- Email: [email protected]
Space charge field enhancement in photorefractive materials by applied sinusoidal fields: An approximate analytical solution
15June
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
139 (1997) 73-76
Space charge field enhancement in photorefractive materials by applied sinusoidal fields: An approximate analytical solution I. Aubrecht * Institute
ofRadio
Engineering
b Holography
und Electronics.
Group, Department
’ Department
a*‘,L. Solymar b, A. Grunnet-Jepsen
of Chemistry
Academy of Sciences of Czech Republic,
of Engineering and Biochemistr?:
Received 28 October
Science, Uniuersio
of Oxford,
UC’SD. 9500 Gilman Drive.
1996; revised 10 February
1997; accepted
’
ChaberskLi 57, 18251 Prague, Parks Road. Oxford Lu Jolla,
CA 92093-0340.
12 February
Czech Republic
OXI 3PJ, UK USA
1997
Abstract The second order temporal differential equation, known to describe the space charge field in the high frequency region, is solved approximately in a closed analytical form. The condition for multiple resonances is found. The maxima and minima in the space charge field are shown to correspond to the roots of the zero and first order Bessel functions respectively. The analytical approximations for the space charge field are compared with numerical results for the complete range of frequencies, and it is shown that there is excellent agreement between the two for the high frequency region which includes all the resonances. Keywords:
Photorefractive
materials;
Space charge field; AC-enhancement
1. Introduction The emergence of space charge fields in photorefractive materials illuminated by two optical waves is well understood. The basic theory is given by papers of the Kiev group [l-3] which has since been solved under a variety of conditions. A particularly interesting topic is the enhancement of the space charge field by an applied ac voltage. The first experiments were performed by Stepanov and Petrov [4] at fairly low frequencies (50 Hz). They found that the enhancement was considerably greater for square wave voltages than for sinusoidal ones. They explained their experimental results by a theory based on the approximate solution of a first order temporal differential equation for the amplitude of the electric field. Their method is known as the averaging technique. For higher applied frequencies ( > 5 kHz) one needs to resort to a second order temporal differential equation first derived by Valley [5] and applied to the enhancement problem by
’ E-mail: [email protected] 0030-4018/97/$17.00 Copyright PII SOO30-40 18(97)00096-5
technique;
High frequency
resonances
Mathey et al. [6], Vachss [7], Aubrecht et al. [8] and Grunnet-Jepsen et al. [9]. The main results of the high-frequency investigations were that (i) a sinusoidal field may be as effective as a square-wave field in enhancing the space charge field, (ii) there are multiple resonances of comparable heights, and (iii) the space charge field tends to a small value corresponding to zero applied field amplitude when the applied frequency exceeds the limit of &, = /.LKE~~,,where p is the electron mobility, I&, is the amplitude of the applied field and K = 2r/A (A is the grating spacing). The solution of the second order differential equation is far from being straightforward since the coefficients contain temporally periodic functions. For square wave voltages it is possible to solve the differential equation analytically by matching the results for every half period but there are no simple solutions for other waveforms. It is clear that for a periodic excitation the space charge field must eventually settle down as a periodic function of time (containing in general many higher harmonics) but any numerical solution that starts at r = 0 must first solve the problem for the transients before reaching the steady-state solution. Such numerical procedure takes an inordinate
0 1997 Elsevier Science B.V. All rights reserved.
74
I. Aubrecht et al. /Optics
Communications
amount of time (changes occur at the scale of the electron life time!) and could also lead to numerical instabilities as pointed out by Mathey et al. [6]. The aim of the present Communication is to find an approximate analytical solution to the relevant differential equation, discuss the validity of the approximations and draw some general conclusions concerning the arising resonant phenomenon.
139 (1997) 13- 76
need the time-averaged value of the imposed periodic solution. Our crucial approximation is similar to that proposed by Stepanov and Petrov [4] in the sense that y is to be replaced by its temporal average, yo. However, in contrast to the method of Stepanov and Petrov we retain here the first and second derivatives in the differential equation which now becomes d’v
1
dy
.
po-7,7d
p+Pl~+Po.vO=m
(
2. The differential equation to be solved
(4)
)
We shall assume the solution in the form dv x = C(r):(r),
Our starting point is a second order temporal differential equation as derived and discussed in Refs. [5-91. The main difference is that we found it more convenient to use the differential equation for the perturbed (periodically varying in space) ionized donor density in contrast to the space charge field. It is of the form
where C(r) is an unknown periodic function, and
d’y dy ~+PI~+PoY=m
Combining Eqs. (4)-(61 we obtain a differential of the following form for the C(r) function
(1)
z(r)-exp(
(5)
-izsinRr).
(6) equation
~+n,c=[m(po-~i-i’oyo]il.
)+I DO
-N&/NJ’,
f&o = /-m&J,
fl,=pKE@
EM=-
1 WK’
ED = YK,
p, =
a,=;+
(2)
if20,cosf2t+
1
e
I+
fJL,
EM/E,
+ O,,
7d
eN!& EQ = &SK
(7)
This is now a first order differential equation with constant coefficients in which the RHS varies with time. It is straightforward to calculate Fourier coefficients of C(r) as functions of v0 from this equation. Using then the fact that the zero order Fourier component of d y/dt must be zero for y to be periodic, we obtain from (51 an equation relating the Fourier coefficients of C(t) to each other whence y, may be determined. After lengthy algebraic manipulations we arrive at the final result m y, = 1
+f’ u
.f= (‘e’d
a;
-
fbE,/EQ)a
+
~&?$,/EQ
nO=slO(N,-N&)~,,
where ND is the density of donor atoms, N& and Ng, are the spatially constant and spatially varying components of the ionized donor density, rd = .sJepn, is the dielectric relaxation time, 7, = l/rNi is the electron life time, E,, EQ and E, are characteristic fields introduced by the Kiev group [l-3], &s is the static dielectric constant, k, is Boltzmann’s constant, T is the temperature, I0 is the average optical intensity, s is the photo-ionization constant, and m is the modulation of the interference pattern.
3. An approximate
solution
Let us reiterate here that it is not our aim to find an exact solution subject to the initial conditions. We only
’
(8) where J, is the nth order Bessel function of the first kind. Having obtained y. we can determine (E,), the temporally averaged value of the perturbed space charge field, from the simple relationship _vo+(.vo-m)--
EM 7, EQ
Td I
EQ,
(9)
which can easily be derived from basic equations of the band-transport model (see Ref. 121). We may now ask the question whether our approximate solution, Eq. (8), is valid in the region l/rd < R < I/T, for which Stepanov and Petrov [4] derived their analytical approximation with the aid of their averaging technique. It may be shown that in the approximation O/O, + 0 the infinite series in (8)
I. Aubrecht et ai. /Optics
Communications
may be summed up (a --f I / \i’I + ( f&,,/fl,)2 ), our expression becomes independent of R and it does reduce to that of Stepanov and Petrov. One peculiarity of the approximate solution is the multiple resonances. We should note here that an investigation of the condition of da/da = 0 would give us a good estimation of the positions of the extrema. Analyzing dn/dfi and da/d0 on the assumption &,.R>> fir_, it is straightforward, albeit laborious, to prove the following simple result: J,,( .1&/f&,, J,( L&,/On,,,)
) = 0 = 0
for maxima. for minima.
(10)
The first resonance (maximum) should then appear for QJR e 2.4048. Higher resonances (i.e. second, third etc.) correspond to further (higher) roots of J, and therefore to lower frequencies. Starting from the first resonance, the argument of J, is high enough for the zero order Bessel function to be replaced with reasonable accuracy by its asymptotic approximation (see e.g. Ref. [IO]). Then the first resonance appears for C$,,,/fi = 37r/4 + 2.36 and the resonance frequencies are given by the simple relation a/n, = (rrn - rr/4)-’ G (3.14~ - 0.79)-‘, which is in a remarkably good agreement with the empirical formula 0/f&,, = (3. I n - 0.7 I )- ’ reported by Grunnet-Jepsen et al. [9]. One has to bear in mind, however, that the lower the resonance frequencies the less reliable results Eq. ( 10) offers in principle.
I39 (1997) 73-76
m-3 , y= 1.9X 10-‘*m3s~‘, and s=2X IO-’ m’ J- ’ ; the value of the mobility was chosen as g = 2 X low6 m2 V-' s-’ in view of the arguments presented in Ref. [9]. The intensity has been taken as I, = 1OmW cm-’ and the temperature so that k,T/e = 26mV. The corresponding curves are plotted in Fig. la-lc for grating spacings of A = 100 pm, 10 pm and 1 ym respectively as a function with foe = &,/2~ as a parameter. The exact of fi/%, curves are given by dashed lines and the approximate ones by solid lines. It may be immediately seen that the approximation is very good for a wide range of applied frequencies. It is in fact a high-frequency approximation which describes perfectly the resonances but fails at low frequencies. Let us ask now the following question. When we have a good resonance in the vicinity of C&,/2 how many further resonances can we have? What is the lowest frequency when a resonance is still discernible? The argument may run as follows. Resonances are observed whenever an electron driven by some effective electric field will traverse n grating spacings in a half period. Mathematically, N
A
=
I()”
pEeffT/2
= n/i,
approximations
Before plotting any curves let us briefly review the high frequency ac resonance as it follows from the work of Pauliat et al. [I I], Vachss [7] and Grunnet-Jepsen et al. [9]. As we increase the applied frequency from 0 first a plateau in Im( E,) is reached which corresponds to the enhancement mechanism of Stepanov and Petrov [4]. For a square wave voltage there might first be a decline in Im( E, > as the frequency further increases and then, provided the applied voltage is high enough, follow the ac resonances. For a sinusoidal voltage the plateau is lower and the ac resonances follow immediately without an intermediate decline. For a square wave voltage the strongest resonance is at f&,/2 and the weaker resonances occur at Q&n where PI is an integer. For a sinusoidal voltage the positions of resonances have been given by Eq. (IO). Our analytical approximation for the full curve is given by Eqs. (8) and (9). The exact solution for v, is obtained by solving numerically a system of linear equations following from a Fourier decomposition of Eq. (I), and ( E, ) is then calculated from Eq. (9). The parameters chosen will be essentially those of an especially grown Fe doped BGO crystal (BGG 2) for which multiple resonances were reported by Pauliat et al. [I I]: ~s/~a = 46. No = 10’” m-j.
(11)
where T = 25-/R is the temporal period. We may now argue that the resonance is still observable when the electron lives just long enough to be able to traverse n grating spacings, i.e. the condition is T/2 = r,,
4. Numerical results and analytical
7s
whence the lowest resonance R = n-/r, = rryNi.
(12) may be expected to occur at (13)
Do the lowest resonant frequencies read from the curves satisfy the above simple criterion? From Fig. la the minimum frequency f,,, = fimin/2.rr may be estimated as 800Hz, from Fig. lb as 2.2 kHz, and from Fig. Ic as 230 kHz. The prediction of Eq. (13) is f= 950 Hz, the right order of magnitude for A = 100 p,m and IO pm but certainly not for A = I p,m. The obvious reason for the inaccuracy of Eq. (13) for short grating spacings is the neglect of diffusion. One may roughly expect resonances to be suppressed below the frequency no = pKE,. Since Ro varies inversely with the square of the grating spacings this is quite large at A = I pm. Its value is fo = 326 kHz, so again we have the right order of magnitude from a simple physical argument. We have been unable to find a rigorous criterion in analytical form for the onset of resonances. A perusal of the quantities given by Eq. (3) in Section 2 would however reveal that the combined importance of 0, and I /T, is reflected merely in the parameter 0,. It is therefore postulated that a,_ is the primary parameter responsible for the suppression of resonances. The values of onL/27r are approximately 1.7 kHz for A= IOOpm, 4.9kHz for A= JOpm and 328kHz for
76
I. Aubrecht et d/Optics
Communications
I39 (1997) 73-76
A = 1 km. Obviously, R,/27r is higher than each particular estimate from the corresponding chart but of the same order of magnitude. Our rule of thumb is then that resonances will occur in the range
(4
fit_ < n < R,/2.
(14)
5. Conclusions The multiple resonance phenomenon first found experimentally by Pauliat et al. [l 1] has been investigated and both numerical and approximate analytical solutions have been found for a BGO crystal. Our analytical approximation has been shown to be excellent for high frequencies. Simple physical arguments for the range of frequencies at which resonant phenomena appear have been presented and they have been shown to give the right order of magnitude.
(b) “1
1
fi >
,/’
yy
,E
(cl
1
___-
Acknowledgements
2o
/’
,_--/
10 LA. wishes to acknowledge a Royal Society Fellowship (1994) making this work possible and the present grant of the Academy of Sciences of the Czech Republic under the contract no. A1067601.
I’
References
’
1
4ookEz
5_ ___.....---
1_____ _._-
200
4l 0.001
0.1 O.O'
1
n/n00
Fig. 1. Variation of Im(E,)/m versus R/f&,, with f200/2~, labeling each curve, as a parameter for (a) A = lOOl.~rn, (b) lOp,m and Cc) 1p,m. The approximate curves are solid, the exact
ones are dashed.
[I] N.V. Kukhtarev, V.B. Markov, S.G. Odulov, Optics Comm. 23 (19771338. [2] N.V. Kukhtarev. V.B. Markov. S.G. Odulov, MS. Soskin, V.L. Vinetskii, Ferroelectrics 22 (1979) 949. [3] N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, V.L. Vinetskii, Ferroelecttics 22 (1979) 961. [4] S.I. Stepanov, M.P. Petrov, Optics Comm. 53 (1985) 292. [51 CC. Valley, IEEE J. Quantum Electron. 19 (1983) 1637. [6] P. Mathey, G. Pauliat, J.C. Launay and Cl. Roosen, Optics Comm. 82 (1991) 101; 83 (19911 390, 171 F. Vachss, J. Opt. Sot. Am. B 11 (1994) 1045. 181I. Aubrecht, A. Grunnet-Jepsen, L. Solymar, Optics Comm. 117 (1995) 303. I. Aubrecht, L. Solymar, Optics Lett. 20 [91 A. Grunnet-Jepsen, (19951 819. [lOI Handbook of Mathematical Functions, Eds. M. Abramowitz, LA. Stegun (National Bureau of Standards, Washington. 1964) Ch. 9.2, p. 364. [ill G. Pauliat. A. Villing, J.C. Launay, G. Roosen, J. Opt. Sot. Am. B 7 (1990) 1481.
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
139 (1997) 73-76
Space charge field enhancement in photorefractive materials by applied sinusoidal fields: An approximate analytical solution I. Aubrecht * Institute
ofRadio
Engineering
b Holography
und Electronics.
Group, Department
’ Department
a*‘,L. Solymar b, A. Grunnet-Jepsen
of Chemistry
Academy of Sciences of Czech Republic,
of Engineering and Biochemistr?:
Received 28 October
Science, Uniuersio
of Oxford,
UC’SD. 9500 Gilman Drive.
1996; revised 10 February
1997; accepted
’
ChaberskLi 57, 18251 Prague, Parks Road. Oxford Lu Jolla,
CA 92093-0340.
12 February
Czech Republic
OXI 3PJ, UK USA
1997
Abstract The second order temporal differential equation, known to describe the space charge field in the high frequency region, is solved approximately in a closed analytical form. The condition for multiple resonances is found. The maxima and minima in the space charge field are shown to correspond to the roots of the zero and first order Bessel functions respectively. The analytical approximations for the space charge field are compared with numerical results for the complete range of frequencies, and it is shown that there is excellent agreement between the two for the high frequency region which includes all the resonances. Keywords:
Photorefractive
materials;
Space charge field; AC-enhancement
1. Introduction The emergence of space charge fields in photorefractive materials illuminated by two optical waves is well understood. The basic theory is given by papers of the Kiev group [l-3] which has since been solved under a variety of conditions. A particularly interesting topic is the enhancement of the space charge field by an applied ac voltage. The first experiments were performed by Stepanov and Petrov [4] at fairly low frequencies (50 Hz). They found that the enhancement was considerably greater for square wave voltages than for sinusoidal ones. They explained their experimental results by a theory based on the approximate solution of a first order temporal differential equation for the amplitude of the electric field. Their method is known as the averaging technique. For higher applied frequencies ( > 5 kHz) one needs to resort to a second order temporal differential equation first derived by Valley [5] and applied to the enhancement problem by
’ E-mail: [email protected] 0030-4018/97/$17.00 Copyright PII SOO30-40 18(97)00096-5
technique;
High frequency
resonances
Mathey et al. [6], Vachss [7], Aubrecht et al. [8] and Grunnet-Jepsen et al. [9]. The main results of the high-frequency investigations were that (i) a sinusoidal field may be as effective as a square-wave field in enhancing the space charge field, (ii) there are multiple resonances of comparable heights, and (iii) the space charge field tends to a small value corresponding to zero applied field amplitude when the applied frequency exceeds the limit of &, = /.LKE~~,,where p is the electron mobility, I&, is the amplitude of the applied field and K = 2r/A (A is the grating spacing). The solution of the second order differential equation is far from being straightforward since the coefficients contain temporally periodic functions. For square wave voltages it is possible to solve the differential equation analytically by matching the results for every half period but there are no simple solutions for other waveforms. It is clear that for a periodic excitation the space charge field must eventually settle down as a periodic function of time (containing in general many higher harmonics) but any numerical solution that starts at r = 0 must first solve the problem for the transients before reaching the steady-state solution. Such numerical procedure takes an inordinate
0 1997 Elsevier Science B.V. All rights reserved.
74
I. Aubrecht et al. /Optics
Communications
amount of time (changes occur at the scale of the electron life time!) and could also lead to numerical instabilities as pointed out by Mathey et al. [6]. The aim of the present Communication is to find an approximate analytical solution to the relevant differential equation, discuss the validity of the approximations and draw some general conclusions concerning the arising resonant phenomenon.
139 (1997) 13- 76
need the time-averaged value of the imposed periodic solution. Our crucial approximation is similar to that proposed by Stepanov and Petrov [4] in the sense that y is to be replaced by its temporal average, yo. However, in contrast to the method of Stepanov and Petrov we retain here the first and second derivatives in the differential equation which now becomes d’v
1
dy
.
po-7,7d
p+Pl~+Po.vO=m
(
2. The differential equation to be solved
(4)
)
We shall assume the solution in the form dv x = C(r):(r),
Our starting point is a second order temporal differential equation as derived and discussed in Refs. [5-91. The main difference is that we found it more convenient to use the differential equation for the perturbed (periodically varying in space) ionized donor density in contrast to the space charge field. It is of the form
where C(r) is an unknown periodic function, and
d’y dy ~+PI~+PoY=m
Combining Eqs. (4)-(61 we obtain a differential of the following form for the C(r) function
(1)
z(r)-exp(
(5)
-izsinRr).
(6) equation
~+n,c=[m(po-~i-i’oyo]il.
)+I DO
-N&/NJ’,
f&o = /-m&J,
fl,=pKE@
EM=-
1 WK’
ED = YK,
p, =
a,=;+
(2)
if20,cosf2t+
1
e
I+
fJL,
EM/E,
+ O,,
7d
eN!& EQ = &SK
(7)
This is now a first order differential equation with constant coefficients in which the RHS varies with time. It is straightforward to calculate Fourier coefficients of C(r) as functions of v0 from this equation. Using then the fact that the zero order Fourier component of d y/dt must be zero for y to be periodic, we obtain from (51 an equation relating the Fourier coefficients of C(t) to each other whence y, may be determined. After lengthy algebraic manipulations we arrive at the final result m y, = 1
+f’ u
.f= (‘e’d
a;
-
fbE,/EQ)a
+
~&?$,/EQ
nO=slO(N,-N&)~,,
where ND is the density of donor atoms, N& and Ng, are the spatially constant and spatially varying components of the ionized donor density, rd = .sJepn, is the dielectric relaxation time, 7, = l/rNi is the electron life time, E,, EQ and E, are characteristic fields introduced by the Kiev group [l-3], &s is the static dielectric constant, k, is Boltzmann’s constant, T is the temperature, I0 is the average optical intensity, s is the photo-ionization constant, and m is the modulation of the interference pattern.
3. An approximate
solution
Let us reiterate here that it is not our aim to find an exact solution subject to the initial conditions. We only
’
(8) where J, is the nth order Bessel function of the first kind. Having obtained y. we can determine (E,), the temporally averaged value of the perturbed space charge field, from the simple relationship _vo+(.vo-m)--
EM 7, EQ
Td I
EQ,
(9)
which can easily be derived from basic equations of the band-transport model (see Ref. 121). We may now ask the question whether our approximate solution, Eq. (8), is valid in the region l/rd < R < I/T, for which Stepanov and Petrov [4] derived their analytical approximation with the aid of their averaging technique. It may be shown that in the approximation O/O, + 0 the infinite series in (8)
I. Aubrecht et ai. /Optics
Communications
may be summed up (a --f I / \i’I + ( f&,,/fl,)2 ), our expression becomes independent of R and it does reduce to that of Stepanov and Petrov. One peculiarity of the approximate solution is the multiple resonances. We should note here that an investigation of the condition of da/da = 0 would give us a good estimation of the positions of the extrema. Analyzing dn/dfi and da/d0 on the assumption &,.R>> fir_, it is straightforward, albeit laborious, to prove the following simple result: J,,( .1&/f&,, J,( L&,/On,,,)
) = 0 = 0
for maxima. for minima.
(10)
The first resonance (maximum) should then appear for QJR e 2.4048. Higher resonances (i.e. second, third etc.) correspond to further (higher) roots of J, and therefore to lower frequencies. Starting from the first resonance, the argument of J, is high enough for the zero order Bessel function to be replaced with reasonable accuracy by its asymptotic approximation (see e.g. Ref. [IO]). Then the first resonance appears for C$,,,/fi = 37r/4 + 2.36 and the resonance frequencies are given by the simple relation a/n, = (rrn - rr/4)-’ G (3.14~ - 0.79)-‘, which is in a remarkably good agreement with the empirical formula 0/f&,, = (3. I n - 0.7 I )- ’ reported by Grunnet-Jepsen et al. [9]. One has to bear in mind, however, that the lower the resonance frequencies the less reliable results Eq. ( 10) offers in principle.
I39 (1997) 73-76
m-3 , y= 1.9X 10-‘*m3s~‘, and s=2X IO-’ m’ J- ’ ; the value of the mobility was chosen as g = 2 X low6 m2 V-' s-’ in view of the arguments presented in Ref. [9]. The intensity has been taken as I, = 1OmW cm-’ and the temperature so that k,T/e = 26mV. The corresponding curves are plotted in Fig. la-lc for grating spacings of A = 100 pm, 10 pm and 1 ym respectively as a function with foe = &,/2~ as a parameter. The exact of fi/%, curves are given by dashed lines and the approximate ones by solid lines. It may be immediately seen that the approximation is very good for a wide range of applied frequencies. It is in fact a high-frequency approximation which describes perfectly the resonances but fails at low frequencies. Let us ask now the following question. When we have a good resonance in the vicinity of C&,/2 how many further resonances can we have? What is the lowest frequency when a resonance is still discernible? The argument may run as follows. Resonances are observed whenever an electron driven by some effective electric field will traverse n grating spacings in a half period. Mathematically, N
A
=
I()”
pEeffT/2
= n/i,
approximations
Before plotting any curves let us briefly review the high frequency ac resonance as it follows from the work of Pauliat et al. [I I], Vachss [7] and Grunnet-Jepsen et al. [9]. As we increase the applied frequency from 0 first a plateau in Im( E,) is reached which corresponds to the enhancement mechanism of Stepanov and Petrov [4]. For a square wave voltage there might first be a decline in Im( E, > as the frequency further increases and then, provided the applied voltage is high enough, follow the ac resonances. For a sinusoidal voltage the plateau is lower and the ac resonances follow immediately without an intermediate decline. For a square wave voltage the strongest resonance is at f&,/2 and the weaker resonances occur at Q&n where PI is an integer. For a sinusoidal voltage the positions of resonances have been given by Eq. (IO). Our analytical approximation for the full curve is given by Eqs. (8) and (9). The exact solution for v, is obtained by solving numerically a system of linear equations following from a Fourier decomposition of Eq. (I), and ( E, ) is then calculated from Eq. (9). The parameters chosen will be essentially those of an especially grown Fe doped BGO crystal (BGG 2) for which multiple resonances were reported by Pauliat et al. [I I]: ~s/~a = 46. No = 10’” m-j.
(11)
where T = 25-/R is the temporal period. We may now argue that the resonance is still observable when the electron lives just long enough to be able to traverse n grating spacings, i.e. the condition is T/2 = r,,
4. Numerical results and analytical
7s
whence the lowest resonance R = n-/r, = rryNi.
(12) may be expected to occur at (13)
Do the lowest resonant frequencies read from the curves satisfy the above simple criterion? From Fig. la the minimum frequency f,,, = fimin/2.rr may be estimated as 800Hz, from Fig. lb as 2.2 kHz, and from Fig. Ic as 230 kHz. The prediction of Eq. (13) is f= 950 Hz, the right order of magnitude for A = 100 p,m and IO pm but certainly not for A = I p,m. The obvious reason for the inaccuracy of Eq. (13) for short grating spacings is the neglect of diffusion. One may roughly expect resonances to be suppressed below the frequency no = pKE,. Since Ro varies inversely with the square of the grating spacings this is quite large at A = I pm. Its value is fo = 326 kHz, so again we have the right order of magnitude from a simple physical argument. We have been unable to find a rigorous criterion in analytical form for the onset of resonances. A perusal of the quantities given by Eq. (3) in Section 2 would however reveal that the combined importance of 0, and I /T, is reflected merely in the parameter 0,. It is therefore postulated that a,_ is the primary parameter responsible for the suppression of resonances. The values of onL/27r are approximately 1.7 kHz for A= IOOpm, 4.9kHz for A= JOpm and 328kHz for
76
I. Aubrecht et d/Optics
Communications
I39 (1997) 73-76
A = 1 km. Obviously, R,/27r is higher than each particular estimate from the corresponding chart but of the same order of magnitude. Our rule of thumb is then that resonances will occur in the range
(4
fit_ < n < R,/2.
(14)
5. Conclusions The multiple resonance phenomenon first found experimentally by Pauliat et al. [l 1] has been investigated and both numerical and approximate analytical solutions have been found for a BGO crystal. Our analytical approximation has been shown to be excellent for high frequencies. Simple physical arguments for the range of frequencies at which resonant phenomena appear have been presented and they have been shown to give the right order of magnitude.
(b) “1
1
fi >
,/’
yy
,E
(cl
1
___-
Acknowledgements
2o
/’
,_--/
10 LA. wishes to acknowledge a Royal Society Fellowship (1994) making this work possible and the present grant of the Academy of Sciences of the Czech Republic under the contract no. A1067601.
I’
References
’
1
4ookEz
5_ ___.....---
1_____ _._-
200
4l 0.001
0.1 O.O'
1
n/n00
Fig. 1. Variation of Im(E,)/m versus R/f&,, with f200/2~, labeling each curve, as a parameter for (a) A = lOOl.~rn, (b) lOp,m and Cc) 1p,m. The approximate curves are solid, the exact
ones are dashed.
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