Radiation generation at multiple and Raman frequencies accompanying nonlinearreflection of two light pulses incident at different angles and consisting of a smallnumber of vibrations

S. A. Kozlov and V. K. Turkova�

St. Petersburg State University of Information Technologies, Mechanics, and Optics, St. Petersburg�Submitted May 14, 2010�Opticheski� Zhurnal 77, 89–91 �November 2010�

This paper discusses the simultaneous nonlinear reflection of two paraxial light pulses made upof a small number of vibrations with different central frequencies when they are incident onthe interface of media at small and, in general, different angles. The dependences on thespatiotemporal characteristics of the incident radiation are obtained for the parameters of pulsesreflected from a nonlinear medium at multiple and Raman frequencies. © 2010 Optical Societyof America.

INTRODUCTION

The propagation regularities of light pulses made up of asmall number of vibrations, which are also often called ul-trashort pulses �USPs�, have been considered in many theo-retical and experimental papers, a review of which is given,for example, in a recent monograph.1 Significantly less re-search has been devoted to the features of nonlinear reflec-tion of USPs. Problems of the reflection of homogeneousultrashort planar light waves have been considered from the-oretical positions in Refs. 2 and 3. Reference 4 studied thenonlinear reflection of transversely inhomogeneous paraxialultrashort waves and modelled both the case of one incidentparaxial USP and the case of a superposition of the lightfields of two paraxial USPs with different central frequencieswhen they are normally incident on an interface. The presentpaper presents a study of the nonlinear reflection of twoparaxial USPs of radiations with different central frequencieswhen they are incident on the interface of media at smalland, in general, different angles.

THE NONLINEAR FRESNEL FORMULA FOR SMALLANGLES OF INCIDENCE OF WAVES ON THE INTERFACEOF DIELECTRIC MEDIA

The dependence of the spatiotemporal spectrum Gref ofreflected radiation on the spectrum Ginc of a paraxial linearlypolarized USP incident on the interface of linear and nonlin-ear homogeneous and isotropic dielectric media at a smallangle has the form

Gref =n1 − n2

n1 + n2�1 +

c2

n1n2

kx2 + ky

2

�2 �Ginc −gcS

3�n1 + n2�, �1�

where

Gref,inc��,kx,ky� =� � � Eref,inc�t,x,y�

�exp�i��t + kxx + kyy��dtdxdy , �2�

Eref and Einc are the fields of the reflected and incidentwaves, respectively; n1��� and n2��� are the linear refractiveindices of the bounding media; �, kx, and ky are the temporaland spatial frequencies; c is the speed of light in vacuum;

729 J. Opt. Technol. 77 �11�, November 2010 1070-9762/2010/

and g=6�� /cN0 is a parameter that describes the noninertialnonlinearity of the polarization response of the reflecting me-dium Pnl=�E3, where � is the nonlinear susceptibility of thismedium, N0 is its refractive index at the central wavelengthof the incident pulse, and

S��,kx,ky� =� � � Einc3 exp�i��t + kxx + kyy��dtdxdy .

THE SPATIOTEMPORAL SPECTRUM OF REFLECTEDRADIATION AT TRIPLED FREQUENCIES

Let us consider the reflection of a single linearly polar-ized pulse incident on the interface of media �z=0� at a smallangle �, whose field has the form

Einc�t,x,y� = E1 exp�− �x/�1�2�exp�− �y/�2�2�

�exp�− ��t − x�/v�/�1�2sin �1�t − x�/v� ,

�3�

while the spectrum, accordingly, can be written as

Ginc��,kx,ky� = �E1/2i��1�12�3/2 exp�− ��1�kx − ��/v�/2�2

�exp�− ��1ky/2�2��exp�− ��1�� + �1�/2�2

− exp�− ��1�� − �1�/2�2� , �4�

where E1 is the amplitude of the incident wave, �1 is thetransverse size of the light beam, �1 is the pulse width, �1 isits central frequency, and v is the phase velocity in the linearmedium.

Substitution of Eq. �4� into Eq. �1� makes it possible toobtain an expression for the spatiotemporal spectrum gener-ated when the wave of radiation given by Eqs. �3� and �4� isreflected at tripled frequency, giving

Gref��,kx,ky� =i

24��/3�3/2 gc

�n1 + n2�E1

3�12�1

�exp�− ��1�kx − ��/v�/23�2

�exp�− ��1ky/23�2�

2

�exp�− ��1�� + 3�1�/2 3�

729110729-02$15.00 © 2010 Optical Society of America

− exp�− ��1�� − 3�1�/23�2� . �5�

It can be seen from this equation that both the spatial andthe temporal spectra of the radiation generated during reflec-tion at the tripled frequency are 3 wider than the spectra ofthe incident radiation. The angular characteristics of the ra-diation at the fundamental and tripled frequencies coincide inthis case.

THE SPATIOTEMPORAL SPECTRUM OF THE REFLECTEDRADIATION AT RAMAN FREQUENCIES

Let us consider the reflection of two identically linearlypolarized waves incident on the interface of media �z=0� inthe same plane but at different small angles � and �, whosefield can be written as

Einc�t,x,y� = E1 exp�− �x/�1�2�exp�− �y/�1�2�

�exp�− ��t − x�/v�/�1�2sin �1�t − x�/v� + E2

�exp�− �x/�2�2�exp�− �y/�2�2�

�exp�− ��t − x�/v�/�2�2sin �1�t − x�/v� , �6�

while the spectrum, accordingly, can be written in the form

Ginc��,kx,ky� = �E1/2i��1�12�3/2 exp�− ��1�kx

− ��/v�/2�2exp�− ��1ky/2�2�

��exp�− ��1�� + �1�/2�2

− exp�− ��1�� − �1�/2�2� + �E2/2i��2�22�3/2

�exp�− ��2�kx − ��/v�/2�2

�exp�− ��2ky/2�2��exp�− ��2�� + �2�/2�2

− exp�− ��2�� − �2�/2�2� , �7�

where E1 and E2 are the amplitudes of the incident waves, �1

and �2 are the transverse sizes of the light beams incident onthe interface, �1 and �2 are their pulse widths, and �1 and �2

are their central frequencies.Substituting Eq. �7� into Eq. �1� makes it possible to

obtain an expression for the spectrum generated when waveshaving Raman frequencies and given by Eqs. �6� and �7�interact, of the form

Gref��,kx,ky� =i

24�3/2 gc

�n1 + n2�E1

2E2�2� exp�− ��1ky/2�2�

��exp�− ���kx − �1/v�/2�2

�exp�− ���� + �2�1 + �2��/22�

− exp�− ���kx − �2/v�/2�2exp�− ����

− �2�1 + �2��/22� − exp�− ���kx

− �1/v�/2�2exp�− ���� + �2�1

− �2��/22� − exp�− ���kx − �2/v�/2�2

� exp�− ���� − �2�1 − �2��/22�� , �8�

where

� = �1�2/�2�12 + �2

2�1/2, � = �1�2/�2�12 + �2

2�1/2,

730 J. Opt. Technol. 77 �11�, November 2010

1 = �−1��� + �2�1 + �2����2/�212 � − �2�1� + �2�� ,

2 = �−1��� − �2�1 + �2����2/�212 � + �2�1� + �2�� ,

1 = �−1��� + �2�1 − �2����2/�212 � − �2�1� − �2�� ,

2 = �−1��� − �2�1 − �2����2/�212 � + �2�1� − �2�� ,

�21 = �1�2/�2�22� + �1

2��1/2. �9�

As can be seen from Eq. �8�, the radiation at the Ramanfrequencies generated when there is nonlinear reflection alsohas wider spatial and temporal spectra than the original ra-diation. Its angular characteristics are determined by theangles of incidence of the light waves and by the ratio oftheir widths and the central frequencies. We should point outthat, in the particular case �=�, the angular characteristics ofthe radiation reflected at the Raman frequencies coincidewith the angular characteristics of the radiation reflected atthe fundamental frequencies, and 1=2=1=2=�.

CONCLUSION

Relationships have been derived in this paper for thespatiotemporal spectra of the waves of multiple and Ramanfrequencies that are generated when there is nonlinear reflec-tion of two light waves having Gaussian spatial and temporalprofiles and simultaneously incident in the same plane atsmall but in general different angles, whose temporal spec-trum can be superwide, and that differ in their central fre-quencies. It has been shown that the width of the temporalemission spectra at the Raman frequencies is greater thanthat of the incident radiation and is determined by the ratio ofthe widths of the interacting pulses. The width of the spatialspectra of the radiation generated at the Raman frequenciesis also larger than that of the incident radiation. Its angularcharacteristics depend on the angles of incidence of the lightwaves, the ratio of their frequencies, and the pulse widths.

This work was supported by the program “Developmentof the Scientific Potential of Higher-Level Schools,” byGrant RNP.2.1.1.4923, and by Grant No. 08-02-00902 of theRussian Foundation for Basic Research.

a�Email: vadim.turkov@gmail.com

1S. A. Kozlov and V. V. Samartsev, Principles of Femtosecond Optics�FIZMATLIT, Moscow, 2009�.

2S. A. Kozlov, Yu. A. Shpolyanski�, and N. V. Yastrebova, “Nonlinearreflection of pulses composed of a small number of vibrations of the lightfield from the antireflection-coated interface of media,” Opt. Zh. 71, No.6, 78 �2004� �J. Opt. Technol. 71, 401 �2004��.

3N. N. Rozanov, “Reflection of ultrashort pulses from the boundary of aDrude–Lorentz medium,” Opt. Spektrosk. 94, 449 �2003� �Opt. Spectrosc.94, 396 �2003��.

4S. A. Kozlov and O. A. Mokhnatova, “Nonlinear reflection of a femtosec-ond spectral supercontinuum,” Zh. Eksp. Teor. Fiz. 133, 260 �2008� �JETP106, 218 �2008��.

730S. A. Kozlov and V. K. Turkov