Optics & Laser Technology 43 (2011) 95–101

Contents lists available at ScienceDirect

Optics & Laser Technology

0030-39

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlastec

Dependence of space charge field and gain coefficient on the applied electricfield 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 i n f o

Article history:

Received 1 January 2010

Received in revised form

18 May 2010

Accepted 18 May 2010Available online 1 July 2010

Keywords:

Space charge field

Two-beam coupling in photorefractive

materials and applied electric field

92/$ - see front matter & 2010 Elsevier Ltd. A

016/j.optlastec.2010.05.010

esponding author. Tel.: +91 9452497623; fax

ail addresses: rayadav@bhu.ac.in, ray1357@gm

a b s t r a c t

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.

1. Introduction

Recently, two-wave mixing gain has been examined inversatile photorefractive crystals [1–7] by several researchers. Intwo-wave mixing, a signal beam is mixed with a pump beaminside the photorefractive crystal, where the two beams caninteract and exchange energy in desirable ways [8]. Two-wavemixing gain and phase conjugation in photorefractive materialsshow high amplification and reflectivities [9]. These processeshave been successfully demonstrated and utilized in variousconfigurations, including ring resonators [10–12] where anoscillating beam arising from scattered noise experiences largeamplification.

Kwak et al. [12,13] obtained intensity dependent space chargefield (SCF) from Kukhtarev’s material rate equations for the singlecharge carrier model. In order to justify their work on two-wavemixing, the gain as a function of both the modulation depth andthe input intensity for various beam ratio was measured and theexperimental results were compared with the theory [14].However, the above authors have not considered the effect ofapplied electric field on the SCF and the gain coefficient. In thepresent paper, we have taken a rigorous approach to deal with thetwo-beam coupling under the influence of externally applied dcelectric field. The material rate equations have been solvedanalytically for SCF in the presence of an externally applied dcelectric field and expressions for the gain coefficient as a function

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ail.com (R.A. Yadav).

of modulation depth and input intensity have been derived.Computed results for SCF and gain coefficient using presentlyderived expressions have been compared with the resultsobtained by Kwak et al. [14] in the absence of applied dc electricfield.

2. Mathematical description

2.1. Non-linear SCF

Non-linear differential equation for SCF has been derived usingKukhtarev’s material rate equations [14] which are given by

@NþD@t¼ ðND�NþD ÞðsIþbÞ�gRnNþD ð1Þ

@n

@t¼@NþD@tþ

1

e

� �@J

@zð2Þ

@E

@z¼

e

e0e

� �ðNþD �NA�nÞ ð3Þ

J¼ emnEþkBTm @n

@zð4Þ

where n, ND, NþD and NA stand for the densities of the electrons,neutral donor traps, ionized donor traps and neutral acceptortraps, respectively. The electrons are generated at the rate ofðND�NþD ÞðsIþbÞ, where s is the photo-ionization cross section, I isthe intensity of the incident light, b is the rate of thermal

R. Singh et al. / Optics & Laser Technology 43 (2011) 95–10196

generation. The rate of trap capture is gRnNþD , where gR is thecarrier ionized trap recombination rate, e is the electronic charge,J is the current density, E is the electrostatic field, e0 is the freespace permittivity, e is the relative static dielectric constant, m isthe mobility, kBT is the thermal energy which is the product of theBoltzmann’s constant kB and absolute temperature T. Theseequations are valid for any range of illuminating intensity. Inorder to explain the intensity dependent SCF one may consideronly single charge carrier model and neglect the electron holerecombination.

When two monochromatic light beams of intensities Ip(0) andIs(0) interact in a photorefractive medium, an interference patternis formed which creates refractive index grating via Pockel’seffect [15]. The refractive index grating moves if the two beamshave differing frequencies by very small amount and remainsstationary if the two beams have the same frequency.

The intensity distribution for the refractive index grating isgiven by [16]

IðzÞ ¼ I0þ1

2I1 expðikgzÞþc:c: ð5Þ

where I0¼ Ip(0)+ Is(0), I1 ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIpð0ÞIsð0Þ

p, m¼ ðI1=I0Þ is the modula-

tion depth or intensity ratio, kg is the magnitude of the gratingwave vector and c.c. represents complex conjugate. It is assumedthat all the physical variables used in Eqs. (1)–(4) are also of theform given by Eq. (5) and can be written in the following generalform:

Yðz,tÞ ¼ Y0þ1

2Y1ðtÞexpðikgzÞþc:c: ð6Þ

Using the general forms of all the physical variables given byEqs. (5) and (6) and comparing the zero and first order coefficientsof exp(ikgz), the following equations are obtained:

@NþD0

@t¼ ðND�NþD0ÞðsI0þbÞ�gRn0NþD0 ð7aÞ

@NþD1

@t¼ ðND�NþD0ÞsI1�NþD1ðsI0þbÞ�gRn1NþD0�gRNþD1n0 ð7bÞ

@n0

@t¼@NþD0

@tð8aÞ

@n1

@t¼@NþD1

@tþ

i

e

� �kgJ1 ð8bÞ

J0 ¼ emn0E0

J1 ¼ emðn0E1þn1E0Þþ iKBTmkgn1 ð9bÞ

NþD0 ¼NAþn0 ð10aÞ

E1 ¼ie

e0ekg

� �n1�NþD1

� �ð10bÞ

Eliminating all the physical variables from Eqs. (7)–(10) exceptthe SCF E1 and after rearranging one finally gets the second orderdifferential equation for the intensity dependent SCF E1 as

@2E1

@t2þAðtÞ

@E1

@tþBðtÞE1 ¼ CðtÞ ð11Þ

where the coefficients A(t), B(t) and C(t) are defined in theappendix Eqs. (A.6)–(A.8).

In deriving Eq. (11), the average electron density n0 is assumedto be constant with time in accordance with the steady stateapproximation and is given by Eq. (A.13):

n0 ¼NA

2

� ��ð1þ f Þþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ f Þ2þ4f ðr�1Þ

q� �ð12Þ

where r¼ND/NA is the concentration ratio and f¼ I0/Isat is theintensity ratio. The saturation intensity Isat¼gRNA/s is defined byneglecting the low thermal excitation rate b when it compareswith the average photo excitation rate sI0.

2.2. Intensity dependence of SCF on externally applied dc electric

field

2.2.1. In the absence of externally applied dc electric field

In the absence of the externally applied dc electric field, thephotoelectrons are transported by diffusion (gradient of theelectron density). When the electrons are generated by illumina-tion of incident light, current is produced by the diffusion andcharges are separated. The static electric field produced by thebuildup of space charges will move these electrons in the oppositedirection [17]. In the absence of the external field E0¼0 andð@E=@tÞ ¼ 0 and therefore using Eq. (A.9) one has

E1ðI0Þ ¼�iEdm

ð1þEd=E*qÞx

ð13Þ

The effective limiting SCF appearing in Eq. (13) can be written as

E*q ¼

eNA

e0ekg1þ n0

NA

r�1� n0

NA

þ

n0NA

rn o

rð14Þ

Substituting the value of E*q from Eq. (14) in Eq. (13), one finally

gets the expression for the steady state SCF as

E1ðI0Þ ¼�iEdm

1þ rð1þðn0=NAÞÞðr�1�ðn0=NAÞÞ

n0NAþ

Ed

Eq

ð15Þ

In the absence of the light beams, the average light intensityreduces to zero (I0¼0) and therefore from Eq. (12) n0¼0, thusEq. (15) becomes

E1ð0Þ ¼�iEdm

1þ rr�1

� � EdEq

ð16Þ

The intensity dependent factor normalized to the low intensityvalue is obtained after dividing Eq. (15) by Eq. (16), i.e.

ImE1ðI0Þ

ImE1ð0Þ¼

1þ rr�1

� � EdEq

1þ r1þðn0=NAÞð Þ r�1�ðn0=NAÞð Þ

n0NAþ

EdEq

ð17Þ

2.2.2. In the presence of externally applied dc electric field

In the case when there is an externally applied dc electric field(E0a0), the amplitude of the SCF is given by Eq. (A.10)

Eu1ðI0Þ ¼�ðE0þ iEdÞm

1þðEd�iE0Þ=E*q

n ox

ð18Þ

Using Eqs. (14) and (18), the expression for E10(I0) is given by

Eu1ðI0Þ ¼�ðE0þ iEdÞm

1þ rð1þðn0=NAÞðr�1�ðn0=NAÞÞ

n0NAþ

Ed�iE0

Eq

ð19Þ

In the presence of the externally applied dc electric field, theintensity independent SCF is given by the expression

Eu1ð0Þ ¼�ðE0þ iEdÞm

1þ rr�1

Ed�iE0

Eq

ð20Þ

Dividing Eq. (19) by Eq. (20), multiplying the numerator anddenominator by the complex conjugate of the denominator of Eq.(19), one has the expression for the intensity dependent factornormalized to the low intensity value as

Zu¼ ImEu1ðI0Þ

ImEu1ð0Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðuxþvyÞ2þðuy�vxÞ2

qx2þy2

ð21Þ

R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101 97

where

u¼ 1þr

r�1

Ed

Eq

� �, v¼

r

r�1

E0

Eq

� �,

x¼ 1þr

1þn0

NA

� �r�1�

n0

NA

� � n0

NAþ

Ed

Eq

� �,

y¼r

1þn0

NA

� �r�1�

n0

NA

� � E0

Eq

Substituting E0¼0 in Eq. (21), it reduces to Eq. (17) whichcorresponds 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 signalbeam in the presence of the pump beam to that in the absence ofthe pump beam and is given by [14]

g¼ ð1þb0Þexpð�aLÞ

1þb0 expð�GLÞð22Þ

where a is the linear absorption coefficient, L is the mediumthickness, b0 ¼ ðIpð0Þ=ðIsð0ÞÞ is the incident beam intensity and G isthe gain coefficient. Eq. (22) which is based on the pumpdepletion 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

G¼2pn3

r reff

lcosyImðE1Þ

mð23Þ

where nr is the refractive index, l is the wavelength of light, y isthe half angle between the two incident beams inside themedium, reff is the electro optic coefficient depending on theincident beam polarization and crystal orientation and m isthe modulation depth. Im(E1) is proportional to m and hence, G isindependent of m by Eq.(23). The measured gain [14] decreasessignificantly at large modulation depth which means that the gaincoefficient is no more constant with m. An empirical correctionfunction f(m)is used in place of the modulation depth m in the SCFexpression in order to explain the decrement in the gain with m

and it is given by

f ðmÞ ¼1

a

� �1�expð�amÞ½ � ð24Þ

where a is a fitting parameter depending on the experimentalconditions. For m51, the RHS of Eq. (24) reduces to the linearmodulation theory, value of f(m), i.e. f(m)¼m. Eqs. (19) and (23)together with Eq. (24) lead to the following expression for thenon-linear gain coefficient that depends on both the modulationdepth and the incident intensity under the influence of theexternally applied dc electric field:

G�GðI0,mÞ ¼Go oZuf ðmÞ=m ð25Þ

where Go o is the limiting value of the gain coefficient in the limitI0-0 and m-0 which also depends on the experimentalconditions such as the state of polarization of the light beam,wavelength, temperature and incident angles [14].

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.

3. Results and discussion

For the computations of the physical parameters such as f¼ I0/Isat, EdN ¼ ðED=EQ Þ, E0N ¼ ðE0=EQ Þ, r¼ ðND=NAÞ and b0 ¼ ðIpð0Þ=Isð0ÞÞthe values of the various constants appearing in these equationshave been taken from the work of Kwak et al. [14]. Hence, ourcalculated results are applicable for the experimental data forBaTiO3 crystals.

The SCF as functions of intensity ratio have been computed inthe absence and presence of the externally applied dc electric fieldusing Eqs. (17) and (21), respectively. Figs. 1(a) and (b) depict thevariation of SCF against intensity ratio (I0/Isat) for variousnormalized diffusion field EdN at r¼100 in the absence andpresence of applied dc electric field, respectively. In the absence ofapplied electric field it is seen that Im(E1) increases rapidly,reaches a maximum value and gradually decreases with intensityratio due to carrier saturation. On the other hand, in the presenceof applied dc electric field the nature of the curves remains thesame as for as the variation with intensity is concerned. The SCFdecreases with the increasing EdN for E0N44EdN while forE0NrEdN 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 dcelectric field (E0N) for the different values of diffusion field (EdN). Itis found that for the lower diffusion fields the SCF dependsappreciably on the externally applied dc electric field. However,for the higher diffusion field, the SCF appears to be independent ofthe externally applied dc electric field.

Variation of the SCF with the intensity ratio (I0/Isat) for thevarious applied dc electric fields is shown in Fig. 3 for EdN¼30 and

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. Singh et al. / Optics & Laser Technology 43 (2011) 95–10198

r¼100. It is obvious from Fig. 3 that the imaginary part of theSCF increases rapidly, reaches a maximum value and thengradually decreases. It is seen that with the increasing applieddc electric field, for a given value of intensity ratios the SCFincreases with the shifting peak positions on the higher intensityratio side.

Figs. 4(a) and (b) present variation of the SCF with the intensityratio (I0/Isat) for the different values of concentration ratio (r) inthe absence and presence of the applied dc electric field,respectively. Comparing these two figures. it is found that theSCF increases more rapidly under the influence of the applied dcelectric field.

Figs. 5(a) and (b) show variation of the gain coefficient G usingEqs. (17) and (21) with Eq. (25) as a function of the input intensityI0 for different beam ratios b0 without and with the externalelectric field in the low intensity region, respectively. From thesefigures it is clear that the enhancement in the gain coefficint ismore in the presence of the applied dc electric field and reaches toa 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 afunction of the input intensity I0 for different beam ratios b0

without and with the external electric field in the high intensityregion, respectively. In the absence of the applied electric field itis seen that the maximum value of the gain coefficient is inthe intensity range 100–200 mW/cm2 while in the presence of theapplied electric field the maximum gain is achieved in theintensity range 100–500 mW/cm2.

Fig. 7 shows variation of the gain coefficient G with the inputintensity I0 for different values of the externally applied electricfield E0N. It is obvious that for each applied electric field, thesteady state gain coefficient increases upto a certain values,reaches a maximum and decreases with the increasingintensity.The peak positions of the gain coefficient along withthe corresponding input intensity are given in Table 1.

Fig. 8(a) shows variation of gain coefficient G with thenormalized electric field E0N for different values of f (fixedr¼100, b0¼100, EdN¼30) and Fig. 8(b) for different value of EdN

(fixed r¼100, f¼10, b0¼100). From Fig. 8(a), it is noticed that thegain decreases with the increasing intensity ratio f while itremains 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 electricfield.

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.

R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101 99

From Fig. 8(b) it is obvious that the gain G increases with theapplied electric field E0N for the lower value of diffusion field EdN.On increasing the diffusion field the effect of applied electric fieldbecomes less effective and for EdN4200, the electric field haspractically no effect on the gain coefficient. On the other hand, forthe applied electric field E0N450, the diffusion field has no effecton the gain coefficient.

4. Conclusion

In the present paper, an analytic expression of the intensitydependent SCF and the gain coefficient under the externallyapplied dc electric field have been derived. The theory developedin this paper could be applicable to the other photorefractivematerials. From the present work, one could conclude thefollowing:

1.

Fig. 7. Variation of G with I0 for different values of applied dc electric field.

For the lower diffusion fields the SCF depends appreciably onthe externally applied dc electric field E0N. However, for thehigher diffusion field EdN, the SCF appears to be independent ofthe externally applied dc electric field.

Table 1Gain coefficients vs input intensity corresponding to the peak positions.

Applied electric

field E0N

Input intensity I0

(mW/cm2)

Gain coefficient

G (cm�1)

10 148.9 9.58

25 174.5 11.31

50 225 15.1

75 264.3 18.38

100 300.8 20.81

200 425 25.37

500 482.7 27.64

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.

R. Singh et al. / Optics & Laser Technology 43 (2011) 95–101100

2.

With the increasing applied dc electric field for a given value ofintensity ratios f, the SCF increases with the shifting peakpositions on the higher intensity ratio side.

3.

The enhancement in the gain coefficient G is more in thepresence of the applied dc electric field and reaches to asaturation 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 intensityratio while it increases with the increasing externally applieddc electric field.

5.

The gain coefficient G increases with the applied dc electricfield for the lower value of diffusion field. For EdN4200, theelectric field has practically no effect on the gain coefficientwhereas for the applied electric field E0N450, the diffusionfield has no effect on the gain coefficient.

Acknowledgement

Ruchi Singh, M.K. Maurya and T.K. Yadav are thankful to theUniversity Grants Commission (UGC), New Delhi for providing thefinancial support in the form of fellowships.

Appendix

Differentiating Eq. (10b) with respect to t and making use ofEq. (8b), we get

@E1

@t¼�

J1

e0eðA:1Þ

Differentiating Eq. (A.1) partially with respect to t and usingEqs. (7b), (8a) and (9b), one has the following expression:

@2E1

@t2¼�

eme0e

n0@E1

@tþE1

@n0

@tþn1

@E0

@tþ E0þ

iKBTmkg

e0e

� �@n1

@t

� �ðA:2Þ

From Eq. (9b) the value of n1 is given by

n1 ¼ �e0e@E1

@t�emn0E1

� �= emE0þ iKBTmkg

� �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:

@2E1

@t2þ

emn0

e0e�

1

E0þ iED

@E0

@t�imkgðE0þ iEDÞþgRN 1þ

n0ND

NNþ

� �� �@E1

@t

þ gRNn0 1þn0ND

NNþ

� �eme0eþðE0þ iEDÞ

eme0e

gRNn0ND

NNþ

� �e0ekg

ie

�

�emn0

e0e1

E0þ iED

@E0

@t

�E1

þemn0

e0egRNðE0þ iEDÞ

sI0

sI0þbm¼ 0 ðA:4Þ

Eq. (A.4) can be written in the following simpler form

@2E1

@t2þAðtÞ

@E1

@tþBðtÞE1 ¼ CðtÞ ðA:5Þ

where the time dependent coefficients are given by

AðtÞ ¼1

t*1þ

ED

E*M

�iE0

E*M

þt*

td�

t*

E0þ iED

@E0

@t

!ðA:6Þ

BðtÞ ¼1

tdt*1þ

ED

E*q

�iE0

E*q

�t*

E0þ iED

@E0

@t

!ðA:7Þ

CðtÞ ¼�1

tdtðE0þ iEDÞ

sI0

sI0þbm ðA:8Þ

where t¼ 1=gRN is the photoelectron lifetime, td ¼ e0e=emn0 is theMaxwell relaxation time, t* ¼ t=x is the effective photoelectronlifetime, t*

d ¼ tdx is the effective Maxwell relaxation time,x¼ 1þðn0ND=NNþ Þ, N¼NA+n0, N+

¼ND�NA�n0, ED ¼ KBTkg=e, isthe diffusion field, E*

M ¼ ðgRN=mkgÞx is the effective drift field, andE*

q ¼ ðe=ðe0ekgÞÞðNNþ =NDÞx 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) 95–101 101

given by

1

tdt*1þ

ED

E*q

�iE0

E*q

�t*

E0þ iED

@E0

@t

!" #E1ðI0Þ ¼�

1

tdtðE0þ iEDÞ

sI0

sI0þbm

ðA:9Þ

It is noted that the intensity dependent factor owing to thethermal excitation is neglected in Eq. (A.9) and hence,the intensity dependent steady state SCF in the presence of theapplied dc electric field ðð@E0=@tÞ ¼ 0Þ is given by

Eu1ðI0Þ ¼�ðE0þ iEdÞm

1þðEd�iE0Þ=E*q

n ox

ðA:10Þ

Under the steady state approximation, the average electrondensity n0 is assumed to be constant with the time and soEqs. (7a) and (8a) lead to the following relations:

n20þn0NA 1þ

sI0þbgRNA

� ��NA ND�NAð Þ

sI0þbgRNA

¼ 0 ðA:11Þ

Since the above equation is quadratic in n0, there are twovalues of n0. Further n0 is positive and one of the two solutions ofEq. (A.11) is negative, the negative solution is unacceptable andthe positive solution is given by

n0 ¼

�NA 1þ sI0 þbgRNA

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN2

A 1þ sI0þbgRNA

2þ4NA ND�NAð Þ

sI0þbgRNA

r( )

2ðA:12Þ

As b is negligibly small compared to sI0, one has the followingexpression for n0 as

n0 ¼ ðNA=2Þ � 1þsI0

gRNA

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

sI0

gRNA

� �2

þ4ðND=NA�1ÞsI0

gRNA

s8<:

9=;

ðA:13Þ

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