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The effects of optical activity and absorption on two-wave mixing in Bi12SiO20
Volume 83, number 3,4
OPTICS COMMUNICATIONS
1 June 1991
Full length article
The effects of optical activity and absorption on two-wave mixing in Bil2SiO2o D.J. W e b b a n d L. S o l y m a r Holography Group, Department of Engineering Science, Oxford University, ParksRd., Oxford OX1 3PZ UK Received 29 October 1990; revised manuscript received 14 January 1991
We consider non-degenerate tworwave mixing in photorefractive Bil2SiO2o. It is shown theoretically that the presence of absorption and optical activity in the photorefractivemedia may result in a number of maximafor the gain as the frequencydetuning between the two beams is varied. Further, when the beam interaction is used for optical amplification, there may also exist an optimum crystal length beyond which there is a reduction in the useful gain obtainable. Experimental results are presented in confirmation of the theory.
1. Introduction
The basic phenomenon of two-wave mixing in photorefractive media is well described by the Band Transport Model of Kukhtarev [ 1 ]. The photorefractive effect results in one beam experiencing gain at the expense of the other. It has been shown that when Bi~2SiO2o (BSO) is used as the photorefractive media, the gain may be considerably enhanced by detuning one of the incident beams with respect to the other, so as to produce a running grating [2 ]. Simple theory shows that the graph of the gain against frequency difference between the incident beams the so called detuning curve - has a single m a x i m u m at a particular value of the detuning frequency [ 2 ]. In order to obtain a large gain, the length of the crystal should be as large as possible. However, when lengths in excess of a couple of millimetres are used, it is necessary to take into account the effects of optical activity and absorption within the material. Consequences of the presence of optical activity in the absence of absorption have been discussed previously [ 3-8 ], and analytic solutions to the coupled wave equations have been obtained when absorption alone is considered [ 9 ]. In this paper we show that the presence of absorption and optical activity in BSO leads to multiple peaks in the detuning curve, and
also to an optimum crystal length beyond which there is a reduction in useful gain.
2. Theory A qualitative explanation of the effects of absorption and optical activity is as follows: Owing to the anisotropic nature of the photorefractive effect within BSO, with the crystal arranged as shown in fig. l, if the p u m p and signal beams were to be plane polarised parallel to the z-axis, there would initially be no coupling between them. I f the two beams were plane polarised parallel to the y-axis, as shown, then the coupling constant would be initially maximiscd. This is one of the usual experimental arrangements for two-wave mixing in BSO with a transmission geometry. Whatever the initial direction of the plane of polarisation, under the influence of optical activity, the plane will rotate through the crystal and, for a sufficiently long crystal, there will be alternating regions of high and low coupling constant. It is well known that the two beam gain is optimised at a particular value of detuning frequency which is proportional to the total optical intensity [ 2 ]. Due to the presence of absorption, the optical intensity decays exponentially through the crystal, and therefore the gain in each of the regions of high coupling
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
287
Volume 83, number 3,4
OPTICS COMMUNICATIONS
1 June 1991
Fig. 1. The system under consideration.
constant will be optimised at different detuning frequencies. The net result of this is a detuning curve with several maxima. To obtain a quantitative result, we begin with the vector wave equation for the optical electric field, E, applicable to an absorbing, optically active and electro-optic medium: VXVXE=V(V-E)-V~E=~~~W~(E+~E),
(1)
where ff is the susceptibility tensor for the medium. Assuming the arrangement shown in fig. 1 and small inter-beam angles (par-axial approximation), we have E=E+ +E_ E,,+ exp( -ik+x) = ( E,, exp( -ik+x)
eiS”‘+E,_ exp( -ik_x) eih’+Ez_ exp( -ik_x)
>’ (2)
where k is the wave vector, and the subscripts + and - refer to the pump and signal beams respectively. In this approximation we also have V*E=O, and with the geometry illustrated, 2 is given by i=
(
2,
-l+iei
iv E,-l+iei > ’
(3)
where v, the gyration constant, describes the optical activity of the medium, ti, the imaginary part of the 288
relative permittivity, describes the effects of absorption within the medium, t, is the relative permittivity of the medium in the absence of any electric fields and C~ is given by t,,,=t,+Aq+Ae,,
exp( -iK*r)+Ar,+
exp(iK*r) . (4)
Here, AC,= c:r.,,E, is the change in relative permittivity due to the externally applied electric field, E,, and r4,, the relevant coefftcient of the electro-optic tensor, K is the grating wave vector and AC,,= e ?r4,E, is the complex amplitude of the periodic variation in the relative permittivity resulting from the space charge field, E, exp( - iK*r), set up by the interference between the two light beams. This space charge field may be shown to satisfy the following material equation [ 2 1, 3E, A at+;&=
!u!
!?I
zd
2’
(5)
where A, B and EQ are parameters of the material and experimental geometry given by A=(llD)(l+E,IEo+iEolEo),
(6)
B=(llD)(-E,IE,+iE,IE,),
(7)
D= 1+E,/EM +iEo/EM,
(8)
Volume 83, number 3,4
eN T,
EQ= IKl¢s'
OPTICS COMMUNICATIONS
IKI Tka ED=--, e
y~N T~
E M = - -IKI# ,
(9)
where e is the electronic charge, kB is Boltzmann's constant, T is the temperature, /z is the mobility, NZ is the density of ionised acceptors, Yr is the recombination coefficient, es is the static field dielectric constant and IKI is the magnitude of the grating vector. The dielectric relaxation time, zd, and the complex modulation of the optical interference pattern, m, are given by
e~yrN~, 2E*+ "E_ zd= egsloND ' m= Io '
(10)
where ND is the donor density, s is the photoionisation constant and Io is the total intensity. Eq. ( 1 ) may be solved by separately equating the sums of terms with wave vectors k+ and k_ to zero. Making the usual slowly varying amplitude approximation (VEE_+<< k2+E+ ), we get the following:
r+-i -Qr_e
or+__ OX
'
(11) OE_ ( . . . ox =
Eo)
r_-ir
. ~.E~¢ r+
ei~O t
,
(12)
where E+=(EE:++ ),
E_=(E:-),
(13,
and ti describe the effects of optical activity and absorption within the medium, respectively, and are given by 0
Y=( - y
0)=°)2g°e°(-0~2k
0)'
(14)
,l,, /~ describes the electro-optic effects and is given by
:)
,16,
Esc may be eliminated from eqs. (11 ) and (12) using eq. (5). If we only consider the steady state, the result is
0E~_0x ,,(
1 June 1991
Eo £X
.
= ~--6t+i--FE.Q _')E+ + i
B/~E* (E~ .E_) (A--iSmZd)Io
(17)
OE_ Ox
=
( #-
Eo,~ p) E _ - i 6 t - i ~--~.
BI~E+ (A-i~o) ( E*+za)Io "E- )
.
(18) Here it should be remembered that Io (and Td which is inversely proportional to Io) will vary exponentially with x. These equations admit of an analytical solution only if the optical activity is disregarded, and the amplitude of one beam is supposed to be much larger than (and hence undepleted by) the other. In the general case, the equations must be solved numerically, and a fourth order Runge-Kutta technique was used with the following initial (x = 0) conditions; Ez_ =Ez+ =0; (Ey_)2+ (Ey+)2=I~n; Ey+/Ey_ =x/~, where/in is the total input intensity and fl is the input beam ratio. Implicit in the derivation of eq. (5) is the assumption that the beam ratio is very large, which permits consideration of only the first harmonic in the Fourier expansion of products of the spatial variables occurring in the material equations. For smaller beam ratios, it is found that there is a sub-linear dependence of E~¢ with m, and in order to have a more generally useful simulation, we made use of the phenomenological approach pioneered by Refregier et al. [2], who obtained good agreement between experiment and theory at low beam ratios by using an effective modulation m* given by the empirical equation: m* = (1/2.8) [ 1 - e x p ( - 2 . 8 m) ]
(19)
in place of m in eq. (5). The form of this relationship has more recently been confirmed by Au and Solymar by solving numerically the material equations in the presence of a high modulation [ 10 ] . For comparison with experiments, the intensity gain was determined as a function of &o. The gain was obtained by finding the intensity of the signal beam after it has propagated the required distance, L, and dividing this by the value obtained with no inter-beam coupling (obtained numerically by letting &o go to infinity). Unless stated otherwise, the following parameters, typical for BSO were used in the numerical calculations: N 7, = 1022 m - 3; No = 1025 289
Volume 83, number 3,4
OPTICS COMMUNICATIONS
m-3; ~s=56, grating spacing=20 ~tm; source wavelength=514.5 nm; T = 3 0 0 K; refractive index= 2.62; r4~=3.5×10 -12 m/V; / t = 1 0 -5 m2/Vs; s = 2N10 -5 m2/J; 7r=1.65×10 -t7 m3/s; Iin=2 m W / cm2; Eo= - 8 kV/cm; r = 1000, ot = 55.5/m, 7= 39 ° / mm. 3. R e s u l t s a n d d i s c u s s i o n
Using the parameters described above, detuning curves were obtained for crystal lengths of 1, 3, 5, 8 and 12 ram. These are plotted in fig. 2, which clearly shows the appearance of additional maxima in the detuning curve as the crystal length is increased to include further regions of high coupling constant. The undepleted pump beam approximation may be modelled numerically by choosing a value of fl much greater than the gain obtained via two-wave mixing. Numerical calculations performed for this regime using a value offl of l0 ts and plotted in fig. 3a show that as the crystal length, L, is increased, the maximum gain obtainable also tends to increase. However, the useful gain obtained from the crystal is the ratio of the power in the signal beam at the end of the crystal to that at the beginning. This quantity differs from our definition of photorefractive gain by
1 June 1991
a factor exp ( - 2c~L), the absorption occurring within the crystal. A plot of the useful gain against crystal length (shown in fig. 3b) shows that there is an optimum crystal length, beyond which the useful gain actually decreases. From fig. 3 it may also be seen that varying the applied electric field does not greatly influence the optimum length of crystal. An increasing field does, however, increase both the maximum obtainable useful gain and the crystal length corresponding to unity useful gain. It is interesting to consider the consequences of trying to improve on the maximum obtainable useful gain by reducing the absorption. The absorption may be considered as being composed of two components. The useful component for photorefraction consists of those absorptions of photons which result in an electron being excited from a donor site to the conduction band. The ratio of this component to the total absorption, the quantum efficiency, has been measured at 86% at 514.5 nm for one particular crystal of BSO [ 11 ]. The non-useful component of absorption might possibly be reduced considerably with appropriate crystal growth techniques, but we would still be left with that component necessary for the photorefractive process, Olp, given by a v = ½S(ND-N3)hv~
½sNohv,
(20)
400-
300-
200"
100'
! I/\ / o12
oN
t t
o~s
I
0.8
I 1.2
I 1.4
Detuning Frequency (Hz) Fig. 2. Detuning curves for crystals of different lengths: . . . . . 8 ram, 12 mm. 290
I mm,
3 ram,
4 mill,
5 ram,
Volume 83, number 3,4
OPTICS COMMUNICATIONS
~
101° ! 109
.
a
lo8
• • •
~
107
108
>= 106
1 June 1991
•
O O0 '~ 104 =
•
•
105
Am
102
104
0 0
00
•
•
•
•
103
@
102 100
000 0 0 •• Og• •ram
0
1
1'o
~
~oo
D QI~DD []
•
lol 13
Crystal Length (cm)
1°° .1
1
10
Crystal
•~IBIB •
14 '=
• A&
O@O@00
• o° 100
...................
• __@.,,_~ r ? r . . . . . . . . . i i
Unity Gain
10
1i i
164
Length (cm)
Fig. 4. Useful gain plotted as a function of donor density. Open squares: Nd=2X 1025,filled squares: Na= 1× 102s, open circles: Na = 7 × 1024,filled circles: Na = 5 × 102`=.
• •
W~ 102
100
.:_-'
10
100
Crystal Length (em)
Fig. 3. Photorefractivegain (a) and usefulgain (b) as a function of crystallength, for various applied electricfields. Circles: 5 kV/ cm; triangles: 8 kV/cm; squares: 10 kV/cm. Dashed lines in (b) indicate crystal lengths corresponding to unity gain at these electric fields. where h is Planck's constant and v is the frequency of the incident radiation. This quantity might best be varied by adjusting the dopant levels during the crystal growth phase and thereby controlling ND. The effect of this is shown in fig. 4 where we have plotted the optimum photorefractive gain versus crystal length for several values of No. Unity quantum efficiency was assumed, with t~ therefore given by eq. (20). It can be seen that, as the donor density is decreased, the m a x i m u m obtainable gain increases, as does the optimum crystal length. The limit on the m a x i m u m obtainable gain would in practice be reached when ND became comparable in magnitude to N ~ , and the donor sites became saturated. It is worth noting that if the useful gain is in-
creased by reducing ND, then there will be concomitant increase in the crystal response time, which is inversely proportional to ND. This may be disadvantageous for certain applications, in which case an improvement in useful gain may be obtained by moving to a longer wavelength (and correspondingly smaller photoionisation constant) and thereby reducing otp from eq. (20).
4. Experimental To provide some confirmation of the theory, experimental detuning curves were obtained for a one centimetre long crystal of BSO at several source wavelengths in the range 476.5-514.5 rim. These are illustrated in fig. 5. In addition, the absorption and gyration constants of the crystal were measured and are listed in table 1 along with the values of the experimental beam ratio used at each wavelength. Theoretical curves corresponding to these wavelengths were obtained in the following way: N o and s only appear in the theory as a product which was determined from the experimental absorption constant using eq. (20), and assuming unity quantum efficiency. Table 1 lists the values of s found in this way, where we have made the usual assumption ND = 1025 m - 3. The experimentally determined values of absorption constant, gyration constant and 291
Volume 83, number 3,4
OPTICS COMMUNICATIONS
1 June 1991
50 00
A
40
30
20
10
0 1
2
Detuning Frequency (Hz) F-
7
(.'3 S-'
30 C Q
B
D
5-
20 4-
310
Detuning Frequency (Hz i 3
Detuning Frequency (Hz) 5 al 4-
3.
2'
4
v
O
2 0
Detuning Frequency (Hz)
Oetuning Frequency (Hz)
Fig. 5. Experimental detuning curves for BSO. A 514.5 nm, B 501.7 nm, C 496.5 nm, D 488.0 nm, E 476.5 nm. beam ratio were used. Other parameters were adjusted to provide the best match between theory and experiment. Account was also taken o f the variation o f k and K with wavelength. The parameters that were different to those quoted earlier were: r41 = 4 . 4 1 ×
Table 1 Experimentally determined values of amplitude absorption constant ( a ) , gyration constant (7) and beam ration (//) and calculated photoionisation constant (s) at various wavelengths.
Source wavelength
a
~,
lO-~2m/V;#=3×lO-6m2/VsandEo=_4.75kV/
(nm)
(m - l )
(°/m)
cm. The low value o f the electric field was needed to ensure that the peaks in the theoretical detuning curves occurred at the same values o f detuning frequency as those in the experimental data. This lends credence to the argument that the static electric field
514.5 501.7 496.5 488.0
55.5 89.5 108.5 i 51.5
39000 42100 43500 46000
292
476.5
233.5
49800
B
s (m2/J)
500 700 200 200 60
2.9x 10-5 4.6x10 -5 5.6× 10-5 7.8× 10-5
12.1X 10 -5
Volume 83, number 3,4
40.
OPTICS COMMUNICATIONS
1 June 1991
Gain ~
30,
A
20,
10,
DetunlngFrequency(Hz) o'.s i'. i~5 2.. 2:5 3'. 3'.5 4'. 20•. Gain 17 ~5'
Gain D
15.. 12.5 i0 7.5 5¢
~n.
cy (Hz B)
2.5
12,
0:5 I'. i~5 2.. 2:s 3'. 3:5 4'. Gain
Detunlng Frequency (Hz) I
0.5 5¢
4.5'
10,
i.
1.5
2.
2.5
3.
3.5
I
4.
Gain
4.'
8,
3.5. 3.'
6.
2.5" 4.
2.'
2,
0:5
i'. 1:5 2'. 2:5
cy (Hz) 3'. 3.5'
1.5,
4'.
Detuning Frequency [Hz) 0 ~5
1'•
' 1.5
' 2.
2 .' 5
' 3.
I
3'.5
4.
Fig. 6. Theoretical detuning curves for BSO. A 514.5 nm, B 501.7 nm, C 496.5 nm, D 488.0 nm, E 476.5 nm.
within the crystal is often less than the applied field as a result of potential drops at the electrode-crystal interface. It may be seen that the theoretical curves shown in fig. 6 are a good approximation to the experimental curves - in particular we obtain values of gain which closely match experiment, even though we are using a realistic value for the electro-optic coefficient, rather than a reduced effective value which is often done to account for the effects of optical activity and absorption. Only by the inclusion within the field equations of terms accounting for optical activity and absorption can the correct form of detuning curve be obtained.
Acknowledgement The authors would like to thank the United Kingdom Science and Engineering Research Council for supporting this work.
References [ 1 ] N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin and V.L. Vinetskii, Ferroelectri¢s 22 (1979) 949. [ 2 ] Ph. Refregier, L. Solymar, H. Rajbenbach and J.P. Huignard, J. Appl. Phys. 58 (1985) 45. [3 ] N.V. Kukhtarev, G.E. Dovgalenko and V.N. Starkov, Appl. Phys. A 33 (1984) 227.
293
Volume 83, number 3,4
OPTICS COMMUNICATIONS
[4] P.D. Foote and T.J. Hall, Optics Comm. 57 (1986) 201. [5] A.E. Mandel, S.M. Shanderov and V.V. Shepelevich, 2nd. Conf. on Photorefractive materials, effects and devices, 1990, Aussois, France. [ 6 ] A. Marrakchi, R.V. Johnson and A.R. Tanguay, J. Opt. Soc. Am.B 3 (1986) 321. [7]T.J. Hall, A.K. Powell and C. Stace, Optics Comm. 75 (1990) 159.
294
1 June 1991
[8] A. Marrakchi, R.V. Johnson and A.R. Tanguay, IEEE J. Quant. Electron. 23 ( 1987 ) 2142. [9] D.C. Jones, D. Phil. Thesis, Department of Engineering Science, University of Oxford, 1989. [ 10 ] L.B. Au and L. Solymar, Optics Lett. 13 (1988) 660. [ 11 ] R.A. Mullen and R.W. Hellwarth, J. Appl. Phys. 58 (1985) 40.
OPTICS COMMUNICATIONS
1 June 1991
Full length article
The effects of optical activity and absorption on two-wave mixing in Bil2SiO2o D.J. W e b b a n d L. S o l y m a r Holography Group, Department of Engineering Science, Oxford University, ParksRd., Oxford OX1 3PZ UK Received 29 October 1990; revised manuscript received 14 January 1991
We consider non-degenerate tworwave mixing in photorefractive Bil2SiO2o. It is shown theoretically that the presence of absorption and optical activity in the photorefractivemedia may result in a number of maximafor the gain as the frequencydetuning between the two beams is varied. Further, when the beam interaction is used for optical amplification, there may also exist an optimum crystal length beyond which there is a reduction in the useful gain obtainable. Experimental results are presented in confirmation of the theory.
1. Introduction
The basic phenomenon of two-wave mixing in photorefractive media is well described by the Band Transport Model of Kukhtarev [ 1 ]. The photorefractive effect results in one beam experiencing gain at the expense of the other. It has been shown that when Bi~2SiO2o (BSO) is used as the photorefractive media, the gain may be considerably enhanced by detuning one of the incident beams with respect to the other, so as to produce a running grating [2 ]. Simple theory shows that the graph of the gain against frequency difference between the incident beams the so called detuning curve - has a single m a x i m u m at a particular value of the detuning frequency [ 2 ]. In order to obtain a large gain, the length of the crystal should be as large as possible. However, when lengths in excess of a couple of millimetres are used, it is necessary to take into account the effects of optical activity and absorption within the material. Consequences of the presence of optical activity in the absence of absorption have been discussed previously [ 3-8 ], and analytic solutions to the coupled wave equations have been obtained when absorption alone is considered [ 9 ]. In this paper we show that the presence of absorption and optical activity in BSO leads to multiple peaks in the detuning curve, and
also to an optimum crystal length beyond which there is a reduction in useful gain.
2. Theory A qualitative explanation of the effects of absorption and optical activity is as follows: Owing to the anisotropic nature of the photorefractive effect within BSO, with the crystal arranged as shown in fig. l, if the p u m p and signal beams were to be plane polarised parallel to the z-axis, there would initially be no coupling between them. I f the two beams were plane polarised parallel to the y-axis, as shown, then the coupling constant would be initially maximiscd. This is one of the usual experimental arrangements for two-wave mixing in BSO with a transmission geometry. Whatever the initial direction of the plane of polarisation, under the influence of optical activity, the plane will rotate through the crystal and, for a sufficiently long crystal, there will be alternating regions of high and low coupling constant. It is well known that the two beam gain is optimised at a particular value of detuning frequency which is proportional to the total optical intensity [ 2 ]. Due to the presence of absorption, the optical intensity decays exponentially through the crystal, and therefore the gain in each of the regions of high coupling
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
287
Volume 83, number 3,4
OPTICS COMMUNICATIONS
1 June 1991
Fig. 1. The system under consideration.
constant will be optimised at different detuning frequencies. The net result of this is a detuning curve with several maxima. To obtain a quantitative result, we begin with the vector wave equation for the optical electric field, E, applicable to an absorbing, optically active and electro-optic medium: VXVXE=V(V-E)-V~E=~~~W~(E+~E),
(1)
where ff is the susceptibility tensor for the medium. Assuming the arrangement shown in fig. 1 and small inter-beam angles (par-axial approximation), we have E=E+ +E_ E,,+ exp( -ik+x) = ( E,, exp( -ik+x)
eiS”‘+E,_ exp( -ik_x) eih’+Ez_ exp( -ik_x)
>’ (2)
where k is the wave vector, and the subscripts + and - refer to the pump and signal beams respectively. In this approximation we also have V*E=O, and with the geometry illustrated, 2 is given by i=
(
2,
-l+iei
iv E,-l+iei > ’
(3)
where v, the gyration constant, describes the optical activity of the medium, ti, the imaginary part of the 288
relative permittivity, describes the effects of absorption within the medium, t, is the relative permittivity of the medium in the absence of any electric fields and C~ is given by t,,,=t,+Aq+Ae,,
exp( -iK*r)+Ar,+
exp(iK*r) . (4)
Here, AC,= c:r.,,E, is the change in relative permittivity due to the externally applied electric field, E,, and r4,, the relevant coefftcient of the electro-optic tensor, K is the grating wave vector and AC,,= e ?r4,E, is the complex amplitude of the periodic variation in the relative permittivity resulting from the space charge field, E, exp( - iK*r), set up by the interference between the two light beams. This space charge field may be shown to satisfy the following material equation [ 2 1, 3E, A at+;&=
!u!
!?I
zd
2’
(5)
where A, B and EQ are parameters of the material and experimental geometry given by A=(llD)(l+E,IEo+iEolEo),
(6)
B=(llD)(-E,IE,+iE,IE,),
(7)
D= 1+E,/EM +iEo/EM,
(8)
Volume 83, number 3,4
eN T,
EQ= IKl¢s'
OPTICS COMMUNICATIONS
IKI Tka ED=--, e
y~N T~
E M = - -IKI# ,
(9)
where e is the electronic charge, kB is Boltzmann's constant, T is the temperature, /z is the mobility, NZ is the density of ionised acceptors, Yr is the recombination coefficient, es is the static field dielectric constant and IKI is the magnitude of the grating vector. The dielectric relaxation time, zd, and the complex modulation of the optical interference pattern, m, are given by
e~yrN~, 2E*+ "E_ zd= egsloND ' m= Io '
(10)
where ND is the donor density, s is the photoionisation constant and Io is the total intensity. Eq. ( 1 ) may be solved by separately equating the sums of terms with wave vectors k+ and k_ to zero. Making the usual slowly varying amplitude approximation (VEE_+<< k2+E+ ), we get the following:
r+-i -Qr_e
or+__ OX
'
(11) OE_ ( . . . ox =
Eo)
r_-ir
. ~.E~¢ r+
ei~O t
,
(12)
where E+=(EE:++ ),
E_=(E:-),
(13,
and ti describe the effects of optical activity and absorption within the medium, respectively, and are given by 0
Y=( - y
0)=°)2g°e°(-0~2k
0)'
(14)
,l,, /~ describes the electro-optic effects and is given by
:)
,16,
Esc may be eliminated from eqs. (11 ) and (12) using eq. (5). If we only consider the steady state, the result is
0E~_0x ,,(
1 June 1991
Eo £X
.
= ~--6t+i--FE.Q _')E+ + i
B/~E* (E~ .E_) (A--iSmZd)Io
(17)
OE_ Ox
=
( #-
Eo,~ p) E _ - i 6 t - i ~--~.
BI~E+ (A-i~o) ( E*+za)Io "E- )
.
(18) Here it should be remembered that Io (and Td which is inversely proportional to Io) will vary exponentially with x. These equations admit of an analytical solution only if the optical activity is disregarded, and the amplitude of one beam is supposed to be much larger than (and hence undepleted by) the other. In the general case, the equations must be solved numerically, and a fourth order Runge-Kutta technique was used with the following initial (x = 0) conditions; Ez_ =Ez+ =0; (Ey_)2+ (Ey+)2=I~n; Ey+/Ey_ =x/~, where/in is the total input intensity and fl is the input beam ratio. Implicit in the derivation of eq. (5) is the assumption that the beam ratio is very large, which permits consideration of only the first harmonic in the Fourier expansion of products of the spatial variables occurring in the material equations. For smaller beam ratios, it is found that there is a sub-linear dependence of E~¢ with m, and in order to have a more generally useful simulation, we made use of the phenomenological approach pioneered by Refregier et al. [2], who obtained good agreement between experiment and theory at low beam ratios by using an effective modulation m* given by the empirical equation: m* = (1/2.8) [ 1 - e x p ( - 2 . 8 m) ]
(19)
in place of m in eq. (5). The form of this relationship has more recently been confirmed by Au and Solymar by solving numerically the material equations in the presence of a high modulation [ 10 ] . For comparison with experiments, the intensity gain was determined as a function of &o. The gain was obtained by finding the intensity of the signal beam after it has propagated the required distance, L, and dividing this by the value obtained with no inter-beam coupling (obtained numerically by letting &o go to infinity). Unless stated otherwise, the following parameters, typical for BSO were used in the numerical calculations: N 7, = 1022 m - 3; No = 1025 289
Volume 83, number 3,4
OPTICS COMMUNICATIONS
m-3; ~s=56, grating spacing=20 ~tm; source wavelength=514.5 nm; T = 3 0 0 K; refractive index= 2.62; r4~=3.5×10 -12 m/V; / t = 1 0 -5 m2/Vs; s = 2N10 -5 m2/J; 7r=1.65×10 -t7 m3/s; Iin=2 m W / cm2; Eo= - 8 kV/cm; r = 1000, ot = 55.5/m, 7= 39 ° / mm. 3. R e s u l t s a n d d i s c u s s i o n
Using the parameters described above, detuning curves were obtained for crystal lengths of 1, 3, 5, 8 and 12 ram. These are plotted in fig. 2, which clearly shows the appearance of additional maxima in the detuning curve as the crystal length is increased to include further regions of high coupling constant. The undepleted pump beam approximation may be modelled numerically by choosing a value of fl much greater than the gain obtained via two-wave mixing. Numerical calculations performed for this regime using a value offl of l0 ts and plotted in fig. 3a show that as the crystal length, L, is increased, the maximum gain obtainable also tends to increase. However, the useful gain obtained from the crystal is the ratio of the power in the signal beam at the end of the crystal to that at the beginning. This quantity differs from our definition of photorefractive gain by
1 June 1991
a factor exp ( - 2c~L), the absorption occurring within the crystal. A plot of the useful gain against crystal length (shown in fig. 3b) shows that there is an optimum crystal length, beyond which the useful gain actually decreases. From fig. 3 it may also be seen that varying the applied electric field does not greatly influence the optimum length of crystal. An increasing field does, however, increase both the maximum obtainable useful gain and the crystal length corresponding to unity useful gain. It is interesting to consider the consequences of trying to improve on the maximum obtainable useful gain by reducing the absorption. The absorption may be considered as being composed of two components. The useful component for photorefraction consists of those absorptions of photons which result in an electron being excited from a donor site to the conduction band. The ratio of this component to the total absorption, the quantum efficiency, has been measured at 86% at 514.5 nm for one particular crystal of BSO [ 11 ]. The non-useful component of absorption might possibly be reduced considerably with appropriate crystal growth techniques, but we would still be left with that component necessary for the photorefractive process, Olp, given by a v = ½S(ND-N3)hv~
½sNohv,
(20)
400-
300-
200"
100'
! I/\ / o12
oN
t t
o~s
I
0.8
I 1.2
I 1.4
Detuning Frequency (Hz) Fig. 2. Detuning curves for crystals of different lengths: . . . . . 8 ram, 12 mm. 290
I mm,
3 ram,
4 mill,
5 ram,
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~
101° ! 109
.
a
lo8
• • •
~
107
108
>= 106
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•
O O0 '~ 104 =
•
•
105
Am
102
104
0 0
00
•
•
•
•
103
@
102 100
000 0 0 •• Og• •ram
0
1
1'o
~
~oo
D QI~DD []
•
lol 13
Crystal Length (cm)
1°° .1
1
10
Crystal
•~IBIB •
14 '=
• A&
O@O@00
• o° 100
...................
• __@.,,_~ r ? r . . . . . . . . . i i
Unity Gain
10
1i i
164
Length (cm)
Fig. 4. Useful gain plotted as a function of donor density. Open squares: Nd=2X 1025,filled squares: Na= 1× 102s, open circles: Na = 7 × 1024,filled circles: Na = 5 × 102`=.
• •
W~ 102
100
.:_-'
10
100
Crystal Length (em)
Fig. 3. Photorefractivegain (a) and usefulgain (b) as a function of crystallength, for various applied electricfields. Circles: 5 kV/ cm; triangles: 8 kV/cm; squares: 10 kV/cm. Dashed lines in (b) indicate crystal lengths corresponding to unity gain at these electric fields. where h is Planck's constant and v is the frequency of the incident radiation. This quantity might best be varied by adjusting the dopant levels during the crystal growth phase and thereby controlling ND. The effect of this is shown in fig. 4 where we have plotted the optimum photorefractive gain versus crystal length for several values of No. Unity quantum efficiency was assumed, with t~ therefore given by eq. (20). It can be seen that, as the donor density is decreased, the m a x i m u m obtainable gain increases, as does the optimum crystal length. The limit on the m a x i m u m obtainable gain would in practice be reached when ND became comparable in magnitude to N ~ , and the donor sites became saturated. It is worth noting that if the useful gain is in-
creased by reducing ND, then there will be concomitant increase in the crystal response time, which is inversely proportional to ND. This may be disadvantageous for certain applications, in which case an improvement in useful gain may be obtained by moving to a longer wavelength (and correspondingly smaller photoionisation constant) and thereby reducing otp from eq. (20).
4. Experimental To provide some confirmation of the theory, experimental detuning curves were obtained for a one centimetre long crystal of BSO at several source wavelengths in the range 476.5-514.5 rim. These are illustrated in fig. 5. In addition, the absorption and gyration constants of the crystal were measured and are listed in table 1 along with the values of the experimental beam ratio used at each wavelength. Theoretical curves corresponding to these wavelengths were obtained in the following way: N o and s only appear in the theory as a product which was determined from the experimental absorption constant using eq. (20), and assuming unity quantum efficiency. Table 1 lists the values of s found in this way, where we have made the usual assumption ND = 1025 m - 3. The experimentally determined values of absorption constant, gyration constant and 291
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OPTICS COMMUNICATIONS
1 June 1991
50 00
A
40
30
20
10
0 1
2
Detuning Frequency (Hz) F-
7
(.'3 S-'
30 C Q
B
D
5-
20 4-
310
Detuning Frequency (Hz i 3
Detuning Frequency (Hz) 5 al 4-
3.
2'
4
v
O
2 0
Detuning Frequency (Hz)
Oetuning Frequency (Hz)
Fig. 5. Experimental detuning curves for BSO. A 514.5 nm, B 501.7 nm, C 496.5 nm, D 488.0 nm, E 476.5 nm. beam ratio were used. Other parameters were adjusted to provide the best match between theory and experiment. Account was also taken o f the variation o f k and K with wavelength. The parameters that were different to those quoted earlier were: r41 = 4 . 4 1 ×
Table 1 Experimentally determined values of amplitude absorption constant ( a ) , gyration constant (7) and beam ration (//) and calculated photoionisation constant (s) at various wavelengths.
Source wavelength
a
~,
lO-~2m/V;#=3×lO-6m2/VsandEo=_4.75kV/
(nm)
(m - l )
(°/m)
cm. The low value o f the electric field was needed to ensure that the peaks in the theoretical detuning curves occurred at the same values o f detuning frequency as those in the experimental data. This lends credence to the argument that the static electric field
514.5 501.7 496.5 488.0
55.5 89.5 108.5 i 51.5
39000 42100 43500 46000
292
476.5
233.5
49800
B
s (m2/J)
500 700 200 200 60
2.9x 10-5 4.6x10 -5 5.6× 10-5 7.8× 10-5
12.1X 10 -5
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40.
OPTICS COMMUNICATIONS
1 June 1991
Gain ~
30,
A
20,
10,
DetunlngFrequency(Hz) o'.s i'. i~5 2.. 2:5 3'. 3'.5 4'. 20•. Gain 17 ~5'
Gain D
15.. 12.5 i0 7.5 5¢
~n.
cy (Hz B)
2.5
12,
0:5 I'. i~5 2.. 2:s 3'. 3:5 4'. Gain
Detunlng Frequency (Hz) I
0.5 5¢
4.5'
10,
i.
1.5
2.
2.5
3.
3.5
I
4.
Gain
4.'
8,
3.5. 3.'
6.
2.5" 4.
2.'
2,
0:5
i'. 1:5 2'. 2:5
cy (Hz) 3'. 3.5'
1.5,
4'.
Detuning Frequency [Hz) 0 ~5
1'•
' 1.5
' 2.
2 .' 5
' 3.
I
3'.5
4.
Fig. 6. Theoretical detuning curves for BSO. A 514.5 nm, B 501.7 nm, C 496.5 nm, D 488.0 nm, E 476.5 nm.
within the crystal is often less than the applied field as a result of potential drops at the electrode-crystal interface. It may be seen that the theoretical curves shown in fig. 6 are a good approximation to the experimental curves - in particular we obtain values of gain which closely match experiment, even though we are using a realistic value for the electro-optic coefficient, rather than a reduced effective value which is often done to account for the effects of optical activity and absorption. Only by the inclusion within the field equations of terms accounting for optical activity and absorption can the correct form of detuning curve be obtained.
Acknowledgement The authors would like to thank the United Kingdom Science and Engineering Research Council for supporting this work.
References [ 1 ] N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin and V.L. Vinetskii, Ferroelectri¢s 22 (1979) 949. [ 2 ] Ph. Refregier, L. Solymar, H. Rajbenbach and J.P. Huignard, J. Appl. Phys. 58 (1985) 45. [3 ] N.V. Kukhtarev, G.E. Dovgalenko and V.N. Starkov, Appl. Phys. A 33 (1984) 227.
293
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OPTICS COMMUNICATIONS
[4] P.D. Foote and T.J. Hall, Optics Comm. 57 (1986) 201. [5] A.E. Mandel, S.M. Shanderov and V.V. Shepelevich, 2nd. Conf. on Photorefractive materials, effects and devices, 1990, Aussois, France. [ 6 ] A. Marrakchi, R.V. Johnson and A.R. Tanguay, J. Opt. Soc. Am.B 3 (1986) 321. [7]T.J. Hall, A.K. Powell and C. Stace, Optics Comm. 75 (1990) 159.
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[8] A. Marrakchi, R.V. Johnson and A.R. Tanguay, IEEE J. Quant. Electron. 23 ( 1987 ) 2142. [9] D.C. Jones, D. Phil. Thesis, Department of Engineering Science, University of Oxford, 1989. [ 10 ] L.B. Au and L. Solymar, Optics Lett. 13 (1988) 660. [ 11 ] R.A. Mullen and R.W. Hellwarth, J. Appl. Phys. 58 (1985) 40.