Studies on the Bistability of a Dissipative Medium Inside a Resonator

Transcription

1 Egypt. J. Solids, Vol. (9, No. (, (6 4 Studies on the istability of a Dissipative Medium nside a Resonator M. M. E-Nicklawy, A.. Hassan, A. T. Matar, A. A. Hemeda, and A.. Ali Physics Department, aculty of Science, Helwan University, Cairo, Egypt Through the differential equation describing the behavior of the nonlinear polarization of a medium with respect to the incident field, the Maxwell field equations and the boundary conditions of the field inside a resonator. Relation between the output field and the incident field, describing bistability phenomena, is obtained.the effect of the spectral profile of the incident field and its spectral half width on the bistability phenomena is also studied.. ntroduction: The high power densities available in coherent laser beams have made possible the experimental observation of nonlinear effects of a medium. ranken and Co-workers ] demonstrated second-harmonic generation of light, Kaiser and Garrett ] two-photon absorption, Giordmaine 3] mixing of light beams and Woodburg discovered stimulated Raman scattering 4] and Maker and Co-workers 5] demonstrated also third harmonic generation. Nonlinear optical effects may be also used to make all optical switches. The optical phase modulation in the Kerr medium may be converted into intensity modulation by placing the medium in one leg of an interferometer, so that as the control light is turned on and off, the transmittance of the interferometer is switched between and. A bistable (or two-state system has an output that can take only one of the two distinct stable values, no matter what input is applied. Switching between these values may be achieved by a temporary change of the level of the input. istable devices are important in the digital circuits and in communications, signal processing, and computing. They are used as switches, logic gates and memory elements (flip-flops 6].

2 M. M. El-Nicklawy, et al. 4 Two features are required for making a bistable device: nonlinearity and feedback. f the output of a nonlinear optical medium is fed back by optical resonator and used to control the transmission of light through the medium itself, bistable behavior can be exhibited. A abry-perot interferometer containing a nonlinear dielectric medium that can be regarded as producing an intensity dependent refractive index and absorption coefficient can become bistable or multi-stable at sufficient high incident intensity 7-]. Here the bistability phenomenon is described from an interferometric point of view through the nonlinearity of the refractive index and the absorption coefficient of a dielectric medium. The effects of the spectral half width of the deriving wave and the finess of a abry-perot resonator on the phenomenon are represented.. Theory: The properties of a dielectric medium through which an electromagnetic wave propagates are completely described by the relation between the polarization density vector and the electric field vector. The mathematical relation between them defines the system and is governed by the characteristics of the medium. The medium is said to be nonlinear if this relation is nonlinear. Consider a dielectric medium for which the dynamic relation between the polarization density P and the electric field E is described by the following nonlinear second order differential equation, d P dp 3 σ ω P P P E( z, o β γ dt dt ( E ω ( zexpi( ωt kz CC ( a b Ne P Nex,, β, γ Ne (Ne m N is the atom density E is the electron charge M is the electron mass ω is the natural oscillation frequency of the dipole with { }

3 Egypt. J. Solids, Vol. (9, No. (, (6 43 σ is the damping coefficient of the dipole a, b are material constants ex is the dipole moment Representing the polarization density as a function of space and time as follows: P( z, Po ( z 3ω { P ( zexpi( ωt kz CC} { P ( zexp i( ωt kz CC} ω { P ( zexp3 i( ωt kz CC} ω (3 P o (z, P ω (z, P ω ( and P ω ( are to be determined z 3 z As a first approximation, the terms P 3 and β P in equation ( were neglected, yielding: d P dp σ ω P E( z, o γ (4 dt dt The solution of Eq.(4 is: γ P exp( iθ E( z, χe( z, (5 t ( ω 4 ] ω σ ω γ χ exp( iθ (6 ω ω 4σ ω ] ( σω and tan θ ω ω Substituting for P 3 and β P in equation ( by P (z,, we get : d P dp 3 3 σ ω P E( z, E E o γ χ βχ (7 dt dt

4 M. M. El-Nicklawy, et al. 44 Employing Eq.(3 on Eq.(7, we get: P 3 3 γ βχ ( z z 4 ( z (8 ω ω iσω ω ( The present work is restricted on the frequency ω of the deriving field, therefore we are concerned only with (z. Equation (8 can be written in the P ω form 3 4 β ( z Γ P ( { exp( 4 ω z Γ iθ exp(iθ } ( z (9 γ where γ Γ ( ( ω ω σ ω ] 4 Now setting the medium inside A fabry-perot resonator and assume that the amplitude (z various slowly compared with the carrier wave expi(ω t-kz, this justifies the following inequalities, d ( z dz << k ( z and d ( z d ( z << k dz dz The forward and backward field equations inside the resonator under the previous inequalities can be deduced through Maxwell equation in the medium in the form, Where: E E iω C P t z E E iω C P t z ( ( E { ( zexpi( ωt kz CC} (3 E { ( zexpi( ωt k( z CC} (4

5 Egypt. J. Solids, Vol. (9, No. (, (6 45 P { P ( zexpi( ω t kz CC} (5 P { P ( z expi( ω t k( z CC} (6 Analogy to Eqn. (9, we can consider: 3 4 β ( z Γ P ( { exp( 4 z Γ iθ exp(iθ } ( z (7 γ 3 4 β ( z Γ P ( { exp( 4 z Γ iθ exp(iθ } ( z (8 γ To solve the forward field equation ( through equations (3, (5, and (7, we firstly consider that P (z Γ exp( iθ (z and neglecting the nonlinear term which depends on (Z. Thus, ω z iω Γ exp( iθ ( z C The solution of this Eqn. can be given in the form, iωγ ω ( z ( exp( (cosθ i sinθ z C the above equation gives, ( z ωγsinθ ( exp( z (9 C Now substitute for (z into Eqn. (7, it follows, ωγsinθ P (z ( Λ D exp( z ( z ( C

6 M. M. El-Nicklawy, et al β ( Γ where Λ Γ exp( iθ, D 4 exp(iθ γ ( Now solving the forward field, Eq.(, after considering equations (3, (5, ( we get ; ω iω ωγsinθ ( Λ D exp( z ( z ( z C C Considering Eqn. (, the solution of Eqn. ( is: ( z iω ω d ωγsinθ ln( Γ cosθ Z Γ sinθ Z i exp( z cos θ ( C C Γ sinθ zoc d ωγ sinθ d d exp( zsin θ i cos θ sin θ Γ sinθ zoc Γsinθ Γ sinθ 3 4 β ( Γ 4 d (3 γ Thus the forward wave inside the resonator E can be written in the form: z E ( exp( expi( ωt k z (4 ω d ωγsinθ Γsinθ sin θ ( exp( z C zγsinθ C and ω Γ cosθ d Γ C ω sinθ k cosθ ( exp( z C o zγsinθ ω C There are two terms determine the value of the absorption coefficient, as shown above. The first term is independent of the intensity of the field while the second term depends on the field intensity. The absorption coefficient should be low enough to sufficient number of reflected beams inside the resonator to happen.these reflections will provide sufficient feedback for the existence of the bistable phenomena. or more explanation let us consider the following example.

7 Egypt. J. Solids, Vol. (9, No. (, (6 47 Consider the medium inside the resonator has an absorption coefficient and thickness. The reflection coefficient of the resonator s mirrors is R. Then, the number of effective reflected beams will be Re xp(. or number of beam reflections of, while R.99 then should be.95. ωγsinθ Thus we can consider Z<<, therefore, we get C ω d Γsinθ C cosθ Γ the above expression for can be shortened and get its absolute value as, with: and: k ω C (5 c ωγsinθ s C C c ( (W/cm s 4 s 4 c ω σγ (cm - (6 C( ω ω 4σ ω ] γ 3 Γ β cosθ (W/cm ( ω ω 4δ ω ] c γ β ( ω ω (7 (8 (W/cm (9 3 3 βγ {( ω ω 4σ ω } ( γ ( ω ω ω 4 {( ω ω 4σ ω } {( ω ω 4σ ω 3 C s K is the intensity inside the laser cavity is the saturation intensity (atomic constan is the wave vector (3

8 M. M. El-Nicklawy, et al. 48 Similarly the backward field is given by: and: ( z E ( exp( expi(ω t- k (-z (3 ω Γsinθ - C d ωγsinθ sin θ ( exp( ( z ( Z Γsinθ C ω Γcosθ k - C d cosθ C ωγ sinθ (- exp( ( z ] ( z Γsinθ ω C The boundary conditions of the fields inside the resonator are: E (, T E (, R E (, (33 E T (, T E (, (34 E (, R E (, (35 E (z, E (, exp( ( z exp( ik ( z (36 T R is the transmission coefficient is the reflection coefficient rom (3 and (35 one gets: E (, E (, exp( exp( ik R E (, exp( exp( ik (37 Through the boundary conditions and equations (33 and (37 one gets: E (, T E (, R E (, exp( exp( ik E (, T E (, R T ET (, exp( exp( ik (38

9 Egypt. J. Solids, Vol. (9, No. (, (6 49 Considering equation (34 and the above obtained results it follows: E (z, T E (, exp( ik z R T ET (, exp( exp( ik ] exp( z E T (, T T E (, T R ET (, exp( exp( ik ] exp( exp( ik TE (, exp( exp( ik E T (, Re xp( exp i( k k T E T (, T in exp( (39 Re xp( cos( k k R exp ( To get ( cavity : (z, cavity T E (, exp( zexp( ik z cavity Re xp( exp i( k k T RE (, exp( exp( ik exp( ( Zexp( ik ( z exp( iφ Re xp( exp i( k k φ is the phase gained by backward wave. cavity int{exp( z Re xp( exp( ( z R exp( ( z exp( ( z Cos(( K Re xp( cos( k k R exp ( or R, consider and k k K ( z}

10 M. M. El-Nicklawy, et al. 5 cavity int{exp( z Re xp( exp( ( z R exp( cos( φ Cos(K Re xp( cos(k R exp ( ( z} Taking the average the average over z we get, cavity ( exp( ( Re xp( R exp( cos( φ int Re xp( cos(k R exp ( (4 Case (: for a orentz spectral line profile exp( T E T (, ET (, dω is orentz distribution. υ υ ( πδυ Δυ / T T in Re xp( cos k R exp( let A R exp(, and R exp( & k k k Then, T T in exp( π A cos( n υ C d ω d ω (4 n n n and due to the complexity of integration (4, we choose another equivalent relation that can be integrated easily, πυ A cos(n C /( A υ υ r ( δυ /

11 Egypt. J. Solids, Vol. (9, No. (, (6 5 δυ C πn Re Re & υ r mc n π A cos( n υ C d ω / πδυ υ υ ( Δυ / /( A υ υ r ( δυ / d υ (4 Equation (4 can be written in the form: / πδυ υ υ ( Δυ / /( A υ υr ( δυ / d υ ( δυ / ( Δυ / π ( Δυ( A a υ bυ c a υ bυ c d υ a, b -υ and c ( Δ υ / υ a, b -υ r and c ( δυ / υr Using partial fraction, ( δυ / ( Δυ / π ( Δυ( A a υ bυ c a υ bυ c d υ ( δυ / ( Δυ / π ( Δυ( A Aυ a υ b υ c A υ a υ b υ c d υ To get the coefficients A, A, and, we make the following: ( A υ ( aυ bυ c ( A υ ( aυ bυ c 3 and equating the coefficient of υ, υ, υ, υ, we get: a A a A b A a b A a

12 M. M. El-Nicklawy, et al. 5 c A b c A b c c n case b b or υ υ r then: A A & - - c c ( Δυ / ( δυ / Since a a, then: ( δυ / ( Δυ / π ( Δυ( A ( δυ / ( Δυ / π ( Δυ( A υ bυ c bυ c υ bυ c υ υ bυ c d υ d υ ( δυ / ( Δυ / π ( Δυ( A ( δυ / ( Δυ / π ( Δυ( A d υ υ bυ c υ bυ c ( δυ / ( Δυ / ( π / { }- π ( Δυ( A 4c b 4c b b b { arctan arctan 4c b 4c b 4c b 4c b }]. T in exp( δυ Then, T ( Re xp( π π Δυ / ( ( Δυ / ( δυ / δυ / ( Δυ / ( δυ / ] υ Δυ / ( arctan( ( Δυ / ( δυ / Δυ / δυ / ( Δυ / ( δυ / (43 υ arctan δυ /

13 Egypt. J. Solids, Vol. (9, No. (, (6 53 Case (: for Gaussian distribution: T T in exp( π Re xp( cos(n υ R exp( C G 4ln υ υ exp( 4ln ( π ( Δυ Δυ is Gaussian distribution. Thus: T T in exp( G d υ 4ln π ( Δυ exp( τx π a cos(n ( x υ C 4ln υ x, a R exp( & τ ( Δυ ω is considered to be a resonance frequency of the ebry-perot resonator. a dx T T in exp( 4ln π ( Δυ exp( ( τ / k y k a cos y a dy τ (n/cx kx y, dx dy/k, and ρ k T T in exp( T T in exp( T T in exp( R exp( 4ln π ( Δυ 4ln π ( Δυ { exp( ( ρ y k a cos y k π ρ a { a dy i i a exp( 4 i ρ } ( Δυ i i k (Re xp( exp( (44 i 6ln

14 M. M. El-Nicklawy, et al. 54 To get cavity for these two cases: G cavity ( exp( ( Rexp( R exp( cos( φ int{ } Rexp( Cos( δ R exp( G d υ cavity ( exp( ( Rexp( Rexp( cos( φ int{ } Rexp( Cos( δ R exp( G d υ is orentz distribution. υ υ ( πδυ Δυ / 4ln G π ( Δω ϖ ϖ exp( 4ln ( Δϖ is Gaussian distribution ( exp( ( Re xp( R exp( cos( φ G cavity G { T } T exp( (45 ( exp( ( Re xp( R exp( cos( φ cavity { T } T exp( 3. Calculations: rom Equations (5, (43, (44 and (45 assuming a value for o and c in given values for corresponding values for are calculated from Equation s s (45, after dividing its both sides by s. Substituting in Equation (43 and (44, after dividing its both sides by s, for the values of c s and the corresponding

15 Egypt. J. Solids, Vol. (9, No. (, (6 55 calculated values of out for R. 99, and o.,. 5 and. 7. in s, out s can be calculated. The calculations are carried c s is given values from. up to 5 in steps of. 5 leading to varying the nonlinear absorption coefficient. in out The behavior of the relative intensities of against is shown in figures. s s The phase shift φ suffered by the backward wave with respect to the forward one is considered to take the value π. to 8 The above figures can carried out for half width range from,. 8. Hz in step of..igures (, and 3 show the behavior of out in against for the case of Gaussian line profile. igures (4, 5 and 6 are s s the corresponding figures to figures (, and 3 of the orentzian line profile. 4. Discussion: rom the igures (- 6, we observe that: The bistability phenomena are more pronounced as the spectral half width of the interacting field with the medium is decreased. t can be attributed to the bistability phenomena that arise through interference phenomena which requires temporal coherent interacting field and in turn a field of small spectral half width. The nteracting fields of Gaussian line profile are more effective for initiating bistability than a field of orentz profile. t is due to the fact that effective spectral range is in case of orentz profile greater than that in case of Gaussian profile.,, Q V ig( gaussian distribution for R.99 and.,, 7 V

16 M. M. El-Nicklawy, et al. 56, Q, V ig( gaussian distribution for R.99 and.5,, 7 V, Q, 4 8 V ig(3 gaussian for R.99 and.7,, 7 V

17 Egypt. J. Solids, Vol. (9, No. (, (6 57,, Q V ig(4 lorentz distribution for R.99 and.,, 7 V,, Q V ig(5 lorentz distribution for R.99 and.5,, 7 V

18 M. M. El-Nicklawy, et al. 58, Q V ig(6 lorentz distributionfor R.99 and.7,, 7 V There are two parameters controlling the bistability, the reflection coefficient R and the nonlinear absorption coefficient 5]. R is responsible for the interference phenomena and also for increasing the life time of the interacting photon inside the resonator, which in turn leading to increase the probability of interaction of the photon with the medium. On the other hand, is responsible for a direct interaction of the photon with the medium. We can conclude that to get bistability it is sufficient that either must be large enough and R can be of small values or vice versa. References:. T. H. Maiman, Nature, 87, 493 (96.. ranken, A. E. Hill, C. W. Peters and G. Weinreich, Phys. Rev. lett. 7, 8 ( W. Kaiser and G. C.. Garrett, Phys. Rev. ett. 7, 9, 3( E. J. Woodburg and W. K. Ng, Proc. EEE 5, 367( J. A. Giordamaine, Phys. Rev. ett. (8, 9- (96.

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