentally were obtained in the numerical simulations, namely the existence of the low-frequency peak and a frequency shift with the increase of the modulation. The calculations also demonstrated the resonance behaviour of the photoinduced current, which was also observed experimentally. Both calculated (Fig. 4) and measured dependences (Fig. 11) demonstrate an electric current damping at the resonance condition by a factor of about 2 with respect to its value far from the resonance. The results are in good coincidence with recent reports [9,10]. We note, however, that in contrast to the nonlinear equation derived in Ref. 161,the sign before the dissipation
M. Vusnetsov et d/Optics
Communiccrtions
is positive in our derivation. The described is also very suitable for the determination of the product sN,, (for our sample we got the value about ten times larger than the standard value [8]) and for study of such nonlinear effects as subharmonic generation [6] and high frequency resonances in photorefractive crystals [ 121. term
y
in Q.
experimental
(4.14)
technique
Acknowledgements Sergei Lyuksyutov and Mikhail Vasnetsov acknowledge support from the Danish Technical Research Council Grant No 9400102. We thank Floris Kharadaylo for assistance during the experiments.
References [I] A. Ashkin, G.D. Boyd, J.M. Dziedzic, R.G. Smith, A.A. Ballman, H.J. Levinstein and K. Nassau, Appl. Phys. Lett. 9 ( 19661 9.
137 (1997) 181-191
191
121N.V. Kukhtarev, Sov. Techn. Phys. Lett. 2 (1976) 438. [31 N.V. Kukhtarev, V.B. Markov, S.G. Odoulov, M.S. Soskin and V.L. Vinetskii, Ferroelectrics 22 (1979) 949. t41J.P. Huignard and A. Marrakchi, Optics Comm. 38 (1981) 249. El S.I. Stepanov, V.V. Kulikov and M.P. Petrov, Optics Comm. 44 (1982) 19. [61B.I. Sturman, A. Bledowski, J. Otten and K.H. Ringhofer, J. Opt. Sot. Am. B 9 (1992) 672; B.I. Sturman, M. Mann, J. Otten and K.H. Ringhofer, J. Opt. Sot. Am. B 10 (1993) 1919. [71 S. Bian and J. Frejlich, Optics Lett. 19 (1994) 1702. [81 Ph. Refregier, L. Solymar, H. Rajbenbach and J.P. Huignard, J. Appl. Phys. 58 (1985) 45. [91 T.E. McClelland, D.J. Webb, B.I. Sturman, M. Mann and K.H. Ringhofer, Optics Comm. 113 (1995) 371. [lOI G.A. Brost, K.M. Magde, J.J. Larkin and M.T. Harris, J. Opt. Sot. Am. B 11 (1994) 1764. t111 P. Buchhave, S.F. Lyuksyutov and M.V. Vasnetsov, Optics Lett. 20 (1995) 2363. [I21A. Gnmnet-Jepsen, C.H. Kwak, I. Richter and L. Solymar, J. Opt. Sot. Am. B 11 (1994) 124.
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications 137 (1997) 181- 191
Full length article
Phase modulation spectroscopy of space-charge wave resonances in Bi,,SiO, Mikhail Vasnetsov ‘, Preben Bud-have *, Sergei Lyuksyutov Depurtment
ofPhysics,
’
Optics Group, Technical University of Denmark, Bid. 309, DK-2800 Lyngby, Denmark
Received 2 July 1996; revised 24 October 1996; accepted 26 November 1996
Abstract A new experimental method for the study of resonance effects and space-charge wave excitation in photorefractive Bi I2SiO,, crystals by using a combination of frequency detuning and phase modulation technique has been developed. The accuracy of the method allows a detection of resonance peaks of diffraction efficiency within 0.5 Hz. Numerical simulations of the nonlinear differential equations describing the behaviour of the space-charge waves in photorefractive crystals have been performed and found to be in a good agreement with experiment. We have measured the photocurrent through the crystal and revealed its resonance dependence. A minimum of electric current through the sample corresponds to the main resonance detected by phase modulation technique. PACS: 42.65
1. Introduction The photorefractive effect has been intensively investigated since the first experiments were reported demonstrating the refractive index change in a LiNbO, crystal under influence of light illumination [l]. Free carrier generation due to the excitation from traps in illuminated areas and charge migration into dark regions lead to the formation of the internal space-charge field which modulates the crystal refractive index via the electro-optic effect. The specific nonlinear material response to the spatially inhomogeneous illumination in photorefractive crystals permits a recording of phase volume holograms which may possess high diffraction efficiency. One of the most prominent features of the photorefractive crystals is the possibility to record and read the holograms in real time (dynamic holography) [2,3]. The light-induced gratings which are the result of
* Corresponding author. Fax: t(45) 45 93 16 69; E-mail: [email protected]. ’ On leave of absence from the Institute of Physics, National Ukrainian Academy of Sciences, 252 650, Prospekt Nauki 46, Kiev, Ukraine.
small variations of the material refractive index in bulk crystals can follow the moving light pattern within the limitations defined by the response time of the material. High speed materials such as Bi ,2Si02,, (BSO) crystals are very promising for dynamic holography applications. The running grating technique in combination with the application of an external dc electric field significantly increases the amplitude of the space-charge field and therefore the grating strength [4,5]. The resonance bchaviour of the running grating strength measured by its diffraction efficiency with respect to the velocity of the grating in a crystal was the reason for the introduction of the idea of a space-charge field in the form of a running space-charge wave [6]. Our goal is to investigate both experimentally and by numerical simulations the main features of the resonance space-charge wave excitation in photorefractive BSO crystals. In this paper we introduce a new experimental technique for the study of space-charge wave excitation and resonance phenomena based on the combination of a frequency shift between the two coherent beams writing a grating and phase modulation of one of the beams. The phase modulation technique [7] permits the production of
0030~4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00754-7
182
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running light fringes with variable contrast, which is important in order to study the dependence of the space-charge wave excitation on the light modulation. We detected a shift of about 60% of the resonance frequency as a result of variations of the modulation. The resonance behaviour of the photocurrent through the crystal was also established experimentally. In the numerical calculations a new technique that gives us the possibility to determine the resonance features and spatial structure of a space-charge wave was used. The structure of this paper is as follows: Section 2 describes the experimental arrangement used for the investigation of the running gratings in the BSO crystal. Section 3 deals with the detailed description of the phase modulation technique. Section 4 gives the theoretical description of the running resonant photorefractive grating excitation by a moving light interference pattern. In Section 5 the experimental results are discussed, and conclusions are summarised in Section 6.
2. Experimental
setup
The experimental setup is shown in Fig. 1. The light source is a single mode Ar-ion laser operating at a wavelength of 514.5 run. The light is brought from the laser to the optical table through a single mode optical fibre. A radial holographic grating serves as a combined beamsplitter and frequency shifter. When the grating rotates, the first and minus first order diffracted beams are frequency shifted by equal amounts of So, but with opposite sign. This arrangement allows precise control of the frequency shift over a wide range without phase jumps. The direct transmitted beam with unshifted frequency w is blocked, and the two diffracted beams of equal intensities and with frequency detuning 60= f(2?r/a)~, where v is the
powt?r meter / or photodetector
137 (1997) IN-19I
angular velocity of the grating and (Yis the angular grating spacing, are collimated, expanded and directed to intersect in the BSO crystal. The BSO crystal with dimensions 10 X 10 X 3 mm was antireflection coated on the input and output surfaces. Gold electrodes were plated onto the side faces of the crystal. A dc electric field E, = 6 kV/cm was applied perpendicular to the (1 IO) surface. The intersection angle between the writing beams 28 was 1.47” resulting in a fringe spacing A = 20.1 pm, The total Ar-laser power incident on the crystal was 3 mW/cm’. Both incident waves were polarised perpendicularly to the plane of incidence. Phase modulation was applied to one of the writing beams via a piezoelectric mirror driven by a sinusoidal voltage from a frequency generator. The accuracy of the frequency control was about 0.1 Hz. Low power He-Ne laser radiation was used as a probe beam to test the diffraction efficiency of the running gratings. The angle of incidence of the probe beam was Bragg-matched to the spatial frequency of the grating recorded in the crystal. The intensity of the diffracted wave which serves as a relative measure of the diffraction efficiency of the grating was picked up by a photodetector and digitised by an oscilloscope. An electric current through the crystal in the direction of the wave vector of the running grating has been measured. With an applied electric voltage of 6 kV and for an incident light intensity of 3 mW/cm2 typical values of the current were of the order of 30 pA.
3. The phase modulation technique The general scheme for analysing the running grating excitation in the BSO crystal is as follows. Two coherent light waves of electric field strength amplitudes E, and E, intersect in the crystal to record a grating in a symmetrical scheme with the intersection angle 20. The frequency shift 260 is created between the waves by the rotating diffraction grating (Fig. 1). the first wave has positive shift &IJ,
(3.1) where k is the wavevector modulus 27r/h where A is the wavelength of the incident light. The other wave has negative shift - 60, and may be phase modulated by a piezoelectric mirror with the amplitude of modulation A and frequency R: E,(x,z,t)=E,exp(i[k(xsin0-zcos0) He-Ne
laser
mlrmrs
Fig. 1. Experimental setup for phase modulation spectroscopy. Inset: rotating diffraction grating producing frequency shift 260 betweenwriting beams.
-6wt+Asin
at]},
(3.2)
where E, and E, are real amplitudes and the phase factor describing the light frequency w has been omitted. The distribution of the resulting light illumination I(x,t) on the
M. Vusnetsov et d/Optics
Communicdons
crystal input face ( ;: = 0) has, apart from a constant of proportionality, the form: I( XJ) = lE,( XJ) + E,(XJ)12.
(3.3)
After evaluation we have I( x,?) = Ef + Ez + [E, E, exp[ i(2kx sin 0 -2Sor+Asin
fir)] +c.c.].
(3.4)
Further we shall use the following notations: m = 2E,E,/(Ef + Ez) (modulation); 1, = E: + Ez (total light illumination); K = 2k sine (grating wavenumber); and decomposition exp[i(A sin fir)] = Cy= _m J,(A) exp(ilRt), where J,(A) are Bessel functions of order 1 with argument A. Inserting into (3.4) results in I(x,f)=I,
1 +T i
i
J,(A)(exp[i(Kx
,= --Ix
.
-260t+lRt)]+c.c.}
I
(3.5)
Light intensity distribution in the form (3.5) is a decomposition into an infinite set of running light fringes with the same spatial frequency K but different velocities V, = (260 - lfi)/K. For a negative sign of the term 2 SW - I R, the fringes move backwards with respect to the x axis. The contrast of the light fringe pattern running with the velocity V, is a product of m and the Bessel function of the corresponding order 1 with argument A. Below we analyse three combinations of frequency shift 2 60 and the frequency 0 produced by phase modulation: (i) no phase modulation, A = 0; (ii) both frequency shift and phase modulation are applied to the writing beams; (iii) frequency shift 26~ is zero, phase modulation is active. 3.1. Space-charge wave excitation without phase modulation
= 1 + m cos( Kx - 260 t) = 1 +mcos[K(x-V,,t)]
(3.6)
and only one cosinusoidal light fringe pattern exists running with the velocity V, = 2&0/K. This case is thoroughly investigated theoretically and experimentally [6,8lo]. Periodical running light fringes write a photorefractive grating, which is supposed to have the same fundamental spatial frequency. To describe the grating diffraction efficiency n, we introduce the response function f(m,V,,>, which is the amplitude of the fundamental spatial harmonic of the internal space-charge field as a function of the interference
183
pattern contrast and the velocity of the light fringe motion. The amplitude of the probe wave diffracted in the Bragg diffraction order Ed will be proportional to the grating strength, which is in turn proportional through the electrooptic coefficient to the space-charge field ES, (more precisely to the fundamental component of the space-charge field flm,V,) cos[ K(x - V,t)]: Ed aflm,V,)). The measured diffraction efficiency n of the running grating is thus proportional to the square of the modulus of the response function, rj a If(m.VJl*. As follows from the dependence of diffraction efficiency on the frequency detuning [6,8], the response function has a maximum (resonance) for a certain frequency w,, or the fringe velocity V, = w,/K, which depends on the light intensity, applied voltage, material parameters and the fringe contrast. Far from the resonance the diffraction efficiency is negligible and flm,V,) = 0. The analytical approach to the determination of the response function is discussed in Section 4. In order to determine the main features of the response function experimentally, we may scan the frequency shift 280 from negative values to positive ones and measure the corresponding diffraction efficiency n of the running grating. In this way we can detect the position of the resonance and determine the shape of the resonance curve $2 6~) for the given value of the modulation m and light intensity 1,. 3.2. Combination offrequency
shift and phase modulation
In the case when both frequency shift and phase modulation are applied to the writing beams, a variety of running light patterns with the same spatial period but different velocities and contrasts are present. The resonance will occur for the following combinations: 2s030r
(f=O);
2&w-R=o, 2&0+n=
In the case when the phase modulation is not applied to the writing beam, (3.5) turns into I( x,r)/l,
137 (1997) 181-191
(I-l); 0,
(I-
(3.7) -1);
and analogously for the following I values. To simplify the analysis, we shall take the frequency of phase modulation 0 higher than the resonance frequency o, so that only one frequency combination 260 - In can appear within the region of the resonance. At least three variables may be used to test the diffraction efficiency 7: 0, 2Sw and A. If we take a constant value of fi > w, and scan the frequency shift 2Sw, we shall observe the main resonance corresponding to I= 0 for the grating recorded by fringes running with a contrast ml,(A) and velocity V, = 2&0/K = V, corresponding to the condition 2 So - w,: 1,[1 +m&(A)cos(Kx-26wt)] =I,,{1 +mJ,(A)cos[K(x-V,,r)]}
(3 *8)
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and additional peaks for combinations of the frequency shift and phase modulation frequency producing the running fringes with the same velocity (2 SW- 10 )/K = V,.: I,[1 +mJ,(A)cos(Kx-26ut+lRr)] = IO{ 1 + mJ,( A) cos[ K( x - V/t)]}.
(3.9)
As seen from (3.8), the shape of the main peak will correspond to the modulation depth of the light distribution or fringe contrast, which is equal to ml& A). Varying the amplitude of phase modulation A, we can easily detect the dependence of the resonance curve on the light fringe contrast without any change in total light intensity incident on the crystal. With zero phase modulation (A = 0) we shall have the same resonance curve as discussed in the previous subsection, for J,(O) = 1. For A f 0 additional peaks appear, which are shifted from the resonance frequency 0, by +lR. These new resonances display a reduction in fringe contrast according to the dependence mJ,( A> in relation (3.9). Another way to measure the resonance curve is to keep the frequency shift 260 constant (larger than the resonance frequency mr) and scan the frequency of the phase modulation R. The fringe velocity will be close to the resonance value when the condition 26~ - I.0 = w, is satisfied. Again, a set of peaks occurs with different 1. The first peak will appear for 1= 1 corresponding to the moving fringe pattern: I,[1 +mJ,(A)cos(Kx-26wt+Rr)] =I,{1
+m.J,(A)cos[K(x-V,t)]}.
(3.10)
Changing A from zero to - 1.9, we have the possibility of varying the contrast &,(A) from zero to = 0.58, which is the maximum of the function J,(A) (m = 1 for equal writing beams intensities). From the point of view of experimental convenience, this combination is preferable, because we can control the frequency of the sinusoidal voltage driving the piezoelectric mirror with higher accuracy than the angular velocity of the rotating grating. By measuring the resonance curves of the diffraction efficiency as a function of 0 for different values of the amplitude A, we are able to obtain a de ndence of the p” square of the response function IjfV,)l on the fringe contrast.
137 (1997) 181-191
the stationary grating will be detected. When R is smaller than the resonant frequency of the space-charge excitation w,, several frequency components may be located within the region of the resonance (V, = V,). For the next analysis, two assumptions are made. First, we suppose that the electric space-charge field inside a crystal has the same spatial structure as the light distribution: only running waves of the form exp(iKx + ilot) are involved. (In the general case this is not valid, because higher spatial harmonics with grating vectors 2K, 3K, ... may be excited, and also subharmonic components with grating vectors K/2 and others may occur.> To obey this condition, we need to restrict ourselves to weak contrast of running light fringes, i.e. A +C 1. Secondly, each running component of light distribution is responsible for the corresponding component of the space-charge field: Es,=
f(m,A,l~){exp[i(Kx+l~t)]+c.c.},
i
t= -zz (3.11) where the amplitude of the space-charge field component flm,A,la) is a response function for frequency 10 with parameters m ,and A. This function describes as before the relative values of space-charge field components created by light fringes running with velocities V,. Further, we suppose proportionality between the running refractive-index grating strength g, and the corresponding component of the space-charge field: g,(m,AJn)
(3.12)
af(m,A,lfi),
and the whole time-dependent grating recorded by the phase-modulated wave and the ordinary wave has the form of a composition of running gratings: G(x,t)a
i f(m,A,lR){exp[i(Kx+lflt)]+c.c.}, ,= -r (3.13)
(the permanent variation of dielectric permittivity due to the constant applied field E, is omitted). In the approximation of small diffraction efficiency, the amplitude of the Bragg-matched diffraction order will be proportional to the grating strength. For the incident probe wave the amplitude of the Bragg diffracted order will have time dependence
3.3. Phase modulation spectroscopy
Ed(t) a
In the case when the frequency shift 280 is zero, the set of the light fringes running with velocities V, = -lo/K and contrast ml,(A), corresponding to the frequency of the phase modulation 0 and integer multiples of it, are present in the light distribution. Fringes running towards the x-axis have positive velocities, and vice versa. A stationary light fringe pattern corresponding to I = 0 also exists. When the frequency of phase modulation R is much larger than the resonant frequency, only the response from
and the diffraction efficiency
2 f(m,A,lfI)e-““‘, I= --P
(3.14)
2
Tot
i f(m,A,lLf)e-““’ ,= --r
.
(3.15)
The analysis of the response function flm,A,l.fI) the time dependence of the diffraction efficiency r] general case is rather complicated. To simplify the lem, we neglect all terms but flm,A,O) and flm,A,
from in the prob- 0)
M. Vusnersou er d/Optics
Communications 137 (1997) 181-191
X e’“’ due to the fact that running light fringes with negative velocities (I> 0) cannot be in resonance, and negligible contrast rnJ,( A) for 1II > 1. For two remaining terms flm,A,O) =flO, and flm,A,C2)ein’=fe’“’ we can write (3.15) as 7~a If(O) +feinr12.
(3.16)
The multiplication will give the (asterisk means complex conjugation):
following
r] a ]f(O)l’ + IfI* +f(O)
(0) fe”“.
f” eCir” +f*
no a If(
and a time-dependent qfjaf(0)f*
efficiency
con-
(3.18)
+ Ml’. term oscillating
at frequency
eelR’ +f’(O)fe’“‘.
0: (3.19)
The complex response functions fl0) and f may be determined from the measurement of Q and no. The relation (3.19) can be rewritten as nf)aIf(0)flcos(nr+
cp).
4. Space-charge wave excitation
dN,+ -=d(N,-N&) at
- yRN,fn.
we need to by the light the x-axis differential [2,3]:
(4.1)
aj dP x++=o,
(4.2)
JE E&O- = p,
(4.3)
dX
p=e(N,+-N,-n),
(4.4)
an j = plc,Tz
+ penE,
(4.5)
where e is the absolute charge of the electron, p is the electron mobility, k, is Boltzmann’s constant, T is the absolute temperature and NA is the density of compensating negatively charged centres. To simplify the analysis, we neglect the contribution of free electrons to the spacecharge distribution p and rewrite (4.4) as p=e(Nz
-NA),
orNz
-N*+p/e.
(4.6)
Following the method used in Ref. [6] to obtain the equation for the space-charge field amplitude in a stationary regime of running light fringes, we may rewrite Eqs. (4.1)-(4.6) in a coordinate system moving with the light fringes. Using the variable .$ = x - Vat and neglecting the term responsible for the diffusion, Eqs. (4.1)-(4.6) may now be combined into one differential equation for E( 5 ):
(3.20)
where cp is the phase shift between the running interference fringes and the space-charge field. The amplitude of oscillations of the diffracted wave intensity with the frequency 0 is found to be proportional to IflO)fl, that may be used to find the position of the resonance and its dependence on running fringe contrast (within the small contrast limitation).
To calculate the response function fim,V,), determine the space-charge field E,,( x,t) built interference pattern I = I(x,t) running along (3.6). We start from the well-known set of equations describing the photorefractive effect
section for the active center photoexcitation, ~a is the recombination constant and p and j are the charge density along the x axis and the electric current defined as
terms
(3.17) As follows from (3.17). the diffraction tains a steady-state term
18.5
where E is the internal electric field, which is a sum of the field E, produced by the applied voltage and the spacecharge field ES, created by charge redistribution, No, Ng, are the total density of active centres and density of ionised centres (positively charged) respectively, n is the concentration of free photoexcited electrons, s is the cross
Volvo d2E/a.g 2 + sl( eN, - eN, - &co aE/&g ) PeE
x(
aE + Vo.sxO-- = j,, 86
eN, +
880
aE/%
>
(4.7)
where jo is a constant of integration, having a physical sense of electric current density across the crystal. Eq. (4.7) may be further simplified if the relatively small term in parentheses EEL i)E/i)S could be omitted:
(4.8) where No., = N, - NA_ When the light fringes do not move (V, = 0) the standard solution appears E = (joy, NA)/( peslkdN,_,); for moving fringes Eq. (4.8) becomes a nonlinear second-order differential oscillatory equation. The light distribution I( 5 ) = I,[ 1 + mcos( K 5 )] stands for the forcing term in Eq. (4.8) in the case of moving fringes. For a particular value of V, the amplitude of E( 5) may exhibit resonant behaviour, i.e. a running space-charge wave is excited. We note that the current j(, will depend on the value of the free electron concentration and hence may also show resonance features. To analyse the resonance conditions, we let the modulation m be small, which results in a small value of I E,( 5 )I compared to the applied field E,. First, j, = ~eiiE,, where ii = sle ND_J~a NA is the average free electrons concentration. Without the small nonlinear terms proportional to E,,Er,
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linear approximation for m (< 1 is the amplitude of the space-charge wave (4.10). To analyse the behavior of the space-charge wave I?,,( 5) for the case of arbitrary m the nonlinear secondorder differential equation (4.8) could be calculated numerically. As we seek a stationary solution in the form of a running wave and suppose that this solution does exist for the nonlinear second-order differential equation (4.8). the existence of a stable attractor (limiting cycle) is further postulated. Eq. (4.8) may be rewritten for more convenience as
and mE,, we obtain an equation for a linear oscillator driven by an external force:
~w,Eo”o YRNA +
tL+%A
EzC+ V, .wOE;, +
&eND., YRN.
ES,
YRNA
EOmcos( Kt)
= 0,
where a prime denotes derivative with respect to 5. The solution has a form of a running wave E,,( 5) mEOsIOeND_,cos( Kt+
=
137 (1997) 181-191
(4.14)
yy”+yy’+Hy(l+mcosKl)=s”J,,
‘p)
where
y( 5) = HO/Eat E,, and
Y = ynNA/pEo+
B =
sI,~N,.,/EE,V~ J, = j,-
where the phase shift cp between the light fringe pattern and the space-charge field E,, is determined as cp= arctan
YRNAEEO”OK
PC EE~E~V,,K~ - sI,eN,.,)
The resonant condition is easily derived from (4.10) as the space-charge field amplitude has a maximum when the fringes move with a velocity
V,., (4.12)
The same condition turns phase shift cp to 7r/2. In the analysis used in Ref. [8] the calculated optimum velocity
has a negative sign due to the definition of the charge e; we use the absolute value of the electron charge instead of the negative in Ref. [8]. In a physical sense the drift of electrons in the applied field occurs in the opposite direction with respect to the fringe motion. The resonance frequency may also be determined as or-KV,----
do eN,_
A
.q,EOK
(4.13)
’
The resulting response function flm,V,,> derived in the
0.0 0.0
0.5
1.0
1.5
2 0
) 0
sI, eEOND. A
, 10
1 20
30
40 s
EWE, (a)
El.
is the normalised current, far from the resonance or for very small modulations J, = 1. We used the following numerical values for the coefficients: m = 0.4, K = 2, y = 0.3 (the choice of the value y/K = 0.15 is explained in Section 6), B ranged from 2 to 200, J, was usually less than 1 and was determined by a fitting procedure described below. In numerical simulations the periodical function y( 5) will come closer and closer to the attractor, independent of the initial values of y(O) and y’(0). Starting from arbitrary (with some reasonable choice) initial values, we integrate Eq. (4.14) along the 8 axis and wait until the periodic structure of the space-charge field will not change more with the growth of 5: this is evidence that the steady-state solution is reached. Then Fourier transform is used to calculate the power spectrum of the oscillations with respect to the spatial frequency. The response to the fundamental spatial frequency K and higher harmonics 2K, 3 K, ... (if necessary) is determined, and their dependence on fringe velocity V, and modulation m may be found. The average value of the calculated electric field E( 5 ) must be equal to the applied field E,. To reach the average value ( y( 6 )) = 1 we need to vary the parame-
(4.11)
’
1
YRNA
(b)
0
K 2K 3K 4K 5K 6K
Spatial frequency CC)
Fig. 2. Numerical calculations. (a) Limiting cycle corresponding to the periodic solution of Eq. (4.14) with parameters m = 0.4, y = 0.3, K = 2. (b) Electric field distribution along the t-axis. (c) Power spectrum of the space-charge field (E(c) - &J/E,. Normalised fringe velocity a- ’ = 0.08. Calculated current J, equals 0.65. Excitation of high spatial harmonics of the space-charge field is clearly seen as peaks corresponding to spatial frequencies 2 K, 3 K, ... (the peak width is determined by numerical FFT procedure).
Vusnetsoo er d/Optics
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, 0.0
/
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1.0
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o.oj,. 15
2.0
:, 0
10
I, 20
ter J, until the component in the oscillation spectrum corresponding to zero spatial frequency becomes unity. In our calculations the Fourier transform was applied to the function y([ ) - 1 = E,,(~)/E,, and we reach zero for the average ( E,,( 5 1) value by a proper choice of ./a. Fig. 2 demonstrates the limiting cycle in coordinates E( 5 j/E, = y( 51, E’( 5 I/E, = y’( 5) and the shape of the normalised field E( 5 )/E, combined with a power spectrum of the space-charge field E,,( .$I/E, obtained in the calculations for the nonnalised fringe velocity Es”-’ = 0.08. The calculated diffraction efficiency of the running grating is thus proportional to the magnitude of the response on the fundamental spatial frequency K. Second and higher spatial harmonics are excited in addition to the fundamental component of E,,(t). Similar calculations for the normalised fringe velocity E- ’ = 0.3 1 are shown in Fig. 3. Mostly oscillations of the space-charge field in the fundamental spatial frequency are excited in this case. Further results of the calculations are plotted as a dependence of the diffraction efficiency on the normalised fringe
10
1.0
/-------
A--
1
09 0.8
0.8
m z-. 0.7 E” : $ ” P g
0.6 0.5 0.4
00
01
0.2
03
04
0.5
06
0.7
0.8
0.9
10
181-191
0.0 40
187
11 0
K
2K
3K
4K
5K
6K
Spatial frequency Cc)
W
Fig. 3. The same as in Fig. 2 but for normalised fringe velocity 8-l spatial harmonic of the space-charge field is excited.
$f
I,
5
(a)
3
(1997)
30
EWE,
09.
137
= 0.31. Calculated current J, equals 0.57. Mostly the fundamental
velocity in Fig. 4. The resonance behaviour of the diffraction efficiency around E:“- ’ = 0.4 is evident, the excitation of the running space-charge wave is optimal. Despite this resonance, the diffraction efficiency at low fringe velocities also demonstrates the resonance peaks. According to the theoretical model proposed in Ref. [6] the excitation of the fundamental harmonic of the space-charge field wave may occur also via excitation of its higher spatial harmonics. As they have spatial frequencies twice, triple and larger (2 K, 3K, ...) than the fundamental spatial frequency, the resonance fringe velocity for them will be four, nine etc. times lower that for the main resonance. It is seen from Fig. 4, that the excitation through the second spatial harmonic has the maximum around ? ’ = 0.1, what is four times lower than that for the main resonance. We detected in numerical calculations a hysteresis around the value s-’ = 0.1, which may be the consequence of a strong interaction between the fundamental and the second spatial harmonic of the space-charge field. The calculated dependence of the normalised current through the crystal Jo is also shown in Fig. 4. The normalised current tends to unity outside the resonance region and decreases at the resonance, both around the peaks of the main resonance and at higher harmonic resonances. Using the method of numerical simulation of the steady-state space-charge field, we calculated the normalised diffraction efficiency for a small value of modulation (m = 0.01) and for a high value (m = 0.99). In the small-modulation regime the effects connected with the resonance wave excitation through high spatial harmonics disappear and only the main resonance exists. High modulation results in the appearance of nonlinear effects: the excitation of the space charge wave through higher harmonics becomes even more efficient than the direct excitation, also the shift of the main resonance position was detected.
1.1 1.2
Normalisedfringevelchty5'
Fig. 4. Numerical calculations. Dependence of the diffraction efficiency (solid line) and current J, (dashed line) on the normalised velocity of interference fringes 5- ’ .
5. Experimental
results and discussion
In our experiments we keep equal intensities of the writing beams to avoid change of the total light intensity
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0
Frequency
10
20
detuning
I37 (1997) 181-191
30
40
26~ (Hz)
Fig. 5. Experiment. Dependence of diffraction efficiency on frequency detuning 260. Total intensity of light is 3 mW/cm’.
when changing the beam intensity ratio. The first set of experiments was performed without application of the phase modulation to the writing beam. The frequency shift 2Sw produced by the rotating grating was measured from the beatings between writing beams. By changing the direction of the grating rotation, we were able to scan the frequency shift 260~ from negative to positive values with accuracy within 0.5 Hz. Fig. 5 represents the dependence of the diffracted signal with a maximum located around 1 Hz. This peak is known as a low-frequency maximum [9]. The shift of its position from exact zero detuning was specially checked. The measured diffraction efficiency shows a decrease at the region of negative frequency shift, while there is another maximum for positive values around 9 Hz, which we shall denote as the main resonance. No other maxima were detected. The second set of measurements included the phase modulation of one of the writing beams by a piezoelectric mirror. We controlled the amplitude of the phase modulation A in the following manner. Keeping the frequency of the phase modulation far from the resonance position, we detected the maximum of the diffraction efficiency at the resonance and changed the amplitude of the driving voltage I/ of the frequency generator. The contrast of the running fringes equals mJ,(A) and becomes zero when J,(A) = 0: no grating is recorded. As the amplitude of the piezoelectric mirror displacement is proportional to the applied voltage, we define the proportionality coefficient between A and Cr. The value of U = U. when diffraction efficiency drops to zero corresponds to the first zero of the J,(A) Bessel function: A = 2.4. Thus the driving voltage U was calibrated in terms of the amplitude of phase modulation: A = 2.4U/Uo. Fig. 6 demonstrates the resonances obtained. The applied frequency of the piezoelectric mirror vibration 0 was 35 Hz and the frequency shift produced by rotating grating 2Sw was varied. The used value of A was 1,
which gives J,(A)-0.77, J,(A) -0.44 and J,(A) = 0.11. Some changes in the central resonance curve in comparison with the one plotted in Fig. 5 are observed: The decrease of the modulation from 1 to 0.77 leads to the relative suppression of the low-frequency maximum. The position of the main resonance is now around 7 Hz which is somewhat less than in Fig. 5. Side resonances corresponding to I = &-1 are shifted at the value of the frequency D = 35 Hz, also resonances for 1= + 2 are resolved. The shifted peaks with the depths of modulation 0.44 and 0.11 do not exhibit low-frequency maxima at all. To resolve the small changes in the position of the main resonance, we varied the amplitude of the phase modulation A from zero to the value A = 2.4 corresponding to the first zero of the .I,( A) Bessel function. For each value of A the position of the main resonance maximum was detected. The data obtained are plotted in Fig. 7 as the resonance frequency w, as a function of the value of
Frequency detuning 280 (Hz)
Fig. 6. Experiment. Dependence of diffraction efficiency on frequency dehming 260. Phase modulation is imparted on one of the beams at frequency D = 35 Hz. Side resonance peaks are shifted with respect to the main resonance by 35 Hz.
M.
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Vasnetsov
et al./Optics
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137 (1997) 181-191
7-400 2 d E
300
s 'G E .E
200
5 J! cl 100
. 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-6 -4 -2 0
Fringe contrast
2
4
6
8 1012141618202224262830
Frequency difference 260 - C?(Hz)
Fig. 7. Experiment. Dependence of resonance frequency on the contrast of the interference pattern (light modulation depth).
modulation of the running light fringes. The observed shift exceeds the experimental error and amounts to about 3.6 Hz from the lowest to highest values of fringe contrast. We note that the position of the resonance is very sensitive to the light power incident on the crystal. Therefore the absolute values of the resonant frequency may differ in different sets of measurements due to the laser output power variations, but the total shift remains the same. In addition to the measurement of the maximum of the diffraction efficiency position we also detected its absolute value, the corresponding dependence is plotted in Fig. 8. The origin of the low-frequency peak was observed in the
Fig. 9. Experiment. Dependence of diffraction efficiency on the frequency difference between constant frequency shift 280 and variable frequency of phase modulation R. Open circles correspond to contrast 0.13; full circles correspond to contrast 0.58.
set of measurements with constant frequency shift 260~ and varied frequency of phase modulation 0. Two resonance curves are shown in Fig. 9 for the modulations 0.58 (maximum for J,(A) Bessel function) and 0.13 as the dependence of the diffraction efficiency on the difference between frequency shift 280 and the frequency of phase modulation 0. The curve for small contrast (0.13) does not contain the low-frequency peak, whereas the appearance of the peak is resolved for a contrast of 0.58.
120 110 100
. 400
l
. .
s
90
G
80-
6
s
d g
2 1
.
300
.E $
.
.Fj 200
6L
s t
40-
::
30.
z
20-
,/’
/
50
02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fringe contrast
Fig. 8. Experiment. Dependence of diffraction efficiency on the contrast of interference pattern.
”
,A”
/I’
’
0.04
/”
/’
0
,,,I’,,,,,‘,,
I,
0.1
,/
/’
. 0.0
0.08
/’
”
10
. 0
, / -‘k L
Contrast 0.15
,’
.
100
g $ s
4
/
.
5 e g
70 -
5
10
15
20
25
30
Frequency of phase modulation Q
35
40
(Hz)
Fig. IO. Experiment. Amplitude of Fourier component of the diffracted signal for a particular value of phase modulation fl as function of 0 for three values of light fringes contrast.
190
M. Vasnetsov et al./Optics 1.0
Communications
. l
. .
0.9
$ g
0.7-
a 0 5 E
0.6 -*
.
05-
(lr
. l
. 0.4
,
(
,
,
/
,
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0 1.1 Velocity
of fringes
(mm/s)
Fig. 1I. Experiment. Dependence of photocurrent through the crystal j normalised to its maximum value j,,,,, on velocity of interference fringes. Applied electric voltage is 6 kV and fringe contrast is 0.4. The electric current minimum corresponds to the main resonance of space-charge wave excitation, 260~ = 8.5 Hz.
To complete the measurements of the dependence of the main resonance maximum on the value of modulation, we used phase modulation without frequency shift (2 60 = 0) between the writing beams. In accordance with the description of the phase modulation technique given in Subsection 3.3, an amplitude of phase modulation was chosen to be small. The resonance curves shown in Fig. 10 correspond to the modulation depths 0.04, 0.08 and 0.15. The value of the response was measured as the Fourier component of the time-dependent signal from the photodetector corresponding to the frequency of phase modulation 0. No frequency shift of the resonance discussed above was detected for these small modulation values. There is no contradiction with the measurements of the resonance frequency shift as plotted in Fig. 7: resonance frequency w, seems to be independent of the fringe contrast in the region of small modulations. In this experimental arrangement we also measured an electric current, which is induced in the BSO crystal by the applied voltage and the light illumination. Photoexcited carriers in the crystal move in the applied electric field thus creating a photocurrent (parameter j, in Eqs. (4.7), (4.8)). The value of the current depends on the carrier modulation. The higher the modulation, the lower the current. We have experimentally detected that the minimum of the current exactly corresponds to the resonance of the space-charge wave (Fig. 11). This is in good agreement with our numerical calculations.
6. Conclusions Phase modulation technique was used to study the main features of the resonant excitation of photorefractive running gratings in BSO induced by moving light fringes. The
137 (1997) IN-191
merits of this technique are the possibility to vary the contrast of the running light fringes without changing the total light intensity and easily control the velocity of the fringe motion. When the velocity of the running fringes is close to the resonance velocity of the space charge wave propagation, strong enhancement of the running grating diffraction efficiency is detected. We observed the development of a low-frequency peak with the growth of light fringe contrast. This low frequency peak is related to the nonlinear effect of space charge wave excitation through higher spatial harmonics. The appearance of additional peaks due to space-charge wave excitation by a combination of the frequency shift produced by the rotating grating and a phase modulation generated by vibration of a piezoelectric mirror was detected. This effect explains the periodical diffraction efficiency peaks observed by us earlier when the crystal vibrated [I 11. The high resolution of the phase modulation spectroscopy technique enables us to measure the resonance frequency dependence on the contrast of the light fringes. We detected a shift of the resonance frequency from 6.2 Hz to 9.8 Hz, which amounts to about 60% of the resonance frequency. The results obtained in the experiments are compared to numerical simulations based on the model of a running grating recorded by a moving light interference pattern. We used the following material constants: ya = 1.65 X IO-” m3/s, No = 1O26 rne3, NA = 2 X lO22 rnm3, s = lo-’ m2/J, CL= 10m5 m*/(V s). The calculated resonance velocity, from relation (4.12), corresponding to our experimental parameters K = 3.1 X lo5 m-l, E. = 6 X lo5 V/m and I, = 30 W/m* attains the value V, = 0.17 mm/s, and the resonance frequency is 8.3 Hz, or 52 s-‘. This is in good agreement with our experimental result o, = 6.2 Hz, regardless of some uncertainty in the sNo value. The direct measurements of the fringe velocity for the resonance condition at contrast m = 0.4 give the value V, = 0.18 mm/s (Fig. 1 I>. In the computer simulation of the periodic electric field distribution in the crystal we applied a new technique of calculation based on the assumption of the convergence of the solution to a stable limiting cycle. The choice of the used y/K value can again be verified by substituting the material parameters, which gives y/K = 0.18. The main features of the resonance space-charge wave excitation observed experimentally were obtained in the numerical simulations, namely the existence of the low-frequency peak and a frequency shift with the increase of the modulation. The calculations also demonstrated the resonance behaviour of the photoinduced current, which was also observed experimentally. Both calculated (Fig. 4) and measured dependences (Fig. 11) demonstrate an electric current damping at the resonance condition by a factor of about 2 with respect to its value far from the resonance. The results are in good coincidence with recent reports [9,10]. We note, however, that in contrast to the nonlinear equation derived in Ref. 161,the sign before the dissipation
M. Vusnetsov et d/Optics
Communiccrtions
is positive in our derivation. The described is also very suitable for the determination of the product sN,, (for our sample we got the value about ten times larger than the standard value [8]) and for study of such nonlinear effects as subharmonic generation [6] and high frequency resonances in photorefractive crystals [ 121. term
y
in Q.
experimental
(4.14)
technique
Acknowledgements Sergei Lyuksyutov and Mikhail Vasnetsov acknowledge support from the Danish Technical Research Council Grant No 9400102. We thank Floris Kharadaylo for assistance during the experiments.
References [I] A. Ashkin, G.D. Boyd, J.M. Dziedzic, R.G. Smith, A.A. Ballman, H.J. Levinstein and K. Nassau, Appl. Phys. Lett. 9 ( 19661 9.
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