propagation in rubidium vapor
by
M.C. Yaneth Marcela Torres García
A dissertation submitted to the program in Optics, Optics Department
in partial fulfillment of the requirements for the degree of
DOCTOR IN OPTICS
at the
National Institute for Astrophysics, Optics and Electronics July 2015
Santa María Tonantzintla, Puebla
Advisor
Dr. Nikolai Korneev Zabello
Optics DepartmentINAOE
© INAOE 2015 All rights reserved
The author hereby grants to INAOE permission to reproduce and to distribute copies of this thesis document in whole or in part
The nonlinear propagation of light in alkali vapors is a well known phenomenon and has been used on multiple applications including magnetometry, slow-light, quantum applications and atomic clocks, among many others. In this thesis, the transverse optical pattern generation and nonlinearity enhance-ment by optical pumping in rubidium vapor are studied.
Optical switches are essential components in optical networks. In such net-works they are used for supplying lightpaths, protection switching and as the elements for high speed packet transmission in switched networks. In this the-sis, the switching of two optical patterns, generated for a laser beam tuned at the 87 Rubidium D2 line, is used as an optical memory mechanism. Two linearly polarized beams are counter-propagated within a rubidium vapor cell in the presence of a longitudinal magnetic field. Optical patterns composed by a set of well defined points are observed in the far field. Hysteresis due to time-varying magnetic field and intensity is observed for a bistable state, com-posed by two well defined optical patterns. Three mechanisms are used to pro-duce optical switching: Mechanical switching, intensity modulation of pump beam and intensity modulation of an additional beam. A rapid switching in a time below 10 microseconds is performed with a low-power additional beam of approximately 2 microwatts .
The self-rotation to absorption ratio, is a parameter used in squeezing ex-periments to measure the characteristic achievable nonlinear phase shift. This ratio is used in this work as a figure of merit in experiments of polarization plane self-rotation, for a signal beam tuned at the 87 Rb D2 transition line, with Fg = 1, and an additional pump beam tuned at the 87 Rb D2 transition line, with Fg = 1, Fe = 0 and Fg = 2, Fe = 2. It is found an enhancement of the self-rotation to absorption ratio in approximately 2 times its value with respect
to the case without additional pump beam.
An experiment of three-wave mixing was realized by intersecting two signal beams inside a rubidium vapor cell. Due to the effect of an additional pump beam, it is observed a strong amplification of approximately 100 times in a ge-nerated conjugated beam, and optical patterns derived from a break-up effect on the signal beam in the far field. The pump beam laser is tuned for this ex-periment, at the transition with Fg = 2, Fe= 2.
El fenómeno de propagación no lineal de la luz a través de los vapores alcali-nos es bien conocido y ha sido utilizado en diversas aplicaciones que incluyen magnetometría, luz lenta, experimentos en mecánica cuántica y relojes atómi-cos, entre otras. En esta tesis se estudia la generación de patrones ópticos transversales y el incremento en el valor de algunas propiedades no lineales en el vapor de rubidio.
La generación de patrones ópticos transversales se realiza para un haz láser sintonizado en la línea de transición D2 del 87Rb y es utilizado como un sistema de memoria óptica cuando dos haces se propagan en direcciones op-uestas dentro de una celda de vapor de rubidio. Adicionalmente, un campo magnético longitudinal es aplicado sobre la celda de rubidio. En la conmutación de dos patrones que componen un estado biestable del sistema se observa histéresis debida a la variación temporal del campo magnético longitudinal aplicado. La conmutación óptica se realiza utilizando tres mecanismos: Con-mutación mecánica, modulación de la intensidad del haz principal y modu-lación de la intensidad de un haz adicional de baja potencia. Se reporta una rápida conmutación de los patrones en un tiempo menor a 10 microsegundos para un haz adicional con una potencia de aproximadamente 2 microvatios.
El incremento de las propiedades no lineales en el vapor de rubidio se rea-liza utirea-lizando un haz de bombeo adicional sintonizado en la línea de transición D2 del87Rb con Fg = 1, Fe= 0 y Fg = 2, Fe= 2. En esta configuración se realizan experimentos de auto-rotación del plano de polarización para un haz señal es-caneado en Fg = 1, utilizando como figura de mérito la razón entre el valor de la auto-rotación y la absorción. Esta razón se incrementa aproximadamente dos veces respecto a su valor sin haz de bombeo para las dos transiciones estudiadas.
En un experimento de mezcla de tres ondas, se observa una amplificación mayor a 100 veces en el haz conjugado y la generación de patrones ópticos en el campo lejano, derivados del rompimiento del haz señal. Para este exper-imento, el haz de bombeo es sintonizado en la transición Fg = 2, Fe= 2.
2.1 Hyperfine structure of 87 Rb D2 transition line . . . 5
2.2 Some typical four-wave mixing processes . . . 13
2.3 Generation of phase conjugated waves by induced holographic gratings . . . 14
2.4 Two counter-propagating beams E1and E2interact within a non-linear medium . . . 19
2.5 Representation of polarization ellipse. . . 21
2.6 Nonlinear Faraday rotation . . . 24
2.7 Generated light emitted along cones . . . 26
2.8 Time-response of the pattern generated in an optical switching experiment . . . 27
2.9 Multi-wave interaction wave vectors geometry . . . 30
3.1 Experimental setup for pattern-based optical switching . . . 44
3.2 Absorption spectrum in the observational plane of an additional beam generated in an optical switching experiment . . . 46
3.3 Optical transverse patterns generated in the far field for an op-tical switching experiment . . . 47
3.4 Intensity dependence of the generated spot in an optical swit-ching experiment on the external magnetic field . . . 48
3.5 Examples of temporal behavior of signal for different generated patterns in an optical switching experiment . . . 50
3.6 Hysteresis due to longitudinal magnetic field . . . 52
3.7 Intensity of the generated spot when the magnetic field is applied and the Dove prism is used . . . 52
3.8 Mechanical switching results . . . 53
3.9 Hysteresis generated in the signal due to modulation intensity pump beam . . . 54 3.10 Hysteresis generated in the signal due to a low power additional
beam . . . 55
4.1 Experimental setup for nonlinearity enhancement in rubidium vapor by using additional pump beam . . . 61 4.2 Pump beam action over the signal absorption and nonlinearity
curves . . . 62 4.3 Effect on self-rotation curve when the pump beam is fixed at
di-fferent frequencies inside Fg = 2 transition line . . . 63 4.4 Self-rotation and rotation to absorption ratio as a function of
signal intensity without pump . . . 64 4.5 Rotation to absorption ratio as a function of signal intensity with
and without pump . . . 65 4.6 Absorption and self-rotation curves in counter-propagation for
different pump beam intensities . . . 66 4.7 Absorption and self-rotation curves in co-propagation for diff
e-rent pump beam intensities . . . 67 4.8 Comparative curves of diffraction efficiency for the pump beam
tuned at Fg = 2, Fe= 1 and Fe= 2 . . . 69 4.9 Diffraction efficiency curves in a conjugated beam without and
with pump beam . . . 71 4.10 Diffraction efficiency curves in a conjugated beam when signal
intensity is changed . . . 72 4.11 Diffraction efficiency curves in a conjugated beam when pump
intensity is changed . . . 73 4.12 Diffraction efficiency curves in a conjugated beam when signal
intensity is 1.16mW/mm2 . . . 74 4.13 Patterns generated in the far field for co- and counter-propagating
schemes due the gain in a three-wave mixing experiment . . . . 76 4.14 Absorption and self-rotation curves for a counter-propagating
pump beam tuned at Fg= 1 with polarization parallel to that of the signal . . . 78
4.15 Absorption and self-rotation curves for a counter-propagating pump beam tuned at Fg = 1 with polarization perpendicular to that of the signal . . . 79 4.16 R as a function of signal intensity for a counter-propagating pump
beam tuned at Fg = 1 with Fe= 0, 1 and 2 . . . 80 4.17 Self-rotation as a function of signal intensity for co- and
counter-propagating pump laser in the transition Fg = 1, Fe= 0. . . 81 4.18 Comparative curves of R as a function of signal intensity without
and with the pump beam in the transition Fg = 1, Fe= 0 . . . 82 4.19 Absorption and self-rotation curves for a co-propagating pump
beam in transition Fg = 1, Fe = 0 for different values of pump beam intensity . . . 83 4.20 Absorption and self-rotation curves for a counter-propagating
pump beam in transition Fg = 1, Fe = 0 for different values of pump beam intensity . . . 84
Quisiera agradecer a todas las personas e instituciones que de algún modo u otro, hicieron posible la consecución de esta tesis.
En primer lugar, quisiera agradecer al gobierno mexicano, por permitirme realizar mis estudios de doctorado en este país a través de la beca-apoyo número 324611 del Consejo Nacional de Ciencia y Tecnología (CONACYT). Esta investigación fue conducida en el marco del proyecto de investigación 156891 del CONACYT.
Al Doctor Nikolai Korneev, mi asesor de tesis, por su constante paciencia y entusiasmo durante la orientación de esta investigación. Su perseverancia y conocimiento son una fuente continua de admiración para mí.
A los miembros del jurado de esta tesis, por sus valiosos aportes y suge-rencias que enriquecieron este texto.
A los profesores del INAOE, a quienes he tenido la fortuna de conocer y de los cuales recibí su amistad, apoyo y orientación durante mi doctorado. De igual forma, quiero agradecer al personal del instituto, sin cuyo trabajo diario no sería posible el desarrollo de las actividades de este centro de inves-tigación.
A mi familia, en especial a mi madre, por su amor, apoyo y comprensión incondicional sin importar la distancia y el tiempo.
A la familia Ruelas Urías, mi familia adoptiva en México, por su hospitalidad y respaldo, quienes me han hecho sentir este hermoso país como un segundo hogar.
A Adrián, mi cómplice en toda esta aventura, quién me ha acompañado, apoyado y animado todo el tiempo, y a quién amo y admiro profundamente.
Finalmente, a tod@s las personas que he conocido, y con las que he com-partido mi estancia en este país.
Abstract i
Resumen iii
List of Figures v
Agradecimientos vii
1 Introduction 1
2 Nonlinear optics in rubidium vapor 4
2.1 Characteristics of rubidium vapor . . . 4
2.2 Optical pumping . . . 6
2.3 Nonlinear susceptibility . . . 7
2.4 Intensity-dependent refractive index . . . 10
2.5 Nonlinear optical processes in atomic gases . . . 11
2.5.1 Four-wave mixing . . . 13
2.5.2 Self-action effects . . . 15
2.5.2.1 Laser beam breakup and pattern formation . . . 15
2.5.2.2 Self-rotation of the polarization plane . . . 20
2.5.3 Nonlinear Faraday rotation . . . 23
2.5.4 Optical bistability and optical switching . . . 25
2.5.5 Optical wave propagation in volume gratings . . . 27
2.6 Theoretical description of alkali atoms . . . 32
2.6.1 Two-level approximation . . . 32
2.6.2 Multi-level density matrix formulation . . . 37
2.7 Concluding remarks . . . 42
3 Pattern-Based optical switching 43
3.1 Experimental setup and pattern generation . . . 44
3.2 Stable generation . . . 48
3.3 Bistability and optical switching . . . 51
3.3.1 Bistability . . . 51
3.3.2 Optical switching . . . 51
3.3.2.1 Mechanical closing of a pump beam . . . 53
3.3.2.2 Modulation of a pump beam intensity . . . 53
3.3.2.3 Modulation of an additional beam intensity . . . 54
3.4 Concluding remarks . . . 56
4 Nonlinearity enhancement in Rb vapor by using an additional pump beam 57 4.1 Experimental setup . . . 59
4.2 Action of the pump beam . . . 61
4.3 Pump beam resonant with Fg = 2 . . . 63
4.3.1 Polarization self-rotation experiment . . . 63
4.3.2 Three-wave mixing experiment . . . 67
4.3.3 Beam amplification and breakup . . . 74
4.4 Pump beam resonant with Fg = 1 . . . 77
4.5 Concluding remarks . . . 85
5 Conclusions 86
Bibliography 90
Introduction
The invention of laser in 1960 allowed the study of matter interaction with high-intensity illumination [1]. Second harmonic generation was first demon-strated by Franken and colleagues in a quartz crystal by using a ruby laser beam in 1961 [2]. Thenceforth, nonlinear optics has been used in applications such as frequency conversion, two-photon absorption, stimulated Raman and Brillouin scattering and self-action effects, among others.
A nonlinear medium can be solid, liquid or gas. However, the nature of the interaction between light and matter restricts the allowed incident light flux on a sample, as occurs for example, in the crystalline medium where infrared and ultraviolet radiation absorption is limited. In the case of atomic gases and vapors, these do not suffer irreversible damage at high intensities of incident radiation, conversely, a strong electromagnetic field must be incident on these materials to obtain stronger nonlinear responses. Before 1970, third harmonic generation had been observed in several atomic and molecular gases but with harmonic conversion efficiencies lower than those achieved with crystals [3– 5]. Later, the nonlinearity in metal vapors was enhanced by using a buffer gas [6] and laser frequencies resonant with electronic transitions [7].
Among alkali vapors, rubidium vapor is a well known nonlinear optical me-dium near the atomic transition frequency. The rubime-dium transition lines are available with inexpensive diode laser light, and moderate temperatures are required to obtain substantial vapor pressures. In a two-level model, the ru-bidium nonlinearity is often described as a scalar Kerr type effect, where the nonlinear coefficient is inversely proportional to the cube of the laser frequency
detuning [8]. This approximation is valid outside the Doppler-broadened ab-sorption line, and fails near to resonance. The theoretical description at re-sonance conditions is more complex because the complete atomic transition structure must be considered, requiring a vectorial model [9].
The applications of rubidium vapor nonlinearity include laser cooling [10], slow light [11, 12], nonlinear magneto-optical effects [13, 14], precise mag-netometry [15], cross-dispersion effects [16], optical switching [17], pattern formation [8], squeezing vacuum generation [18–22] and electromagnetic in-duced transparency [23], among others.
In this thesis, a strong vectorial positive nonlinearity, obtained deep inside the Doppler-broadened 87 Rubidium D2 line, is experimentally studied for a low power beam. This nonlinearity for rubidium was studied previously in our group in reference [9]. The experimental results of this thesis, include an opti-cal memory system based on the generation of optiopti-cal transverse patterns in presence of a longitudinal magnetic field. Also, an enhancement of the non-linearity in experiments of polarization self-rotation and three-wave mixing is obtained when an additional pump beam tuned at the 87 Rb D2 transition line, withFg = 1 and 2 is considered.
The text is organized as follows. Chapter 2 describes the characteristics of rubidium vapor and some of the principal nonlinear phenomena that occur in a third-order nonlinear medium. An emphasis is realized on those relevant to this thesis, such as modulational instability, transverse optical pattern gene-ration, optical switching and nonlinear self-rotation of the polarization plane. Finally, a brief description of two of the theoretical approximations usually em-ployed to describe the nonlinear phenomena in rubidium vapor is made.
Chapter 3 is dedicated to the experimental results obtained regarding the development of an transverse patterns-based optical memory. For this pur-pose, two beams are counter-propagating inside a rubidium vapor cell, and a longitudinal magnetic field is applied. For a well defined value of magnetic field, beam polarization, temperature and angle between counter-propagating beams, a bistable state composed by two transverse optical patterns is ob-served in the far field. The switching between the two states is realized by using different mechanisms, one of which includes an additional beam with a modulated intensity. A time of switching below 10 microseconds is obtained by using an additional intensity-modulated beam of approximately 2 microwatts.
The results obtained were published in reference [24].
Chapter 4 describes the experimental study of the elliptically polarized beam polarization self-rotation by using an additional optical pump beam. It was found that the nonlinear phase shift, given by the product of nonlinearity and absorption length, can be enhanced approximately 2 times, with respect to the case when the pump beam is absent, when the additional pump beam is co-or counter-propagated. An increment of diffraction efficiency is obtained with optical pumping in a three-wave mixing experiment as well as beam amplifica-tion and pattern breakup. The results of these experiments, were published in references [25, 26].
Nonlinear optics in rubidium vapor
2.1
Characteristics of rubidium vapor
The fast and strong nonlinear effects produced at the resonance lines of alkali vapors are well known. These transition lines are accessible by using inexpen-sive tunable semiconductor lasers in the NIR. One of the alkali elements most commonly used in nonlinear optics experiments is rubidium. Rubidium (Rb) is an element of the alkali metal group with atomic number 37 and electron con-figuration [Kr] 5s1. Natural rubidium is composed by 72%of isotope 85 Rb and 28%of isotope 87 Rb.
The fine structure of Rb, i.e., the coupling between the angular orbital mo-mentumL with the spin angular momentumS of the outer electron, is repre-sented by the transition lines D1 (52S1{2 Ñ52P1{2) and D2 (52S1{2 Ñ52P3{2). This doublet of the D line corresponds to the transitionL = 0ÑL = 1, where the total electron angular momentum is given byJg = 1/2 in the ground state andJe= 1/2 or 3/2 in the first excited state1. The line of interest in this thesis is 87 Rb D2 line. This line is of much relevance in atomic and quantum op-tics experiments, because it has a cyclic transition that is used for cooling and trapping.
The hyperfine structure is the result of the coupling betweenJ and the total nuclear angular momentumI. The total atomic angular momentum is given by
F =J +I. For the 87 Rb ground state,Jg = 1/2 andIg = 3/2, thusFg = 1 or 2. For 1The transitions are represented in the form n2S 1L
J, withnthe quantum principal number
while the letters SandP corresponds toL= 0 andL= 1 respectively.
the first excited state in the D2 line of 87 Rb (52P3{2),Je= 1/2 or 3/2, and thus
Fe= 0, 1, 2, 3.
The hyperfine structure of rubidium’s energy levels is used in atomic clocks, making rubidium useful for high-precision timing. Rubidium is used as the main component of secondary frequency references (rubidium oscillators), to main-tain frequency accuracy in electronic transmitting, networking, and test equip-ment. Its use as a frequency standard in telecommunication industry, has a greater accuracy and is less expensive than caesium standards.
Simple absorption techniques do not have the necessary resolution to ob-serve hyperfine splitting in atomic gases, due to Doppler broadening. This is realized by using the saturated absorption spectroscopy technique [27, 28]. The hyperfine structure of 87 Rb (52P3{2) is shown in figure 2.1.
Figure 2.1: Saturated absorption spectrum (left) and hyperfine structure (right) of 87 Rb D2 transition line. The spectrum was obtained in our laboratory.
Spectroscopy provides information on the atomic structure through the excitation of atomic transitions with laser beams tuned at resonant frequen-cies. In a two-level model, typically used to describe a Kerr-type material, the transitions are made from the ground level E0to an excited state E1by the ab-sorption of a quantity of energy∆E = E1 - E0. The transition from the excited state to the ground level either takes place by emitting a photon that has the same frequency, direction and polarization than those of incident radiation (stimulated emission), or by emission in an arbitrary direction (spontaneous emission).
The atomic spectral lines have a natural non-zero linewidth that is related to finite lifetime of the atomic excited states. There are many other pheno-mena that cause the broadening of spectral lines, among those stand out the power broadening, caused by the intensity of incident radiation and the Doppler broadening, due to the random thermal motion of the atoms. For ru-bidium D lines, the lifetime broadening is approximately 6MHz and the Doppler broadening is approximately 500MHz at a temperature of about 300K.
A simple absorption line with Doppler broadening can be observed when a laser beam is propagated through a rubidium vapor cell. Saturated absorption spectroscopy uses two beams (pump and probe), derived from a single laser, that are counter-propagated through the vapor. The pump action produces a transition to the excited state E1 and, later, the atom returns to the ground state E0 by spontaneous emission. For a pump beam with sufficiently high in-tensity, the ground state is depleted and therefore the absorption of the probe beam is reduced related to the case without a pump beam. This induces the appearance of a spike in the transmission spectrum of the probe beam. This spike gives information about the hyperfine structure of the atom and it is nar-rower than Doppler width in the transmission spectrum, as can be seen in the left part of figure 2.1.
2.2
Optical pumping
The transitions corresponding to lines D1 and D2 of rubidium, are electrical dipole transitions. They are only possible if the selection rules: ∆mF = 0, or ∆mF=1, are fulfilled for the magnetic sublevels of the total atomic angular momentum F. Photons have also an angular momentum J, which depends on their polarization. For a right circular polarizationσ with respect to, e.g., the z-axis, Jz = +~. For a left circular polarizationσ with respect to the z-axis, Jz
= -~. For a linear polarization πparallel to the z-axis, Jz = 0. When an atom
absorbs a photon, it gains the angular momentum of the absorbed photon and its angular moment changes in a quantity equal to angular momentum of the absorbed photon.
If a cell filled with rubidium vapor is irradiated withσ circularly-polarized light, the energy levels which are higher by∆mF = 1 are excited. However, the
excited states decay spontaneously to the ground state and emit light withπ, σ orσpolarization, in accordance with the∆mF= 0,1 selection rule. In this form, the incident light polarizes the atomic vapor in the cell. Without optical irradiation, the difference between populations of the sublevels is small, due to its low-energy spacing in thermal equilibrium. The incident light generates a new population distribution in the equilibrium between optical pumping and relaxation processes; For example, consider a transition connecting a ground state, with an total angular momentum Fg = 1, to an excited state with and angular momentum Fe= 0. In this case, there are three ground sublevels with mFg = -1, 0, 1, and one excited sublevel with mFe = 0. If one excites such an atom with σ polarized light, only the transition with mFg = -1 to mFe = 0 is possible, because it corresponds to∆mF= +1. Once the atom has been excited to the sublevel with mFe = 0, it can fall back by spontaneous emission either in mFg = -1, in which case it can repeat the same cycle, or, e.g., in mFg = +1 by emission of a σ-polarized photon. In the last case, the atoms remains trapped in mFg = +1 because there is noσ transition starting from mFg = +1. This gives rise to an optical pumping cycle transferring atoms from mFg = -1 to mFg = +1 through mFe = 0.
The magnetic atomic sublevels are shifted under the action of an external magnetic (Zeeman effect), or electric field (Stark effect). The combined action of external fields and optical pumping on an atom, gives rise to effects such as magneto-optical rotation.
2.3
Nonlinear susceptibility
Interaction of laser radiation with matter includes both microscopic and macro-scopic objects. Interaction with micromacro-scopic objects (atoms, molecules, ions and electrons), leads to polarization, excitation and ionization phenomena. These phenomena can be nonlinear in the intensity of external field or multi-photon in the number of multi-photons involved. The result of this interaction is defined not only by the individual effect produced by the light on an element of the system (atom, molecule or ion), but by more general characteristics, such as, absence or presence of resonance between the radiation frequency and some of the transition frequencies in the system [29].
The nonlinear optics effects, that emerge when intense light interacts with atomic gases such as the rubidium vapor, are the consequence of the non-linear effects produced by the radiation on an isolated atom. The most im-portant of these is the nonlinearity of the atomic susceptibility. This is the electronic susceptibility, which is the result of the transitions of an electron between different one-electron states in the atomic spectrum. Some parame-ters of the system, such as the diffractive index and radiation absorption, are related to atomic susceptibility.
The optical response of any material to the interaction with an incident ra-diation, is often represented by the atomic polarizationPinduced by the elec-tric field E in the medium. If the incident field E is weaker than the atomic electric fieldEatom, which binds together the electrons, this polarization can be
expanded in powers of electric field in the form, (perturbative approximation):
P 0rχp1qE χp2qE2 χp3qE3 ...s, (2.1)
where 0 is the free-space permittivity. In general, the electronic atomic
sus-ceptibilityχpkq is a tensor with rankk 1and represents transitions between different one-electron states in the atomic spectrum. The real and imaginary parts of a susceptibility may describe physically distinct phenomena; for ex-ample, the real and imaginary parts of linear susceptibilityχp1q, describe the refractive index and linear absorption, respectively. The real and imaginary parts of third-order susceptibilityχp3q, can describe diverse processes as, e.g., optical Kerr effect and stimulated Raman scattering, respectively [30].
In the case ofχp1q, the frequency of the scattered light may (Rayleigh scat-tering), or may not coincide (spontaneous linear Raman scatscat-tering), with the atomic transition frequency. In the nonlinear case, i.e., χpkq with k¡1, the frequency spectrum is broader because it can contain harmonics of the in-cident light (multiphoton absorption), or the sums or differences of those ini-tial, when the incident radiation is composed by several external fields. While χp1qis a single-photon absorption process, the terms of nonlinear susceptibil-ity, χp2q, χp3q ..., represent more complex processes of absorption and emis-sion of several photons [31]. The second-order term χp2q is nonzero only in non-centrosymmetrical media, for instance, in some crystalline structures. In atomic gases the lowest-order non-zero nonlinear susceptibility isχp3q.
If the applied field is a superposition of monochromatic waves with frequen-ciesω1,ω2, ... ,ωn, the polarization and susceptibility can be represented in the
frequency-domain. When the total field is expanded in terms of its Fourier com-ponents, the nonlinear polarization, i.e., the terms withk¡1in equation 2.1, is composed by several terms oscillating at diverse combination of frequen-cies [32]. For example, if the total field is composed by three waves oscillating at frequenciesω1,ω2andω3, the third order can oscillate to 3ω1, 3ω2, 3ω3,ω1
+ω2 +ω3,ω1+ω2 -ω3, etc. The i-th cartesian component of the polarization
amplitude, withω4=ω1 +ω2 +ω3 is:
Pip3qpω4q 0Dp3q
¸
jkl
χpijkl3qpω4;ω1, ω2, ω3qEjpω1qEkpω2qElpω3q, (2.2)
where pi,j,k,lq run through values of coordinatesx, yandz. The degeneracy factorDis defined as:
Dp3q $ ' ' ' & ' ' ' %
1 all fields are indistinguishable
3 two fields are indistinguishable
6 all fields are distinguishable
where two fields of the same frequency are physically distinguishable if, e.g., they travel in different directions.
The specific functional form of χp3q, can be deduced from the classical model of the anharmonic oscillator or via density matrix formalism [33]. The tensor χp3q is related to the value of ω4 - ωn, with n = 1, 2 and 3, in a such form thatχp3q is undefined in resonance conditions, i.e., when the incident ra-diation frequency matches some atomic transition frequency. In this case, an additional factor of phenomenological dampingiΓ, must be considered in the equation of motion. As a consequence, the resonant susceptibility is a com-plex coefficient; its real part describes the change in the refractive index of the medium due to the interaction with the incident field, and the imaginary part is related to processes of absorption and spontaneous emission of photons.
2.4
Intensity-dependent refractive index
The situation described in this section is one in which an intense monochro-matic beam of frequency ω, interacts with a nonlinear medium with a third-order susceptibilityχp3q (ω=ω+ω-ω). The nonlinear polarizationPp3q, modi-fies the propagation properties of the light. The refractive index of the medium is [33, 34]:
nn0 2 ¯n2|Epωq|2, (2.3)
where the coefficientn0is the linear refractive index of the medium,Epωqis the
amplitude of the optical field andn¯2gives the rate at which the refractive index
is modified when the incident optical intensity increases. This is a correction factor of the refractive index due to the nonlinear effects in the material and it is called the optical Kerr effect, by analogy of the Kerr electro-optic effect in which the refractive index of a material changes by an amount proportional to the squared amplitude of an applied static electric field. The correction in the refractive indexn¯2is zero at resonance frequencies and suffers a sign change
in the resonance neighbourhood when a two-level model is used, as will be seen in section 2.6 (equations 2.10 and 2.60).
The scalar nonlinear polarization for a material like rubidium vapor can be described as:
Ppωq 0rχp1q 3χp3q|Epωq|2sEpωq 0χeffEpωq, (2.4)
where the effective susceptibility is defined asχeff χp1q 3χp3q|Epωq|2. In a
non-resonant case, the refractive indexnis related to the nonlinear suscepti-bility in the form
n21 χeff1 χp1q 3χp3q|Epωq|2. (2.5)
If the square root of refractive index in eq. 2.3 is extracted, and it is introduced in the last equation, the linear and nonlinear refractive indices are related to
the susceptibility in the form
n0
b
1 χp1q (2.6)
¯
n2
3<rχp3qs
4n0
. (2.7)
Usually the intensity-dependent refractive index is defined as:
npIq n0 n2I, (2.8)
whereIcorresponds to the time-average intensity of the optical field given by
I 20n0c|Epωq|2, (2.9)
withcbeing the velocity of light. The coefficientsn2andn¯2are related by
n2
¯
n2
n00c
3<rχp3qs
4n200c
. (2.10)
This nonlinear change in the refractive index, induced by the intensity of an applied field, is responsible for the so called self-action effects, some of which are described in the next section. A more detailed theoretical description of the susceptibility behavior in resonant conditions is presented in section 2.6.
2.5
Nonlinear optical processes in atomic gases
Nonlinear processes can be classified considering a variety of criteria. Ac-cording to the frequency of incident radiation, they can be resonant and non-resonant. According to the frequency and wave vector of emitted radiation, they can be coherent and incoherent, and according to the final state of the material can be parametric and non-parametric:
1. Resonant and non-resonant process: A process is resonant if the in-cident radiation frequency matches some atomic transition frequency. This interaction produces real transitions from one quantum state to another through one-photon or two-photon absorption processes. When the incident frequency is non-resonant the interaction perturbs weakly
the normal distribution or motion of internal electric charges within atoms [35].
2. Coherent and incoherent process: If the radiation generated in a non-linear process has the same frequency and the same wave vector direc-tion, as the incident illuminadirec-tion, the process is named coherent. Other-wise, it is incoherent.
3. Parametric and non-parametric process:If the initial and final quantum-mechanical states are the same after the interaction with the incident ra-diation, the process is parametric. The processes that involve the trans-fer of population from one level to another are known as non-parametric.
In a coherent process the phase-matching condition is fulfilled. This con-dition assures a proper phase relationship between the interacting waves that is maintained along the propagation direction and the conservation of energy and momentum in the medium is guaranteed. The solution of Maxwell equa-tions for a polarized monochromatic incident light field,Epωq, restricts the in-cident,Kω, and generated,Kν, wave vectors in the form: Kω =Kν.
In a more general process, where the incident radiation is composed by a number of waves with frequencies (ω1,ω2, ...ωm) and a generated set of waves with frequencies (ν1,ν2, ...νn), the phase-matching condition is defined as
Kω1 Kω2 ... Kωm Kν1 Kν2 ... Kνn. (2.11)
In this section, some of the third-order nonlinear processes that occur in ru-bidium vapor are described. Those involved in the experimental phenomena studied in this thesis are described in more detail, as is the case of self-action effects, optical switching and wave propagation in volume gratings.
Although these phenomena generally involve the occurrence of electro-magnetic fields with frequenciesω1,ω2 andω3, most of the phenomena
des-cribed in this section are for the case of a monochromatic wave with a fre-quencyωincident on a nonlinear medium with a susceptibilityχp3q(ω=ω+ω -ω). If necessary, the relevant details will be made in each case.
2.5.1
Four-wave mixing
Four-wave mixing (FWM) is a parametric interaction between four photons in third-order nonlinear media [35]. In this case, it is considered an incident radi-ation of waves oscillating at three different frequencies and a radiation emit-ted containing one, two or three frequencies, depending on the process in-volved, as is shown in figure 2.2. In this figure, the frequency phase-matching condition is indicated for each case.
The first case illustrated in figure 2.2 (a)), is third-order nonlinear sum-frequency generation. It can be understood as a process in which three in-cident photons of frequencies ω1, ω2 andω3 are annihilated, while the atom
makes a transition to an intermediate state, that is represented as a dashed line in the figure. Later, the atom returns to the ground state and a new photon is created. In all the cases illustrated in the figure, the initial and final state of the atom are the same so that the energy and momentum are conserved.
In the case of third-harmonic generation (figure 2.2(b)), three photons of the same frequency are annihilated and then, a photon of the third-harmonic wave is created. The other two processes shown in figures 2.2(c) and 2.2(d), can be described in a similar way.
Figure 2.2: Some typical four-wave mixing processes. From left to right: Sum-frequency generation, third harmonic generation and four photon parametric interactions [35].
When all the waves in the interaction have the same frequency, the process is called degenerate. Degenerate four-wave mixing (DFWM) can yield phase
conjugation. In phase conjugation, two counter-propagating pump beams with amplitudes A1 and A2, interfere with a probe beam of amplitude A3. This pro-cess can be explained in the following way: Two beams induce a volume grating and the subsequent reconstruction by a third wave generates a backward pro-pagating wave. This reconstructed wave can be generated in two forms: First, the pump wave A1and the probe wave A3 can produce nearly parallel interfer-ence fringes along the bisector direction of the crossing angleθ, figure 2.3(a). This results in a complex volume hologram due to a spatial distribution of the refractive index (Kerr-like media), or absorption (saturated absorber). The an-tiparallel pump wave A2 is then diffracted under Bragg conditions by the dy-namic volume hologram and the diffracted beam A4 is created. The wave A4 will bring the spatial information carried by the incident wave A3, i.e., the waves A4 and A3are phase-conjugated. The second form in which A4 can be genera-ted is through the diffraction of the pump wave A1 on the holographic grating formed by the waves A2and A3, figure 2.3(b). This process is also called back-ward four-wave mixing (BFWM).
Figure 2.3: (a) and (b): Generation of phase conjugated waves by induced holo-graphic gratings [35]. (c): Forward four-wave mixing.
In the two cases described above, the frequency and polarization state of all the waves are the same, but phase conjugated waves can also be obtained if the polarization or frequency of some of the waves is different. However, in this type of process, the waves forming the holographic grating must have the same polarization state in order to interfere. If the reading wave has a polarization orthogonal to that of the writing waves, the generated wave will also be orthogonal in polarization to that of the writing beams.
mix-ing (FFWM). In this case, a strong pump wave A1 and the probe wave A3 in-terfere and induce the grating where the pump wave A2 is diffracted and the diffracted beam A4is generated, figure 2.3(c).
2.5.2
Self-action e
ff
ects
Self-action effects are processes in which a beam of light modifies its own pro-pagation by means of the nonlinear response of a material medium, repre-sented by the nonlinear dependence of the refractive index with the intensity of the incident light (equation 2.3 ). The alkali vapors are characterized by very large self-action effects, and its optical properties can be calculated far from transition by using the two-level model. Some of the most important self-action effects are:
• Self-focusing:It occurs when an intense beam converges during its pro-pagation through a medium with a positive indexn2(eq. 2.8).
• Self-trapping: Is the beam tendency to propagate with a constant dia-meter as a consequence of an exact balance between self-focusing and diffraction effects.
• Self-phase modulation: Is a self-induced nonlinear phase-shift expe-rienced by a beam. This is proportional to the beam intensity.
Laser beam breakup and self-rotation of polarization plane, are another important self-action effects. These are relevant for this thesis and, thus, are described in more detail.
2.5.2.1 Laser beam breakup and pattern formation
A beam with a high enough incident power, propagating through a nonlinear media with a positive refractive index, experiences a breakup into many com-ponents or filaments. This is the result of a transverse modulational instability process, and can be described by the model developed by Bespalov and Ta-lanov [36]. This model considers a uniform plane wave, unstable in presence of amplitude and/or phase perturbations. As a consequence of this, the wave is break up into multiple "small beams".
The modulational instability (MI) is one of the most fundamental effects associated with the propagation of a beam in a nonlinear medium [37, 38]. This mechanism implies the exponential growth of a weak perturbation in the wave amplitude while it is propagating. The gain leads to amplification of side-bands, which break up the wavefront and generate fine localized structures (filamentation). In the far field, the effect of the instability can be observed as conical rings, pairs of spots, or an array of spots, sometimes with hexa-gonal symmetry depending on the frequency and the intensity of the primary beams. Sometimes the term modulational instability is related to the pertur-bation of the wave over its propagation direction, and the perturpertur-bation in the transverse direction is named filamentation, which is time-independent and has been studied in what is called transverse nonlinear optics [39]. In this the-sis, the MI describes the transversal effect.
The observation of pattern formation in atomic vapors, occurs in systems that involve only the propagation of a simple laser beam through a nonli-near medium [40], or those with some sort of optical feedback, such as, a single feedback mirror [41–43], resonators [44], or wave mixing of counter-propagating beams [45–50].
In the theoretical description of the beam breakup, based on the model of Bespalov and Talanov [36], the incident field is a monochromatic beam of frequencyω, interacting with a nonlinear medium with a third-order suscepti-bilityχp3q (ω =ω +ω-ω). The field within the nonlinear medium is described as [33, 51]
Epr, tq Epr, tqeiωt Epr, tqeiωt. (2.12) This field can be represented by the sum of three components E0, E1 and E1given by
Epr, tq rA0pzq A1prq A1prqseikz
A0pzq a1pzqeiqr a1pzqeiqr
eikz, (2.13)
wherek(|k|=n0ω{c), is the wave vector in thezdirection,qis a wave vector in
the transverse direction andA0,A1 andA1 are the amplitudes ofE0,E1 and
andE1are two weak (signal) off-axis field components located symmetrically
with respect to the pump.
The nonlinear polarization in the material is given by
P 30χp3q|E|2EP0 P1 P1, (2.14)
where
P030χp3q|A0|2A0eikzp0eikz (2.15)
and in a similar form,
P130χp3q 2|E0|2E1 E02E1
p1eikz. (2.16)
The amplitudeA0satisfies the equation
2ikBA0
Bz ∇2KA0
ω2 0c2
p0, (2.17)
but for the on-axis component∇2KA00, so the solution is
A0pzq A00eiγz, (2.18)
where
γ3ωχ
p3q
2n0c |A00| 2n
2kI. (2.19)
This solution implies that the central component undergoes a nonlinear phase shift after it has been propagated. The quantity A00 is assumed as real. In
terms ofa1, the polarization in eq. 2.16 can be now expressed as:
P130χp3q
2|A00|2a1eiqr A200ei2γza1e iqr . (2.20)
The amplitudesa satisfy an equation similar to 2.17. The final set of equa-tions to be solved is:
da dz
iq2
2kaiγ
2a1 a1e2iγz . (2.21)
term and solve it more easily. By takingaa1eiγz, the resulting equation is:
d dza
1
1i γq2{k
a11 iγa11, (2.22) which has exponential solutions in the formapzq ap0qeΛz, with
Λ
d q2
2k
2γq
2
2k
. (2.23)
This system of equations produces gain (<Λ > 0), if γ ¡ q2
4k, so n2 must be positive in order to produce beam breakup (focusing media). The gain vani-shes for all values ofqgreater thanqmax = 2
a
kγ, and reaches its maximum value for qopt = qmax{
?
2. There is a characteristic angle for which the beam breakup occurs given by θopt = qopt{k and represents the direction in which the phase-matching condition is performed in the four wave mixing process formed when the nonlinear contributions of wave vectors on- and off-axis are taken into account. The breakup occurs within the interaction region and can always be induced by using a high enough laser power.
The instabilities can also be found when two or more beams are propa-gated through the nonlinear medium. For a strong interaction in this type of configuration, the mirrorles parametric self-oscillation gives rise to stationary, periodic, or chaotic behavior of intensity and/or polarization [45,52]. The ana-lysis of this type of systems is based in the four wave mixing process [53–57]. Consider two optical beams, with wavenumber k and amplitudes E1 and E2,
which are propagating in opposite directions inside a Kerr medium of length L; figure 2.4. For low values of incident intensities, a stationary plane wave is found in the media [51]. The perturbative analysis can be done by introdu-cing of additional probe beamsEa,Eb,EcandEd with small amplitudes. These beams impinge at an angleθ with respect toE1. The evolution of Ea can be described by the equation [47, 51, 55]:
BEa
Bz iξ
p1 aqEc qEbeikθ
2z
p1 aqEdeikθ
2z
qEa
, (2.24)
where ξ is proportional to the productχp3qE1E2, q= |E1|{|E2| and 0¤a¤ 1.
vari-ation are washed out by the motion inside the medium and it takes the value of one if all the gratings are retained independently of its spatial variation.
Figure 2.4: Two counter-propagating beams E1 and E2 interact within a nonli-near medium and can generate spontaneous coherent emission beams Ea, Eb, Ecand Ed [51].
Each term on the right side of eq. 2.24 represents a four-wave mixing (FWM) contribution to the evolution ofEa. The first term is associated with the BFWM gain and it is related to the absorption of one photon in each pump beam and the emission of photons inEaandEc. The second term corresponds to FFWM gain and it is related to absorption of two photons from beamE1 and emission
of photons into beamsEaandEb. The third term is associated with parametric interaction between the beams counter-propagating through the media, and it is related to the absorption of one photon from beams E1 and Ed and the
emission of one photon into beamsE2 andEa. The last term in the equation, corresponds to the nonlinear modification of the susceptibility of the beam Ea. The equation 2.24 and similar equations for Eb, Ec and Ed, define a set of linear differential equations that is valid only when higher-order nonlinear contributions can be neglected, as well as any parametric interaction where more than two probe beams are involved.
The dependence of instabilities with the angleθ, increases the net gain for an unique value of the angle θ given by the phase-matching condition. For this angle, the backward- and forward-four wave mixing processes are added coherently. This type of mirrorless parametric oscillation occurs with conical emission, i.e., the emitted radiation is in the backward and forward direction, respect to that of the incident radiation [55]. The threshold for mirrorless self-oscillation is infinite forθ= 0, thus no oscillation is possible on the pump beam axis.
The symmetry of the patterns resulting from this amplification processes is often hexagonal. This has been confirmed experimentally [42, 58]. As ex-plained by Grynberg [47], this is possibly a result of the involved four-wave mixing processes, and each spot of the hexagon is the result of a parametric process where one off-axis photon and one pump photon are absorbed and two off-axis photons are created both with wave vectors corresponding to the two next-nearest spots in the hexagon [59].
2.5.2.2 Self-rotation of the polarization plane
The eq. 2.1 represents the average dipole moment of the atom induced by the electromagnetic fieldE. The polarization vector ofE, is characterized by the field orientation, given by the relation between amplitudes and phases of its components. This can be altered when the field propagates through a nonli-near medium. A completely polarized electromagnetic wave, with real-valued fieldEcan be represented as [31]
EE˜ E˜, (2.25)
where the vectorE˜ corresponds to a monochromatic electromagnetic wave of frequencyω, and wave vectork, with propagation direction along the direc-tionz:
˜
EEωexppiωtikzq. (2.26) The complex-valued vectorEω lies in thexy-plane, perpendicular to the direc-tion of propagadirec-tion of the electromagnetic wave. It is convenient to take as a basis the following linear combinations of unit vectorsex andey:
e ?1
2 ex iey
e ?1
2 exiey
. (2.27)
In this baseEω can be expressed as
whereAandBare complex-valued expansion coefficients or, in a more specific form,
EωEωrcospθ π{4qexppiφqe cospθπ{4qexppiφqes. (2.29) Here the angle θ defines the ratio between the polarization ellipse semiaxes (ellipticity, figure 2.5). Ifθ= 0, the field is linearly polarized in thexy-plane. At θ=π/4 the field is circularly polarized clockwise (+) or counterclockwise (-). The angleφ determines the orientation of the ellipse semiaxes in relation to
x-y coordinates.
Figure 2.5: Representation of polarization ellipse [31].
For an elliptically polarized electromagnetic wave passing through a non-linear medium (such as an alkali vapor), and in the unperturbed field approxi-mation, it is found that the polarization ellipse retains its shape (∆θ= 0). The change of wave vector,∆k, and hence, of the refractive index, depends on the radiation intensity,Eω2, and the number of atoms per unit volume of the gas N. The change in orientation of the ellipse, which is related to the change in the refractive index, is given by the angleφ. This is proportional to the length tra-versed by the incident wave into the medium z, the wave vector k, the intensity Eω2 and ellipticityθ: ∆φ2πkzNEω2sin 2θ. The change∆φis known as self-rotation of the polarization ellipse. The complete deduction of these results can be found in reference [31].
For linearly polarized light,θ= 0, the polarization plane remains fixed, and there is no self-rotation,∆φ= 0. The self-rotation effect is maximum for a cir-cularly polarized wave,θ = π{4, however, the rotation of a circle cannot be observed in principle, so for this case∆φ= 0 again. While the ellipse is not de-formed,θ= constant, the angle of rotation∆φgrows linearly with coordinate z [31].
The effect of self-rotation of the polarization ellipse is equivalent to the effect of a wave-induced optical anisotropy on an initially isotropic medium. The refractive index of the medium becomes dependent on the transmitted light polarization due to the interaction. For and elliptically polarized incident beam, each of its components of circularly polarized waves, σ andσ, have a different refractive index associated, n and n (induced gyrotropy), and a different absorption coefficient (induced circular dichroism). The change in refractive index,∆n = n - nis proportional to the rotation angle∆φ.
The polarization self-rotation effect can be used to measure Kerr nonlin-earities, and it is a systematic effect in optical rotation measurements, thus, it must be considered in optical rotation magnetometry measurements. A re-cent application of this effect, is the generation of squeezed light in resonant atomic vapors, where the nonlinearity to absorption ratio is a parameter that should be enhanced [60]. It can also be helpful in the use of atomic coherences to control the polarization state of a relatively weak probe beam, by using an initial strong resonant pump beam; see [61] and references therein.
The mechanisms producing self-rotation in atomic vapors are optical pump-ing and the ac-Stark shift effect [61]. In optical pumping, an incident elliptically-polarized beam interacts with each atom in the vapor. The elliptic polariza-tion of the incident beam, is composed by two unbalanced circular-polarizapolariza-tion components. Due to the interaction with photons, the angular momentum of the atom changes and the initially isotropic atomic lower level is polarized. As a consequence, a difference in the populations of the atomic sublevels is pro-duced as well as, a phase shift between the circularly polarized components left-σ, and right-σ , of the propagating light in the medium. As a result, there exist a difference∆n, between the refractive indices for the left- and right- cir-cular components of polarization. This is observed as a change of the angleφ, which determines the orientation of the polarization ellipse plane with respect to plane x,y in figure 2.5. In the particular case of the transition with Jg = 1/2
to Je = 1/2, the value of ∆φ is independent of the beam intensity, as it was showed in reference [62].
In the case of the ac-Stark effect, a shift between the transition frequencies for left- and right- circularly polarized components of the light is induced by an elliptically polarized wave. A difference in the refraction indices of both cir-cular polarization components is produced, and the polarization ellipse plane rotates.
2.5.3
Nonlinear Faraday rotation
Atoms have permanent magnetic dipole moments and so they experience a force in an external magnetic field. This interaction produces a precession of the atomic angular momentum around the axis of applied magnetic field, and the spectral lines of atom are splitted (Zeeman effect). The energy of zeeman sublevels, is a function of field’s amplitude and the magnetic quantum number of angular momentum. The magnetic field introduces an axis of symmetry into the atom, that initially has an spherical symmetry, and has an effect on the polarization of the radiation emitted or absorbed in relation to this axis.
The motion of an electron in the atom can be resolved into a motion along the field axis, e.g. the z-axis, and a circular motion in the x, y plane. Due to the quantum mechanics selection rules, the z-component is zero unless the diff e-rence of sublevels of total atomic angular moment mF = 0. This is identified with a linear motion parallel to the magnetic field for an observer in the x, y plane, this is, light linearly polarized. The component in the x, y plane is zero unless mF=1. This appears as a circular motion in the x, y plane around the z-axis. Thus, transitions with mF =1, give circularly polarized lightσ (right) orσ(left), for an observer in the z-axis.
Magneto-optical effects arise when light interacts with a medium in the pre-sence of a magnetic field. Most prominent among the magneto-optical effects are the Faraday and the Voigt effects, i.e., rotation of light’s polarization plane as it propagates through a medium placed in a longitudinal or transverse mag-netic field, respectively. Magneto-optic rotation of light beams tuned near res-onant transitions of alkali atoms exhibits nonlinear features, depending on the intensity of the probing light beam.
circularly-polarized components of a linearly circularly-polarized beam, gain different phase shifts, due to zeeman effect, leading to optical rotation. A difference in absorption between the two components induces ellipticity in the output light.
The mechanism by which the nonlinear Faraday rotation occurs can be illustrated by the case of Fg = 1 to Fe= 0 transition. In the absence of a mag-netic field, the sublevels mF
g =1 are degenerate and the optical resonance
frequencies for the two circularly polarized components of incident beam are equal. When a magnetic field B is applied, a Zeeman shift in the sublevels µgBmF
g is produced (figure 2.6), with µ the Bohr magneton and g the Landé
factor. It leads to a difference between the resonance frequencies, and a dis-placement of the dispersion curves for the left- and right- circular components of polarization. This last is shown in reference [14].
The dipole moments corresponding to the components of right-σ and left-σcircular polarization, have opposite signs. As a consequence, it is produced a change in the refractive index (circular birefringence), and absorption (circu-lar dichroism), in proportion to the Zeeman shift. The relative phase between the two circular polarization components determines the direction of polariza-tion of the light emerging from the medium and the output beam is elliptically polarized.
Figure 2.6: Nonlinear Faraday rotation. In the presence of a longitudinal mag-netic field, the energy Zeeman sublevels of the ground state are shifted by µgBmF
g [14]. σ andσ represent the right- and left- circular components of
2.5.4
Optical bistability and optical switching
A system is called bistable if, for a given range of input values, there exist two possible output values. To get bistability, an optical system requires a nonli-near medium and a feedback mechanism. The most typical optical bistable configuration is the Fabry-Perot etalon containing a nonlinear optical medium. Bistable systems can be absorptive or dispersive, depending on whether the feedback occurs by way of an dependent absorption or by an intensity-dependent refractive index, respectively. These systems can also be all-optical, or hybrid (mixed optics and electronics). In an all-optical system, the intensity dependence arises from a direct interaction of light with matter. In a hybrid system, the intensity dependence arises from an electric signal in a detec-tor that is monidetec-toring the transmitted intensity and, usually, controls an intra-cavity phase shifter [63]. Current research is advocated to improve the size, switching time and operating power in systems with bistability due the possible applications in optical communications and computing, especially as low-light-level optical switches [59, 64].
The low-light-level switching can be realized in two ways: In the first case, fields and atoms are confined and strongly coupled to an optical cavity with high spectral resolution in a wide spectral range (high-finesse). In the second method, a quantum interference is induced by propagating waves within the nonlinear medium and the nolinearity is improved. A description of some tech-niques used in low-light-level switching is given in reference [59].
The work of Dawes et. al. [17, 59] is cited here to describe the mecha-nism of interest. They use the strong instabilities produced when two beams are counter-propagated through a nonlinear medium (rubidium vapor). This interaction can produce a stationary, periodic, or chaotic behavior of the in-tensity and/or polarization [39, 45, 65]. Additionally, as it was mentioned be-fore, optical transverse pattern formation is possible in this type of interac-tion. These patterns are used as an all-optical switching mechanism. This system is very sensitive to perturbations, due to a combination of intrinsic ins-tabilities with an enhancement of the atom-photon coupling through a near-resonance pumping, and sub-Doppler nonlinear optics (due to the counter-propagation scheme). The theory involved in these systems is that developed in section 2.5.2.1.
The instabilities induced by to two parallel polarized, counter-propagated pump beams, generate new output beams (patterns), only if the pump power is over a certain threshold. The generated spots have the same frequency of pump beams, and a polarization orthogonal to that of them. The power of the output light is increased, if the frequency of the pump beam is near-resonant with the transitions lines of rubidium. In the case of references cited, the 87 Rb D2 transition line, with Fg = 1, Fe= 1).
The symmetry of the generated patterns in the far field is typically hexa-gonal, but it is easily broken due to imperfections in the optical elements, e.g., variations of the glass surface thickness in the rubidium cell. In the genera-ted patterns, this fact is manifesgenera-ted as a preferred azimuthal emission angle, where the spots on the far-field patterns are brighter. Additional to this unavoi-dable but weak asymmetry, another way to change the characteristics of the far-field pattern, is producing a slight misalignment (< 1mrad), between the in-cident counter-propagated pump beams. This misalignment switches the pa-ttern, e.g., from one composed by six spots to another with only two spots.
Figure 2.7: Generated light is emitted along cones (blue), which are centered on the pump beams (red), and forms a transverse pattern (red spots) in the measurement plane. (a) Unperturbed system with two-spot pattern induced by pump-beam misalignment, (b) Two-spot pattern in the far-field. (c) A weak switch beam (yellow) rotates the pattern in the direction defined by the azi-muth where it incides, (d) Rotated two-spot pattern in the far-field. Drawing and description reproduced from reference [59].
In order to implement an optical switch, the preferred emission direction, figure 2.7(a), is used as one of the two output states of the switch (off-state),
and the other (the on-state), is defined by an additional-pulsed weak beam that incides at another azimuthal angle of the hexagonally symmetric pattern, fi-gure 2.7(b). When the switch beam is absent, the system is in the off-state and when this beam incides, the system changes to the on-state (figure 2.8). The patterns can be produced by using a pump beams with a power less to 1mW, and with a switch beam power of approximately 2nW. The system is isolated of any transverse or logitudinal magnetic field, thus the switching is influenced only by the weak switch beam.
Figure 2.8: Time-response of the pattern in Dawes et.al. experiments. The switch is turning on and off and the power is decreased. (a) two detectors simultaneously measure the spatial regions indicated on the right frames (c, d) by circles which encompass bright and dark areas of the output pattern. The output signals are shown in (b, e) and corresponds to the spatial location of the detector within the pattern. (b) shows the on-state and (e) shows the off-state. Drawing and description reproduced from reference [59].
2.5.5
Optical wave propagation in volume gratings
Volume holographic gratings are diffractive elements created in a medium due to a periodic spatial modulation of absorption, refractive index, or both. These elements have been used for data storage, optical correlators, optical information encryption, fiber optic communication and spectroscopy. Phase gratings are the most widely used volume holographic gratings due to their
low absorption and they are fabricated in photorefractive crystals, polymers, dichromated gelatins and photosensitive glasses. Usually, these gratings are classified as transmission or reflection. In a transmission grating, the incident and the diffracted beams are at opposite sides of the grating while in a reflec-tion grating both beams are at the same side of the grating [66].
Additional to volume holograms there are dynamic holograms, i.e., holo-grams that do not need an initial recording procedure in a physical or elec-tronic media to be reconstructed later. The diffraction in this type of grams can be observed during its recording. In nonlinear optics, dynamic holo-grams are described in the frame of "non-linear wave mixing" [67]. This type of real-time holograms can be recorded in materials such as photorefractive crystals, some type of semiconductor structures, plasmas, atomic vapors and gases, among others.
The diffraction properties of the thick and dynamic holograms are analysed with the coupled-wave theory of Kogelnik [68]. This theory restricts the ana-lysis to the holographic recording of sinusoidal fringe patterns and assumes that only two waves are present: One is the incoming monochromatic wave R (reference), that is polarized perpendicular to the plane of incidence and impinges at or near the Bragg angle. The other one is the outgoing wave S (signal), that satisfies the Bragg condition whereupon all the higher diffraction orders are neglected. The theory solves a set of coupled differential equations for R and S, by assuming that the energy interchange between R and S is slow (by neglecting the second derivatives terms). Reference and signal waves have complex amplitudes R(z) and S(z) (withzthe propagation direction). This vari-ation inz can be due to the energy interchange between beams, or by energy loss due to absorption. A dephasing measure is incorporated to the descrip-tion, as a measure of the deviation from Bragg condition so, if this grows, the interaction between R and S waves decreases. The coupling between these waves determines the diffraction efficiency of the grating, so that an efficient wavefront reconstruction is obtained when the reading beam is incident at or near the Bragg angle.
A wider approximation to multi-wave diffraction in volume holograms was developed by Magnusson&Gaylord [69], in which the presence of all possible diffracted orders and gratings with arbitrary shape are considered (not only si-nusoidal). No restriction is made on wavelength or incidence angle. The
theo-retical development shows that it is possible to have several high-order diff rac-tion beams simultaneously with an appreciable diffraction efficiency (>20%). This multiple-wave diffraction efficiency is determined by the interaction that occurs between the waves in the grating. Changes in grating shape or strength may influence this process, and affect the interchange of energy between the diffraction orders.
For a small enough non-Bragg angle between the grating writing beams, higher diffraction orders appear in a photorefractive crystal if an external field is applied or if a strong photovoltaic effect exists [69]. Additionally, a signal beam can be amplified due to its interaction with a strong pump beam for an unshifted phase grating [70]. The mechanism of this gain is similar to the mo-dulational instability amplification described before: Two weak signal beams, located symmetrically around a strong pump beam, grow exponentially as they propagate through the medium. In a similar way, if only one weak beam and a strong pump beam intersect at a small angle within a nonlinear medium, with a local nonlinear response, the weak beam is amplified and a symmetrical beam appears. The condition to obtain gain, is that the difference of wave vec-tors in the propagation direction, z, be comparable with the gain parameter of the medium. This type of gain is not described in the Kogelnik theory.
A thick hologram can be divided in many thin holograms. Each of these gratings diffract the incident wave S0into several well-defined orders with am-plitudes Sj, polarizationsˆiand propagation vectorσi. The total electric field in
the grating is given by [69, 70]:
E
8
¸
i8
ˆ
siSipzqexppjσirq, (2.30)
where j =?1,r= (x, y, z). The wave vectors are related by
σiσ0iK, (2.31)
whereσ0 is the wave vector of 0th diffraction order andK is the grating wave vector. The geometry of multi-wave interaction is shown in figure 2.9.
