2 Pulsed Optics C.Hirlimann With 23 Figures 2.1 Introduction Optics is the field of physics which comprises knowledge on the interaction between light and matter.When the superposition principle can be applied to electromagnetic waves or when the properties of matter do not depend on the intensity of light,one speaks of linear optics.This situation occurs with regular light sources such as light bulbs,low-intensity light-emitting diodes and the sun.With such low-intensity sources the reaction of matter to light can be characterized by a set of parameters such as the index of refraction, the absorption and reflection coefficients and the orientation of the medium with respect to the polarization of the light.These parameters depend only on the nature of the medium.The situation changed dramatically after the development of lasers in the early sixties,which allowed the generation of light intensities larger than a kilowatt per square centimeter.Actual large-scale short-pulse lasers can generate peak powers in the petawatt regime.In that large-intensity regime the optical parameters of a material become functions of the intensity of the impinging light.In 1818 Fresnel wrote a letter to the French Academy of Sciences in which he noted that the proportionality between the vibration of the light and the subsequent vibration of matter was only true because no high intensities were available.The intensity dependence of the material response is what usually defines nonlinear optics.This distinction between the linear and nonlinear regimes clearly shows up in the polynomial expansion of the macroscopic polarization of a medium when it is illuminated with an electric field E: 号=X四.E Linear optics,index,absorption +X(2):EE Nonlinear optics,second-harmonic generation,parametric effects (2.1) +x(3):EEE third-harmonic generation,nonlinear index
2 Pulsed Optics C. Hirlimann With 23 Figures 2.1 Introduction Optics is the field of physics which comprises knowledge on the interaction between light and matter. When the superposition principle can be applied to electromagnetic waves or when the properties of matter do not depend on the intensity of light, one speaks of linear optics. This situation occurs with regular light sources such as light bulbs, low-intensity light-emitting diodes and the sun. With such low-intensity sources the reaction of matter to light can be characterized by a set of parameters such as the index of refraction, the absorption and reflection coefficients and the orientation of the medium with respect to the polarization of the light. These parameters depend only on the nature of the medium. The situation changed dramatically after the development of lasers in the early sixties, which allowed the generation of light intensities larger than a kilowatt per square centimeter. Actual large-scale short-pulse lasers can generate peak powers in the petawatt regime. In that large-intensity regime the optical parameters of a material become functions of the intensity of the impinging light. In 1818 Fresnel wrote a letter to the French Academy of Sciences in which he noted that the proportionality between the vibration of the light and the subsequent vibration of matter was only true because no high intensities were available. The intensity dependence of the material response is what usually defines nonlinear optics. This distinction between the linear and nonlinear regimes clearly shows up in the polynomial expansion of the macroscopic polarization of a medium when it is illuminated with an electric field E: P ε0 = χ(1) · E Linear optics, index, absorption + χ(2) : EE Nonlinear optics, second-harmonic generation, parametric effects + χ(3) · · · EEE third-harmonic generation, nonlinear index + ··· (2.1)
26 C.Hirlimann In this expansion,the linear first-order term in the electric field describes linear optics,while the nonlinear higher-order terms account for nonlinear optical effects (eo is the electric permittivity). The development of ultrashort light pulses has led to the emergence of a new class of phase effects taking place during the propagation of these pulses through a material medium or an optical device.These effects are mostly related to the wide spectral bandwidth of short light pulses,which are affected by the wavelength dispersion of the linear index of refraction.Their analytical description requires the Taylor expansion of the light propagation factor k as a function of the angular frequency w, k(u)=k(wo)+kK(u-O)+"(w-wo)2+.. (2.2) Contrary to what happens in "classical"nonlinear optics,these nonlinear effects occur for an arbitrarily low light intensity,provided one is dealing with short (<100 fs)light pulses.Both classes of optical effects can be classified under the more general title of "pulsed optics." 2.2 Linear Optics 2.2.1 Light If one varies either a magnetic or an electric field at some point in space,an electromagnetic wave propagates from that point,which can be completely determined by Maxwell's equations. If the magnetic field has an amplitude which is negligible when compared to the electric field,the propagation equation for light can be written as 102E 1 VE- C2=40e0, (2.3) where c is a parameter,usually called the velocity of light,depending on the electric and the magnetic permittivity,so and uo respectively,of the mate- rial medium in which the wave is propagating.Equation 2.3 is a second-order differential equation,for which retarded plane waves are the simplest propa- gating solutions, E,=Re(Eet-/o）. (2.4) This particular solution 2.4 describes the propagation of a transverse electric field Ey along the positive x axis.The amplitude of the electric field varies periodically as a cosine function in time with angular frequency w and in space with wavelength A=2nc/w.At any given point x along the propagation axis the amplitude has the same value as it had at the earlier time t-z/c,when it was at the origin (x =0). A rewriting of 2.4 as
26 C. Hirlimann In this expansion, the linear first-order term in the electric field describes linear optics, while the nonlinear higher-order terms account for nonlinear optical effects (ε0 is the electric permittivity). The development of ultrashort light pulses has led to the emergence of a new class of phase effects taking place during the propagation of these pulses through a material medium or an optical device. These effects are mostly related to the wide spectral bandwidth of short light pulses, which are affected by the wavelength dispersion of the linear index of refraction. Their analytical description requires the Taylor expansion of the light propagation factor k as a function of the angular frequency ω, k(ω) = k(ω0) + k (ω − ω0) + 1 2 k(ω − ω0) 2 + .... (2.2) Contrary to what happens in “classical” nonlinear optics, these nonlinear effects occur for an arbitrarily low light intensity, provided one is dealing with short (< 100 fs) light pulses. Both classes of optical effects can be classified under the more general title of “pulsed optics.” 2.2 Linear Optics 2.2.1 Light If one varies either a magnetic or an electric field at some point in space, an electromagnetic wave propagates from that point, which can be completely determined by Maxwell’s equations. If the magnetic field has an amplitude which is negligible when compared to the electric field, the propagation equation for light can be written as ∇2E = 1 c2 ∂2E ∂t , 1 c2 = μ0ε0, (2.3) where c is a parameter, usually called the velocity of light, depending on the electric and the magnetic permittivity, ε0 and μ0 respectively, of the material medium in which the wave is propagating. Equation 2.3 is a second-order differential equation, for which retarded plane waves are the simplest propagating solutions, Ey = Re E0 eiω(t−x/c) . (2.4) This particular solution 2.4 describes the propagation of a transverse electric field Ey along the positive x axis. The amplitude of the electric field varies periodically as a cosine function in time with angular frequency ω and in space with wavelength λ = 2πc/ω. At any given point x along the propagation axis the amplitude has the same value as it had at the earlier time t − x/c, when it was at the origin (x = 0). A rewriting of 2.4 as
2 Pulsed Optics 27 R Fig.2.1.Plane-wave propagation Ey=Re(Eoet-kr）, (2.5) allows the introduction of the wave vector k of the light.Figure 2.1 shows the geometry of the propagation of a plane wave.Let us consider the plane P orthogonal to the propagation vector k,at distance x from the origin O.For any point in this plane,at distance r from the origin O,the scalar product k.r=kx =wx/c is constant;therefore,at a given time t,plane P is a plane of equal phase or equal time delay for a plane wave.A plane wave with a wavelength in the visible range(400>A>800 nm)has an angular frequency w equal to a few petahertz. It is to be noticed that a plane wave,being a simple sine or cosine function, has an infinite duration and its spectrum,which contains only one angular frequency w,is a 6 distribution.A plane wave is the absolute opposite of a light pulse！ Up to this point light has been considered as a wave,but its particle behav- ior cannot be ignored.The wave-particle duality has the following meaning. The wave description of light is continuous in both time and space,while the particle description is discrete.In our classical culture these two visions are antithetical and one cannot be reduced to the other.Nowadays physics says and experiments show that light is both continuous and discrete.In the parti- cle description light is made of photons,energy packets equal to the product of the frequency and Planck's constant hy=fw.Photons have never been observed at rest and as they travel at the speed of light,relativity implies that their mass would be zero. Fermat's principle states that the path of a ray propagating between two fixed point,A and B must be a stationary path,which in a mathematical point of view translates as 6k·dl=0, (2.6) 6 standing for a variation of the path integral and dl for an elementary path element anywhere between A and B
2 Pulsed Optics 27 r R Fig. 2.1. Plane-wave propagation Ey = Re E0 ei(ωt−k·r) , |k| = ω c = 2π λ , (2.5) allows the introduction of the wave vector k of the light. Figure 2.1 shows the geometry of the propagation of a plane wave. Let us consider the plane P orthogonal to the propagation vector k, at distance x from the origin O. For any point in this plane, at distance r from the origin O, the scalar product k · r = kx = ωx/c is constant; therefore, at a given time t, plane P is a plane of equal phase or equal time delay for a plane wave. A plane wave with a wavelength in the visible range (400 >λ> 800 nm) has an angular frequency ω equal to a few petahertz. It is to be noticed that a plane wave, being a simple sine or cosine function, has an infinite duration and its spectrum, which contains only one angular frequency ω, is a δ distribution. A plane wave is the absolute opposite of a light pulse! Up to this point light has been considered as a wave, but its particle behavior cannot be ignored. The wave–particle duality has the following meaning. The wave description of light is continuous in both time and space, while the particle description is discrete. In our classical culture these two visions are antithetical and one cannot be reduced to the other. Nowadays physics says and experiments show that light is both continuous and discrete. In the particle description light is made of photons, energy packets equal to the product of the frequency and Planck’s constant hν = ¯hω. Photons have never been observed at rest and as they travel at the speed of light, relativity implies that their mass would be zero. Fermat’s principle states that the path of a ray propagating between two fixed point, A and B must be a stationary path, which in a mathematical point of view translates as δ k · dl = 0, (2.6) δ standing for a variation of the path integral and dl for an elementary path element anywhere between A and B
28 C.Hirlimann Doubling crystal k1,0,20 k1+k2,2o k20u20 Fig.2.2.Energy and momentum conservation in the doubling geometry used to record background-free autocorrelation traces of short optical pulses.Both the en- ergy and the momentum are conserved during the mixing process The path of the light is either the fastest one or the slowest one (in the latter some cases of reflection geometry with concave mirrors). For a material particle with momentum p,Maupertuis's principle states that the integrated action of the particle between two fixed points A and B must be a minimum,which translates as 6/p:dl=0. (2.7) By analogy,the wave vector k for the light can be seen as the momentum of a zero-mass particle travelling along the light rays. When light interacts with matter,both the energy and the momentum are conserved quantities.As an example,let us consider the dual-beam frequency-doubling process used in the background-free autocorrelation tech- nique (Fig.2.2). In this geometry,two beams are incident,at an angle o,on a doubling crystal that can mix two photons with angular frequency w and momentak and k2,and produce one photon with angular frequency w2 and momentum k.Energy conservation implies that w2 =2w,while momentum conservation yields k =k1+k2,k=2k1 or k=2k2.Five rays therefore exit the doubling material:two rays are at the fundamental frequency w in the directions k1 and k2 of the incident rays,and superimposed on them are two rays with doubled frequency 2w and momentum vectors 2k and 2k2.The fifth ray,with angular frequency 2w,is oriented along the bisector of the angle a,corresponding to the geometrical sum k1+k2. 2.2.2 Light Pulses It is quite easy to produce "gedanken"light pulses.Let us start with a monochromatic plane wave,previously defined as (Fig.2.3) Ey=Re(Eoeio). (2.8) The time representation of the field is an unlimited cosine function.Con- structing a light pulse implies multiplying 2.8 by a bell-shaped function.To
28 C. Hirlimann Doubling crystal k1+k2, 2ω k2, ω, 2ω k1, ω, 2ω Fig. 2.2. Energy and momentum conservation in the doubling geometry used to record background-free autocorrelation traces of short optical pulses. Both the energy and the momentum are conserved during the mixing process The path of the light is either the fastest one or the slowest one (in the latter some cases of reflection geometry with concave mirrors). For a material particle with momentum p, Maupertuis’s principle states that the integrated action of the particle between two fixed points A and B must be a minimum, which translates as δ p · dl = 0. (2.7) By analogy, the wave vector k for the light can be seen as the momentum of a zero-mass particle travelling along the light rays. When light interacts with matter, both the energy and the momentum are conserved quantities. As an example, let us consider the dual-beam frequency-doubling process used in the background-free autocorrelation technique (Fig. 2.2). In this geometry, two beams are incident, at an angle α, on a doubling crystal that can mix two photons with angular frequency ω and momenta k 1 and k 2, and produce one photon with angular frequency ω2 and momentum k. Energy conservation implies that ω2 = 2ω, while momentum conservation yields k = k 1 +k 2, k = 2k 1 or k = 2k 2. Five rays therefore exit the doubling material: two rays are at the fundamental frequency ω in the directions k 1 and k 2 of the incident rays, and superimposed on them are two rays with doubled frequency 2ω and momentum vectors 2k 1 and 2k 2. The fifth ray, with angular frequency 2ω, is oriented along the bisector of the angle α, corresponding to the geometrical sum k 1 + k 2. 2.2.2 Light Pulses It is quite easy to produce “gedanken” light pulses. Let us start with a monochromatic plane wave, previously defined as (Fig. 2.3) Ey = Re E0 eiω0t . (2.8) The time representation of the field is an unlimited cosine function. Constructing a light pulse implies multiplying 2.8 by a bell-shaped function. To
2 Pulsed Optics 29 Fig.2.3.Schematic time evolution of the electric field of a monochromatic plane wave.Notice that because of the finite width of the figure this picture is already representative of a light pulse having a rectangular envelope Fig.2.4.Time evolution of the electric field in a Gaussian-shaped pulse.This pulse is built up by multiplying a cosine function by a Gaussian envelope function simplify further calculation,we choose to multiply by a Gaussian function.A Gaussian pulse can be written E,=Re(Eoe←rt+iot (2.9) and its time evolution is shown in Fig.2.4. I is the shape factor of the Gaussian envelope;it is proportional to the inverse of the squared duration to,i.e.It2. Let us now turn to the spectral content of a light pulse.This can be obtained by calculating the modulus of the Fourier transform of the time evolution function of the pulse.As said earlier,a plane wave oscillates with the unique angular frequency wo and its Fourier transform is a Dirac distribution 6(uo)(Fig.2.5). The Fourier transform of a Gaussian pulse is also a Gaussian function (Fig.2.6).Therefore the frequency content of a light pulse is larger than the
2 Pulsed Optics 29 t 1 –1 Fig. 2.3. Schematic time evolution of the electric field of a monochromatic plane wave. Notice that because of the finite width of the figure this picture is already representative of a light pulse having a rectangular envelope t 1 –1 Fig. 2.4. Time evolution of the electric field in a Gaussian-shaped pulse. This pulse is built up by multiplying a cosine function by a Gaussian envelope function simplify further calculation, we choose to multiply by a Gaussian function. A Gaussian pulse can be written Ey = Re E0 e(−Γ t2+iω0t) (2.9) and its time evolution is shown in Fig. 2.4. Γ is the shape factor of the Gaussian envelope; it is proportional to the inverse of the squared duration t0, i.e. Γ ∝ t −2 0 . Let us now turn to the spectral content of a light pulse. This can be obtained by calculating the modulus of the Fourier transform of the time evolution function of the pulse. As said earlier, a plane wave oscillates with the unique angular frequency ω0 and its Fourier transform is a Dirac distribution δ(ω0) (Fig. 2.5). The Fourier transform of a Gaussian pulse is also a Gaussian function (Fig. 2.6). Therefore the frequency content of a light pulse is larger than the