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Sub-cycle pulse propagation in a cubic medium Ajit Kumar Department of Physics, Indian Institute of Technology, Delhi, India @ NONLINEAR PHYSICS. THEORY AND EXPERIMENT. V JUNE 17, 2008 Applications Telecommunications: Multi-terabit per second transfer of information in a single fiber. Industry : Dramatic improvement in the resolution of optical microscopy. Provides the only means of machining and structuring materials (Nanophotonics). Medicine: New diagnostic and therapic tools. Frequency and Time Metrology: Frequency ruler of unprecedented precision. Ultra-fast Metrology: Attosecond metrology allows to observe the motion of electrons on atomic length scale in real time and record the electric field of the visible light. Allow to create coherent light sources in all the way from the infra-red to the soft X-ray regimes. Ultra-short pulses have also been used to study: # Light-matter interaction leading to higher harmonics and attosecond pulse generation # Nonlinear optical phenomena, like, spatio-temporal self-focusing and filamentation # Controlled manipulation of chemical reactions and bond formation # Coherent quantum control of microscopic dynamics etc. Study of propagation characteristics of such pulses: Important EVOLUTION EQUATION ? Problem: SVEA doesn’t hold . What to do? Main attempts: # Brabec and Krausz (PRL,1997): Showed that it may be possible to define envelope for optical pulses in the single-cycle regime. exp[ i t i ] cc, 0 | E ( ) |2 d / | E ( ) |2 d 0 0 0 E A(t ) The phase is defined in such a way that the imaginary part of the complex envelope A(t) is zero at t=0. Equivalently, the center frequency remains unchanged under a phase shift of the electric field: 0 ~ ~ E (t ) E (t )ei Using, then, the SVEA, they derived the following evolution equation (NEE) A 0 2i i 1 2 2 A iDˆ A [1 (i / 0 ) ] ( t A) 2 0 [1 (i / 0 ) ] A A 2 20 n0 Using this NEE they studied the propagation properties of a single-cycle at 0.8 micron wavelength up to a propagation distance close to the characteristic distance of self-steepening. WHAT IS MISSING? # The expected coupling between spatial and temporal dynamics of the pulse (Rothenberg, 1992) in single- and sub-cycle regimes is missing. # The SEWA: requires, besides the envelope A(t), the relative phase not to vary significantly as the pulse covers a distance equal to the wavelength 0 . Leads to applicability condition 0 Lchar 1 , which can not be satisfied always, for instance in the case of strong resonant coupling. # In the sub-cycle regime, the introduced concept of envelope does not hold good. # Schaefer and Wayne (Physica D, 2004): derived a short pulse equation (SPE) 2 ( 3) 2 2u c1u ( u u ), (t x) / , x 2 where u is the real dimensionless electric field. # This equation was derived by approximating the susceptibility function by in the region from 1.6-3.0 microns . (1) ( ) 2 , , const (1) ( ) # This equation is integrable via IST (Sakovich, 2004) and possesses soliton solutions but is valid only for a particular form of susceptibility. # The novel feature of this SPE is the coupling between spatial and temporal dynamics which was pointed out by Rothenberg in 1992. # Recently Tsurumi (J. Phys. Soc. Japan,2006) has investigated Gaussian beam propagation in a linear medium on the basis of a model equation, for the real electric field. In almost all the problems in nonlinear optics and plasma physics, involving ultra-short pulses, the Brabec and Krausz NEE is used. However, in the sub-cycle regime it is difficult to accept the concept of an envelope and hence, it is desirable to have a pulse evolution equation for the real electric field. In the given work a nonlinear pulse evolution equation, for the real electric field, is derived without separating the pulse into a slowly varying envelope and the fast carrier oscillation parts. The wave equation: 2 1 t 2 E ( r , t ) 2 2 (t ' ) E ( r , t t ' ) z c t 0 2 2 dt ' ( 3) 2 t c 2 2 E (r,t )E (r,t ) E (r,t ) (3) where (t ) is the third-order nonlinear susceptibility, (t ) is the linear permittivity of the medium and c is the speed of light in vacuum. Nondimensionlization: where l0 , t0 , E0 r r / l0 , t t / t 0 , E E / E0 are constants having dimensions of length, time and the electric field respectively. The non-dimensional wave equation is 2 2 3 2 ' ' ' 2 E ( x , y, z , t ) 1 E E E ( x , y, z , t ) (t )E ( x , y , z , t t )dt 2 2 t 2 2 t 2 z c c 0 t c c 0 , l 0 where t , 0 3 E 2 3, 0 1 i t d , k i (t ) ( )e 2 0 Assuming the spectrum to be localized near zero frequency, we expand k ( ) 0 around to obtain 2 n 2 E ( x , y, z , t ) 1 k i n 2 n! t z 0 n 0 where k l k, 0 2 3 2 E E E ( x , y, z , t ) E ( x , y, z , t ) c 2 t 2 x x/l , y y/l , z z /l 0 0 0 2 2 / x 2 2 / y 2 As a result, we obtain 2 1 n k n 2 (1) 2 2 ˆ 2 ˆ i 0 D(T ) D (T ) 2i n T T n! n0 0 t where the dispersion operator Dˆ i 2 m3 m 0 Dˆ (T ) is given by 2 m 2 2 m 3 0 2 m 3 2 m 3 0 2 m 3 2 m 2 (2m 3)! T 2(2m 2)! T If now we go over to the moving frame by introducing new variables T 0(1) z ' T 1 z' , vg z' where is a constant and v g is the group velocity which is identical with the phase velocity at 0, we obtain. E ( R , , ) 2 2a 2 R (1) 0 2 2 2 2 2 E ( R , , ) E ( R , , ) 2i 3 0(1) Dˆ ( ) E ( x , y , z , T ) 2 3 2 ˆ 2 2 E ( R , , ) E ( R , , )E ( R , , ) D ( ) E ( x , y , z , T ) 2 c 2 4 To account for dispersion, dissipation and diffraction consistently, we introduce a a R x, y, z z , T t We get 1 nk 2a 2 2 R E ( x , y , z , T ) n i 2 z n! n 0 0 T 2 Note that i k ( ) ( ), 2 2 3 2 2 E E E ( x , y, z , T ) E ( x , y , z , T ) 2 2 c T cos , c c Im tan 1 . Re ( ) Re sin , c c ( ) Im and the electric field is real. As a consequence ( 2m) 0 for m=0,1,2,3,… and d 2m o, 2m d 0 ( 0) 0, ( 2 m 1) 0 d 2 m1 o. 2 m 1 d 0 ( 0) 0. The dispersion operator Dˆ ( ) is given by 2 m 3 2 m 2 ( 2 m 3) 2 m 2 ˆ 2 m3 2 m 1 2m 0 0 D i ( 2 m 3 )! 2 ( 2 m 2 )! m 0 Consider now the case when diffraction and dispersion are comparable . This leads to 2 in the leading order in . Further, for a weakly nonlinear and weakly dissipative medium, we assume the dissipative and nonlinear effects to be of the order of and 2 , respectively: 0 (2) 0( 2 ) , As a result, in the leading order in 3 2 3. , we obtain 2 E 0(3) 4 E 0( 2) 3 E 3 2 2 E E E E 2 4 3 2 6 4 c 2 t (1) 0 The above partial differential equation describes the spatio-temporal evolution of the real electric field in a cubic medium. If we include the delayed response of the medium, we have t2 1 2 2 ) t , r ( E (t ') E ( r ,t t ') dt ' c 2 t 2 0 z 2 3 2 c t 2 2 E ( r ,t )E ( r ,t ) E ( r ,t ) (1 ) 3 2 ' dt ) ' t t , r ( E ) ' t t , r ( E ) ' t ( g ) t , r ( E R t 2 c2 0 where is a numerical constant representing the fraction of the fast electronic response and g R (t ) 1 2 22 1 2 expt / 2 sin t / 1 is the Raman Response function for the case when one vibrational mode with linewidth 1/ and eigenfrequency 1/ is important. 2 1 Going through the same steps of calculations, we arrive at 2 (1 ) t E 2 0 ( 3) ( 2) 3 2 2 E 0 4 E 0 3E 2 c 2 6 4 4 3 (1 ) 3 2 2 2 E 0 g R ( ) E ( ) E ( ) d c E E E Three features of this equation # There is a coupling between the spatial and temporal evolution. # Dispersion comes from the third-order derivative of the wave vector with respect to frequency. # Dissipation emanated from the second-order derivative of the wave vector with respect to frequency. Results of Numerical Simulations: For half a-cycle pulse at 0.8 m. 2 E 0(3) 4 E 0(2) 3 E 3 2 2 0(1) E E E 4 3 2 2 6 4 c Parameters used in numerical calculations: # # # 3 0.05 corresponding to fs3 3 0.0274 , m l0 ld 10 V E0 1.496 10 m , 2 2 3 2.234 10 22 m n 0.6 10 22 m 2 V2 V2 fs2 2 0.03592 ( Diddams and Diels ,1996) m ( p / 1.67 ) 3 3 18 .58 m , Lnl z s 0.39 Lnl 0 T0 7.23 m , # Input pulse shape: At 0 1 1 2 9.85 m , P0 0 n0 cAeff E 0 P0 2 2 n20 0 n0 c 2 Aeff is taken to be 2 E exp cos( ) 0 T 0 0 Fig.1: Dimensionless Electric field as a function of time. The solid curve corresponds to the pulse after 3.7 micron propagation. The dashed curve corresponds to the input pulse shape. Fig.2: Dimensionless intensity as a function of dimensionless frequency. The solid curve corresponds to the pulse after 3.7 micron propagation. The dashed curve corresponds to the input pulse shape. Fig.3: The dimensionless Electric Field of the pulse, after7.4 microns of propagation as a function of dimensionless time. Fig.3: The dimensionless Electric Field of the pulse, after16.7 microns of propagation as a function of dimensionless time. Fig.5: The dimensionless Electric Field of the pulse, after 22.3 microns of propagation as a function of dimensionless time. Fig.6: Dimensionless intensity as a function of dimensionless frequency, after 22.3 microns of propagation. Conclusions: # We have presented a novel nonlinear evolution equation for a fewand sub-cycle pulses, which has been derived from Maxwell’s equations for the real electric field. Novel Features # In the leading order, the dispersion effects come from the fourth-order derivative of the electric field with the coefficient equal to the third-order derivative of the propagation constant with respect to frequency at 0. # Dissipation arises from the third-order derivative of the electric field with the coefficient equal to the second-order derivative of the propagation constant with respect to frequency at 0. # By numerical calculations, we have shown that the given evolution equation is capable of describing pulse propagation up to and beyond the critical distance of self-steepening without any pulse break-up taking place.