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    Course Objectives

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    COURSE OBJECTIVES : BB0


    This course is an introduction to nonlinear optical interactions that occurs when an optical beam at high intensity propagates through a material medium. In a nonlinear regime, the response of the medium is no longer proportional to the amplitude of the excitation field. In a perturbative regime, the nonlinear response of the material is then described as a power expansion of the applied electromagnetic field, related to the so-called 2nd, 3rd… order nonlinearities. Nonlinear interactions between a field and a material medium give rise to a wide variety of phenomena and applications: generation of 2nd, 3rd harmonic for the doubling (even tripling) the frequency of a laser beam, parametric amplification and oscillation with the realization of widely tunable coherent oscillators (OPO), generation of frequency comb, supercontinuum sources, self-phase modulation effects, self-focusing or de-focusing of beams...

    Starting from basic concepts in electromagnetism, the course will focus on introducing useful tools for understanding the basic concepts of nonlinear optics, for deriving standard models to describe the mostly used nonlinear interactions (2nd and 3rd orders), and finally for evaluating the efficiency and order of magnitudes in specific cases.


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    LEARNING OUTCOMES

    Nonlinear optics course : learning outcomes

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    LECTURE NOTES

    You will find below the lecture notes for the nonlinear optics course.
    These extensive notes cover all the course topics. However, some parts in the last chapter still remind to be completed. They concern in particular the Raman and Brillouin scattering effects.

    Do not hesitate to send me your comments, possible corrections, which will help me to improve the contents of these notes. Thank you !
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    Lecture 1 : an introduction to nonlinear optics

    The objective of the first lecture is to introduce nonlinear optics by discussing the signification of optics in nonlinear regime, the related effects and applications . A first description of nonlinear interactions will be given enabling to identify the various nonlinear effects that will be studied during the course. Finally, the origin of the nonlinearities will be discussed using a classical model.


    By the end of this first course, students will be able to ...

    • derive the relation between the macroscopic polarization and the electrice field (the so-called constitutive relations of nonlinear optics) (K1)
    • cite nonlinear effects that arise in a 2nd and 3rd order nonlinear materials (K2)
    • provide a classical description for the origin of the nonlinear susceptibilities (K3)
    By the end of this first course, students will start to understand  ...
    • the capability of light matter interactions in modifying light properties : frequency generation, optical rectification… (Q1)
    • how a perturbative approache enables in describing and deriving a NON LINEAR problem in physics (Q2)

    • the link between the microscopic and macroscopic terms in Maxwell’s equations (induced dipole, macroscopic polarization and fields) (Q3)


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    Lecture 2 - Nonlinear wave equation & Phase matching condition

    Following the introduction of the tensorial notation for the nonlinear susceptibilities, the nonlinear wave equation for a stationnary wave propagating through an isotropic material is derived. The importance of the phase matching condition will be underlined and discussed in details.

    By the end of this lecture, students will be skilled at...
    • Deriving and solving the nonlinear wave equation in a parametric situation under the undepleted pump approximation (S3)

    By the end of this lecture, students will understand  ...

    • Nonlinear effects are subject to phase matching conditions (U5)

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    Lecture 3 : Nonlinear susceptibilities & Constitutive relations of nonlinear optics

    During this lecture, we will discuss the limits of the models presented in introduction. Reminding the relation between the linear suceptiblity and the impulse response of a linear medium, a tensorial description for the nonlinear susceptibilities will be provided. The constitutive relations of nonlinear optics will be demonstrated :
    P=\epsilon_0 \chi^{(1)} E + \epsilon_0 \chi^{(2)} E E + \epsilon_0 \chi^{(2)} E E E+\cdots

    The second part of the lecture the  properties of nonlinear susceptibility tensors will be presented in details, with a focus on their symmetry properties.

    LEARNING OUTCOMES OF THE LECTURE

    By the end of this lecture, students will know...

    • the constitutive relations of nonlinear optics ( D= \epsilon_0 E +P and P=\epsilon_0 \chi^{(1)} E + \epsilon_0 \chi^{(2)} E E + \epsilon_0 \chi^{(2)} E E E+\cdots ) (K1)

    • the basic properties of nonlinear susceptibility tensors (K4)

    By the end of this course, students will be skilled at ...

    • Manipulating the nonlinear susceptibility tensor components and, with given incident fields, calculate the components of nonlinear polarisation vector
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    Tutorial n°1 - Three-wave mixing interactions

    This tutorial concerns the generation of a sum-frequency beam after the propagation of two linearly polarized beams through a 2nd order nonlinear material, and in a second part the efficiency of the second-harmonic generation effect (under the undepleted pump approximation).

    By the end of this activity, students will be skilled at ...

    •S3 - Solving the nonlinear equation in parametric situations and derive analytical solutions under the undepleted pump approximation
    •S4 - Calculating nonlinear interaction performances/efficiencies in situations governed by analytical solutions or expressions

    By the end of this activity, students will start to understand  ...

    • U3 - Nonlinear interactions lead to energy transfer between optical beams, and/or between matter and beams, enabling in some cases the realization of nonlinear optical amplification and/or oscillation.

    U4 - Nonlinear optics is an essential tool to create novel optical frequencies generated through the interaction of incident beams within nonlinear materials


    By the end of this activity, students will start to be able to  ...

    •T1 - Model any 2nd or 3rd order nonlinear effects, including numerical simulations
    •T2 - Evaluate nonlinear interaction performances/efficiencies under approximations that should be specified, explained and justified

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    Lecture 4 : 2nd order nonlinear interactions (PART 1)

    Second order nonlinearities will be introduced starting with a derivation of the Manley-Rowe relations, which give physical insights in the 2nd order nonlinear interactions to be studied in the course.
    We will consider the case of the Second Harmonic Generation (SHG) effect (under the undepleted pump approximation) and discuss about the phase matching condition.  
    Following a short reminder on optics in anisotropic materials, we will show how the birefringent properties can be used to fulfill the phase matching condition. An application in the case of SHG will be provided.

    By the end of this course, students will be skilled at ...

    • S2 - Determining the phase matching conditions for a given nonlinear interaction and achieving/fulfilling this condition by exploiting birefringence properties of materials and/or QPM technique
    •S3 - Solving the nonlinear equation in parametric situations and derive analytical solutions under the undepleted pump approximation
    •S4 - Calculating nonlinear interaction performances/efficiencies in situations governed by analytical solutions or expressions
    •

    By the end of this course, students will understand  ...

    • Link between the frequency relation and the law of energy conservation, the phase matching relation and the law of momentum conservation
    • Nonlinear effects are a key points in the development of many applications in photonics (especially in relation with laser physics)
    • Nonlinear interactions lead to energy transfer between optical beams, and/or between matter and beams, enabling in some cases the realization of nonlinear optical amplification and/or oscillation.
    • Nonlinear optics is an essential tool to create novel optical frequencies generated through the interaction of incident beams within nonlinear materials
    • Nonlinear effects are subject to phase matching conditions

    By the end of this course, students will start to be able to  ...

    • Model 2nd order nonlinear effects, 
    • Evaluate nonlinear interaction performances/efficiencies under approximations that should be specified, explained and justified
    • Slides on 2nd order nonlinear interactions (PART 1) Archivo PDF
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    • Nonlinear susceptibility tensors - Ex-1 Cuestionario
      No disponible hasta que: se pertenezca al grupo Cohort 2024-25
    • Nonlinear susceptibility tensors - Ex 2 Cuestionario
      No disponible hasta que: se pertenezca al grupo Cohort 2024-25
    • Nonlinear susceptibility tensors - Ex 3 Cuestionario
      No disponible hasta que: se pertenezca al grupo Cohort 2024-25
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    Tutorial n°2 - Second Harmonic Generation

    Optimzation of the SHG efficiency in a KDP crystal : nonlinear crystal orientation, polarization states of the interacted waves...
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    Lecture 5 : 2nd order nonlinear interaction (PART 2)

    This lecture is dedicated to parametric interactions in 2nd order nonlinear material, which leads to the generation of novel frequencies through the optical parametric fluoresence effect, the amplification and optical parametric oscillation (OPO).
    Finally, the quasi-phase matching technique is described.

    By the end of this course, students will be skilled at ...

    • S2 - Determining the phase matching conditions for a given nonlinear interaction and achieving/fulfilling this condition by exploiting birefringence properties of materials and/or QPM technique
    •S3 - Solving the nonlinear equation in parametric situations and derive analytical solutions under the undepleted pump approximation
    •S4 - Calculating nonlinear interaction performances/efficiencies in situations governed by analytical solutions or expressions
    •

    By the end of this course, students will understand  ...

    • Link between the frequency relation and the law of energy conservation, the phase matching relation and the law of momentum conservation
    • Nonlinear effects are a key points in the development of many applications in photonics (especially in relation with laser physics)
    • Nonlinear interactions lead to energy transfer between optical beams, and/or between matter and beams, enabling in some cases the realization of nonlinear optical amplification and/or oscillation.
    • Nonlinear optics is an essential tool to create novel optical frequencies generated through the interaction of incident beams within nonlinear materials
    • Nonlinear effects are subject to phase matching conditions

    By the end of this course, students will start to be able to  ...

    • Model 2nd order nonlinear effects, 
    • Evaluate nonlinear interaction performances/efficiencies under approximations that should be specified, explained and justified


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    Tutorial n° 3 & 4 : Nonlinear optics Problems

    This Problem-Based-Learning activity is devoted to solve or optimize a practical case or problem involving a nonlinear optical interaction. Each group will treat one specific subject and prepare the corresponding  deliverable(s).

    The deadline to submit your report is set for January 19, 23:59.

    • Student learning outcomes survey Archivo
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    • Proposal for the organization of the working-sessions Archivo
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    • Reports - GROUP A Tarea
      No disponible hasta que: se pertenezca al grupo Cohort 2024-25
    • Reports - GROUP B Tarea
      No disponible hasta que: se pertenezca al grupo Cohort 2024-25
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    Lecture 6: 3rd order nonlinearities (part 1)

    This chapter is dedicated to the study of 3rd order nonlinear optical effects, through which 4 waves at \omega_1, \omega_2, \omega_4, and \omega_4= \omega_1+ \omega_2 + \omega_3 could interact in a material with a non-zero third order susceptibility \chi^{(3)}.

    We will start by presenting the Optical Kerr Effect related to the capability of an intense beam to modify the refractive index of the material. The consequence on the propagation of optical wave-packets (either temporal or spatial) will be analyzed in details, showing the interdependence between optical Kerr effect and dispersion (in time domain) or diffraction (in spatial domain).


    LEARNING OUTCOMES OF THE LECTURE

    By the end of this lecture, students will know...

    • the nonlinear effects that arise in a 3rd order nonlinear materials

    By the end of this course, students will be skilled at ...

    • Solving the nonlinear equation in parametric situations and derive analytical solutions under the undepleted pump approximation
    • Calculating nonlinear interaction performances/efficiencies in situations governed by analytical solutions or expressions

    By the end of this course, students will understand  ...

    • the interplays between linear and nonlinear effects

    • Nonlinear effects are a key points in the development of many applications in photonics (especially in relation with laser physics)

    • Nonlinear interactions lead to energy transfer between optical beams, and/or between matter and beams, enabling in some cases the realization of nonlinear optical amplification and/or oscillation.
    • Nonlinear optics is an essential tool to create novel optical frequencies generated through the interaction of incident beams within nonlinear materials
    • Nonlinear effects are subject to phase matching conditions

    By the end of this course, students will be able to  ...

    • Model 3rd order nonlinear effects, 
    • Evaluate nonlinear interaction performances/efficiencies under approximations that should be specified, explained and justified
    • Determine in a situation involving the propagation of at least one light beam through a dielectric material medium (solid or guided), whether non-linear effects can be neglected.
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    Lecture 7 : 3rd order nonlinearities (part 2)

    In a first part, we will describe the so-called Four-Wave Mixing (FWM) interactions occurring in lossless materials. Similarly to  parametric interactions in \chi^{(2)} materials, third order optical parametric amplification process can be exploited to amplify a signal beam by means of its interaction with a pump beam.

    Finally, we end the lecture by illustrating resonant nonlinear interactions involving a two-photon transition on the material. It focuses on Raman and Brillouin scattering effects involving the interaction of light respectevely with optical and acoustic phonons in materials.
    We first introduce a classical description of spontaneous Raman scattering which originates from the polarizibility fluctuations of molecules due to vibrating modes and provide a quantum description insight of Raman scattering. We will show that Raman scattering can be stimulated by through a double excitation at \omega_L and \omega_S, providing that \omega_L-\omega_S match with the Raman shift \omega_v of the molecular material. Under a stimulated regime, the Stokes beam at \omega_S is amplified at the expense of the pump beam at at \omega_L, leading to a cascading regime with the generation of successive Stokes peaks at \omega_{S1}=\omega_L-\omega_v, \omega_{S2}=\omega_{S1}-\omega_v, ..., . Such a Raman cascading effect is used in Raman fiber lasers.


    LEARNING OUTCOMES OF THE LECTURE

    By the end of this lecture, students will know...

    • the nonlinear effects that arise in a 3rd order nonlinear materials

    By the end of this course, students will be skilled at ...

    • Solving the nonlinear equation in parametric situations and derive analytical solutions under the undepleted pump approximation
    • Calculating nonlinear interaction performances/efficiencies in situations governed by analytical solutions or expressions

    By the end of this course, students will understand  ...

    • Nonlinear effects are a key points in the development of many applications in photonics (especially in relation with laser physics)

    • Nonlinear interactions lead to energy transfer between optical beams, and/or between matter and beams, enabling in some cases the realization of nonlinear optical amplification and/or oscillation.
    • Nonlinear optics is an essential tool to create novel optical frequencies generated through the interaction of incident beams within nonlinear materials
    • Nonlinear effects are subject to phase matching conditions

    By the end of this course, students will be able to  ...

    • Model 3rd order nonlinear effects, 
    • Evaluate nonlinear interaction performances/efficiencies under approximations that should be specified, explained and justified
    • Determine in a situation involving the propagation of at least one light beam through a dielectric material medium (solid or guided), whether non-linear effects can be neglected.
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    Tutorial n°5 & 6 - Optical Kerr Effect

    Self-phase modulation effects and the numerical simulation of the nonlinear Schrödinger equation.

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    FORUM

    Forum for the course and tutorials.