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Coherent Control of FourWave Mixing
Coherent Control of FourWave Mixing
Yanpeng Zhang, Zhiqiang Nie, Min Xiao
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Coherent Control of FourWave Mixing discusses the frequency, temporal and spatial domain interplays of fourwave mixing (FWM) processes induced by atomic coherence in multilevel atomic systems. It covers topics in five major areas: the ultrafast FWM polarization beats due to interactions between multicolor laser beams and multilevel media; coexisting RamanRayleighBrillouinenhanced polarization beats due to colorlocking noisy field correlations; FWM processes with different kinds of dualdressed schemes in ultrathin, micrometer and long atomic cells; temporal and spatial interference between FWM and sixwave mixing (SWM) signals in multilevel electromagnetically induced transparency (EIT) media; spatial displacements and splitting of the probe and generated FWM beams, as well as the observations of gap soliton trains, vortex solitons, and stable multicomponent vector solitons in the FWM signals.The book is intended for scientists, researchers, advanced undergraduate and graduate students in Nonlinear Optics.Dr. Yanpeng Zhang is a professor and Zhiqiang Nie is a Ph. D. student at the Key Laboratory for Physical Electronics and Devices of the Ministry of Education, Xi'an Jiaotong University, China. Dr. Min Xiao is a professor of physics at the University of Arkansas, Fayetteville, U.S.A.
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3642191142
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9783642191145
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fwm^{1128}
fig^{983}
dressing^{754}
beam^{711}
exp^{680}
beams^{643}
frequency^{628}
polarization^{585}
atomic^{554}
dressed^{479}
laser^{473}
phase^{462}
wave mixing^{462}
processes^{455}
nonlinear^{441}
raman^{423}
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spatial^{412}
probe^{383}
intensity^{374}
curve^{370}
swm^{368}
multi^{333}
rayleigh^{329}
induced^{317}
respectively^{310}
interference^{303}
dressing ﬁeld^{263}
signals^{262}
resonant^{260}
versus^{259}
photon^{253}
kerr^{236}
coherence^{230}
solitons^{225}
phys rev^{225}
aspb^{224}
fwm signal^{220}
correlation^{220}
coupling^{217}
atomic system^{216}
eit^{214}
optical^{212}
mhz^{206}
signal intensity^{201}
transition^{200}
polarization beats^{200}
enhancement^{195}
suppression^{194}
stochastic^{190}
mixing processes^{183}
level atomic^{179}
spectra^{179}
brillouin^{174}
enhanced fwm^{169}
amplitude^{161}
laser beams^{160}
shown in fig^{156}
ghz^{156}
wave mixing processes^{153}
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Yanpeng Zhang Zhiqiang Nie Min Xiao Coherent Control of FourWave Mixing Yanpeng Zhang Zhiqiang Nie Min Xiao Coherent Control of FourWave Mixing With 190 ﬁgures Authors: Prof. Yanpeng Zhang Zhiqiang Nie Key Laboratory for Physical Electronics Department of Electronics Science and and Devices of the Ministry of Education Technology Xi’an Jiaotong University Xi’an Jiaotong University Xi’an 710049, China Xi’an 710049, China Email: ypzhang@mail.xjtu.edu.cn Email: 01051138@163.com Prof. Min Xiao Department of Physics University of Arkansas Fayetteville Arkansas 72701, U.S.A. Email: mxiao@uark.edu ISBN 9787040313390 Higher Education Press, Beijing ISBN 9783642191145 eISBN 9783642191152 Springer Heidelberg Dordrecht London New York c Higher Education Press, Beijing and SpringerVerlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acidfree paper Springer is part of Springer Science + Business Media (www.springer.com) Preface The ﬁeld of nonlinear optics has developed rapidly since the invention of the ﬁrst laser exactly 50 years ago. Many interesting scientiﬁc discoveries and technical applications have been made with nonlinear optical eﬀects in al; l kinds of nonlinear materials. There are already several excellent general textbooks covering various aspects of nonlinear optics, including Nonlinear Optics by Robert W. Boyd, Nonlinear Optics by YuenRon Shen, Quantum Electronics by Amnon Yariv, Nonlinear Fiber Optics by Govind P. Agrawal, etc. These textbooks have provided solid foundations for readers to understand various (secondand thirdorder) nonlinear optical processes in atomic gas and solid media. The earlier monograph by the authors, Multiwave Mixing Processes, published last year, has presented experimental and theoretical studies of several topics related to multiwave mixing processes (MWM) previously done in the authors’ group. The topics covered in that monograph include ultrafast polarization beats of fourwave mixing (FWM) processes; heterodyne detections of FWM, sixwave mixing (SWM), and eightwave mixing (EWM) processes; Raman and Rayleigh enhanced polarization beats; coexistence and interactions of MWM processes via electromagnetically induced transparency (EIT). The monograph shows the eﬀects of highorder correlation functions of diﬀerent noisy ﬁelds on the femto and attosecond polarization beats, and heterodyne/homodyne detections of ultrafast thirdorder polarization beats. It has also shown the coexistence of FWM and SWM processes in several multilevel EIT systems, as well as interactions between these two diﬀerent orders of nonlinearities. This new monograph builds on and extends the previous works, and presents additional and new works done in recent years in the authors’ group. Many newly obtained results, extended detail calculations, and more discussions are provided, which can help readers to better understand these interesting nonlinear optical phenomena. Other than showing more results on controls and interactions between MWM processes in hot atomic media, several novel types of spatial solitons in FWM signals are presented and discussed, which are new phenomena in multilevel atomic systems. Chapter 1 reviews some basic concepts to be used in later chapters, such as the nonlinear susceptibility, coherence functions and doubledressing schemes. Chapter 2 extends the previous results on polarization beats to include both diﬀerencefrequency femtosecond and sumfrequency attosecond beats in the multilevel vi Preface media depending on the specially arranged relative time delays in the multicolored laser beams. Chapter 3 gives results on Raman, RamanRayleigh, RayleighBrillouin, and coexisting RamanRayleighBrillouinenhanced polarization beats. Chapter 4 presents multidressing FWM processes in conﬁned and nonconﬁned atomic systems with speciallydesigned spatial patterns and phasematching conditions for laser beams. Chapter 5 shows enhancement and suppression in FWM processes in multilevel atomic media, generated FWM signals can be selectively enhanced and suppressed via an EIT window. The evolution of dressed eﬀects can be from pure enhancement into pure suppression in degenerateFWM processes. Chapter 6 demonstrates the modiﬁcation and control of MWM processes by manipulating the darkstate or EIT windows with polarization states of laser beams via multiple Zeeman sublevels. Chapter 7 shows spatial dispersion properties of the probe and generated FWM beams which can lead to spatial shift and splitting of these weak laser beams. Chapter 8 presents the observations of several novel types of solitons, such as gap, dipole, and vortex solitons, for generated FWM beams in diﬀerent experimental parametric regions. Authors believe that this monograph treats some special topics of coherent controls of FWM and MWM and can be useful to researchers interested in related nonlinear MWM processes. Several features presented here are distinctly diﬀerent and advantageous over previously reported works. For example, authors have shown evolutions of enhancement and suppression of FWM signals due to various dressing schemes by scanning the dressing ﬁeld detuning. Also theoretical calculations are in good agreement with experimentally measured results in demonstrating enhancement and suppression of MWM processes. Eﬃcient spatialtemporal interference between FWM and SWM signals generated in a fourlevel atomic system has been carefully investigated, which exhibits controllable interactions between two different (third and ﬁfth) order nonlinear optical processes. Such controllable highorder nonlinear optical processes can be used for designing new schemes for alloptical communication and quantum information processing. Authors also experimentally demonstrate that by arranging the strong pump and coupling laser beams in speciallydesigned spatial conﬁgurations (to satisfy phasematching conditions for eﬃcient FWM processes), generated FWM signals can be spatially shifted and split easily by the crossphase modulation (XPM) in the Kerr nonlinear medium. Moreover, when the spatial diﬀraction is balanced by XPM, spatial beam proﬁles of FWM signals can become stable to form spatial optical solitons. For diﬀerent input orientations and experimental parameters (such as laser powers, frequency detunings, and temperature), novel gap, vortex, and dipole solitons have been shown to exist in the multilevel atomic systems in vapour cell. These studies have opened the door for achieving rapid responding alloptical controlled spatial switch, routing, and soliton communications. This monograph serves as a reference book intended for scientists, researchers, advanced undergraduate and graduate students in nonlinear op Preface vii tics. We take this opportunity to thank many researchers and collaborators who have worked on the research projects as described in this book. Yanpeng Zhang Zhiqiang Nie Min Xiao October 2010 Contents 1 Introduction· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 1 Nonlinear Susceptibility · · · · · · · · · · · · · · · · · · · · · · · · · · Coherence Functions · · · · · · · · · · · · · · · · · · · · · · · · · · · · Suppression and Enhancement of FWM Processes · · · · · · · Double Dressing Schemes of Probe and FourWave Mixing Fields · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 1.5 Spatial Optical Modulation via Kerr Nonlinearities · · · · · · 1.6 Formations and Dynamics of Novel Spatial Solitons · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· ·· ·· 2 3 6 ·· ·· ·· ·· 8 10 15 18 Ultrafast Polarization Beats of FourWave Mixing Processes · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 23 1.1 1.2 1.3 1.4 2 3 2.1 Fourlevel Polarization Beats with Broadband Noisy Light · · · 2.1.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 2.1.2 FLPB in a Dopplerbroadened System· · · · · · · · · · · · · 2.1.3 Photonecho· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 2.1.4 Experiment and Result· · · · · · · · · · · · · · · · · · · · · · · · 2.2 Ultrafast Sumfrequency Polarization Beats in Twin Markovian Stochastic Correlation · · · · · · · · · · · · · · · · · · · · · 2.2.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 2.2.2 Secondorder Stochastic Correlation of ASPB · · · · · · · 2.2.3 Fourthorder Stochastic Correlation of ASPB· · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 44 45 50 62 76 Raman, Rayleigh and Brillouinenhanced FWM Polarization Beats · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 81 3.1 Attosecond Sumfrequency Ramanenhanced Polarization Beats Using Twin Phasesensitive Color Locking Noisy Lights · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.1.1 Basic Theory of Attosecond Sumfrequency REPB · · · · 81 83 23 25 32 34 39 x Contents 3.1.2 Homodyne Detection of Sumfrequency REPB · · · · · · · 3.1.3 Heterodyne Detection of Diﬀerencefrequency REPB · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.2 Competition Between Raman and Rayleighenhanced FourWave Mixings in Attosecond Polarization Beats · · · · · · · · · · · 3.2.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.2.2 Stochastic Correlation Eﬀects of Rayleigh and Ramanenhanced FWM · · · · · · · · · · · · · · · · · · · · · · · 3.2.3 The Raman and Rayleighenhanced Nonlinear Susceptibility in cw Limit· · · · · · · · · · · · · · · · · · · · · · 3.2.4 Homodyne Detection of ASPB · · · · · · · · · · · · · · · · · · 3.2.5 Heterodyne Detection of ASPB · · · · · · · · · · · · · · · · · · 3.2.6 Discussion and Conclusion · · · · · · · · · · · · · · · · · · · · · 3.3 Coexisting Brillouin, Rayleigh and Ramanenhanced FourWave Mixings · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.3.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.3.2 Homodyne Detection of ASPB · · · · · · · · · · · · · · · · · · 3.3.3 Heterodyne Detection of ASPB · · · · · · · · · · · · · · · · · · 3.3.4 Phase Angle· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 3.3.5 Discussion and Conclusion · · · · · · · · · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 4 89 104 112 113 116 124 126 132 140 144 145 148 152 162 164 166 MultiDressing FourWave Mixing Processes in Conﬁned and Nonconﬁned Atomic System · · · · · · · · · · · · · · · · · · · · · 169 4.1 Temporal and Spatial Interference Between FourWave Mixing and SixWave Mixing Channels · · · · · · · · · · · · · · · · · 4.2 Intermixing Between FourWave Mixing and SixWave Mixing in a Fourlevel Atomic System · · · · · · · · · · · · · · · · · · 4.2.1 Interplay Between FWM and SWM· · · · · · · · · · · · · · · 4.2.2 Discussion · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 4.3 Coexistence of FourWave, SixWave and EightWave Mixing Processes in Multidressed Atomic Systems · · · · · · · · 4.3.1 Parallel and Nested Dressing Schemes · · · · · · · · · · · · · 4.3.2 Interplay Among Coexisting FWM, SWM and EWM Processes· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 4.4 Controlled MultiWave Mixing via Interacting Dark States in a Fivelevel System· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 4.4.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 4.4.2 Numerical Results · · · · · · · · · · · · · · · · · · · · · · · · · · · 169 176 177 183 183 185 193 198 199 208 Contents 4.4.3 Discussion · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 4.5 Polarization Interference of MultiWave Mixing in a Conﬁned Fivelevel System · · · · · · · · · · · · · · · · · · · · · · · · · · 4.5.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 4.5.2 MWM in Long Cells· · · · · · · · · · · · · · · · · · · · · · · · · · 4.5.3 MWM in Ultrathin and Micrometer Cells · · · · · · · · · · 4.5.4 Discussion · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5 221 221 223 232 238 246 247 Enhancement and Suppression in FourWave Mixing Processes · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 253 5.1 Interplay among Multidressed FourWave Mixing Processes · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.2 Observation of Enhancement and Suppression of FourWave Mixing Processes · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5.3 Controlling Enhancement and Suppression of FourWave Mixing via Polarized Light · · · · · · · · · · · · · · · · · · · · · · · · · · 5.3.1 Theoretical Model and Analysis · · · · · · · · · · · · · · · · · 5.3.2 Experimental Results · · · · · · · · · · · · · · · · · · · · · · · · · 5.4 Enhancing and Suppressing FourWave Mixing in Electromagenetically Induce Transparency Window · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6 xi 253 260 266 266 269 273 280 MultiWave Mixing Processes in Multilevel Atomic System · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 283 6.1 Modulating MultiWave Mixing Processes via Polarizable Dark States · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.2 Polarization Spectroscopy of Dressed FourWave Mixing in a Threelevel Atomic System · · · · · · · · · · · · · · · · · · · · · · · · 6.2.1 Various Nonlinear Susceptibilities for Diﬀerent Polarization Schemes · · · · · · · · · · · · · · · · · · · · · · · · · 6.2.2 Nonlinear Susceptibilities for Zeemandegenerate System Interacting with Polarized Fields · · · · · · · · · · · 6.2.3 Thirdorder Densitymatrix Elements in Presence of Dressing Fields · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.3 Controlling FWM and SWM in MultiZeeman Atomic System with Electromagnetically Induced Transparency· · · · · · · · · · · 6.3.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.3.2 Dualdressed EIT · · · · · · · · · · · · · · · · · · · · · · · · · · · 6.3.3 FourWave Mixing · · · · · · · · · · · · · · · · · · · · · · · · · · · 284 298 300 302 306 314 315 319 323 xii Contents 6.3.4 SixWave Mixing · · · · · · · · · · · · · · · · · · · · · · · · · · · · 328 References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 330 7 Controlling Spatial Shift and Spltting of FourWave Mixing · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 333 7.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 7.2 Electromagneticallyinduced Spatial Nonlinear Dispersion of FourWave Mixing Beams · · · · · · · · · · · · · · · · · · · · · · · · · 7.3 Spatial Dispersion Induced by Crossphase Modulation · · · · · 7.4 Experimental Demonstration of Optical Switching and Routing via FourWave Mixing Spatial Shift · · · · · · · · · · · · · 7.4.1 Theoretical Model and Experimental Scheme · · · · · · · · 7.4.2 Optical Switching and Routing via Spatial Shift· · · · · · 7.5 Controlled Spatial Beamsplitter Using FourWave Mixing Images· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 7.6 Spatial Splitting and Intensity Suppression of FourWave Mixing in Vtype Threelevel Atomic System · · · · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 8 333 337 346 351 352 354 358 365 370 Spatial Modulation of FourWave Mixing Solitons · · · · · · · 373 8.1 Basic Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 8.1.1 Calculation of Double Dressed CrossKerr Nonlinear Index of Refraction · · · · · · · · · · · · · · · · · · · · · · · · · · 8.1.2 Calculation of Analytical Solution of Onedimensional Bright and Dark Spatial Solitons · · · · · · · · · · · · · · · · 8.2 Novel Spatial Gap Solitons of FourWave Mixing · · · · · · · · · · 8.3 Dipolemode Spatial Solitons of FourWave Mixing · · · · · · · · 8.4 Modulated Vortex Solitons of FourWave Mixing · · · · · · · · · · References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 373 374 380 384 391 398 408 Index · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 411 1 Introduction The subjects of this book focus on mainly around two topics. The ﬁrst topic (Chapters 2 and 3) covers the ultrafast fourwave mixing (FWM) polarization beats due to interactions between multicolored laser beams and multilevel media. Both diﬀerencefrequency femtosecond and sumfrequency attosecond polarization beats can be observed in multilevel media depending on the specially arranged relative time delays in multicolored laser beams. The polarization beat signal is shown to be particularly sensitive to the statistical properties of the Markovian stochastic light ﬁelds with arbitrary bandwidth. Also, the Raman, RamanRayleigh, RayleighBrillouin, coexisting RamanRayleighBrillouinenhanced polarization beats due to colorlocking noisy ﬁeld correlations have been studied. Polarization beats between various FWM processes are among the most important ways to study transient properties of media. The second topic (Chapters 4 – 8) relates to the frequency domain and spatial interplays of FWM processes induced by atomic coherence in multilevel atomic systems. FWM processes with diﬀerent kinds of dualdressed schemes in ultrathin, micrometer and long atomic cells, selectively enhanced and suppressed FWM signals via an electromagnetically induced transparency (EIT) window are described, coexisting FWM and sixwave mixing (SWM) processes, especially temporal and spatial interference between them in multilevel EIT media are presented in Chapter 4. Furthermore, These eﬀects of spatial displacements and splitting of the probe and generated FWM beams, as well as the observations of gap soliton trains, vortex solitons of FWM, stable multicomponent vector solitons consisting of two perpendicular FWM dipole components induced by nonlinear crossphase modulation (XPM) in multilevel atomic media, are shown and investigated in Chapters 5 – 8. Experimental results will be presented and compared with the theoretical calculations throughout the book. Also, emphasis will be given only to the works done by the authors’ groups in the past few years. Some of the works presented in this book are built upon our previous book (Multiwave Mixing Processes published by High Education Press & Springer 2009), where we have mainly discussed the coexistence and interactions between eﬃcient multiwave mixing (MWM) processes enhanced by atomic coherence in multilevel atomic systems. Before starting the main topics of this book, some basic physical concepts and mathematical techniques, which are useful and needed in the later chapters, will be brieﬂy introduced and discussed in Y. Zhang et al., Coherent Control of FourWave Mixing © Higher Education Press, Beijing and SpringerVerlag Berlin Heidelberg 2011 2 1 Introduction this chapter. 1.1 Nonlinear Susceptibility For over a decade, one of useful nonlinear optical techniques is the socalled “noisy” light spectroscopy. The ultrashort time resolution of material dynamics has been accomplished by the interferometric probing of wave mixing with broadband, nontransformlimited noisy light beams. The time resolution is determined by the ultrafast correlation time of noisy light and not by its temporal envelope, which is typically a few nanoseconds [1 – 6]. Such the noisy light source is usually derived from a dye laser modiﬁed to permit oscillation over almost the entire bandwidth of the broadband source. The typical bandwidth of the noisy light is about 100/cm, and has a correlation time of 100 fs [7]. In fact, the multimode broadband light has an autocorrelation time similar to the autocorrelation time of the transformlimited femtosecond laser pulse of equivalent bandwidth, although the broadband light can, in principle, be a continuous wave (cw). In order to describe more precisely what we mean by optical nonlinearity, let us consider how the dipole moment per unit volume, or polarization P , of a material system depends on the strength E of the applied optical ﬁeld. The induced polarization depends nonlinearly on the electric ﬁeld strength of the applied ﬁeld in a manner that can be described by the relation P = PL +PN L [8, 9]. Here, PL = P (1) = ε0 χ(1) · E and PN L = P (2) + P (3) + · · · = ε0 (χ(2) : . EE + χ(3) .. EEE + · · · ). When we only consider the atomic system (which are isotropic and have inversion symmetry), we can write the total polarization as P = ε0 χE in general, where the total eﬀective optical susceptibility can be described by ∞ 2j χ(2j+1) E . The terms with evenorder a generalized expression of χ = j=0 powers in the applied ﬁeld strength vanish [8]. The lowest order term χ(1) (j = 0) is independent of the ﬁeld strength and is known as the linear susceptibility. The next two terms in the summation, χ(3) and χ(5) , are known as the thirdand ﬁfthorder nonlinear optical susceptibilities, respectively. FWM refers to nonlinear optical processes with four interacting electromagnetic waves (i.e., with three applied ﬁelds to generate the fourth ﬁeld). In the weak interaction limit, FWM is a pure thirdorder nonlinear optical process and is governed by the thirdorder nonlinear susceptibility [8]. Let us consider a special case of FWM processes. The thirdorder nonlinear polarization governing the process has, in general, three components with diﬀerent wave vectors k1 , k2 , and k2 . E1 (ω1 ), E2 (ω2 ), and E2 (ω2 ) denote the three input laser ﬁelds. Here, ωi and ki are the frequency and propagation wave vectors of the ith beam. We can choose to have a small angle θ between the 1.2 Coherence Functions 3 input pump laser beams k2 and k2 . The probe laser beam (beam k1 ) propagates along a direction that is almost opposite to that of the beam k2 (see Fig. 1.1). The corresponding nonlinear atomic polarization P (3) (ω1 ) along the i(i = x, y) direction, from ﬁrstorder perturbation theory, is given by [10] (3) (3) ∗ χijkl E1j (ω1 )E2k (ω2 )E2l (ω2 ), (1.1) Pi (ω1 ) = ε0 jkl Fig. 1.1. (a) Schematic diagram for the phaseconjugate FWM process. (b) Energylevel diagram for FWM in a closecycled threelevel cascade system. where the thirdorder susceptibility contains the microscopic information (3) about the atomic system. The susceptibility of the nonlinear tensor χijkl (ωF ; ω1 , −ω2 , ω2 ) is also related to polarization components of incident and generated ﬁelds. For an isotropic medium, as in the atomic vapor, only four elements are not zero, and they are related to each other by χxxxx = χxxyy + χyxxy +χyxyx . For the generated SWM signal ES (ﬁelds E2 and E3 propagate along the direction of beam 2 and E2 and E3 propagate along beam 3), the ﬁfthorder nonlinear polarization P (5) (ω1 ) along the i(i = x, y) direction is then given by (5) (5) ∗ ∗ Pi (ω1 ) = ε0 χijklmn E1j (ω1 )E2k (ω2 )E2l (ω2 )E3m (ω3 )E3n (ω3 ), (1.2) jklmn (5) where χijklmn is the ﬁfthorder nonlinear susceptibility. For an isotropic medium, there are sixteen nonzero components and only ﬁfteen of them are independent because they are related to each other by 3χxxxxxx = χyyxxxx + χyxyxxx + χyxxyxx + χyxxxyx + χyxxxxy + χxyyxxx + χxyxyxx + χxyxxyx + χxyxxxy + χxxyyxx + χxxyyxx + χxxyxyx + χxxyxxy + χxxxyxy + χxxxxyy . (1.3) 1.2 Coherence Functions Lasers are inherently noisy devices, in which both phase and amplitude of the ﬁeld can ﬂuctuate. There are many diﬀerent stochastic models to describe laser ﬁelds. However, since many models of ﬂuctuating laser ﬁelds 4 1 Introduction have identical secondorder ﬁeld correlations, diﬀerences among them will become important only if observable eﬀects depend on higherorder ﬁeld correlations. Noisy light can be used to probe atomic and molecular dynamics, and it oﬀers an unique alternative to the more conventional frequencydomain spectroscopies and ultrashort timedomain spectroscopies [1, 11 – 14]. When the laser ﬁeld is suﬃciently intense that multiphoton interactions occur, the laser spectral bandwidth or spectral shape, obtained from the secondorder correlation function, is inadequate to characterize the ﬁeld. Rather than using higherorder correlation functions explicitly, three diﬀerent Markovian ﬁelds are considered: i.e., (a) the chaotic ﬁeld, (b) the phasediﬀusion ﬁeld, and (c) the Gaussianamplitude ﬁeld. If laser sources have Lorentzian line shape, we have the secondorder coherence function ui (t1 )u∗i (t2 ) = exp(−αi t1 − t2 ) (i.e., ui (t)2 = 1 when t = t1 = t2 ). Here, αi = δωi /2, with δωi being the linewidth of the laser with frequency ωi . On the other hand, if laser sources are assumed to have Gaussian line shape, then we have ui (t1 )u∗i (t2 ) = exp √ 2 − αi (t1 − t2 )/2 ln 2 . In the following, we only consider the former case. In fact, the form of the secondorder coherence function shown above, which is determined by the laser line shape, is the general feature of stochastic models [15]. In this section, three Markovian noise stochastic models, the chaotic ﬁeld model (CFM), the phasediﬀusion model (PDM), and the Gaussianamplitude model (GAM) are considered at a high enough intensity level to fully appreciate the subtle features of FWM spectroscopy [16, 17]. First, in CFM, we assume that the pump laser is a multimode thermal source and u(t) = a(t)eiφ(t) , where a(t) is the ﬂuctuating modulus and φ(t) is the ﬂuctuating phase. In this case, u(t) has Gaussian statistics with its fourthorder coherence function satisfying [18] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 )CF M = ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 )+ui(t1 )u∗i (t4 )ui (t2 )u∗i (t3 ) = exp[−αi (t1 −t3 + t2 − t4 )] + exp[−αi (t1 − t4  + t2 − t3 )]. In fact, all higher order coherence functions can be expressed in terms of products of secondorder coherence functions. Thus any given 2nthorder coherence function may be decomposed into the sum of n! terms, each consisting of the product of n secondorder coherence functions. The general expression can be obtained as, ui (t1 ) · · · ui (tn )u∗i (tn+1 ) · · · u∗i (t2n )CF M = ui (t1 )u∗i (tp )ui (t2 )u∗i (tq ) · · · ui (tn )u∗i (tr ), where π denotes a summation over the n! possible per π mutations of (1, 2, . . ., n). Second, in PDM the dimensionless statistical factor can be written as u(t) = eiφ(t) (i.e., u(t) = 1) with φ̇(t) = ω(t), ωi (t)ωi (t ) = 2αi δ(t − t ), ωj (t)ωj (t ) = 2αj δ(t − t ) and ωi (t)ωj (t ) = 0. The secondorder co 1.2 Coherence Functions 5 herence function for a beam with Lorentzian line shape is given by [19] u(t1 )u∗ (t2 ) = exp iΔφ t1 −t2 = exp − (t1 − t2 − t)ω(t1 )ω(t1 − t)dt = exp(−αt1 − t2 ). 0 Here, Δφ = φ(t1 ) − φ(t2 ) = t1 ω(t)dt has Gaussian statistics, so that t2 2 /2). By the classical relation of linear ﬁltering, we exp iΔφ = exp(−σΔφ have t1 −t2 2 σΔφ = L(t1 − t2 ) = 2 (t1 − t2 − t)ω(t1 )ω(t1 − t)dt. 0 Now, the fourthorder coherence function can be calculated, which can be written as ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 )P DM = exp{−[L(t1 − t3 ) + L(t1 − t4 ) + L(t2 − t3 ) + L(t2 − t4 ) − L(t1 − t2 ) − L(t3 − t4 )]} = exp[−αi (t1 − t3  + t1 − t4  + t2 − t3  + t2 − t4 )] × exp[αi (t1 − t2  + t3 − t4 )] ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 )ui (t1 )u∗i (t4 )ui (t2 )u∗i (t3 ) . = ui (t1 )u∗i (t2 )ui (t3 )u∗i (t4 ) Furthermore, we have the general expression for the secondorder coherence function as ui (t1 ) · · · ui (tn )u∗i (tn+1 ) · · · u∗i (t2n )P DM n n ui (tp )u∗i (tn+q ) = p=1 q=1 n n . ui (tp )u∗i (tq )ui (tn+p )u∗i (tn+q ) p=1 q=p+1 Finally, in GAM, one has u(t) = a(t), where a(t) is real and Gaussian, and ﬂuctuates about a mean value of zero. The fourthorder coherence function of u(t) satisﬁes [20] ui (t1 )ui (t2 )ui (t3 )ui (t4 )GAM = ui (t1 )ui (t3 )ui (t2 )ui (t4 ) + ui (t1 )ui (t4 )ui (t2 )ui (t3 ) + ui (t1 )ui (t2 )ui (t3 )ui (t4 ) = ui (t1 )ui (t2 )ui (t3 )ui (t4 )CF M + ui (t1 )ui (t2 )ui (t3 )ui (t4 ) = exp[−αi (t1 − t3  + t2 − t4 )] + exp[−αi (t1 − t4  + t2 − t3 )] + exp[−αi (t1 − t2  + t3 − t4 )]. 6 1 Introduction In fact, according to the moment theorem for real Gaussian random variables, we have the general expression for the 2nthorder coherence function, as ui (t1 ) · · · ui (tn )ui (tn+1 ) · · · ui (t2n )GAM = 2n ui (t1 )ui (tk ), P j=k where indicates the summation over all possible distinct combinations of P the 2n variables in pairs. 1.3 Suppression and Enhancement of FWM Processes In presence of a strong dressing ﬁeld G2 , the dressed states + and − can be generated with the separation Δ± = 2G2 , as shown in Fig. 1.2. When scanning the frequency of the dressing ﬁeld, we can obtain the EIT for the probe ﬁeld and a suppressed FWM signal [Fig. 1.2 (b)], or electromagnetically induced absorption (EIA) for the probe ﬁeld and an enhanced FWM [Fig. 1.2 (c)]. For the probe ﬁeld propagating through the medium, we deﬁne the baseline versus the dressing ﬁeld detuning Δ2 to be at the probe ﬁeld intensity without the dressing ﬁeld. Thus, this baseline is just the Dopplerbroadened absorption signal of the material. With G2 beam on, we can obtain one EIT peak at Δ1 + Δ2 = 0, where the transmitted intensity is largest comparing to the baseline. Since there is no energy level in the original position of 1 and the probe ﬁeld is no longer absorbed by the material, the degree of transmission (or suppression of absorption) of the probe ﬁeld is highest. 2 An EIA dip is obtained at Δ1 + Δ2 = G2  /Δ1 , where the transmitted intensity is smallest compared to the baseline. The reason is that the dressed state + or − is resonant with the probe ﬁeld which is absorbed by the material and the degree of transmission (or enhancement of absorption) of the Fig. 1.2. (a) The diagram of the threelevel laddertype system with a dressing ﬁeld G2 (and detuning Δ2 ). The dressedstate pictures of the (b) suppression (EIT) and (c) enhancement (EIA) of FWM Gf (or probe ﬁeld Gp with detuning Δ1 ) for the twolevel system, respectively. 1.3 Suppression and Enhancement of FWM Processes 7 probe ﬁeld is lowest. Moreover, at Δ1 = 0, since the transparent degree (or suppression of absorption) of the probe ﬁeld G1 is largest, the EIT peak for G2 is highest. While at certain detuning Δ1 , the induced transparent degree decreases and the EIT peak changes lower. When the detuning Δ1  becomes much larger, the degree of transparency decreases and the suppression of absorption changes to an enhancement of absorption. Then, the enhancement and suppression of FWM signals can be understood as follows: The background versus the dressing ﬁeld detuning Δ2 represents the signal intensity of the FWM without dressing ﬁeld. The dips lower than the background (or suppression peaks at Δ1 + Δ2 = 0) and the peaks 2 higher than the background (enhancement peaks at Δ1 + Δ2 = G2  /Δ1 ) represent that FWM signals are suppressed and enhanced, respectively. Figure 1.3 presents suppressed and enhanced FWM signals and the corresponding probe transmission signal versus the probe ﬁeld detuining Δ1 and versus the dressing ﬁeld detuning Δ3 . In this system, two pump ﬁelds E2 and E2 induce the EIT satisfying the twophoton resonant condition Δ1 + Δ2 = 0 [see the general baseline in Fig. 1.3 (a) versus Δ1 ] and generate the FWM signal [see the general background in Fig. 1.3 (b) versus Δ1 ]. The dressing ﬁeld E3 induces each EIT peak versus Δ3 (satisfying the condition Δ1 + Δ3 = 0) in Fig. 1.3 (a) and generates the enhancement and suppression of FWM signals versus Δ3 as shown in Fig. 1.3 (b). Moreover, Fig. 1.3 (c) shows the normalized enhancement and suppression of FWM signals by dividing the background (i.e., the signal intensity of the FWM without dressing ﬁeld) versu Δ3 . Fig. 1.3. (a) The probe transmission signal; (b) the enhancement and suppression of the FWM signal versus the dressing ﬁeld detuning Δ3 for the diﬀerent value of the probe ﬁeld detuining Δ1 ; (c) the FWM signal normalized by the doublepeak FWM signal [dashed curve in (b)]. In Fig. 1.3 (a) one can see that the EIT peaks at large detuning Δ1 are higher than those near Δ1 = 0. This phenomenon can be explained as follows: although E2 EIT window cannot be observed directly when scanning the detuning Δ3 , the E2 dressing at Δ1 = −Δ3 = −Δ2 ≈ 0 results in an uplift of the probe transmission baseline, and the suppression of the probe transmission. At the large detuning of Δ1 [Fig. 2 (b)], the E2 ﬁeld basically 8 1 Introduction cannot aﬀect the E3 EIT peaks. Moreover, at certain large detunings of Δ1 , the enhancement of absorption of probe beams will show up. For the enhancement and suppression of FWM signals in Figs. 1.3 (b) and (c), the curves show enhancement and suppression evolution behaviors when increasing Δ1 . Speciﬁcally, FWM signals change from allenhanced to halfsuppressed and halfenhanced, then to allsuppressed around the resonant point, and then to halfsuppressed and halfenhanced, and ﬁnally to allenhanced. On the other hand, the suppression of FWM signals at the detunings Δ1 corresponding to the FWM signal peaks is more obvious than those suppression or enhancement at large Δ1 in Fig. 1.3 (b), while the normalized allenhancement of FWM signals at large Δ1 in Fig. 1.3 (c) is much larger than the cases with halfsuppression, halfenhancement and allsuppression signals at small Δ1 . 1.4 Double Dressing Schemes of Probe and FourWave Mixing Fields Recently, Investigations about dressed and doublydressed states in multilevel atomic systems interacting with multiple electromagnetic ﬁelds have attracted many interests [21 – 24]. The interaction of doubledark states (equivalent to nested scheme of doublydressing) and splitting of dark states (equivalent to the secondarilydressed state) in fourlevel atomic systems were studied theoretically [25]. Then doublydressed states in cold atoms were experimentally observed, in which the triplephoton absorption spectrum exhibits a constructive interference between excitation paths of two closelyspaced, doublydressed states [26]. Similar results were obtained in the invertedY system [27] and doubleΛ system [28]. Similar to the probe ﬁeld, the generated FWM beam (between states 0 and 1) has the same doublydressing behavior. Figure 1.4 shows three kinds of schemes in three fourlevel systems, respectively, i.e., the nestedcascade scheme in the fourlevel laddertype system [Fig. 1.4 (a)], parallelcascade scheme in the fourlevel Vtype system [Fig. 1.4 (b)], and sequentialcascade scheme in the fourlevel Ytype system [Fig. 1.4 (c)]. For the nestedcascade doublydressing scheme of the probe or FWM ﬁeld, the two dressing ﬁelds (with frequencies ω2 and ω3 , and Rabi frequencies G2 and G3 , respectively) connect three neighboring levels (1, 2 and 3) and the outer dressing ﬁeld G3 is based on the inner dressing ﬁeld G2 , while this inner ﬁeld dresses the state of FWM processes. In the perturbation the chain G G G∗ G∗ 3 2 2 3 ρ10 −−→ ρ20 −−→ ρ30 −−→ ρ20 −−→ ρ10 [16], the two dressing ﬁelds G2 and G3 are intertwined tightly with each other which gives the strongest interaction. In this case, only one term in the FWM expression (3) ρ10 ∝ Ga {Γ10 + iΔp + G22 /[Γ20 + i(Δp + Δ2 )+ G23 /(Γ30 + i(Δp + Δ2 + Δ3 ))]}−1 1.4 Double Dressing Schemes of Probe and FourWave Mixing Fields 9 Fig. 1.4. Sketches of the (a) nestedcascade scheme in the fourlevel laddertype system; (b) parallelcascade scheme in the fourlevel Vtype system; and (c) sequentialcascade scheme in the fourlevel Ytype system. Gf (Gp , G2 and G3 ) are the Rabi frequencies of FWM (the probe and the coupling) ﬁelds, respectively. is modiﬁed by the dressing ﬁelds and if the Rabi frequency of the inner dressing ﬁeld G2 is zero, the term for the outer dressing ﬁeld disappears also, where Ga = Gp for the doublydressed probe (DDP) ﬁeld and Ga = −1 −1 Gp G1 G∗ for the doublydressed FWM (DDFWM). Γij is 1 Γ00 (Γ10 + iΔp ) the transverse relaxation rate between i and j, and Δp , Δ2 , and Δ3 are the frequency detunings of the probe and the coupling ﬁelds, respectively. On the other hand, for parallelcascade double dressing scheme, the dressing ﬁelds G2 and G3 dress the two diﬀerent states (0 and 1, respectively) directly and independently, as shown in Fig. 1.4 (b). In the perturbation the chain G G∗ G G∗ 2 3 2 3 ρ10 −−→ ρ20 −−→ ρ10 and ρ10 −−→ ρ13 −−→ ρ10 , these two dressing processes are separated and the corresponding terms of dressing ﬁelds are independent in the expression of (3) ρ10 ∝ Gb [Γ10 + iΔp + G22 /(Γ20 + iΔp + iΔ2 )]−1 [Γ10 + G23 /(Γ13 + iΔp − iΔ3 )]−1 , −1 where Gb = Gp for DDP ﬁeld and Gb = Gp G1 G∗ 1 Γ00 for DDFWM. Finally, for the sequentialcascade doublydressing scheme, the dressing ﬁelds also dress the states directly and independently [both on level 1 as shown in Fig. 1.4 (c)], however, two dressing processes are conjoined by the same G G∗ G 2 2 3 densitymatrix element in the perturbation the chain ρ10 −−→ ρ20 −−→ ρ10 −−→ G∗ 3 ρ10 , so only one term in FWM is modiﬁed by the dressing ﬁeld in ρ30 −−→ the expression: (3) ρ10 ∝ Gc [Γ10 + iΔp + G22 /(Γ20 + iΔp + iΔ2 ) + G23 /(Γ30 + iΔp + iΔ3 )]−1 , −1 −1 for where Gc = Gp for DDP ﬁeld and Gc = Gp G1 G∗ 1 Γ00 (Γ10 + iΔp ) DDFWM. The interaction between two dressing ﬁelds in the nestedcascade scheme is the strongest, and it is the weakest in the parallelcascade scheme. The sequentialcascade scheme is an intermediate case between the other two 10 1 Introduction cases. Also, from the dressedstate picture, the dressing ﬁelds of the nestedcascade scheme are intertwined tightly with each other. Only the inner dressing ﬁeld can create primary dressed states, and the outer dressing ﬁeld can only create secondary dressed states [29, 30]. While for the parallelcascade scheme, the dressing ﬁelds have a weaker interaction, and they can directly create two independent dressed states [31, 32]. However, for the sequentialcascade scheme the dressing ﬁeld can also directly create dressed states but they have a strong interaction to create primary and secondary dressed states [30]. On the other hand, forms of the nested and parallelcascade DDFWM expressions can be converted into forms of the sequentialcascade DDFWM case under the conditions that the outer dressing ﬁeld is weak or its detuning is large for nestedcascade DDFWM and the two dressing ﬁelds are weak for parallelcascade DDFWM, respectively [30]. Investigations of those diﬀerent doublydressing schemes in multilevel atomic systems can help us to understand the underlying physical mechanisms and to eﬀectively optimize the generated multichannel nonlinear optical signals. Controlling these processes can have important applications in designing novel nonlinear optical devices in multistate systems. 1.5 Spatial Optical Modulation via Kerr Nonlinearities The Kerr eﬀect is a special kind of nonlinear optical phenomenon occurring when intense light beams propagate in crystals, glasses, or gases. Its physical origin is a thirdorder nonlinear polarization generated in the medium. For selfKerr nonlinearity, the intense light modiﬁes its own propagation properties, while for the crossKerr nonlinearity the propagation properties of a light beam are modiﬁed by the interaction with another overlapping beam in a Kerr medium. Actually, the Kerr eﬀect originates from an instantaneously occurring thirdorder nonlinear response, which can be described as a modiﬁcation of the refractive index. The refractive index of many optical materials depends on the intensity of the light beam due to special thirdorder nonlinear responses, which can be written as n = n0 + n2 I. Here, n2 is the Kerr nonlinear index proportional to χ(3) . If higherorder (such as ﬁfth) nonlinearity is considered, the nonlinear index n2 will be inﬂuenced by the intensity of the light beam [33]. With weak cw diode lasers in threelevel systems sharp dispersion of n0 can be induced due to the EIT [34], which can slow down the optical pulse propagation. Also, the thirdorder nonlinear optical Kerr coefﬁcient n2 of the threelevel EIT system has been measured which is greatly enhanced comparing to its twolevel subsystem [35]. Since the Kerr nonlinear dispersion in such EIT medium has been shown to have an opposite sign (anomalous dispersion) from the linear dispersion (Fig. 1.5), and they change dramatically near the EIT resonance, the cavity transmission linewidth with such and EIT medium can be greatly modiﬁed due to the modiﬁed group 1.5 Spatial Optical Modulation via Kerr Nonlinearities 11 index: ng = (n0 + n2 Ip ) + ωP (∂n0 /∂ωP + Ip ∂n2 /∂ωP ), where Ip is the probe beam intensity, and ωP is the probe laser beam frequency [36]. The linear and nonlinear dispersion terms (the derivatives) dominate in ng . Since the two derivatives have opposite signs [8], ng can take either positive or negative values, depending on the frequency detuning and probe intensity. Fig. 1.5. (a) Linear and (c) nonlinear refractive indices and their derivatives (b) and (d), respectively, as a function of Δp . Adopted from Ref. [36]. Eﬀects of self and crossKerr nonlinearities also induce phenomena of the selfphasemodulation (SPM) and crossphasemodulation (XPM) [37] which modulate spatial optical beams. Depending on the sign of the nonlinear refractive index, such an intensitydependent refractive index can produce either a converging or a diverging wave front to change the transverse beam proﬁle during beam propagation. With SPM a single beam modulates itself during its propagation through medium. When two copropagating or counterpropagating beams modulate each other via nonlinear interaction, it is due to XPM. When n2 > 0, the converging wave front counteracts against diﬀractioninduced spatial spreading, which can focus the optical 12 1 Introduction beam to demonstrate the selffocused or the crossfocused beam when the beam power exceeds a critical value. Similarly, when n2 < 0 the diverging wave front increases the natural diverging, which gives the selfdefocused or crossdefocused beam. In 1990 Agrawal reported the phenomenon of induced focusing occurring in the selfdefocusing nonlinear media as a result of XPM. When a weak optical beam copropagates with an intense pump beam, the XPMinduced interaction between two beams can focus the weak beam, even though the pump beam exhibits selfdefocusing. [37] Also, the electromagneticallyinduced focusing (EIF) phenomenon was reported in the threelevel atomic system [38]. In the threelevel EIT system the radial intensity proﬁle of the strong pump laser can generate a modiﬁed spatial refractive index proﬁle which is experienced by the weak probe laser as it tunes through the transparency window near resonance. It leads to spatial focusing and defocusing of the probe beam [38]. Equations (1.4 – 1.6) are the MaxwellBloch equations under rotatingwave and slowlyvaryenvelope approximations, which give the mathematical description of the SPM and XPMinduced spatial interactions among the probe and two FWM beams for the system as shown in Fig. 1.4. ∂Ep i∂ 2 Ep ∂Ep i∇2⊥ Ep + − − ∂z c∂t 2∂t2 2kp ikp 2 X1 2 X2 2 = n1 + nS1 Ep + 2 Ep  + 2n2 E1  + 2n2 E2  n0 η1 E1 (E1 )∗ EF 1 + η2 E2 (E2 )∗ EF 2 , (1.4) ∂EF 1 i∂ 2 EF 1 ∂EF 1 i∇2 EF 1 + − − ⊥ 2 ∂z c∂t 2∂t 2kF 1 ikF 1 2 X3 2 = n1 + nS2 EF 1 + 2 EF 1  + 2n2 E1  n0 η3 E1 (E1 )∗ Ep + η4 E2 (E2 )∗ EF 2 , (1.5) i∇2 EF 2 ∂EF 2 ∂EF 2 i∂ 2 EF 2 − ⊥ + − 2 ∂z c∂t 2∂t 2kF 2 ikF 2 2 X4 2 n1 + nS3 EF 2 + = 2 EF 2  + 2n2 E2  n0 η5 E1 (E1 )∗ EF 1 + η6 E2 (E2 )∗ Ep . (1.6) Here, on the left side of these equations, the ﬁrst terms describe the beam propagation, the second terms give the dispersion ones, the third terms are for the secondorder dispersion, and the fourth terms describe the diﬀraction of the beams diverging propagation. On the right hand, the ﬁrst terms are the linear response, the second terms are for the nonlinear selfKerr eﬀects, the third terms [the third and fourth terms for Eq. (1.4)] describe nonlinear crossKerr eﬀects, the fourth and ﬁfth terms [the ﬁfth and sixth terms for Eq. (1.4)] represent the phasematched coherent FWM process. z is the longitudinal 1.5 Spatial Optical Modulation via Kerr Nonlinearities 13 coordinate in the propagation direction and kp = kF 1 = ω1 n0 /c. n0 and n1 are the linear refractive index at ω1 in vacuum and medium, respectively. S2 nS1 2 is the selfKerr nonlinear coeﬃcient of the ﬁeld E3 , n2 is the selfKerr nonlinear coeﬃcient for the generated FWM ﬁeld EF 1 , and nS3 2 is the selfKerr nonlinear coeﬃcient for the generated FWM ﬁeld EF 2 . nX1 2 is the crossKerr nonlinear coeﬃcient of the ﬁeld E3 induced by the strong pump ﬁeld E1 , nX2 is the crossKerr nonlinear coeﬃcient of the ﬁeld E3 induced by the 2 strong pump ﬁeld E2 , nX3 is the crossKerr nonlinear coeﬃcient of the ﬁeld 2 EF 1 induced by the strong pump ﬁeld E1 , nX4 is the crossKerr nonlinear 2 coeﬃcient of the ﬁeld EF 2 induced by the strong pump ﬁeld E2 . In general the Kerr nonlinear coeﬃcients can be deﬁned as n2 = Re χ(3) /(ε0 cn0 ), where (3) the thirdorder nonlinear susceptibility is given by χ(3) = Dρ10 with D = 2 2 3 2 N μp μi0 /( ε0 Gp Gi ). μp (μi0 ) is the dipole matrix element between the states coupled by the probe beam Ep (between i and 0). ηi are the constants. (3) ρ10 can be determined from the densitymatrix equations for the multilevel medium. The strong pump beam distorts the phase proﬁles of the probe and FWM beams through XPM, which induces spatial modiﬁcations of the probe and FWM beams, including spatial displacement and splitting, and produces spatial solitons. Thus, we can neglect the dispersion, linear term, and coherent FWM processes in the equations, for the moment for simplicity. Actually, these simpliﬁed diﬀerential equations are still diﬃcult to solve analytically. By assuming Gaussian proﬁles for input ﬁelds, we can use a numerical approach (i.e., the splitstep Fourier method [37]) to solve Eqs. (1.4 – 1.6). However, the numerical solution of threedimensional equations requires a considerable computing resource with both x and y directions. For simplicity, we only consider one dimension in the ydirection. For example, one can consider Eq. (1.5) and obtain ikF 1 ∂EF 1 (z, y) ∂ i 2 X3 2 + = nS2 EF 1 (z, y). 2 EF 1  + 2n2 E1  2 ∂z 2kF 1 ∂y n0 (1.7) The solution of this equation is approximately EF 1 (z + h, y) ≈ Exp[ihD̂] · Exp[ihN̂ ]EF 1 (z, y) [37]. Here h is the steplength, D̂ = (2kF 1 )−1 ∂/∂y 2 is the 2 X3 2 diﬀraction functor and N̂ = kF 1 (nS2 2 EF 1  + 2n2 E1  )/n0 is the SPM and XPM functor. Finally we can use the splitstep Fourier method to obtain the numerical solution. Furthermore, If we also neglect the diﬀraction term and the small SPM contribution, Eqs. (1.4 – 1.6) can be readily solved to obtain the XPMinduced phase shift φN L imposed on the probe and FWM beams by the pump. In this case, Equation (1.7) reduces to i2kF 1 X3 2 ∂EF 1 (z, y) = n E1  EF 1 (z, y), (1.8) ∂z n0 2 · which gives EF 1 (z, y) = EF 1 (0, y) exp(iφN L ) with φN L (z, y) = 2kF 1 nX3 2 2 E1  z/n0 . The additional transverse propagation wavevector is dky = φN L 14 1 Introduction [37]. Here, the strong ﬁeld E has a Gaussian proﬁle, like the solid line in Fig. 1.6 (a). In this case, when nX3 2 > 0, φN L has a positive Gaussian proﬁle [see the thick solid line in Fig. 1.6 (a)] and dky is shown by the dash line in Fig. 1.6 (a). The arrows in Fig. 1.6 (a) represent the direction of dky . The direction of dky is always towards the beam center of the pump ﬁeld, and therefore, the weak Ep,F 1,F 2 ﬁelds [the thin solid lines in Fig. 1.6 (a)] are shifted to the pump ﬁeld center. When nX3 2 < 0, φN L has a negative Gaussian proﬁle [see the thick solid line in Fig. 1.6 (b)] and the direction of dky [the dash line in Fig. 1.6 (b)] is outward from the beam center of the pump ﬁeld, thus Ep,F 1,F 2 is shifted away from the pump ﬁeld [see Fig. 1.6 (b)]. Fig. 1.6. Instantaneous nonlinear phase shift induced by a Gaussian beam in a (a) focusing and (b) defocusing nonlinear medium and the corresponding contribution to the onedimensional component of the propagation vector. Recently, spatial displacements of the probe and generated FWM beams have been observed in a threelevel Vtype, and twolevel atomic systems near resonance [39]. The observed spatial shift curves as a function of frequency detuning reﬂect the typical enhanced crossKerr nonlinear dispersion properties in the EIT system. This dispersionlike spatial deﬂection is named as electromagneticallyinduced spatial dispersion (EISD). The spatial beam displacements can be controlled by the strong control laser beam and the atomic density. Such EISD can be used as a single way to measure the Kerrnonlinear refractive indices for the multilevel atomic media. Also, it can be used for controllable alloptical spatial switching and routing of optical signals [40]. The spot shifts of the FWM and probe laser beams can be used as the “on” and “oﬀ” states of the spatial alloptical switch. The extinction ratio for the on/oﬀ state, as well as the beam shift distances and directions, can be optimized by modulating frequency detunings, intensities, and temperature of the medium. At the same time, beam shifts in opposite directions have been realized simultaneously for diﬀerent FWM beams, which could be employed to construct switching/routing arrays. Then, spatial shifts and splittings of FWM signal beams induced by additional dressing laser beams were investigated which are caused by the enhanced crossKerr nonlinearity due to atomic coherence in the atomic system. The spatial separation and number of the split FWM beam can both be controlled by the 1.6 Formations and Dynamics of Novel Spatial Solitons 15 intensity of the dressing beam, and by the modiﬁed Kerr nonlinearity and atomic density. Although the spatial beam shifting and splitting have been reported in previous works [41], current atomic systems have some advantages: (1) large beam shift and splitting can be achieved due to enhanced Kerr nonlinearity induced by atomic coherence; (2) the “dispersion” curve for the beam displacement has been measured for the probe beam and matched to the calculated crossKerr nonlinear index; (3) displacements and splitting of FWM signal beams are experimentally demonstrated, which have never been done before; (4) speciallydesigned spatial beam conﬁguration was used to achieve the unique phasematching conditions for FWM processes, and for the beam shiftings and splittings at the same time; (5) current multilevel systems have much better experimental controls with additional laser beams; (6) such studies can have important applications in the spatial image storage, spatial entanglement, and spatial quantum correlation of laser beams. 1.6 Formations and Dynamics of Novel Spatial Solitons A spatial soliton can be formed when the diﬀraction of a laser beam is compensated by selffocusing or crossKerr eﬀects in a Kerr nonlinear medium [42, 43]. In recent years, many new spatial soliton eﬀects, such as discrete solitons [44, 45], gap solitons [46], surface gap solitons [47, 48], and vortex solitons [49], have been investigated (both theoretically and experimentally) in waveguide arrays [48], ﬁber Bragg gratings [50], BoseEinstein condensates [51], and photorefractive crystals [44, 45]. In achieving such interesting spatial eﬀects, large refractive index modulations are needed by either ﬁxed periodic structures (such as waveguide arrays and ﬁber Bragg grating) or reconﬁgurable optical lattices by laser beams as in the photorefractive crystals [46]. Gap soliton exists in band gaps of the linear spectra in various structures, and the forward and backwardpropagating waves both experience Bragg scattering and form of the periodic structure, which are coupled nonlinearly [42]. A vortex soliton appears as the selftrapping of a phase singularity and from which a screwtype phase distribution is generated where the real and imaginary parts of the ﬁeld amplitude are zero. Spatially modulated vortex solitons (azimuthons) have been theoretically considered in selffocusing nonlinear media [49]. Transverse energy ﬂow occurs between the intensity peaks (solitons) associated with the phase structure, which is a staircaselike nonlinear function described by the factor exp(imϕ), where ϕ is the azimuthal coordinate and the integer number m is deﬁned as the topological charge. If a phase mask is used to introduce certain phase delay for half of the soliton beam, the soliton can split into two parts with opposite (π) phases between them, called dipolemode vector soliton with a HermiteGaussian mode structure [52]. The dipolemode vector soliton is a vector soliton originated from trapping of a dipolemode beam. In an opticallyinduced twodimensional 16 1 Introduction photonic lattice, dipolemode solitons can be created with either opposite phases or same phase between the two parts [53]. Vector solitons with one nodeless fundamental component and another dipolemode component can couple to each other and be trapped jointly in the photonic lattices [52, 54]. A radially symmetric vortexmode soliton can decay into a radially asymmetric dipolemode soliton that has a nonzero angular momentum, which can survive for a very long propagation distance [52]. Spatial multicomponent soliton has vectorial interaction, mutually selftrapping in a nonlinear medium, and their total intensity proﬁle exhibits multiple humps [55]. Spatial gap solitons, dipole mode spatial solitons, and modulated vortex solitons of FWM in multilevel atomic systems are presented in Chapter 8 with details. For example, Section 8.3 shows the experimental observation of vortex solitons of FWM in the multilevel atomic media created by interference patterns with three or more superposition waves. The modulation eﬀect of vortex solitons is induced by the crossKerr nonlinear dispersion due to atomic coherence in the multilevel atomic system. These FWM vortex patterns are explained via the three, four and ﬁvewave interference topologies. The complex amplitude vectors can be overlaid at the observation plane and give rise to the total complex amplitude vector (CX , CY ) of interfering planewaves [13, 14]. The local structures of optical vortices are given by the polarization ellipse relation 2 2 2 [CX /(TX + TY2 )] sin2 (β + α) + [CY2 /(TX + TY2 )] cos2 (β + α) = 1, (1.9) where β = arctan(TX /TY ), and α is the ellipse orientation. The ellipse axes TX , TY are related to the spatial conﬁguration of laser beams (including the incident beam directions, phase diﬀerences between beams, etc.) and their intensities. Section 3.1 presents experimental results of generating gap soliton trains in FWM signals. Such novel spatial FWM gap soliton trains are induced in the periodically modulated selfdefocusing atomic medium by the crossphase modulation, which can be reshaped under diﬀerent experimental conditions, such as diﬀerent atomic densities, nonlinear dispersions, and dressing ﬁelds. Eﬀects due to the frequency detuning and intensity dependences of the refractive index are considered in addition to its onedimensional (axis ξ) periodic variation by using n(Δ, I, ξ) = n1 (Δ)+n2 (Δ)I+δn(ξ), where I is the dressing ﬁeld intensity. δn = n2 cos(2πξ/Λ) accounts for the periodic index variation inside the grating. The grating period is given by Λ = λ/θ, where θ is the angle between the two pump beams. Section 3.2 describes the formation of a novel type of stable multicomponent vector solitons consisting of two perpendicular FWM dipole components induced by XPM. The formation and steering of the steady dipole solitons and their dynamical (energy transfer) eﬀects have been analyzed. The dipolemode solitons of two FWM processes have horizontal and vertical orientations, respectively, which can coexist in the same atomic system, and their characteristics can be compared directly. In detail, we consider the incoherent superposition of two dipole components, u2 and u3 , as a generalization of a twocomponent dipolemodel 1.6 Formations and Dynamics of Novel Spatial Solitons 17 soliton {u1 , V }. This twocomponent come from a threecomponent solution {u1 , u2 , u3 }. The transformation of the dipole components is V → {u2 , u3 }, where u2 = V cos α and u3 = V sin α (α is a transformation parameter). Such a straightforward generalization is indeed possible for an Ncomponent system [55]. The gap, dipole and vortex solitons have all been observed before in photorefractive crystals [56 – 61]. However, the works presented in Chapter 8 are done in multilevel atomic systems, which have quite diﬀerent nonlinear properties compared to the photorefractive systems used before to observe such as gap, dipole and vortex solitons. As we have demonstrated, the multilevel atomic systems have wellcontrolled linear, as well as nonlinear, absorption and dispersion properties, which are essential in generating such interesting spatial gap, vortex and multicomponent dipole solitons in atomic meida. Without the enhanced Kerr nonlinearities due to atomic choherence [23], it will be hard to reach the needed index contrast for observing these novel spatial soliton phenomena. With several wellcontrolled experimental parameters, one can drive the Kerr medium to diﬀerent parameter regions to investigate richer spatial soliton phenomena (such as formation and dynamics), better explore parametric spaces, and compare with theoretical predictions. Observing such solitons and studying their dynamics in FWM is not a simple extension of previous results, but a signiﬁcant breakthrough to explore diﬀerent nonlinear regions and mechanism for forming such spatial dipole solitons and their evolutions. In solidstate materials, tenable parametric spaces are limited, so certain theoretically predicted phenomena are not reachable in the experiments. However, in multilevel atomic systems, the tenable region for parameters is broadened, which can be used to explore interesting phenomena, such as transition from one type of spatial soliton to another and energy transfer between diﬀerent dipole modes. Also, previous spatial solitons in solid materials were all done in the probe beam, not for FWM beams as in the multilevel atomic media, where Kerrnonlinear FWM processes are greatly enhanced and become more eﬃcient. The tenable parameters, such as atomic density, coupling/pumping ﬁeld intensities, and frequency detunings can be easily and independently controlled experimentally, which are important in reaching diﬀerent regions of the system. Due to the nature of induced atomic coherence in the system, the enhanced Kerr indices change dramatically with experimental parameters and can reach high values. Combining with the use of pulsed laser beams with high beam intensities, the refractive index contrast Δn = n2 I in the multilevel atomic system reaches the high value, so those interesting novel solitons can be observed. 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Opt Lett, 1998, 23: 1444 – 1446. 2 Ultrafast Polarization Beats of FourWave Mixing Processes Fourlevel diﬀerencefrequency polarization beat (FLPB) and attosecond sumfrequency polarization beat (ASPB) with broadband noisy light are investigated using chaotic ﬁeld, phasediﬀusion, and Gaussianamplitude models. The diﬀerencefrequency polarization beat signal is shown to be particularly sensitive to statistical properties of Markovian stochastic light ﬁelds with arbitrary bandwidth. Diﬀerent stochastic models of laser ﬁelds only aﬀect fourthorder coherence functions. The constant background of beat signal originates from the amplitude ﬂuctuation of Markovian stochastic ﬁelds. The Gaussianamplitude ﬁeld shows ﬂuctuations larger than the chaotic ﬁeld, which again exhibits ﬂuctuations much larger than those for the phasediﬀusion ﬁeld with pure phase ﬂuctuations caused by spontaneous emission. It has been also found that asymmetric behaviors of polarization beat signals due to the unbalanced dispersion eﬀects between two arms of interferometer and Dopplerwidth do not aﬀect the overall accuracy in case using FLPB to measure the energylevel diﬀerence between two states, which are dipolar forbidden from the ground state. On the other hand, a Dopplerfree precision in the measurement of the energylevel sum can be achieved with an arbitrary bandwidth. The advantage of ASPB is that the ultrafast modulation period 900 as can still be improved, because the energylevel interval between ground state and excited state can be widely separated. 2.1 Fourlevel Polarization Beats with Broadband Noisy Light Statistical properties of the broadband noisy (nontransform limited) light ﬁeld are of particular importance for nonlinear optical processes since these are often sensitive to higherorder correlations in the ﬁeld. The eﬀects of such correlations have been studied in several nonlinear processes characterized by either Markovian or nonMarkovian ﬂuctuations [1 – 5]. The Markovian ﬁeld is now described statistically in terms of marginal and conditional probability densities [6, 7]. The atomic response to nonMarkovian ﬁelds is much less well understood [4]. This is primarily because the complete hierarchy of conY. Zhang et al., Coherent Control of FourWave Mixing © Higher Education Press, Beijing and SpringerVerlag Berlin Heidelberg 2011 24 2 Ultrafast Polarization Beats of FourWave Mixing Processes ditional probabilities must be known in order to describe a nonMarkovian process. Some nonMarkovian processes can be made Markovian by extension to higher dimensions. The atomic response to Markovian stochastic optical ﬁelds is now largely well understood [1 – 3, 5]. When the laser ﬁeld is suﬃciently intense that many photon interactions occur, the laser spectral bandwidth or spectral shape, obtained from the secondorder correlation function, is inadequate to characterize the ﬁeld. Rather than using higherorder correlation functions explicitly, three diﬀerent Markovian ﬁelds are considered: (a) the chaotic ﬁeld, (b) the phasediﬀusion ﬁeld, and (c) the Gaussianamplitude ﬁeld. The chaotic ﬁeld undergoes both amplitude and phase ﬂuctuations and corresponds to a multimode laser ﬁeld with a large number of uncorrelated modes, or a singlemode laser emitting light below threshold. Since a chaotic ﬁeld does not possess any intensity stabilization mechanism, the ﬁeld can take on any value in a twodimensional region of the complex plane centered about the origin. The phasediﬀusion ﬁeld undergoes only phase ﬂuctuations and corresponds to an intensitystabilized singlemode laser ﬁeld. The phase of the laser ﬁeld, however, has no natural stabilizing mechanism [5]. The Gaussianamplitude ﬁeld undergoes only amplitude ﬂuctuations. Although pure amplitude ﬂuctuations cannot be produced by a nonadiabatic process, we do consider the Gaussianamplitude ﬁeld for two reasons. First, it allows us to isolate those eﬀects due solely to amplitude ﬂuctuations; and second, it is an example of a ﬁeld which undergoes stronger amplitude (intensity) ﬂuctuations than a chaotic ﬁeld. By comparing the results for the chaotic ﬁeld and the Gaussianamplitude ﬁeld, we can determine the eﬀect of increasing amplitude ﬂuctuations [6, 7]. The chaotic ﬁeld, the Brownianmotion phasediﬀusion ﬁeld, and the Gaussianamplitude ﬁeld are considered in parallel with a discussion on fourlevel atom transitions. We develop a uniﬁed theory which involves fourthorder coherencefunction to study the inﬂuence of partialcoherence properties and unbalance dispersion eﬀects of pump beams on polarization beats. Polarization beats, which originate from the interference between the macroscopic polarizations, have attracted a lot of attention recently [8 – 16]. It is closely related to quantum beat spectroscopy. DeBeer et al. performed the ﬁrst ultrafast modulation spectroscopy (UMS) experiment in sodium vapor [17]. Fu et al. [18] then analyzed the UMS with phaseconjugate geometry in a Dopplerbroadened system by a secondorder coherencefunction theory. They found that a Dopplerfree precision in the measurement of the energylevel splitting could be achieved. In this section, we investigated the eﬀects of Markovian ﬁeld ﬂuctuations in fourlevel polarization beats. Based on three types of models described above, we studied the inﬂuence of various quantities, such as light statistics, laser linewidth, Doppler width, and unbalance dispersion. One of relevant problems is the stationary fourwave mixing (FWM) with broadband noisy light, which was proposed by Morita et al. [19], to achieve an ultrafast temporal resolution of relaxation processes. Since they assumed that laser linewidth 2.1 Fourlevel Polarization Beats with Broadband Noisy Light 25 is much longer than transverse relaxation rate, their theory cannot be used to study the eﬀect of the light bandwidth on the Bragg reﬂection signal. Asaka et al. [20] considered the ﬁnite linewidth eﬀect. However, the constant background contribution has been ignored in their analysis. Our higherorder correlation on polarization beats includes the ﬁnite light bandwidth eﬀect, constant background contribution, and controllable dispersion eﬀects [21]. Diﬀerent roles of the phase ﬂuctuation and amplitude ﬂuctuation have been pointed out in the time domain. If the FLPB is employed for the energylevel diﬀerence measurement, there are advantages that the energylevel diﬀerence between two states which are dipolar forbidden from the ground state can be widely separated and a Dopplerfree precision in the measurement can be achieved. the FLPB is closely related to the Dopplerfree twophoton absorption spectroscopy with a resonant intermediate state and the sumfrequency trilevel photonecho when the pump beams are narrow band and broadband linewidth, respectively [16]. However, it possesses the main advantages of these techniques in the frequency domain and in the time domain. 2.1.1 Basic Theory The FLPB is a polarization beat phenomenon originating from the interference between two twophoton processes. Let us consider a fourlevel system (Fig. 2.1) with a ground state 0, an intermediate state 1 and two excited Fig. 2.1. Fourlevel conﬁguration to be treated by FLPB. states 2 and 3. States between 0 and 1 and between 1 and 2(3) are coupled by dipolar transition with resonant frequencies Ω1 and Ω2 (Ω3 ), respectively, while states between 2 and 3 and between 0 and 2(3) are dipolar forbidden. We consider in this fourlevel system a doublefrequency timedelay FWM experiment in which the beams 2 and 3 consist of two frequency components ω2 and ω3 , while beam 1 has frequency ω1 (Fig. 2.2). We assume that ω1 ≈ Ω1 and ω2 ≈ Ω2 (ω3 ≈ Ω3 ), therefore ω1 and ω2 (ω3 ) will drive the transitions from 0 to 1 and from 1 to 2(3), respectively. In this doublefrequency timedelay FWM, the beam 1 with frequency ω1 and the ω2 (ω3 ) frequency component of the beam 2 induce coherence between 0 and 2 (3) by twophoton transition, the which is then probed by the ω2 (ω3 ) frequency component of the beam 3. These are twophoton FWM with a resonant intermediate state and the frequency of the signal (beam 4) 26 2 Ultrafast Polarization Beats of FourWave Mixing Processes equals ω1 . Fig. 2.2. Schematic diagram of the geometry of FLPB. The complex electric ﬁelds of the beam 2, Ep2 (r, t), and the beam 3, Ep3 (r, t), can be written as Ep2 (r, t) = A2 (r, t) exp(−iω2 t) + A3 (r, t) exp(−iω3 t) = ε2 u2 (t) exp[i(k2 · r − ω2 t)] + ε3 u3 (t) exp[i(k3 · r − ω3 t)], (2.1) EP 3 (r, t) = A2 (r, t) exp(−iω2 t) + A3 (r, t) exp(−iω3 t) = ε2 u2 (t − τ ) exp[i(k2 · r − ω2 t + ω2 τ )] + ε3 u3 (t − τ + δτ ) exp[i(k3 · r − ω3 t + ω3 τ − ω3 δτ )]. (2.2) Here, εi , ki (εi , ki ) are the constant ﬁeld amplitude and the wave vector of ωi component in the beam 2 (beam 3), respectively. ui (t) is a dimensionless statistical factor that contains phase and amplitude ﬂuctuations. δτ denotes the diﬀerence in the zero time delay (δτ > 0). We assume that the ω2 (ω3 ) component of Ep2 (r, t) and Ep3 (r, t) comes from a single laser source, and τ is the time delay of the beam 3 with respect to the beam 2. On the other hand, the beam 1 is assumed to be a quasimonochromatic light, the complex electric ﬁelds of beam 1 can be written as EP1 (r, t) = A1 (r, t) exp(−iω1 t) = ε1 u1 (t) exp[i(k1 · r − ω1 t)]. (2.3) Here, u1 (t) ≈ 1, ε1 and k1 are the ﬁeld amplitude and the wave vector of the ﬁeld, respectively. We employ perturbation theory to calculate the density matrix elements. In the following perturbation chains: (0) ω (1) ω (2) −ω (3) (0) ω (1) ω (2) −ω (3) 1 2 2 ρ10 −→ ρ20 −−−→ ρ10 (I) ρ00 −→ 1 3 3 (II) ρ00 −→ ρ10 −→ ρ30 −−−→ ρ10 (2.4) Chains (I) and (II) (2.4) correspond to the processes with twophoton transitions from 0 to 2 and from 0 to 3, respectively. We obtain the (3) thirdorder oﬀdiagonal density matrix element ρ10 which has wave vector (3) (I) (II) k2 − k2 + k1 or k3 − k3 + k1 , ρ = ρ + ρ . Here gR and ρ(II) corresponding 2.1 Fourlevel Polarization Beats with Broadband Noisy Light 27 (3) to ρ10 of the perturbation chains (I) and (II), respectively, are ρ (I) +∞ ∞ ∞ iμ1 μ22 = − 3 exp(−iω1 t) dvw(v) dt3 dt2 × −∞ 0 0 ∞ dt1 H1 (t1 )H2 (t2 )H1 (t3 ) × 0 ρ(II) A1 (t − t1 − t2 − t3 )A2 (t − t2 − t3 )[A2 (t − t3 − τ )]∗ , ∞ +∞ ∞ iμ1 μ2 = − 3 3 exp(−iω1 t) dvw(v) dt3 dt2 × −∞ 0 0 ∞ dt1 H1 (t1 )H3 (t2 )H1 (t3 ) × (2.5) A1 (t − t1 − t2 − t3 )A3 (t − t2 − t3 )[A3 (t − t3 − τ )]∗ . (2.6) 0 Here, H1 (t) = exp[−(Γ10 + iΔ1 )t], H2 (t) = exp[−(Γ20 + iΔ1 + iΔ2 )t], H3 (t) = exp[−(Γ30 + iΔ1 + iΔ3 )t]; μ1 and μ2 (μ3 ) are dipole moment matrix elements between 0 and 1 and between 1 and 2(3), respectively; Δ1 = Ω1 − ω1 , Δ2 = Ω2 − ω2 , Δ3 = Ω3 − ω3 ; Γ10 and Γ20 (Γ30 ) are transverse relaxation rates of the coherence between states 0 and 1 and between 0 and 2(3), respectively. The nonlinear polarization P (3) responsible for the phaseconjugate FWM signal is given by averaging over the velocity distribution function w(v), i.e., P (3) = N μ1 +∞ −∞ (3) dvw(v)ρ10 (v). Here, v is the atomic velocity, N is the density of atoms. For a Dopplerbroadened atomic system, we have 1 w(v) = √ exp[−(v/u)2 ]. πu Thus the total polarization is P (3) = P (I) + P (II) . Here P (I) and P (II) corresponding to polarizations of the perturbation chains (I) and (II), respectively, are ∞ +∞ ∞ dvw(v) dt3 dt2 × P (I) = S1 (r) exp[−i(ω1 t + ω2 τ )] −∞ 0 0 ∞ dt1 exp[−iθI (v)] × H1 (t1 )H2 (t2 )H1 (t3 ) × 0 u1 (t − t1 − t2 − t3 )u2 (t − t2 − t3 )u∗2 (t − t3 − τ ), (2.7) 28 2 Ultrafast Polarization Beats of FourWave Mixing Processes ∞ +∞ ∞ P (II) = S2 (r) exp[−i(ω1 t + ω3 τ − ω3 δτ )] dvw(v) dt3 dt2 × −∞ 0 0 ∞ dt1 exp[−iθII (v)] × H1 (t1 )H3 (t2 )H1 (t3 )u1 (t − t1 − t2 − t3 ) × 0 u3 (t − t2 − t3 )u∗3 (t − t3 − τ + δτ ). (2.8) Here, iN μ21 μ22 ε1 ε2 (ε2 )∗ exp[i(k1 + k2 − k2 ) · r], 3 iN μ21 μ23 ε1 ε3 (ε3 )∗ exp[i(k1 + k3 − k3 ) · r]; S2 (r) = − 3 θI (v) = v · [k1 (t1 + t2 + t3 ) + k2 (t2 + t3 ) − k2 t3 ], θII (v) = v · [k1 (t1 + t2 + t3 ) + k3 (t2 + t3 ) − k3 t3 ]. S1 (r) = − The FWM signal is proportional to average of the absolute square of P (3) over the random variable of the stochastic process P (3) 2 , which involves fourth and secondorder coherence functions of ui (t) in phase– conjugation geometry. While the FWM signal intensity in Debeer’s selfdiﬀraction geometry is related to the sixthorder coherence functions of incident ﬁelds. We ﬁrst assume that the beam 2 (beam 3) is a multimode thermal source. ui (t) has Gaussian statistics with its fourthorder coherence function satisfying [6, 7] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 ) + ui (t1 )u∗i (t4 )ui (t2 )u∗i (t3 ). (2.9) Furthermore, assuming that beam 2 (beam 3) has Lorentzian line shape, then we have ui (t1 )u∗i (t2 ) = exp(−αi t1 − t2 ) (2.10) 1 here αi = δωi with δωi the linewidth of the laser with frequency ωi . The 2 form of the secondorder coherence function, which is determined by the laser line shape, as expressed in Eq. (2.10), is general feature of the three diﬀerent stochastic models [6, 7]. We ﬁrst consider the case that the beams 2 and 3 are a narrow band so that α2 , α3 << Γ10 , Γ20 , Γ30 and Γ20 τ , Γ30 τ  >> 1. Performing the tedious integration, the beat signal intensity then becomes I(τ, r) ∝ P (3) 2 = B1 + η2 B2 + B3 2 exp(−2α2 τ ) + ηB4 2 exp(−2α3 τ − δτ ) + exp(−α2 τ  − α3 τ − δτ ) × {ηB3∗ B4 exp[−iΔk · r − i(ω3 − ω2 )τ + iω3 δτ ]} + exp(−α2 τ  − α3 τ − δτ ) × {η ∗ B3 B4∗ exp[iΔk · r + i(ω3 − ω2 )τ − iω3 δτ ]}. (2.11) 2.1 Fourlevel Polarization Beats with Broadband Noisy Light 29 where Δk = (k2 −k 2 ) − (k3 −k 3 ), Γ210 + Δ21 Γ10 + 2α2 − , B1 = B3 Γ10 [Γ20 − i(Δ1 + Δ2 )] 2Γ10 Γ20 (Γ10 + Γ20 − iΔ2 ) Γ10 + 2α3 Γ210 + Δ21 − Γ10 [Γ30 − i(Δ1 + Δ3 )] 2Γ10 Γ30 (Γ10 + Γ30 − iΔ3 ) 1 B3 = , Γ20 + iΔ1 + iΔ2 1 B4 = , Γ30 + iΔ1 + iΔ3 u2 ε3 (ε3 )∗ . η = 32 u2 ε2 (ε2 )∗ B2 = B4 , Relation (2.10) consists of ﬁve terms. The ﬁrst and third terms, which is the autocorrelation intensity for twophoton transition from 0 to 2, are dependent on the u2 (t) fourthorder coherence function, while the second and fourth terms, which is the autocorrelation intensity for twophoton transition from 0 to 3, are dependent on the u3 (t) fourthcoherence function. The ﬁrst and second terms originating from the amplitude ﬂuctuation of the chaotic ﬁeld are independent of the relative timedelay between the beams 2 and 3. The third and fourth terms indicate an exponential decay of the beat signal as τ  increases. The ﬁfth term depending on u2 (t) and u3 (t) secondorder coherence functions, which is determined by the laser line shape, gives rise to the modulation of the beat signal. Equation (2.11) indicates that beat signal oscillates not only temporally but also spatially with a period 2π/Δk along the direction Δk, which is almost perpendicular to the propagation direction of the beat signal. Here Δk ≈ 2πλ2 − λ3 θ/λ3 λ2 , θ is the angle between the beam 2 and beam 3. Physically, the polarizationbeat model assumes that both the pump beams are plane waves. Therefore two twophoton FWM signals, which propagate along ks1 = k2 − k 2 + k1 and ks2 = k3 − k3 + k1 , respectively, are plane waves also. Since two twophoton FWM propagate along slightly diﬀerent direction, the interference between them leads to the spatial oscillation. Equation (2.11) also indicates that beat signal modulates temporally with a frequency ω3 −ω2 as τ is varied. In this case that ω2 and ω3 are tuned to the resonant frequencies of the transitions from 1 to 2 and from 1 and 3, respectively, then the modulation frequency equals Ω3 − Ω2 . In other words, we can obtain beating between the resonant frequencies of a fourlevel system. A Dopplerfree precision can be achieved in the measurement of Ω3 − Ω2 . We then consider the case that the beams 2 and 3 are broadband, i.e., α2 , α3 >> Γ10 , Γ20 , Γ30 . (i) τ > δτ , the beat signal rises to its maximum quickly and then decays with time constant mainly determined by the transverse relaxation times of 30 2 Ultrafast Polarization Beats of FourWave Mixing Processes the system. Although the beat signal modulation is complicated in general, at the tail of the signal (i.e., α2 τ  >> 1, α3 τ  >> 1) we have I(τ, r) ∝ P (3) 2 = B5 + η2 B6 + B7 2 exp(−2Γ20 τ ) + ηB8 2 exp(−2Γ30 τ − δτ ) + B7 B8 exp(−Γ20 τ  − Γ30 τ − δτ ){η exp[−iΔk · r − i(Ω3 − Ω2 )τ + iΩ3 δτ ] + η ∗ exp[iΔk · r + i(Ω3 − Ω2 )τ − iΩ3 δτ ]}, (2.12) where Γ210 + Δ21 2Γ10 1 2α22 + iα2 Δ1 − iΓ20 Δ2 − × α2 Γ20 (2α2 + iΔ1 )[α22 + (Δ1 + Δ2 )2 ] α2 − i(Δ1 + Δ2 ) 1 Γ10 + Γ20 + iΔ2 − , (2α2 − iΔ1 )[α2 − i(Δ1 + Δ2 )] Γ20 [(Γ10 + iΔ2 )2 − α22 ] 2α23 + iα3 Δ1 1 Γ2 + Δ21 − × B6 = 10 2Γ10 α3 Γ30 (2α3 + iΔ1 )[α23 + (Δ1 + Δ3 )2 ] α3 − i(Δ1 + Δ3 ) 1 Γ10 + Γ30 + iΔ3 − , (2α3 − iΔ1 )[α3 − i(Δ1 + Δ3 )] Γ30 [(Γ10 + iΔ3 )2 − α23 ] 2α2 2α3 B7 = 2 , B8 = 2 . 2 α2 + (Δ1 + Δ2 ) α3 + (Δ1 + Δ3 )2 B5 = Relation (2.12) also consists of ﬁve terms. The ﬁrst and third terms for twophoton transition from 0 to 2 are dependent on the u2 (t) fourthorder coherence function, while the second and fourth terms for twophoton transition from 0 to 3 are dependent on the u3 (t) fourthcoherence function. The third and fourth terms indicate an exponential decay of the beat signal as τ  increases. The ﬁfth term depending on the u2 (t) and u3 (t) secondorder coherence functions, which is determined by the laser line shape, gives rise to the modulation of the beat signal. Equation (2.12) indicates that the temporal modulation frequency of the beat signal equals Ω3 − Ω2 when δτ = 0. The overall accuracy of using FLPB with broadband lights to measure the energylevel diﬀerence between two excited states is limited by the homogeneous linewidths [13]. (ii) 0 < τ < δτ, α2 τ  >> 1 I(τ, r) ∝ P (3) 2 = B5 + η2 B6 + B7 2 exp(−2Γ20τ ) + ηB9 2 exp(−2α3 τ − δτ ) + B7 exp(−Γ20 τ  − α3 τ − δτ ){ηB9 exp[−iΔk · r − i(ω3 − ω2 )τ + iω3 δτ − i(Δ1 + Δ2 )τ ] + η∗ B9∗ exp[iΔk · r + i(ω3 − ω2 )τ − iω3 δτ + i(Δ1 + Δ2 )τ ]}, where B9 = 1 . α3 − i(Δ1 + Δ2 ) (2.13) 2.1 Fourlevel Polarization Beats with Broadband Noisy Light 31 (iii) τ < 0 I(τ, r) ∝ P (3) 2 = B5 + η2 B6 + B10 2 exp(−2α2 τ ) + ηB11 2 exp(−2α3 τ − δτ ) + ∗ exp(−α2 τ  − α3 τ − δτ ){ηB10 B11 exp[−iΔk · r − i(ω3 − ω2 )τ + ∗ exp[iΔk · r + i(ω3 − ω2 )τ − iω3 δτ ]}, iω3 δτ ] + η ∗ B10 B11 (2.14) 1 1 , B11 = . For simplicity, here α2 + i(Δ1 + Δ2 ) α3 + i(Δ1 + Δ3 ) we neglect the Doppler eﬀect only in ﬁnal expressions B1 to B11 . This equation is consistent with Eq. (2.11). Therefore, the requirement for the existence of a τ dependent beat signal for τ < 0 is that the phasecorrelated subpulses in the beams 2 and 3 are overlapped temporally. Since the beams 2 and 3 are mutually coherent, the temporal behavior of the beat signal should coincide with the case when the beams 2 and 3 are nearly monochromatic [13, 18]. where B10 = Fig. 2.3. The beat signal intensity versus relative time delay. The parameters are Ω3 − Ω2 = 254 ps−1 , Ω3 = 3317 ps−1 , Δk = 0, η = 1, Bi = 0.6, Γ20 = 12.5 ps−1 , Γ30 = 14.5 ps−1 ; while δτ = 0 fs for dotted line, δτ = 43 fs for dashed line and δτ = 100 fs for solid line. Adopted from Ref. [22]. Figure 2.3 shows the interferograms of the beat signal intensity versus relative time delay for three diﬀerent values of the reduced oﬀset imbalance δτ, and the parameters are Ω3 − Ω2 = 254 ps−1 , Ω3 = 3317 ps−1 , Δk = 0, η = 1, Bi = 0.6, Γ20 = 12.5 ps−1 , Γ30 = 14.5 ps−1 ; while δτ = 0 fs for dotted line, δτ = 43 fs for dashed line and δτ = 100 fs for solid line. It is noticed that as δτ increases, the peaktobackground contrast ratio of the interferograms diminishes, as anticipated. Interestingly, the phase of the fringe beating also changes sensitively to produce a variety of interferograms including asymmetric ones. δτ expresses the unbalance dispersion eﬀects between the two arms. A simple realistic example is an interferometer having an eﬀective thickness of quartz or glass that diﬀers signiﬁcantly (many mm to a few cm) between its 32 2 Ultrafast Polarization Beats of FourWave Mixing Processes two arms. Changing the thickness in one arm will control the degree of imbalance in the dispersion eﬀects [21]. Physically, δτ corresponds to the separation of the peaks of the third and fourth terms of Eq. (2.11), i.e., the separation between the ω2 only interferogram and the ω3 only interferogram. Furthermore the ﬂuctuations in δτ require phasedependent ﬂuctuations (otherwise δτ cannot change), which may be due to, for example, unbalance amplitude (thermal) ﬂuctuations in air or the optics between the two arms of the Michelson interferometer. 2.1.2 FLPB in a Dopplerbroadened System The beat signal can be calculated from a diﬀerent viewpoint. Under the Dopplerbroadened limit (i.e., k1 u → ∞), we have √ +∞ 2 π dvw(v) exp[−iθI (v)] ≈ (2.15) δ(t1 + t2 + t3 − ξ1 t2 ), k1 u −∞ √ +∞ 2 π δ(t1 + t2 + t3 − ξ2 t2 ). dvw(v) exp[−iθII (v)] ≈ (2.16) k1 u −∞ Here, ξ1 = k2 /k1 , ξ2 = k3 /k1 . We assume ξ1 > 1, ξ2 > 1. When we substitute Eqs. (2.15) and (2.16) into Eqs. (2.7) and (2.8) we obtain I(τ, r) ∝ P (3) 2 = P (I) + P (II) 2 . We ﬁrst consider the case that the beams 2 and 3 are narrow band so that α2 , α3 << Γ20 , Γ30 and Γ20 τ , Γ30 τ  >> 1. Performing the tedious integration, the beat signal intensity is I(τ, r) ∝ P (3) 2 ∝ B5 + η2 B6 + B12 2 exp(−2α2 τ ) + ηB13 2 exp(−2α3 τ − δτ ) + ∗ B13 exp[−iΔk · r − i(ω3 − ω2 )τ + exp(−α2 τ  − α3 τ − δτ )×{ηB12 ∗ ∗ iω3 δτ ] + η B12 B13 exp[iΔk · r + i(ω3 − ω2 )τ − iω3 δτ ]}, (2.17) (ξ1 − 1)(Γ210 + Δ21 )2 (ξ2 − 1)(Γ210 + Δ21 )2 , B13 = a ; Γa = Γ20 + a a 2 2 (Γ20 − Γ10 ) + (Δ2 ) (Γ30 − Γ10 )2 + (Δa3 )2 20 ξ1 Γ10 , Γa30 = Γ30 + ξ2 Γ10 , Δa2 = Δ2 + ξ1 Δ1 , Δa3 = Δ3 + ξ2 Δ1 . This equation is consistent with Eq. (2.11). We now consider the case that beams 2 and 3 are broadband so that α2 , α3 >> Γ10 , Γ20 , Γ30 . (i) τ > δτ, α2 τ  >> 1, α3 τ  >> 1 where B12 = I(τ, r) ∝ P (3) 2 = B14 + η2 B15 + B16 2 exp[−2(Γa20 − Γ10 )τ ] + ηB17 2 exp[−2 × (Γa30 − Γ10 )τ − δτ ] + B16 B17 exp[−(Γa20 − Γ10 )τ  − (Γa30 − Γ10 )τ − 2.1 Fourlevel Polarization Beats with Broadband Noisy Light 33 δτ ]{η exp[−iΔk · r − i(Ω3 − Ω2 )τ − i(ξ2 − ξ1 )Δ1 τ + iξ2 Δ1 δτ ] + η ∗ exp[iΔk · r + i(Ω3 − Ω2 )τ + i(ξ2 − ξ1 )Δ1 τ − iξ2 Δ1 δτ ]}, (2.18) where (ξ1 − 1)[α22 + (Δa2 )2 − 2iα2 Δa2 ] , 2(Γa20 − Γ10 )2 [α22 + (Δa2 )2 ] (ξ2 − 1)[α23 + (Δa3 )2 − 2iα3 Δa3 ] , = 2(Γa30 − Γ10 )2 [α23 + (Δa3 )2 ] 2(ξ1 − 1)α2 τ = 2 , α2 + (Δa2 )2 2(ξ2 − 1)α3 (τ − δτ ) = . α23 + (Δa3 )2 B14 = B15 B16 B17 Equation (2.18) indicates that the temporal modulation frequency of the beat signal equals Ω3 − Ω2 when Δ1 = δτ = 0. The overall accuracy of using FLPB with broadband lights to measure the energylevel diﬀerence between two excited states is limited by the homogeneous linewidth. This equation is analogous to Eq. (2.11). (ii) τ < 0 I(τ, r) ∝ P (3) 2 = B14 + η2 B15 + B18 2 exp(−2α2 τ ) + ηB19 2 exp(−2α3 τ − δτ ) + exp(−α2 τ  − ∗ α3 τ − δτ ){ηB18 B19 exp[−iΔk · r − i(ω3 − ω2 )τ + iω3 δτ ] + ∗ exp[iΔk · r + i(ω3 − ω2 )τ − iω3 δτ ]}, η ∗ B18 B19 (2.19) where ξ1 − 1 , (α2 − iΔa2 )2 ξ2 − 1 = . (α3 − iΔa3 )2 B18 = B19 This equation is consistent with Eq. (2.10). (iii) 0 < τ < δτ and α2 τ  >> 1 I(τ, r) ∝ P (3) 2 = B14 + η2 B15 + B16 2 exp[−2(Γa20 − Γ10 )τ ] + ηB19 2 × exp(−2α3 τ − δτ ) + B16 exp[−2(Γa20 − Γ10 )τ  − α3 τ − δτ ]{ηB19 exp[−iΔk · r − i(ω3 − ω2 )τ + iω3 δτ − ∗ i(Δ1 + Δ2 )τ ] + η∗ B19 exp[iΔk · r + i(ω3 − ω2 )τ − iω3 δτ + i(Δ1 + Δ2 )τ ]}. This equation is analogous to Eq. (2.12). (2.20) 34 2 Ultrafast Polarization Beats of FourWave Mixing Processes 2.1.3 Photonecho It is interesting to understand the underlying physics in FLPB with broadband nontransform limited quasicw (noisy) lights [19, 20]. Much attention has been paid to the study of various ultrafast phenomena by using incoherent light sources recently [21 – 24]. For the phase matching condition k2 − k2 + k1 and k3 − k3 + k1 two sumfrequency trilevel echoes exist for the perturbation the chains (I) and (II), respectively. The chaotic ﬁeld is a complex Gaussian stochastic process. Under the Dopplerbroadened limit (i.e., k1 u → ∞), If assuming that the beams 2 and 3 are broadband so that α2 , α3 >> Γ20 , Γ30 , then we have ui (t1 )u∗i (t2 ) = exp(−αi t1 − t2 ) ≈ 2 δ(t1 − t2 ). αi (2.21) When we substitute Eqs. (2.8) and (2.21) into Eq. (2.16), we obtain as follows: (i) τ > δτ , I(τ, r) ∝ P (3) 2 = A1 + η2 A2 + A3 2 exp[−2(Γa20 − Γ10 )τ ] + ηA4 2 exp[−2 × (Γa30 − Γ10 )τ − δτ ] + A3 A4 exp[−(Γa20 − Γ10 )τ  − (Γa30 − Γ10 )τ − δτ ]{η exp[−iΔk · r − i(Ω3 − Ω2 )τ − i(ξ2 − ξ1 )Δ1 τ + iξ2 Δ1 δτ ] + η ∗ exp[iΔk · r + i(Ω3 − Ω2 )τ + i(ξ2 − ξ1 )Δ1 τ − iξ2 Δ1 δτ ]}, (2.22) where A1 = ξ1 − 1 4[α2 (Γa20 − Γ10 )]2 A2 = ξ2 − 1 4[α3 (Γa30 − Γ10 )]2 A3 = (ξ1 − 1)τ , α2 A4 = (ξ2 − 1)(τ − δτ ) . α3 This equation is consistent with Eq. (2.18). (ii) 0 < τ < δτ I(τ, r) ∝ P (3) 2 = A1 + η2 A2 + A3 2 exp[−2(Γa20 − Γ10 )τ ]. Photonecho only exists for the perturbation the chain (I). (iii) τ < 0 I(τ, r) ∝ P (3) 2 = A1 + η2 A2 . (2.23) 2.1 Fourlevel Polarization Beats with Broadband Noisy Light 35 In this case, photonecho doesn’t exist for the perturbation chains (I) and (II). The requirement for the existence of a τ dependent beat signal for τ < 0 is that the phasecorrelated subpulses in the beams 2 and 3 are overlapped temporally. Since beams 2 and 3 are mutually coherent, the temporal behavior of the beat signal should coincide with the case when the beams 2 and 3 are nearly monochromatic [13, 18]. Therefore, this case is analogous to Eq. (2.10). We have assumed that the laser sources are chaotic ﬁeld in the above calculation. A chaotic ﬁeld, which is used to describe a multimode laser source, is characterized by the ﬂuctuation of both the amplitude and the phase of the ﬁeld. Another commonly used stochastic model is a phasediﬀusion model, which is used to describe an amplitudestabilized laser source. This model assumes that the amplitude of the laser ﬁeld is a constant, while its phase ﬂuctuates as a random process. We substitute Eqs. (2.21) and (2.23) into Eq. (2.16), we obtain as follows: (i) τ > δτ I(τ, r) ∝ P (3) 2 = A3 2 exp[−2(Γa20 − Γ10 )τ ] + ηA4 2 exp[−2(Γa30 − Γ10 )τ − δτ ] + A3 A4 × exp[−(Γa20 − Γ10 )τ  − (Γa30 − Γ10 )τ − δτ ]{η exp[−iΔk · r − i(Ω3 − Ω2 )τ − i(ξ2 − ξ1 )Δ1 τ + iξ2 Δ1 δτ ] + η ∗ exp[iΔk · r + i(Ω3 − Ω2 )τ + i(ξ2 − ξ1 )Δ1 τ − iξ2 Δ1 δτ ]}. (2.24) (ii) 0 < τ < δτ I(τ, r) ∝ P (3) 2 = A3 2 exp[−2(Γa20 − Γ10 )τ ]. (2.25) Photonecho only exists for the perturbation the chain (I). (iii) τ < 0 I(τ, r) ∝ P (3) 2 = 0. (2.26) In this case, photonecho doesn’t exist for the perturbation the chains (I) and (II). Relation (2.24) consists of three terms. The ﬁrst term for twophoton transition from 0 to 2 is dependent on the u2 (t) fourthorder coherence function, while the second term for twophoton transition from 0 to 3 is dependent on the u3 (t) fourthorder coherence functions. The ﬁrst and second terms indicate an exponential decay of the beat signal as τ  increases. The third term depending on the u2 (t) and u3 (t) secondorder coherence functions, which is determined by the laser line shape, gives rise to the modulation of the beat signal. This case is consistent with results of the secondorder coherence function theory P (3) 2 [10, 18]. The constant background contribution has been ignored in their analysis. Therefore, the fourthorder coherence function theory P (3) 2 of chaotic ﬁeld is of vital importance in FLPB. 36 2 Ultrafast Polarization Beats of FourWave Mixing Processes The Gaussianamplitude ﬁeld has a constant phase but its real amplitude undergoes Gaussian ﬂuctuations. If the lasers have Lorentzian line shape, the fourthorder coherence function is [6, 7] ui (t1 )ui (t2 )ui (t3 )ui (t4 ) = ui (t1 )ui (t3 )ui (t2 )ui (t4 ) + ui (t1 )ui (t4 )ui (t2 )ui (t3 ) + ui (t1 )ui (t2 )ui (t3 )ui (t4 ). (2.27) When we substitute Eqs. (2.21) and (2.25) into Eq. (2.16) we obtain as follows: (i) τ > δτ I(τ, r) ∝ P (3) 2 = A1 + η2 A2 + A5 exp[−2(Γa20 − Γ10 )τ ] + ηA6 2 exp[−2(Γa30 − Γ10 ) × τ − δτ ] + A7 exp[−(Γa20 − Γ10 )τ  − (Γa30 − Γ10 )τ − δτ ]{η exp[−iΔk · r − i(Ω3 − Ω2 )τ − i(ξ2 − ξ1 )Δ1 τ + iξ2 Δ1 δτ ] + η ∗ exp[iΔk · r + i(Ω3 − Ω2 )τ + i(ξ2 − ξ1 )Δ1 τ − iξ2 Δ1 δτ ]}, (2.28) where 4(ξ1 − 1)2 τ 2 ξ1 − 1 + , α22 (α2 Δa2 )2 4(ξ2 − 1)2 (τ − δτ )2 ξ2 − 1 + , A6 = α23 (α3 Δa3 )2 (ξ1 − 1)(ξ2 − 1) τ (τ − δτ ). A7 = α2 α3 A5 = (ii) 0 < τ < δτ I(τ, r) ∝ P (3) 2 = A1 + η2 A2 + A5 exp[−2(Γa20 − Γ10 )τ ]. Photonecho only exists for the perturbation the chain (I). (iii) τ < 0 I(τ, r) ∝ P (3) 2 = A1 + η2 A2 . (2.29) (2.30) In this case, photonecho doesn’t exist for the perturbation chains (I) and (II). Relation (2.28) consists of ﬁve terms. The ﬁrst and third terms for twophoton transition from 0 to 2 are dependent on the u2 (t) fourthorder coherence function, while the second and fourth terms for twophoton transition from 0 to 3 are dependent on the u3 (t) fourthcoherence function. The ﬁrst and second terms originating from the amplitude ﬂuctuation of the Gaussianamplitude ﬁeld are independent of the relative timedelay between the beams 2 and 3. The third and fourth terms indicate an exponential decay of the beat signal as τ  increases. The ﬁfth term depending on the u2 (t) and u3 (t) secondorder coherence functions, which is determined by the laser line shape, gives 2.1 Fourlevel Polarization Beats with Broadband Noisy Light 37 rise to the modulation of the beat signal. Equation (2.26) also indicates that beat signal oscillates not only temporally with a period 2π/Ω3 − Ω2  = 25 fs but also spatially with a period 2π/Δk = 0.28 mm along the direction Δk, which is almost perpendicular to the propagation direction of the beat signal. The threedimensional interferogram of the beat signal intensity I(τ, r) versus time delay τ and transverse distance r has the larger constant background caused by the intensity ﬂuctuation of the Gaussianamplitude ﬁeld in Fig. 2.4 (a), (b), and the parameters are Ω3 − Ω2 = 254 ps−1 , Δk = 22.22 mm−1 , η = 1, ξi = 1.5, Δ1 = 0, Ai = 0.6, Γa20 − Γ10 = 12.5 ps−1 , Γa30 − Γ10 = 14.5 ps−1 ; while δτ = 0 fs for (a) and δτ = 43 fs for (b). At zero relative time delay (τ = 0), the twin beams originating from the sam