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Dependence of space charge field and gain coefficient on the applied electric field in photorefractive materials
Optics & Laser Technology 43 (2011) 95–101
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Dependence of space charge field and gain coefficient on the applied electric field in photorefractive materials Ruchi Singh, M.K. Maurya, T.K. Yadav, D.P. Singh, R.A. Yadav n Department of Physics, Banaras Hindu University, Varanasi-221005, India
a r t i c l e in fo
abstract
Article history: Received 1 January 2010 Received in revised form 18 May 2010 Accepted 18 May 2010 Available online 1 July 2010
Intensity dependent space charge field and gain coefficient in the photorefractive medium due to the two interfering beams have been calculated by solving the material rate equations in presence of externally applied dc electric field. The gain coefficient has been studied with respect to variations in the input intensity, modulation depth, concentration ratio and normalized diffusion field in the absence and presence of the externally applied dc electric field. Space charge field has also been computed by varying the intensity ratio in the presence and absence of the externally applied dc electric field. It has been found that the rate of change of the space charge field with the normalized dc field decreases with the increasing intensity ratio for different values of the normalized diffusion field. It has also been found that the externally applied dc electric field has appreciable effect only when it is larger than the diffusion field. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Space charge field Two-beam coupling in photorefractive materials and applied electric field
1. Introduction Recently, two-wave mixing gain has been examined in versatile photorefractive crystals [1–7] by several researchers. In two-wave mixing, a signal beam is mixed with a pump beam inside the photorefractive crystal, where the two beams can interact and exchange energy in desirable ways [8]. Two-wave mixing gain and phase conjugation in photorefractive materials show high amplification and reflectivities [9]. These processes have been successfully demonstrated and utilized in various configurations, including ring resonators [10–12] where an oscillating beam arising from scattered noise experiences large amplification. Kwak et al. [12,13] obtained intensity dependent space charge field (SCF) from Kukhtarev’s material rate equations for the single charge carrier model. In order to justify their work on two-wave mixing, the gain as a function of both the modulation depth and the input intensity for various beam ratio was measured and the experimental results were compared with the theory [14]. However, the above authors have not considered the effect of applied electric field on the SCF and the gain coefficient. In the present paper, we have taken a rigorous approach to deal with the two-beam coupling under the influence of externally applied dc electric field. The material rate equations have been solved analytically for SCF in the presence of an externally applied dc electric field and expressions for the gain coefficient as a function
n
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of modulation depth and input intensity have been derived. Computed results for SCF and gain coefficient using presently derived expressions have been compared with the results obtained by Kwak et al. [14] in the absence of applied dc electric field.
2. Mathematical description 2.1. Non-linear SCF Non-linear differential equation for SCF has been derived using Kukhtarev’s material rate equations [14] which are given by @NDþ ¼ ðND NDþ ÞðsI þ bÞgR nNDþ @t
ð1Þ
@NDþ @n 1 @J ¼ þ @t e @z @t
ð2Þ
@E e ¼ ðNDþ NA nÞ @z e0 e
ð3Þ
J ¼ emnE þ kB T m
@n @z
ð4Þ
where n, ND, NDþ and NA stand for the densities of the electrons, neutral donor traps, ionized donor traps and neutral acceptor traps, respectively. The electrons are generated at the rate of ðND NDþ ÞðsI þ bÞ, where s is the photo-ionization cross section, I is the intensity of the incident light, b is the rate of thermal
96
R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
generation. The rate of trap capture is gR nNDþ , where gR is the carrier ionized trap recombination rate, e is the electronic charge, J is the current density, E is the electrostatic field, e0 is the free space permittivity, e is the relative static dielectric constant, m is the mobility, kBT is the thermal energy which is the product of the Boltzmann’s constant kB and absolute temperature T. These equations are valid for any range of illuminating intensity. In order to explain the intensity dependent SCF one may consider only single charge carrier model and neglect the electron hole recombination. When two monochromatic light beams of intensities Ip(0) and Is(0) interact in a photorefractive medium, an interference pattern is formed which creates refractive index grating via Pockel’s effect [15]. The refractive index grating moves if the two beams have differing frequencies by very small amount and remains stationary if the two beams have the same frequency. The intensity distribution for the refractive index grating is given by [16] IðzÞ ¼ I0 þ
1 I1 expðikg zÞ þ c:c: 2
ð5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where I0 ¼Ip(0)+ Is(0), I1 ¼ 2 Ip ð0ÞIs ð0Þ, m ¼ ðI1 =I0 Þ is the modulation depth or intensity ratio, kg is the magnitude of the grating wave vector and c.c. represents complex conjugate. It is assumed that all the physical variables used in Eqs. (1)–(4) are also of the form given by Eq. (5) and can be written in the following general form: Yðz,tÞ ¼ Y0 þ
1 Y1 ðtÞexpðikg zÞ þc:c: 2
ð6Þ
Using the general forms of all the physical variables given by Eqs. (5) and (6) and comparing the zero and first order coefficients of exp(ikgz), the following equations are obtained: þ @ND0 þ þ ¼ ðND ND0 ÞðsI0 þ bÞgR n0 ND0 @t
ð7aÞ
þ @ND1 þ þ þ þ ¼ ðND ND0 ÞsI1 ND1 ðsI0 þ bÞgR n1 ND0 gR ND1 n0 @t
ð7bÞ
þ @ND0 @n0 ¼ @t @t
ð8aÞ
þ @ND1 @n1 i kg J1 ¼ þ e @t @t
þ ND0
¼ NA þ n0
ie þ n1 ND1 E1 ¼ e0 ekg
2.2.1. In the absence of externally applied dc electric field In the absence of the externally applied dc electric field, the photoelectrons are transported by diffusion (gradient of the electron density). When the electrons are generated by illumination of incident light, current is produced by the diffusion and charges are separated. The static electric field produced by the buildup of space charges will move these electrons in the opposite direction [17]. In the absence of the external field E0 ¼0 and ð@E=@tÞ ¼ 0 and therefore using Eq. (A.9) one has iEd m ð1 þ Ed =E*q Þx
E1 ðI0 Þ ¼
ð13Þ
The effective limiting SCF appearing in Eq. (13) can be written as n o
eNA n0 r1 Nn0A þ Nn0A r e0 ekg 1 þ NA * ð14Þ Eq ¼ r Substituting the value of E*q from Eq. (14) in Eq. (13), one finally gets the expression for the steady state SCF as iEd m
E1 ðI0 Þ ¼
r 1 þ ð1 þ ðn0 =NA ÞÞðr1ðn 0 =NA ÞÞ
n0 NA
þ EEdq
ð15Þ
In the absence of the light beams, the average light intensity reduces to zero (I0 ¼0) and therefore from Eq. (12) n0 ¼0, thus Eq. (15) becomes E1 ð0Þ ¼
iEd m r E
1þ
ð16Þ
d
r1 Eq
The intensity dependent factor normalized to the low intensity value is obtained after dividing Eq. (15) by Eq. (16), i.e. r Ed 1 þ r1 ImE1 ðI0 Þ Eq
¼ ð17Þ ImE1 ð0Þ 1 þ 1 þ ðn =N Þ rr1ðn =N Þ Nn0A þ EEdq ð 0 0 A Þð A Þ 2.2.2. In the presence of externally applied dc electric field In the case when there is an externally applied dc electric field (E0 a0), the amplitude of the SCF is given by Eq. (A.10)
ð9bÞ
Eu1 ðI0 Þ ¼ n
ð10aÞ
ðE0 þ iEd Þm o 1þ ðEd iE0 Þ=E*q x
ð18Þ
Using Eqs. (14) and (18), the expression for E10 (I0) is given by
ð10bÞ
Eliminating all the physical variables from Eqs. (7)–(10) except the SCF E1 and after rearranging one finally gets the second order differential equation for the intensity dependent SCF E1 as @2 E1 @E1 þ BðtÞE1 ¼ CðtÞ þ AðtÞ @t @t 2
2.2. Intensity dependence of SCF on externally applied dc electric field
ð8bÞ
J0 ¼ emn0 E0 J1 ¼ emðn0 E1 þn1 E0 Þ þ iKB T mkg n1
where r¼ ND/NA is the concentration ratio and f ¼I0/Isat is the intensity ratio. The saturation intensity Isat ¼ gRNA/s is defined by neglecting the low thermal excitation rate b when it compares with the average photo excitation rate sI0.
ð11Þ
where the coefficients A(t), B(t) and C(t) are defined in the appendix Eqs. (A.6)–(A.8). In deriving Eq. (11), the average electron density n0 is assumed to be constant with time in accordance with the steady state approximation and is given by Eq. (A.13): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NA ð1 þf Þ þ ð1 þ f Þ2 þ 4f ðr1Þ ð12Þ n0 ¼ 2
Eu1 ðI0 Þ ¼
ðE0 þiEd Þm
r 1þ ð1 þ ðn0 =NA Þðr1ðn 0 =NA ÞÞ
n0 NA
0 þ Ed iE Eq
ð19Þ
In the presence of the externally applied dc electric field, the intensity independent SCF is given by the expression Eu1 ð0Þ ¼
ðE0 þiEd Þm
r 1 þ r1
Ed iE0 Eq
ð20Þ
Dividing Eq. (19) by Eq. (20), multiplying the numerator and denominator by the complex conjugate of the denominator of Eq. (19), one has the expression for the intensity dependent factor normalized to the low intensity value as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðux þ vyÞ2 þðuyvxÞ2 ImEu1 ðI0 Þ ¼ Zu ¼ ð21Þ ImEu1 ð0Þ x2 þy2
R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
where r Ed r E0 , v¼ , u ¼ 1þ r1 Eq r1 Eq r n0 E þ d , x ¼ 1þ n0 n0 NA Eq 1þ r1 NA NA r E 0 y¼ n0 n0 Eq 1þ r1 NA NA Substituting E0 ¼0 in Eq. (21), it reduces to Eq. (17) which corresponds to the case of zero applied electric field. 2.3. Photorefractive gain coefficient The gain g is defined as the intensity ratio of the output signal beam in the presence of the pump beam to that in the absence of the pump beam and is given by [14]
g¼
ð1 þ b0 ÞexpðaLÞ 1 þ b0 expðGLÞ
ð22Þ
where a is the linear absorption coefficient, L is the medium thickness, b0 ¼ ðIp ð0Þ=ðIs ð0ÞÞ is the incident beam intensity and G is the gain coefficient. Eq. (22) which is based on the pump depletion theory is valid only for small modulation depth m51. The gain coefficient G is related to the imaginary part of the SCF E1 by the relation
G¼
2pn3r reff ImðE1 Þ m l cos y
97
The SCF as functions of intensity ratio have been computed in the absence and presence of the externally applied dc electric field using Eqs. (17) and (21), respectively. Figs. 1(a) and (b) depict the variation of SCF against intensity ratio (I0/Isat) for various normalized diffusion field EdN at r ¼100 in the absence and presence of applied dc electric field, respectively. In the absence of applied electric field it is seen that Im(E1) increases rapidly, reaches a maximum value and gradually decreases with intensity ratio due to carrier saturation. On the other hand, in the presence of applied dc electric field the nature of the curves remains the same as for as the variation with intensity is concerned. The SCF decreases with the increasing EdN for E0N 44EdN while for E0N rEdN it increases with the increasing EdN. Fig. 2 shows the variation of the SCF with the intensity ratio (I0/Isat) in the absence and presence of the externally applied dc electric field (E0N) for the different values of diffusion field (EdN). It is found that for the lower diffusion fields the SCF depends appreciably on the externally applied dc electric field. However, for the higher diffusion field, the SCF appears to be independent of the externally applied dc electric field. Variation of the SCF with the intensity ratio (I0/Isat) for the various applied dc electric fields is shown in Fig. 3 for EdN ¼30 and
ð23Þ
where nr is the refractive index, l is the wavelength of light, y is the half angle between the two incident beams inside the medium, reff is the electro optic coefficient depending on the incident beam polarization and crystal orientation and m is the modulation depth. Im(E1) is proportional to m and hence, G is independent of m by Eq.(23). The measured gain [14] decreases significantly at large modulation depth which means that the gain coefficient is no more constant with m. An empirical correction function f(m)is used in place of the modulation depth m in the SCF expression in order to explain the decrement in the gain with m and it is given by 1 ½1expðamÞ f ðmÞ ¼ ð24Þ a where a is a fitting parameter depending on the experimental conditions. For m51, the RHS of Eq. (24) reduces to the linear modulation theory, value of f(m), i.e. f(m)¼m. Eqs. (19) and (23) together with Eq. (24) lead to the following expression for the non-linear gain coefficient that depends on both the modulation depth and the incident intensity under the influence of the externally applied dc electric field:
G GðI0 ,mÞ ¼ Go o Zuf ðmÞ=m
ð25Þ
where Go o is the limiting value of the gain coefficient in the limit I0-0 and m-0 which also depends on the experimental conditions such as the state of polarization of the light beam, wavelength, temperature and incident angles [14].
3. Results and discussion For the computations of the physical parameters such as f¼I0/ Isat, EdN ¼ ðED =EQ Þ, E0N ¼ ðE0 =EQ Þ, r ¼ ðND =NA Þ and b0 ¼ ðIp ð0Þ=Is ð0ÞÞ the values of the various constants appearing in these equations have been taken from the work of Kwak et al. [14]. Hence, our calculated results are applicable for the experimental data for BaTiO3 crystals.
Fig. 1. (a) Variation of SCF with f for the different values of EdN in the absence of the applied dc electric field E0N. (b) Variation of SCF with f for the different values of EdN in the presence of the applied dc electric field E0N.
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R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
Fig. 2. Dependence of SCF on f for different values of EdN in the absence and presence of the applied dc electric field E0N.
Fig. 3. Dependence of SCF on f for different values of the applied dc electric field E0N.
Fig. 4. (a) Variation of SCF with f for different values of r in the absence of the applied dc electric field E0N. (b) Variation of SCF with f for different values of r in the presence of the applied dc electric field E0N.
r ¼100. It is obvious from Fig. 3 that the imaginary part of the SCF increases rapidly, reaches a maximum value and then gradually decreases. It is seen that with the increasing applied dc electric field, for a given value of intensity ratios the SCF increases with the shifting peak positions on the higher intensity ratio side. Figs. 4(a) and (b) present variation of the SCF with the intensity ratio (I0/Isat) for the different values of concentration ratio (r) in the absence and presence of the applied dc electric field, respectively. Comparing these two figures. it is found that the SCF increases more rapidly under the influence of the applied dc electric field. Figs. 5(a) and (b) show variation of the gain coefficient G using Eqs. (17) and (21) with Eq. (25) as a function of the input intensity I0 for different beam ratios b0 without and with the external electric field in the low intensity region, respectively. From these figures it is clear that the enhancement in the gain coefficint is more in the presence of the applied dc electric field and reaches to a saturation value in the optimum intensity range of 100– 200 mW/cm2 with the increasing input intensity. Figs. 6(a) and (b) depict variation of the gain coefficient G as a function of the input intensity I0 for different beam ratios b0
without and with the external electric field in the high intensity region, respectively. In the absence of the applied electric field it is seen that the maximum value of the gain coefficient is in the intensity range 100–200 mW/cm2 while in the presence of the applied electric field the maximum gain is achieved in the intensity range 100–500 mW/cm2. Fig. 7 shows variation of the gain coefficient G with the input intensity I0 for different values of the externally applied electric field E0N. It is obvious that for each applied electric field, the steady state gain coefficient increases upto a certain values, reaches a maximum and decreases with the increasing intensity.The peak positions of the gain coefficient along with the corresponding input intensity are given in Table 1. Fig. 8(a) shows variation of gain coefficient G with the normalized electric field E0N for different values of f (fixed r¼ 100, b0 ¼100, EdN ¼30) and Fig. 8(b) for different value of EdN (fixed r ¼100, f¼10, b0 ¼100). From Fig. 8(a), it is noticed that the gain decreases with the increasing intensity ratio f while it remains constant for low values of the applied electric field (E0N 10) and increases with the increasing applied electric field, afterwards it saturates above certain value of the applied electric field.
R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
Fig. 5. (a) Dependence of G on I0 for different values of b0 without applied dc electric field E0N in the low intensity region. (b) Dependence of G on I0 for different values of b0 with applied dc electric field E0N in the low intensity region.
Fig. 6. (a) Variation of G with I0 for different values of b0 in the presence of the applied dc electric field in the high intensity region. (b) Variation of G with I0 for different values of b0 in the absence of the applied dc electric field in the high intensity region.
From Fig. 8(b) it is obvious that the gain G increases with the applied electric field E0N for the lower value of diffusion field EdN. On increasing the diffusion field the effect of applied electric field becomes less effective and for EdN 4200, the electric field has practically no effect on the gain coefficient. On the other hand, for the applied electric field E0N 450, the diffusion field has no effect on the gain coefficient.
4. Conclusion In the present paper, an analytic expression of the intensity dependent SCF and the gain coefficient under the externally applied dc electric field have been derived. The theory developed in this paper could be applicable to the other photorefractive materials. From the present work, one could conclude the following: 1. For the lower diffusion fields the SCF depends appreciably on the externally applied dc electric field E0N. However, for the higher diffusion field EdN, the SCF appears to be independent of the externally applied dc electric field.
99
Fig. 7. Variation of G with I0 for different values of applied dc electric field.
100
R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
Table 1 Gain coefficients vs input intensity corresponding to the peak positions. Applied electric field E0N
Input intensity I0 (mW/cm2)
Gain coefficient G (cm 1)
10 25 50 75 100 200 500
148.9 174.5 225 264.3 300.8 425 482.7
9.58 11.31 15.1 18.38 20.81 25.37 27.64
5. The gain coefficient G increases with the applied dc electric field for the lower value of diffusion field. For EdN 4200, the electric field has practically no effect on the gain coefficient whereas for the applied electric field E0N 4 50, the diffusion field has no effect on the gain coefficient.
Acknowledgement Ruchi Singh, M.K. Maurya and T.K. Yadav are thankful to the University Grants Commission (UGC), New Delhi for providing the financial support in the form of fellowships.
Appendix Differentiating Eq. (10b) with respect to t and making use of Eq. (8b), we get @E1 J1 ¼ @t e0 e
ðA:1Þ
Differentiating Eq. (A.1) partially with respect to t and using Eqs. (7b), (8a) and (9b), one has the following expression: @2 E1 em @E1 @n0 @E0 iKB T mkg @n1 þE þn þ E ¼ n þ 0 1 1 0 @t @t @t e0 e @t e0 e @t 2 ðA:2Þ From Eq. (9b) the value of n1 is given by @E1 emn0 E1 = emE0 þ iKB T mkg n1 ¼ e0 e @t
ðA:3Þ
Substituting the values of n1 and ð@n0 =@tÞ from Eqs. (8a) and (A.3) respectively, Eq. (A.2) leads to the following expression: @2 E1 emn0 1 @E0 n 0 ND @E1 i þ m k ðE þiE Þ þ g N 1 þ g D 0 R e0 e E0 þ iED @t NN þ @t @t 2 n0 ND em em n N e ek þ gR Nn0 1 þ g N 0 D 0 g þ ðE0 þiED Þ NN þ e0 e ie e0 e R NN þ emn0 1 @E0 E e0 e E0 þ iED @t 1 emn0 sI þ ðA:4Þ g NðE þiED Þ 0 m ¼ 0 e0 e R 0 sI0 þ b Eq. (A.4) can be written in the following simpler form 2
@ E1 @E1 þBðtÞE1 ¼ CðtÞ þ AðtÞ @t @t 2
ðA:5Þ
where the time dependent coefficients are given by ! 1 ED E0 t* t* @E0 AðtÞ ¼ * 1 þ * i * þ t EM EM td E0 þiED @t
Fig. 8. (a) Variation of G with E0N for different values of intensity ratios f. (b) Variation of G with E0N for different values of normalized diffusion field EdN.
2. With the increasing applied dc electric field for a given value of intensity ratios f, the SCF increases with the shifting peak positions on the higher intensity ratio side. 3. The enhancement in the gain coefficient G is more in the presence of the applied dc electric field and reaches to a saturation value in the optimum intensity range of 100– 500 mW/cm2 with the increasing input intensity I0. 4. The gain coefficient G decreases with the increasing intensity ratio while it increases with the increasing externally applied dc electric field.
BðtÞ ¼
1
td t*
CðtÞ ¼
1
td t
ED E0 t* @E0 1 þ * i * Eq Eq E0 þiED @t ðE0 þ iED Þ
sI0 m sI0 þ b
ðA:6Þ
! ðA:7Þ
ðA:8Þ
where t ¼ 1=gR N is the photoelectron lifetime, td ¼ e0 e=emn0 is the Maxwell relaxation time, t* ¼ t=x is the effective photoelectron lifetime, t*d ¼ td x is the effective Maxwell relaxation time, x ¼ 1 þ ðn0 ND =NN þ Þ, N ¼NA +n0, N + ¼ND NA n0, ED ¼ KB Tkg =e, is the diffusion field, E*M ¼ ðgR N=mkg Þx is the effective drift field, and E*q ¼ ðe=ðe0 ekg ÞÞðNN þ =ND Þx is the effective limiting SCF. The SCF E1 is assumed to be constant with time in accordance with the steady state approximation and therefore Eq. (A.5) is
R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
References
given by "
1
1þ
td t*
ED E0 t* @E0 i * E*q Eq E0 þiED @t
!# E1 ðI0 Þ ¼
1
td t
ðE0 þiED Þ
sI0 m sI0 þ b ðA:9Þ
It is noted that the intensity dependent factor owing to the thermal excitation is neglected in Eq. (A.9) and hence, the intensity dependent steady state SCF in the presence of the applied dc electric field ðð@E0 =@tÞ ¼ 0Þ is given by Eu1 ðI0 Þ ¼ n
ðE0 þ iEd Þm o 1 þðEd iE0 Þ=E*q x
ðA:10Þ
Under the steady state approximation, the average electron density n0 is assumed to be constant with the time and so Eqs. (7a) and (8a) lead to the following relations: sI0 þ b sI0 þ b n20 þn0 NA 1 þ NA ðND NA Þ ¼0 ðA:11Þ g R NA g R NA Since the above equation is quadratic in n0, there are two values of n0. Further n0 is positive and one of the two solutions of Eq. (A.11) is negative, the negative solution is unacceptable and the positive solution is given by (
)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 þb sI0 þ b 2 1þ sI0 þ b þ N NA 1 þ sIg0 N þ 4N ð N N Þ D A A g NA g NA A A R
n0 ¼
101
R
R
2 ðA:12Þ
As b is negligibly small compared to sI0, one has the following expression for n0 as 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 < sI0 sI0 2 sI0 = þ 4ðND =NA 1Þ þ 1þ n0 ¼ ðNA =2Þ 1 þ : gR NA g R NA g R NA ; ðA:13Þ
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Dependence of space charge field and gain coefficient on the applied electric field in photorefractive materials Ruchi Singh, M.K. Maurya, T.K. Yadav, D.P. Singh, R.A. Yadav n Department of Physics, Banaras Hindu University, Varanasi-221005, India
a r t i c l e in fo
abstract
Article history: Received 1 January 2010 Received in revised form 18 May 2010 Accepted 18 May 2010 Available online 1 July 2010
Intensity dependent space charge field and gain coefficient in the photorefractive medium due to the two interfering beams have been calculated by solving the material rate equations in presence of externally applied dc electric field. The gain coefficient has been studied with respect to variations in the input intensity, modulation depth, concentration ratio and normalized diffusion field in the absence and presence of the externally applied dc electric field. Space charge field has also been computed by varying the intensity ratio in the presence and absence of the externally applied dc electric field. It has been found that the rate of change of the space charge field with the normalized dc field decreases with the increasing intensity ratio for different values of the normalized diffusion field. It has also been found that the externally applied dc electric field has appreciable effect only when it is larger than the diffusion field. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Space charge field Two-beam coupling in photorefractive materials and applied electric field
1. Introduction Recently, two-wave mixing gain has been examined in versatile photorefractive crystals [1–7] by several researchers. In two-wave mixing, a signal beam is mixed with a pump beam inside the photorefractive crystal, where the two beams can interact and exchange energy in desirable ways [8]. Two-wave mixing gain and phase conjugation in photorefractive materials show high amplification and reflectivities [9]. These processes have been successfully demonstrated and utilized in various configurations, including ring resonators [10–12] where an oscillating beam arising from scattered noise experiences large amplification. Kwak et al. [12,13] obtained intensity dependent space charge field (SCF) from Kukhtarev’s material rate equations for the single charge carrier model. In order to justify their work on two-wave mixing, the gain as a function of both the modulation depth and the input intensity for various beam ratio was measured and the experimental results were compared with the theory [14]. However, the above authors have not considered the effect of applied electric field on the SCF and the gain coefficient. In the present paper, we have taken a rigorous approach to deal with the two-beam coupling under the influence of externally applied dc electric field. The material rate equations have been solved analytically for SCF in the presence of an externally applied dc electric field and expressions for the gain coefficient as a function
n
Corresponding author. Tel.: +91 9452497623; fax: + 91 5422368390. E-mail addresses: [email protected], [email protected] (R.A. Yadav).
0030-3992/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2010.05.010
of modulation depth and input intensity have been derived. Computed results for SCF and gain coefficient using presently derived expressions have been compared with the results obtained by Kwak et al. [14] in the absence of applied dc electric field.
2. Mathematical description 2.1. Non-linear SCF Non-linear differential equation for SCF has been derived using Kukhtarev’s material rate equations [14] which are given by @NDþ ¼ ðND NDþ ÞðsI þ bÞgR nNDþ @t
ð1Þ
@NDþ @n 1 @J ¼ þ @t e @z @t
ð2Þ
@E e ¼ ðNDþ NA nÞ @z e0 e
ð3Þ
J ¼ emnE þ kB T m
@n @z
ð4Þ
where n, ND, NDþ and NA stand for the densities of the electrons, neutral donor traps, ionized donor traps and neutral acceptor traps, respectively. The electrons are generated at the rate of ðND NDþ ÞðsI þ bÞ, where s is the photo-ionization cross section, I is the intensity of the incident light, b is the rate of thermal
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generation. The rate of trap capture is gR nNDþ , where gR is the carrier ionized trap recombination rate, e is the electronic charge, J is the current density, E is the electrostatic field, e0 is the free space permittivity, e is the relative static dielectric constant, m is the mobility, kBT is the thermal energy which is the product of the Boltzmann’s constant kB and absolute temperature T. These equations are valid for any range of illuminating intensity. In order to explain the intensity dependent SCF one may consider only single charge carrier model and neglect the electron hole recombination. When two monochromatic light beams of intensities Ip(0) and Is(0) interact in a photorefractive medium, an interference pattern is formed which creates refractive index grating via Pockel’s effect [15]. The refractive index grating moves if the two beams have differing frequencies by very small amount and remains stationary if the two beams have the same frequency. The intensity distribution for the refractive index grating is given by [16] IðzÞ ¼ I0 þ
1 I1 expðikg zÞ þ c:c: 2
ð5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where I0 ¼Ip(0)+ Is(0), I1 ¼ 2 Ip ð0ÞIs ð0Þ, m ¼ ðI1 =I0 Þ is the modulation depth or intensity ratio, kg is the magnitude of the grating wave vector and c.c. represents complex conjugate. It is assumed that all the physical variables used in Eqs. (1)–(4) are also of the form given by Eq. (5) and can be written in the following general form: Yðz,tÞ ¼ Y0 þ
1 Y1 ðtÞexpðikg zÞ þc:c: 2
ð6Þ
Using the general forms of all the physical variables given by Eqs. (5) and (6) and comparing the zero and first order coefficients of exp(ikgz), the following equations are obtained: þ @ND0 þ þ ¼ ðND ND0 ÞðsI0 þ bÞgR n0 ND0 @t
ð7aÞ
þ @ND1 þ þ þ þ ¼ ðND ND0 ÞsI1 ND1 ðsI0 þ bÞgR n1 ND0 gR ND1 n0 @t
ð7bÞ
þ @ND0 @n0 ¼ @t @t
ð8aÞ
þ @ND1 @n1 i kg J1 ¼ þ e @t @t
þ ND0
¼ NA þ n0
ie þ n1 ND1 E1 ¼ e0 ekg
2.2.1. In the absence of externally applied dc electric field In the absence of the externally applied dc electric field, the photoelectrons are transported by diffusion (gradient of the electron density). When the electrons are generated by illumination of incident light, current is produced by the diffusion and charges are separated. The static electric field produced by the buildup of space charges will move these electrons in the opposite direction [17]. In the absence of the external field E0 ¼0 and ð@E=@tÞ ¼ 0 and therefore using Eq. (A.9) one has iEd m ð1 þ Ed =E*q Þx
E1 ðI0 Þ ¼
ð13Þ
The effective limiting SCF appearing in Eq. (13) can be written as n o
eNA n0 r1 Nn0A þ Nn0A r e0 ekg 1 þ NA * ð14Þ Eq ¼ r Substituting the value of E*q from Eq. (14) in Eq. (13), one finally gets the expression for the steady state SCF as iEd m
E1 ðI0 Þ ¼
r 1 þ ð1 þ ðn0 =NA ÞÞðr1ðn 0 =NA ÞÞ
n0 NA
þ EEdq
ð15Þ
In the absence of the light beams, the average light intensity reduces to zero (I0 ¼0) and therefore from Eq. (12) n0 ¼0, thus Eq. (15) becomes E1 ð0Þ ¼
iEd m r E
1þ
ð16Þ
d
r1 Eq
The intensity dependent factor normalized to the low intensity value is obtained after dividing Eq. (15) by Eq. (16), i.e. r Ed 1 þ r1 ImE1 ðI0 Þ Eq
¼ ð17Þ ImE1 ð0Þ 1 þ 1 þ ðn =N Þ rr1ðn =N Þ Nn0A þ EEdq ð 0 0 A Þð A Þ 2.2.2. In the presence of externally applied dc electric field In the case when there is an externally applied dc electric field (E0 a0), the amplitude of the SCF is given by Eq. (A.10)
ð9bÞ
Eu1 ðI0 Þ ¼ n
ð10aÞ
ðE0 þ iEd Þm o 1þ ðEd iE0 Þ=E*q x
ð18Þ
Using Eqs. (14) and (18), the expression for E10 (I0) is given by
ð10bÞ
Eliminating all the physical variables from Eqs. (7)–(10) except the SCF E1 and after rearranging one finally gets the second order differential equation for the intensity dependent SCF E1 as @2 E1 @E1 þ BðtÞE1 ¼ CðtÞ þ AðtÞ @t @t 2
2.2. Intensity dependence of SCF on externally applied dc electric field
ð8bÞ
J0 ¼ emn0 E0 J1 ¼ emðn0 E1 þn1 E0 Þ þ iKB T mkg n1
where r¼ ND/NA is the concentration ratio and f ¼I0/Isat is the intensity ratio. The saturation intensity Isat ¼ gRNA/s is defined by neglecting the low thermal excitation rate b when it compares with the average photo excitation rate sI0.
ð11Þ
where the coefficients A(t), B(t) and C(t) are defined in the appendix Eqs. (A.6)–(A.8). In deriving Eq. (11), the average electron density n0 is assumed to be constant with time in accordance with the steady state approximation and is given by Eq. (A.13): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NA ð1 þf Þ þ ð1 þ f Þ2 þ 4f ðr1Þ ð12Þ n0 ¼ 2
Eu1 ðI0 Þ ¼
ðE0 þiEd Þm
r 1þ ð1 þ ðn0 =NA Þðr1ðn 0 =NA ÞÞ
n0 NA
0 þ Ed iE Eq
ð19Þ
In the presence of the externally applied dc electric field, the intensity independent SCF is given by the expression Eu1 ð0Þ ¼
ðE0 þiEd Þm
r 1 þ r1
Ed iE0 Eq
ð20Þ
Dividing Eq. (19) by Eq. (20), multiplying the numerator and denominator by the complex conjugate of the denominator of Eq. (19), one has the expression for the intensity dependent factor normalized to the low intensity value as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðux þ vyÞ2 þðuyvxÞ2 ImEu1 ðI0 Þ ¼ Zu ¼ ð21Þ ImEu1 ð0Þ x2 þy2
R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
where r Ed r E0 , v¼ , u ¼ 1þ r1 Eq r1 Eq r n0 E þ d , x ¼ 1þ n0 n0 NA Eq 1þ r1 NA NA r E 0 y¼ n0 n0 Eq 1þ r1 NA NA Substituting E0 ¼0 in Eq. (21), it reduces to Eq. (17) which corresponds to the case of zero applied electric field. 2.3. Photorefractive gain coefficient The gain g is defined as the intensity ratio of the output signal beam in the presence of the pump beam to that in the absence of the pump beam and is given by [14]
g¼
ð1 þ b0 ÞexpðaLÞ 1 þ b0 expðGLÞ
ð22Þ
where a is the linear absorption coefficient, L is the medium thickness, b0 ¼ ðIp ð0Þ=ðIs ð0ÞÞ is the incident beam intensity and G is the gain coefficient. Eq. (22) which is based on the pump depletion theory is valid only for small modulation depth m51. The gain coefficient G is related to the imaginary part of the SCF E1 by the relation
G¼
2pn3r reff ImðE1 Þ m l cos y
97
The SCF as functions of intensity ratio have been computed in the absence and presence of the externally applied dc electric field using Eqs. (17) and (21), respectively. Figs. 1(a) and (b) depict the variation of SCF against intensity ratio (I0/Isat) for various normalized diffusion field EdN at r ¼100 in the absence and presence of applied dc electric field, respectively. In the absence of applied electric field it is seen that Im(E1) increases rapidly, reaches a maximum value and gradually decreases with intensity ratio due to carrier saturation. On the other hand, in the presence of applied dc electric field the nature of the curves remains the same as for as the variation with intensity is concerned. The SCF decreases with the increasing EdN for E0N 44EdN while for E0N rEdN it increases with the increasing EdN. Fig. 2 shows the variation of the SCF with the intensity ratio (I0/Isat) in the absence and presence of the externally applied dc electric field (E0N) for the different values of diffusion field (EdN). It is found that for the lower diffusion fields the SCF depends appreciably on the externally applied dc electric field. However, for the higher diffusion field, the SCF appears to be independent of the externally applied dc electric field. Variation of the SCF with the intensity ratio (I0/Isat) for the various applied dc electric fields is shown in Fig. 3 for EdN ¼30 and
ð23Þ
where nr is the refractive index, l is the wavelength of light, y is the half angle between the two incident beams inside the medium, reff is the electro optic coefficient depending on the incident beam polarization and crystal orientation and m is the modulation depth. Im(E1) is proportional to m and hence, G is independent of m by Eq.(23). The measured gain [14] decreases significantly at large modulation depth which means that the gain coefficient is no more constant with m. An empirical correction function f(m)is used in place of the modulation depth m in the SCF expression in order to explain the decrement in the gain with m and it is given by 1 ½1expðamÞ f ðmÞ ¼ ð24Þ a where a is a fitting parameter depending on the experimental conditions. For m51, the RHS of Eq. (24) reduces to the linear modulation theory, value of f(m), i.e. f(m)¼m. Eqs. (19) and (23) together with Eq. (24) lead to the following expression for the non-linear gain coefficient that depends on both the modulation depth and the incident intensity under the influence of the externally applied dc electric field:
G GðI0 ,mÞ ¼ Go o Zuf ðmÞ=m
ð25Þ
where Go o is the limiting value of the gain coefficient in the limit I0-0 and m-0 which also depends on the experimental conditions such as the state of polarization of the light beam, wavelength, temperature and incident angles [14].
3. Results and discussion For the computations of the physical parameters such as f¼I0/ Isat, EdN ¼ ðED =EQ Þ, E0N ¼ ðE0 =EQ Þ, r ¼ ðND =NA Þ and b0 ¼ ðIp ð0Þ=Is ð0ÞÞ the values of the various constants appearing in these equations have been taken from the work of Kwak et al. [14]. Hence, our calculated results are applicable for the experimental data for BaTiO3 crystals.
Fig. 1. (a) Variation of SCF with f for the different values of EdN in the absence of the applied dc electric field E0N. (b) Variation of SCF with f for the different values of EdN in the presence of the applied dc electric field E0N.
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R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
Fig. 2. Dependence of SCF on f for different values of EdN in the absence and presence of the applied dc electric field E0N.
Fig. 3. Dependence of SCF on f for different values of the applied dc electric field E0N.
Fig. 4. (a) Variation of SCF with f for different values of r in the absence of the applied dc electric field E0N. (b) Variation of SCF with f for different values of r in the presence of the applied dc electric field E0N.
r ¼100. It is obvious from Fig. 3 that the imaginary part of the SCF increases rapidly, reaches a maximum value and then gradually decreases. It is seen that with the increasing applied dc electric field, for a given value of intensity ratios the SCF increases with the shifting peak positions on the higher intensity ratio side. Figs. 4(a) and (b) present variation of the SCF with the intensity ratio (I0/Isat) for the different values of concentration ratio (r) in the absence and presence of the applied dc electric field, respectively. Comparing these two figures. it is found that the SCF increases more rapidly under the influence of the applied dc electric field. Figs. 5(a) and (b) show variation of the gain coefficient G using Eqs. (17) and (21) with Eq. (25) as a function of the input intensity I0 for different beam ratios b0 without and with the external electric field in the low intensity region, respectively. From these figures it is clear that the enhancement in the gain coefficint is more in the presence of the applied dc electric field and reaches to a saturation value in the optimum intensity range of 100– 200 mW/cm2 with the increasing input intensity. Figs. 6(a) and (b) depict variation of the gain coefficient G as a function of the input intensity I0 for different beam ratios b0
without and with the external electric field in the high intensity region, respectively. In the absence of the applied electric field it is seen that the maximum value of the gain coefficient is in the intensity range 100–200 mW/cm2 while in the presence of the applied electric field the maximum gain is achieved in the intensity range 100–500 mW/cm2. Fig. 7 shows variation of the gain coefficient G with the input intensity I0 for different values of the externally applied electric field E0N. It is obvious that for each applied electric field, the steady state gain coefficient increases upto a certain values, reaches a maximum and decreases with the increasing intensity.The peak positions of the gain coefficient along with the corresponding input intensity are given in Table 1. Fig. 8(a) shows variation of gain coefficient G with the normalized electric field E0N for different values of f (fixed r¼ 100, b0 ¼100, EdN ¼30) and Fig. 8(b) for different value of EdN (fixed r ¼100, f¼10, b0 ¼100). From Fig. 8(a), it is noticed that the gain decreases with the increasing intensity ratio f while it remains constant for low values of the applied electric field (E0N 10) and increases with the increasing applied electric field, afterwards it saturates above certain value of the applied electric field.
R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
Fig. 5. (a) Dependence of G on I0 for different values of b0 without applied dc electric field E0N in the low intensity region. (b) Dependence of G on I0 for different values of b0 with applied dc electric field E0N in the low intensity region.
Fig. 6. (a) Variation of G with I0 for different values of b0 in the presence of the applied dc electric field in the high intensity region. (b) Variation of G with I0 for different values of b0 in the absence of the applied dc electric field in the high intensity region.
From Fig. 8(b) it is obvious that the gain G increases with the applied electric field E0N for the lower value of diffusion field EdN. On increasing the diffusion field the effect of applied electric field becomes less effective and for EdN 4200, the electric field has practically no effect on the gain coefficient. On the other hand, for the applied electric field E0N 450, the diffusion field has no effect on the gain coefficient.
4. Conclusion In the present paper, an analytic expression of the intensity dependent SCF and the gain coefficient under the externally applied dc electric field have been derived. The theory developed in this paper could be applicable to the other photorefractive materials. From the present work, one could conclude the following: 1. For the lower diffusion fields the SCF depends appreciably on the externally applied dc electric field E0N. However, for the higher diffusion field EdN, the SCF appears to be independent of the externally applied dc electric field.
99
Fig. 7. Variation of G with I0 for different values of applied dc electric field.
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R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
Table 1 Gain coefficients vs input intensity corresponding to the peak positions. Applied electric field E0N
Input intensity I0 (mW/cm2)
Gain coefficient G (cm 1)
10 25 50 75 100 200 500
148.9 174.5 225 264.3 300.8 425 482.7
9.58 11.31 15.1 18.38 20.81 25.37 27.64
5. The gain coefficient G increases with the applied dc electric field for the lower value of diffusion field. For EdN 4200, the electric field has practically no effect on the gain coefficient whereas for the applied electric field E0N 4 50, the diffusion field has no effect on the gain coefficient.
Acknowledgement Ruchi Singh, M.K. Maurya and T.K. Yadav are thankful to the University Grants Commission (UGC), New Delhi for providing the financial support in the form of fellowships.
Appendix Differentiating Eq. (10b) with respect to t and making use of Eq. (8b), we get @E1 J1 ¼ @t e0 e
ðA:1Þ
Differentiating Eq. (A.1) partially with respect to t and using Eqs. (7b), (8a) and (9b), one has the following expression: @2 E1 em @E1 @n0 @E0 iKB T mkg @n1 þE þn þ E ¼ n þ 0 1 1 0 @t @t @t e0 e @t e0 e @t 2 ðA:2Þ From Eq. (9b) the value of n1 is given by @E1 emn0 E1 = emE0 þ iKB T mkg n1 ¼ e0 e @t
ðA:3Þ
Substituting the values of n1 and ð@n0 =@tÞ from Eqs. (8a) and (A.3) respectively, Eq. (A.2) leads to the following expression: @2 E1 emn0 1 @E0 n 0 ND @E1 i þ m k ðE þiE Þ þ g N 1 þ g D 0 R e0 e E0 þ iED @t NN þ @t @t 2 n0 ND em em n N e ek þ gR Nn0 1 þ g N 0 D 0 g þ ðE0 þiED Þ NN þ e0 e ie e0 e R NN þ emn0 1 @E0 E e0 e E0 þ iED @t 1 emn0 sI þ ðA:4Þ g NðE þiED Þ 0 m ¼ 0 e0 e R 0 sI0 þ b Eq. (A.4) can be written in the following simpler form 2
@ E1 @E1 þBðtÞE1 ¼ CðtÞ þ AðtÞ @t @t 2
ðA:5Þ
where the time dependent coefficients are given by ! 1 ED E0 t* t* @E0 AðtÞ ¼ * 1 þ * i * þ t EM EM td E0 þiED @t
Fig. 8. (a) Variation of G with E0N for different values of intensity ratios f. (b) Variation of G with E0N for different values of normalized diffusion field EdN.
2. With the increasing applied dc electric field for a given value of intensity ratios f, the SCF increases with the shifting peak positions on the higher intensity ratio side. 3. The enhancement in the gain coefficient G is more in the presence of the applied dc electric field and reaches to a saturation value in the optimum intensity range of 100– 500 mW/cm2 with the increasing input intensity I0. 4. The gain coefficient G decreases with the increasing intensity ratio while it increases with the increasing externally applied dc electric field.
BðtÞ ¼
1
td t*
CðtÞ ¼
1
td t
ED E0 t* @E0 1 þ * i * Eq Eq E0 þiED @t ðE0 þ iED Þ
sI0 m sI0 þ b
ðA:6Þ
! ðA:7Þ
ðA:8Þ
where t ¼ 1=gR N is the photoelectron lifetime, td ¼ e0 e=emn0 is the Maxwell relaxation time, t* ¼ t=x is the effective photoelectron lifetime, t*d ¼ td x is the effective Maxwell relaxation time, x ¼ 1 þ ðn0 ND =NN þ Þ, N ¼NA +n0, N + ¼ND NA n0, ED ¼ KB Tkg =e, is the diffusion field, E*M ¼ ðgR N=mkg Þx is the effective drift field, and E*q ¼ ðe=ðe0 ekg ÞÞðNN þ =ND Þx is the effective limiting SCF. The SCF E1 is assumed to be constant with time in accordance with the steady state approximation and therefore Eq. (A.5) is
R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101
References
given by "
1
1þ
td t*
ED E0 t* @E0 i * E*q Eq E0 þiED @t
!# E1 ðI0 Þ ¼
1
td t
ðE0 þiED Þ
sI0 m sI0 þ b ðA:9Þ
It is noted that the intensity dependent factor owing to the thermal excitation is neglected in Eq. (A.9) and hence, the intensity dependent steady state SCF in the presence of the applied dc electric field ðð@E0 =@tÞ ¼ 0Þ is given by Eu1 ðI0 Þ ¼ n
ðE0 þ iEd Þm o 1 þðEd iE0 Þ=E*q x
ðA:10Þ
Under the steady state approximation, the average electron density n0 is assumed to be constant with the time and so Eqs. (7a) and (8a) lead to the following relations: sI0 þ b sI0 þ b n20 þn0 NA 1 þ NA ðND NA Þ ¼0 ðA:11Þ g R NA g R NA Since the above equation is quadratic in n0, there are two values of n0. Further n0 is positive and one of the two solutions of Eq. (A.11) is negative, the negative solution is unacceptable and the positive solution is given by (
)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 þb sI0 þ b 2 1þ sI0 þ b þ N NA 1 þ sIg0 N þ 4N ð N N Þ D A A g NA g NA A A R
n0 ¼
101
R
R
2 ðA:12Þ
As b is negligibly small compared to sI0, one has the following expression for n0 as 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 < sI0 sI0 2 sI0 = þ 4ðND =NA 1Þ þ 1þ n0 ¼ ðNA =2Þ 1 þ : gR NA g R NA g R NA ; ðA:13Þ
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