Four-wave Mixing in a Transparent Medium Based on Electrostriction and Dufour Effect at Large Reflectance

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ScienceDirect Physics Procedia 73 (2015) 26 – 32

4th h International Confereence Photon nics and Info formation Optics, O PhIO O 2015, 28-330 January 2015 2

Fo our-wavee mixingg in a traansparentt medium m based on electtrostrictiion andd Dufou ur effect at large reflectan nce V.V. Ivakhnik, M.V. Saveelyev* Sam mara State Univerrsity, Akademika P Pavlova st. 1, Samara, 443011, Russia

Absttract We analyze a the quaality of wave froont reversal forr the four-wave converter in op ptically transpaarent two-compoonent medium in the appro oximation of laarge reflectance. We obtain thee dependences oof the reflectance and the spatiial spectrum haalf-width of the wave refleccted such a connverter on the inntensity of pum mp waves. © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license ©2015 20 015The TheAuthors. Authorrs. Published byy Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer--review under rresponsibility of the National Research R Nucleear University MEPhI M (Moscow w Engineering PPhysics Institutte). Peer-review under responsibility of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) Keyw words:four-wave cconverter; transpaarent medium; eleectrostriction; Duufour effect.

1. In ntroduction Knowledge K off spatial, spatio-temporal characteristics c s of four-wav ve converters is a prerequiisite for its use u in adap ptive optics, iimage processing systems [Dmitriev (22003)]. Spatiaal, spatio-temp poral characte teristics have been studied in detail for four-wave converters in media witth Kerr and th hermal nonlin nearities, in m media modeleed by enerrgy levels systtem and in revversible photocchromic mateerials [Ivakhniik (2010)]. To T produce waave with wavve front reverssal (WFR) in four-wave mixing m can be used multicoomponent med dia in whicch due to thee presence off nanoparticlees in a mediuum several no onlinearities are simultaneeously realizeed, in partiicular, thermaal diffusion annd electrostricttion [Livashviili et al. (2013 3), Ivakhnik an nd Savelyev ((2013), Vorob byeva et all. (2014), Hem mmerling et al. a (2001), Russconi et al. (22004), Mahiln ny et al. (2006 6)]. In contrasst to the four-wave conv verters in meddia with Kerrr and thermall nonlinearitiees, filtered low w spatial freq quencies of thhe incident (siignal) wave, as shown inn [Ivakhnik annd Savelyev (2013), ( Vorobbyeva et al. (2014)], the fou ur-wave conveerter in a nonllinear dium based on electrostrictioon filters high h spatial frequeencies. med

* Corresponding C autthor. E-mail E address:[email protected]

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) doi:10.1016/j.phpro.2015.09.117

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V.V. Ivakhnik and M.V. Savelyev / Physics Procedia 73 (2015) 26 – 32

Reflectance can significantly influence on the spatial characteristics of the four-wave converter [Akimov et al. (2013), Akimov et al. (2015), Ivakhnik (1983), Akimov et al. (2011)]. At large reflectance number of dynamic gratings influencing the four-wave mixing process increases. Along with dynamic grating arising due to the interference of the wave incident on the nonlinear medium and the first pump wave, it is necessary to consider the grating arising due to the interference of the wave with WFR and a second pump wave. Processes such as selfdiffraction of the pump waves and the energy transfer from the object wave to signal wave begin to influence on the spatial structure of the wave with WFR. The purpose of our work is to study the spatial characteristics of the four-wave converter in a transparent twocomponent medium based on electrostriction and Dufour effect at high reflectance. 2. Derivation of basic equations We consider a plane layer of the optically transparent two-component medium (for example, liquid and nanoparticles having a density equal to the density of liquid) thickness Ɛ in which two opposing pump waves with complex amplitudes A1 and A2 and a signal wave with amplitude A3 propagate. Propagation of the radiation, intensity of which varies with the spatial coordinates, in such a medium leads due to electrostriction to the appearance of nanoparticles concentration flux which due to the Dufour effect changes the temperature (įT), and hence the refractive index of medium (įn=įT(dn/dT)). As a result of the degenerate four-wave interaction Ȧ+Ȧ–Ȧ=Ȧ object wave with complex amplitude A4 propagating toward the signal wave is generated. Initial scalar wave equation describing the four-wave interaction of radiation in a transparent nonlinear medium [Vorobyeva et al. (2014)] is

§ 2 · 2k 2 dn 2 G T ¸  A  A 0, ¨’  k  n0 dT © ¹

(1)

where A=A1+A2+A3+A4, k=Ȧn0/c, n0 is average refractive index. The equation (1) is supplemented with the system of balance equations for the concentration (įC) and temperature variations written in the approximation of linear nonequilibrium thermodynamics [Livashvili et al. (2013), Ivakhnik and Savelyev (2013), Vorobyeva et al. (2014)]

wG C wt

c pQ

D22’ 2G C  J’ 2 I ,

wG T wt

D11’ 2G T  D12’ 2G C.

(2)

(3)

Here, D11, D22, D12 and Ȗ are the coefficients of thermal conductivity, diffusion, Dufour and electrostriction respectively, cp is the specific heat capacity, Ȟ is the density, I=AA*. For steady state, substituting ’2G C from (2) into (3), we obtain

’ 2G T

J D12 D11 D22

’2 I .

The four-wave interaction of radiation is considered under the following conditions: x the approximation of the specified field on the waves of the pump (|A1,2|>>|A3,4|);

(4)

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V.V. Ivakhnik and M.V. Savelyev / Physics Procedia 73 (2015) 26 – 32

x the temperature gratings due to the interference of the first pump and the signal wave, the object wave and the second pump wave are considered; x the self diffraction of the pump waves on the temperature gratings is considered. The intensity of the radiation propagating in a nonlinear medium can be written as I I 0  A1 A3  A1 A3  A2 A4  A2 A4 where I 0 A1 A1  A2 A2 . Then the temperature variation can be represented as vary rapidly (įT31, įT42) or slowly (įT0) in space. the sum G T G T0  G T31  G T31  G T42  G T42 of components & & which & & Let the pump waves be plane A1,2  r A1,2  z exp ik1,2 r , where k1,2 is the wave vectors of the pump waves, & & U  x, y and z are the transverse and longitudinal components of the radius vector r . We expand the signal and object waves into plane waves



f

& Aj  r

&

& &



&

³ A N , z exp  iN U  ik z dN . j

j

j

jz

j

f

& of the jth wave, N j N jx , N jy and kjz are the transverse and longitudinal Here, j=3,4, A j is the spatial spectrum & components of the wave vector k j . Rapidly varying temperature components we expand into harmonic gratings



&

G T31,42  r

f

&

³ G T N 31,42

T 1,2



& & & , z exp  iNT 1,2 U dNT 1,2 .

f

& & Here, G T31 and G T42 are the spatial spectra of temperature gratings, N T 1 and N T 2 are the wave vectors of the gratings. In the approximation of slowly varying amplitudes at quasicollinear propagation of interacting waves (k/k1,3z§1, k/k2,4z§–1) equation (1) splits into a system of equations having the following form

dA1,2 dz dA

ri

k dn G T0 A1,2 n0 dT

0,

k dn k dn  ri G T0 A3,4 r i G T31,42  G T42,31 A1,2 exp ª¬ i  k1,2 z  k3,4 z z º¼ dz n0 dT n0 dT



3,4



& System of equation (5) is written under condition that N T 1 equations of system (5) has the form

A1  z

A10 exp ª¬  P  z º¼ , A 2  z

& NT 2

&

&

N1  N 3

&

(5)

0.

&

N 4  N 2 . The solution of the first two

A 20 exp ª¬  P  "  P  z º¼ .

Here,

A10

A1  z

0 , A20

A2  z

z

" , P  z i

k dn G T0  z1 dz1. n0 dT ³0

& & & & We make the substitutions A 4 N 4 , z A 4c N 4 , z exp ¬ª  P  "  P  z ¼º , A3 N 3 , z A3c N 3 , z exp ª¬  P  z º¼ . Then the equations describing the changes in spatial spectra of the signal and object waves, take the form

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V.V. Ivakhnik and M.V. Savelyev / Physics Procedia 73 (2015) 26 – 32

c dA3,4 dz

ri

k dn  G T31,42  G T42,31 A1,20 exp ¬ª i  k1,2 z  k3,4 z z ¼º 0. n0 dT





(6)

Equation (4) splits into the system of equations

J D12

’ 2G T0

D11D22

’2 I0

§ d2 ·  & J D12  2 A1,2 exp ª¬ i  k1,2 z  k3,4 z z º¼ u ¨ 2  NT 1,2 ¸ G T31,42 NT 1,2 , z  D11D22 © dz ¹ 2º & d ª u « 2i  k1,2 z  k3,4 z  NT21,2   k1,4 z  k3,4 z » A3,4 N 3,4 , z .  dz ¬ ¼

(7)

Differentiating twice with respect to the z coordinate terms in the system of equation (6) and considering (7) we obtain a system of coupled third order differential equations for the spatial spectra of the signal and object waves

c ªN 2   k  k 2 º r 2G  k  k dA3,4  r 1,2 z 3,4 z 1,2 1,2 z 3,4 z «¬ T 1,2 »¼ dz 3 dz 2 dz c dA4,3 2 2 ª º  c riG1,2 «NT 1,2   k1,2 z  k3,4 z » A3,4 #2G  k2,1z  k4,3 z exp  i'z r ¬ ¼ dz 2 c . riG ª«NT2 2,1   k2,1z  k4,3 z º» exp  i'z A4,3 ¬ ¼ c d 3 A3,4

 2i  k1,2 z  k3,4 z

c d 2 A3,4

^

`

(8)

Here, G1,2=kȖD12I1,2(dn/dT)/(n0D11D22), G2=G1G2, I1,2 A1,20 A1,20 are the intensities of the pump waves, ǻ=k1z+k2z– k3z–k4z is the projection of the wave detuning on the Z axis. & & In the paraxial approximation for the pump waves propagating strictly along the Z axis ( N1 N 2 0 ) we have & & 2 k1z–k3z= k4z–k2z=ț /(2k) where N N 3 N 4 . In solving the system of equations (8), we use the boundary conditions

& A3c N 3 , z

0

& c N 3 , A4c  z A30

" 0,

dA3c dz

z 0

dA3c dz

0, z "

dA 4c dz

z 0

dA 4c dz

0.

(9)

z "

Equality of the spatial spectra derivatives of the signal and object waves on the nonlinear layer edges derives from temperature invariability on the edges. Numerical solution of system (8) considering the boundary conditions (9) allows us to analyze the spatial spectrum of the object wave on the front edge of the nonlinear layer considering written temperature gratings įT31 and įT42 and self diffraction of the pump waves. 3. Discussion of the results

We consider the wave generated by a point source located on the front edge of a nonlinear layer as the signal & wave ( A30c N 3 1 ). Fig. 1 shows typical plots of spatial spectra of the object wave obtained by numerical analysis of the system of equations (8) at different intensities of the pump waves. For values G1,2Ɛ less than unity absolute value of the spatial

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V.V. Ivakhnik and M.V. Savelyev / Physics Procedia 73 (2015) 26 – 32

specctrum of the obbject wave with increasing g spatial frequeency (ț) increeases, then reaaches a constaant value. Witth the rise of the intensitty of the pumpp wave spatial frequency att which the sp patial spectrum m reaches a coonstant value shifts s to hiigher frequenccies. Physically P the increasing off absolute valu ue of the spatiial spectrum is i caused by presence p of ellectrostriction term in the t equation (4) J’ 2 I Ÿ J’ 2 A1 A3 . With W decreasinng of the period p of thee written inteerference graatings electtrostriction teerm increases, and thus th he amplitudess of the writtten temperatu ure gratings in increase. The heat cond duction process blurs the gratings. g At low spatial freequencies an electrostrictiv ve recording mechanism of o the gratiings dominatees that lead to increasing off the absolute values of its spatial s spectru um. At large sspatial frequeencies increeasing of the aabsolute valuees of temperature gratings spatial spectraa is compensated by gratinngs blurring due d to the heat h conductioon process [Voorobyeva et all. (2014)]. We W introduce aan amplitude reflectance r off the four-wavee converter

R

A4c N o 0.1k , z c N A30

0

.

Numerical N anaalysis of the system of equations (8) shows that at a high spatiaal frequenciess (țĺ0.1k) terms t conttaining the seccond and thirdd derivatives of o the spatial sspectrum of th he signal and object o waves to the z coord dinate c dz 2  2k dA3,4 c dz c dz 3  N 2 dA3,4 c dz , d 2 A3,4 d 3 A33,4 can be negleected. . Also the cond dition

  c c dA3,4 dz  2k ddA4,3 dz is satisfied. Then n the system oof equations (8), describing g the change iin spatial specctrum of th he signal and oobject waves, takes the form m







c N o 0.1k , z dA3,4 ddz





c N o 0.1k , z r iG1,2 A3,4





c N o 0.1k , z . #iGA4,3

(10)

Solving the sysstem of equations (10) and considering tthe boundary conditions on n the amplitudde of the signaal and pression descrribing the dep pendence of the reflectancee on the nonllinear objeect waves (9) we obtain ann analytic exp med dium parameteers and the inteensity of pum mp wave

Fig. 1. The spatial specttrum of the objectt wave at

2k"

183 , G1" 1 (1, 3), 0.8 (2), I1 I 2 1 (1), 0.44 (2, 3).

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V.V. Ivakhnik and M.V. Savelyev / Physics Procedia 73 (2015) 26 – 32

R

G2 1  exp ª¬i  G1  G2 " º¼ . G1 1  G2 expp ªi G  G " º 2 ¼ ¬  1 G1

At equal e intensitiies of the pum mp waves refleectance has thee form

R

tg  G1" .

(11)

The T expressionn for the refleectance of the four-wave coonverter in the optically tran nsparent two-ccomponent meedium baseed on electrostriction and Dufour effectt (11) formallly coincides with w the expression for thee reflectance of the fourr-wave converrter in the meddium with Keerr nonlinearityy [Ivakhnik (2 2010)]. With W increasinng of the refleectance the sm mooth increassing of absolu ute value of th he spatial specctrum of the object o wav ve with the chhange of the spatial s frequen ncy followed bby a constantt value is chan nged. With the he rise of the spatial s freq quency absoluute value of the spatial sp pectrum increeases, reachess a maximum m, then decreaases and reacches a con nstant value. The T quality oof WFR provvided that A 4c N , z 0 d A 4c N o 0.1k , z 0 is chaaracterized byy a cut out spatial s freq quencies bandd half-width (ǻ ǻț) determined from the conndition A 4c N 'N , z 0 R 2 . Fig. F 2 shows tthe dependencce of the refleectance (curvees 1 and 2) an nd the normaliized spatial fre requency band d halfwid dth (curves 1' aand 2') on the parameter G1. With increaasing intensity of the pump waves w a monootonic increassing in both h reflectance aand the cut ouut spatial frequ uency band haalf-width is ob bserved. Depeendence of thee reflectance on o the inteensity of the ppump waves agrees a qualitattively with thee analogous dependence d for the four-wavve converter on o the therrmal nonlinearrity [Akimov et al. (2011)].. Numerical N analysis of R and a ǻț dependencies on thhe intensities of the pump waves show ws that at I1=II2 and G1Ɛ<0.7 Ɛ the channge in the spaatial frequency y band half-wiidth of the objject wave is associated a withh the reflectan nce of the four-wave connverter by thee following ex xpression

Fig. 2. Dependencies of the reflectaance and the spaatial frequency bband half-width on the normalized intensity of tthe first pump wave w at 2kk " 183 , I1 I 2 1 (1, 1'), 2 (2, 2').

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V.V. Ivakhnik and M.V. Savelyev / Physics Procedia 73 (2015) 26 – 32

'N

'N 0  b ˜ R 2 .

where, ǻț0 is the spatial frequency band half-width at small reflectance (R<<1), b is the proportionally coefficient depending on the nonlinear layer thickness, the wavelength and the ratio of the intensities of the pump waves. Increasing of the nonlinear layer thickness leads to decreasing of the spatial frequency band half-width ǻț0 is directly proportional to the law 1/Ɛ. Increasing of the spatial frequency band half-width cut out by the four-wave converter with the rise of the reflectance or the intensities of the pump waves indicates a deterioration of the WFR quality. Let us give estimates the spatial frequency band half-width of the object wave. As a two-component medium we consider water (n0=1.33) containing nanoparticles with a radius a0=10-8 m [Livashvili et al. (2013)]. Suppose that in such medium thickness Ɛ=10-3 m waves interact with a wavelength of Ȝ=5.32·10-7 m. Then the spatial frequency band half-width obtained from the numerical analysis of the system (8) at small reflectance is ǻț0=3830 m-1, the coefficient b=1190 m-1. 4. Conclusion

As a result of the analysis of the spatial characteristics of the degenerate four-wave converter in a transparent two-component medium based on electrostriction and Dufour effect, we obtain an analytical expression coupling reflectance with the nonlinear medium parameters and the intensity of the pump waves. We show a correlation between the spatial frequency band half-width cut out by the four-wave converter and the reflectance. Increasing of the reflectance leads to increasing of the spatial frequency band half-width which means deterioration of the WFR quality. References Akimov, A.A., Ivakhnik, V.V., Nikonov, V.I., 2011. Four wave interaction on thermal nonlinearity at large reflectance with allowance pumping waves self-diffraction, Computer Opt. 35, 250-255. Akimov, A.A., Ivakhnik, V.V., Nikonov, V.I., 2013. Phase conjugation under four-wave mixing on resonant and thermal nonlinearities at relatively high reflection coefficients, Opt. Spectrosc. 115, 384-390. Akimov, A.A., Ivakhnik, V.V., Nikonov, V.I., 2015. Four-wave interaction on resonance and thermal nonlinearities in a scheme with concurrent pump waves for high conversion coefficients, Radiophys. Quantum Electron. 57, 672-679. Dmitriev, V.G., 2003. Nonlinear optics and wavefront reversal, Fizmatlit Publisher, Moscow. Hemmerling, B., Radi, P., Stampanoni-Panariello, A., Kouzov, A., Kozlov, D., 2001. Novel non-linear optical techniques for diagnostics: laserinduced gratings and two-color four-wave mixing, Comptes Rendus de l’Acad. des Sci. Ser. IV 2, 1001-1012. Ivakhnik, V.V., 1983. Optical radiation filtration with nongenerate four-photon interaction, Russ. Phys. J. 25, 765-767. Ivakhnik, V.V., 2010. Wavefront reversal at four-wave interaction, Samarskiy Universitet Publisher, Samara. Ivakhnik, V.V., Savelyev, M.V., 2013. Spatial selectivity of four-wave radiation converter based on thermodiffusion and electrostriction mechanisms of nonlinearity, Fiz. Voln. Proc. i Radiotech. Syst. 16 , 6-11. Livashvili, A.I., Kostina, G.V., Yakunina, M.I., 2013. Temperature dynamics of a transparent nanoliquid acted on by a periodic light field, J. Opt. Technol. 80, 124-126. Mahilny, U.V., Marmysh, D.N., Stankevich, A.I., Tolstik, A.L., Matusevich, V., Kowarschik, R., 2006. Holographic volume gratings in a glasslike polymer material, Appl. Phys. B 82, 299-302. Rusconi, R., Isa, L., Piazza, R., 2004. Thermal-lensing measurement of particle thermophoresis in aqueous dispersions, JOSA B 21, 605-616. Vorobyeva, E.V., Ivakhnik, V.V., Savelyev, M.V., 2014. Spatial and temporal characteristics of a four-wave radiation converter in a transparent medium based on electrostriction and Dufour effect, Computer Opt. 38, 223-228.