The dispersive properties of materials, i.e., their frequency-dependent response to the interaction with light, in most situations determines whether an optical process can be observed. Although one can always search for a specific material with the sought-after properties, this material might be far from optimum or might not even exist. Therefore, it is of great interest to develop methods that could tune the dispersive properties of a medium independently of the working frequency band. Pulses with angular dispersion, or pulse-front tilt, precisely allow us to achieve this goal. In this tutorial, we show the basics of how angular dispersion can manage to tune the dispersion parameters that characterize the propagation of light in a medium, thus permitting the observation and application of various optical processes in nonlinear and quantum optics that could not be realized otherwise. To keep the focus on first principles, the list of topics addressed is not exhaustive. More specifically, we consider the role of angular dispersion for pulse stretching and compression, broadband second-harmonic generation, the generation of temporal solitons in nonlinear media, the tunable generation of terahertz waves by means of optical rectification of femtosecond pulses, and the tuning of the frequency correlations and of the bandwidth of entangled paired photons.
© 2010 Optical Society of America
When light interacts with a material, the atoms and molecules that constitute the medium respond differently depending on the frequency of the light, ω. For low intensities, the relationship between the induced polarization and the electric field is linear, , where the constant of proportionality is the frequency-dependent susceptibility and is the permittivity of free space. The frequency dependence of the susceptibility is the so-called dispersion of the medium. One easily observable manifestation of the dispersive nature of materials was described in 1704 by Isaac Newton in his book Opticks: or a Treatise of the Reflections, Refractions, Inflections and Colours of Light . Newton showed that when a prism deflects a white-light optical beam, each of the colors that make up the white light leaves the prism in a different direction (see Fig. 1). The refractive index of the medium is wavelength dependent, and therefore the angles of refraction governed by Snell’s law correspondingly change at each interface in accordance with wavelength.
Dispersion and diffraction are the main linear effects that describe the propagation of a light beam in a medium. In the linear regime, the propagation of the light beam for a distance z can be easily described by multiplying the electric field amplitude at , , by a phase factor , where q is the transverse wavenumber, ω is the angular frequency, and k is the longitudinal wavenumber. Normally, the bandwidth of a light pulse is much smaller than its central frequency, so the wavenumber k can be expanded in a Taylor series about the central frequency. The number of terms of the expansion to be included increases with the bandwidth, but in most situations one can safely keep the expansion to second order. In this case, the group velocity and the group-velocity dispersion (GVD) are the two parameters that characterize the linear dispersion of the medium.
With the invention of the laser, it was possible to observe optical phenomena that generate material polarizations that are no longer linearly proportional to the electric field amplitude, but that depend on its higher powers. In this case, the strength of the different nonlinear terms is determined, among other things, by the nonlinear susceptibilities of the medium, , with . Examples of these nonlinear effects are the generation of harmonics [2, 3] and self-phase modulation (SPM) .
In order to observe any of these processes, the dispersive properties of the material have to be taken into account. In certain situations, the presence of dispersion is harmful and one must look for materials that exhibit negligible dispersion over a sufficiently large bandwidth. This is the case, for instance, for the generation of ultrashort pulses by means of second-harmonic generation (SHG): owing to its intrinsic dispersion, the material can severely limit the bandwidth of the generated second-harmonic pulse, imposing an effective lower limit on the duration of the pulse that can be generated. In other situations, the dispersion is not harmful, but, to the contrary, a certain amount of dispersion is required, and in addition must be of the appropriate sign. This is the case for the observation of quadratic temporal solitons in media and for the stretching and compression of pulses.
Although one can always search for a specific material with the sought-after linear dispersion properties, this material might be far from optimum or might not even exist. Moreover, certain applications need to be implemented in specific frequency bands. Even if an appropriate material can exist in one frequency band, it might be useless in another band. Therefore, it is of great interest to develop methods that could tune the dispersive properties of a medium independently of the material and the working frequency band.
In this tutorial, we will review how the introduction of angular dispersion into pulses, or pulse-front tilt, can tailor the dispersive response of any material in any frequency band. By applying angular dispersion to pulses, one can effectively generate a new group velocity and group-velocity dispersion as required by the specific application in mind. This capability has been used in different areas of nonlinear and quantum optics, but the different approaches, the emphases on different aspects of the process, and the different language used make it difficult to see what is the role of pulses with angular dispersion with a single view that might encompass all the different applications. The goal in this tutorial is to give an overall view of the role of angular dispersion that could be applied in the various applications of nonlinear and quantum optics that use these techniques.
Although the dispersive properties of a light beam that propagates in any given material can be changed in many different ways—for instance, by changing the temperature or by considering a different direction of propagation inside the nonlinear medium—these changes are related to modifications of the material properties of the media. Our interest here are the modifications of the dispersive properties introduced by changing the spatiotemporal structure of the light beam itself with appropriately designed dispersive elements, such as prisms or gratings.
In Section 2, we analyze the role of angular dispersion, or pulse-front tilt, in the propagation of light beams in dispersive media and derive the main equations that will be used throughout the tutorial. Sections 3, 4 will describe five optical process and applications that use pulses with angular dispersion.
In Section 3 we consider applications in the field of nonlinear optics. We begin with techniques used to obtain pulse stretching and pulse compression to show that, even in free space, with no dispersive medium present, one can induce an effective anomalous group-velocity dispersion that can be used to broaden or compress optical pulses. Next we analyze how pulses with pulse-front tilt can enlarge the bandwidth of the nonlinear process of second-harmonic generation, which allows frequency upconversion of ultrashort pulses and the generation of ultrashort pulses in new frequency bands. We also analyze why the use of angular dispersion allows the observation of quadratic temporal solitons. Without angular dispersion, the currently used materials exhibit neither the appropriate amount of dispersion nor the appropriate sign. Furthermore, the group-velocity mismatch (GVM) between all the interacting waves is to large to let us observe temporal solitons with reasonable intensities. Finally, we consider the generation of tunable terahertz (THz) waves by means of changing the amount of angular dispersion present.
Section 4 reviews how it is possible to tune the frequency correlations and the bandwidth of entangled photon pairs generated during the process of spontaneous parametric frequency downconversion, which is achieved by tuning the group velocity of the interacting waves by means of angular dispersion.
2. Angular Dispersion: How Does it Work?
The aim in this tutorial is to demonstrate that angular dispersion can be used as a tool in nonlinear and quantum optics. The key point of this demonstration is to show that, in an appropriate configuration, dispersive elements allow us to modify the dispersive response of materials; in particular, they allow us to control the group velocities and group-velocity dispersion of light beams traveling through them.
In this section, the effects of angular dispersion in a medium flanked by two dispersive elements will be discussed. We will derive analytical expressions for the effective group velocities and effective group-velocity dispersion of materials placed between two gratings. These expressions will be the key points used throughout the rest of the tutorial to exemplify how angular dispersion can be considered a tool in nonlinear and quantum optics.
The tutorial will be centered mainly on angular dispersion introduced by diffraction gratings. At the beginning of this section, the effects of a grating on a pulse beam impinging on it will be considered. In particular, it will be shown that angular dispersion produces pulse-front tilt; i.e., it tilts the front of a pulse by a certain angle. The value of the angle that describes this tilt will be used to quantify the effect of dispersive elements. Moreover, it will be shown that the tilt angle is indeed the parameter that will allow us to tune the dispersive properties of different media. For completeness, the derivation of the pulse-front tilt angle produced by a prism is also included.
Finally, a brief discussion about the terms angular dispersion and pulse-front tilt is presented. Although in most parts of the tutorial both terms will be used without distinction, we will see that indeed they represent different physical effects. In most situations they appear together, but one can find scenarios where there are pulses with pulse-front tilt but without angular dispersion and vice versa.
2.1. Effect of a Diffraction Grating on an Optical Beam
A reflection diffraction grating (see Fig. 2) is formed by a periodically corrugated reflecting surface, where the reflecting elements, the grooves, are separated by a distance comparable with the wavelength of light. An incident beam of light is separated, on reflection, into different waves; i.e., several reflected waves come off at different angles that mark the individual diffraction orders. The amount of energy that is actually reflected in each diffraction order depends on the specific shape of the grooves on the grating surface and can be appropriately designed . On the other hand, the directions of propagation of the reflected waves are independent of the specific shape of the grating surface and are given by the grating equation 
Let us consider an incident wave with a spectral distribution centered at angular frequency (that corresponds to a wavelength in the medium) impinging on a grating. In this case, the expression for the angular dispersion, γ, produced by a grating can be found by expanding Eq. (1) up to first order. Let us assume that the grating is oriented in such a way that the angular dispersion occurs in the x direction. The angles of incidence and diffraction at the grating can be expressed as and , where and are the central angles of incidence and diffraction, and θ and ϵ are small deviations from the corresponding central values. Analogously, . If we take into account that , , and , Eq. (1) can be written up to first order as [7, 8]
In the slowly varying envelope approximation, the amplitude of an electric field at a spatial position and at time t can be written as
Equation (2) allows us to find a relationship between the wave vectors of the incident and the diffracted waves. For clarity, we will use the subindex 1 to refer to the variables of the beam before the grating and the subindex 2 for the variables after the diffraction at the grating. When impinging on a grating, the incident field is diffracted, and each frequency component is dispersed in a different direction. Consider a grating that introduces angular dispersion in the x coordinate, as depicted in Fig. 3. In this case, the variables of interest become the x component of q and the x component of the transverse wavenumber after diffraction at the grating, . By using geometrical considerations and the fact that θ and ϵ are small, it is possible to show that and . For a grating immersed in a vacuum, the index of refraction is equal to one.
Considering that , can be rewritten as8) is independent of the specific element that introduces the angular dispersion, and it is valid for a functional form such as the one given by Eq. (2).
Equation (7) relates the transverse wavenumber of the diffracted and incident beams. The transformation of the optical beam due to the presence of the grating placed at can then be written as
To further illustrate the effects of a grating on the optical beam, let us consider the amplitude of the electric field at a distance from the grating. Neglecting the temporal and spatial broadening of the optical pulse due to diffraction and dispersion effects, propagation over a distance corresponds to the transformation9), the electric field amplitude at a distance after diffraction at the grating is
Equation (12) describes the transformation of the shape of the electric field amplitude by the grating. This provides us with a further physical insight into the grating effect. For the transverse and longitudinal components are independent; therefore, at any time, we observe that the front of the pulse is perpendicular to the propagation direction . The left-hand part of Fig. 3 depicts, for a fixed time, an input beam whose intensity is spatially described by a Gaussian function with a beam waist in the x direction . On the other hand, after the beam has been reflected by the grating, Eq. (12) tells us that the transverse and longitudinal variables are no longer independent. As a consequence, the pulse-front is no longer perpendicular to the propagation direction, as shown in the right-hand part of Fig. 3.
To clarify this idea, consider the loci of the peak intensities9, 10]2, 8) we obtain that the tilt angle for a grating is given by
From Eq. (13), we also see that for a fixed distance the maximum of the field arrives at different times, , for each transverse coordinate . Therefore, it becomes evident that the angle Φ is the angle of the loci of the peak intensities in the plane, as shown in Fig. 4.
In what follows, we will see that the tilt angle can be used as a control parameter to modify the dispersive response of a material at will, and in that way angular dispersion can become a tool in nonlinear and quantum optics.
2.2. How to Use Angular Dispersion to Control Material Dispersive Properties
Angular dispersion can be effectively and expediently used to control the dispersive properties of materials. Let us consider a medium flanked by two gratings or prisms as shown in Fig. 5. As we will see below, this configuration allows us to tune at will the group velocities, group-velocity dispersion (GVD), and higher dispersive terms of the fields propagating in the medium. It is precisely this point that makes angular dispersion an enabling tool in many areas of optics. We will see particular examples in Sections 3 and 4.
The effect of a diffraction grating on the propagation of a pulse is to tilt the front of the pulse by an angle ν, as depicted in Fig. 3. Alternatively, we can also see that the line of the loci of peak intensities is tilted by an angle Φ, as shown in Fig. 4.
To see how the particular configuration depicted in Fig. 5 can help tune the dispersive properties of the material, let us start by considering what happens when a pulse with pulse-front tilt enters a medium with a different refractive index. Let us assume that the beam enters the material normally. Making use of Snell’s law to first order in the angles and wavelength, we obtain8), changes when the beam with pulse-front tilt enters the medium. For normal incidence, the amplitude of the field at the boundary after entering the material is still .
Let us now consider the medium to be dispersive; i.e., let us let its refractive index vary with frequency. Further, we assume that the medium fills all the space between the two gratings, as shown in Fig. 5. From now on, we will work mostly with the diffracted beam, and, in order to simplify the notation, we will drop the use of the subindex 2 used to refer to the diffracted beams, so that . The electric field amplitude at any propagation distance z inside the medium can be written as11].
To understand the consequences of light with pulse-front tilt traveling through a dispersive media, let us expand up to second order about . Defining the inverse group velocity and the inverse GVD , it follows that18) into Eq. (17), we obtain the field amplitude at :
To see the effect of the second grating, let us consider a grating characterized by the parameters and . This second grating introduces angular dispersion in the x direction as well, and it is oriented in such a way that it satisfies and . With these parameters, the transformation of Eq. (9) for the second grating becomes
To simplify this expression and see more clearly the physics behind it, let us assume that the input beam has an elliptical spatial shape with a large beam width in the x direction, . Then , and the field of Eq. (21) becomes22) is restricted to propagation distances . For example, for and a walk-off angle , the propagation length should be .
From Eq. (22), we can see that the evolution of the spatial shape in the y transverse coordinate is the well-known expression for the diffraction of a beam. In the time domain, the effect of the presence of pulse-front tilt can be described by the introduction of two effective dispersive parameters [12, 13, 14]: an effective inverse group velocity,
It is important to remark that Eq. (23) refers to the effective group velocity along the propagation direction z, . In this case, the change of group velocity requires the existence of spatial walk-off . For this reason, optical beams with pulse-front tilt propagating in vacuum or in noncritical directions (with ) in birefringent crystals do not modify their group velocity when placed between two gratings. On the other hand, the situation is different for the effective GVD: even when no dispersive material is present, it is possible to observe an effective anomalous dispersion. This can be clearly seen by setting in Eq. (24) so that the effective anomalous dispersion is .
The fact that an effective anomalous dispersion always accompanies angular dispersion in a vacuum was shown in . It was demonstrated that the optical transfer function for a beam traversing a pair of gratings, arranged in tandem, contains a quadratic frequency term that is responsible of the appearance of anomalous dispersion. This fact allows GVD-free propagation to be achieved in dispersive media as demonstrated in .
Equations (23, 24) are the key results of this section. They demonstrate that by means of the tilt angle Φ, i.e., by the amount of pulse-front tilt introduced by diffraction gratings, it is possible to modify the inverse group velocity and the GVD parameters of a material placed between the two grating as depicted in Fig. 5. It is precisely this capability that will become the crucial point to make angular dispersion a tool in nonlinear and quantum optics.
The propagation of pulses with pulse-front tilt Φ produced by angular dispersion in a dispersive medium can be described by an effective inverse group velocity and an effective inverse group-velocity dispersion (GVD) given by
2.3. Angular Dispersion Produced by a Prism
Diffraction gratings are not the only optical devices that can introduce angular dispersion [17, 18]. One outstanding example is the prism . Recalling Eq. (8), the tilt angle is determined solely by the amount of angular dispersion present. For completeness, here we derive the pulse-front tilt introduced by prisms following the general procedure used previously for the grating.
Let us consider a prism with an apex angle C and refractive index . Light impinges on the prism at an angle and exits at an angle after suffering refraction at the two interfaces of the prism, as depicted in Fig. 6. Using Snell’s law to describe refraction at both surfaces, one has6.
The light impinging on the prism has a spatial and spectral distribution centered at and , respectively. The spatial and spectral variables can be conveniently written as , , and .
Notice that Eq. (27) is equal to Eq. (2), which was obtained for a diffraction grating, although each equation was derived in a different way. For the grating, Eq. (2) comes from the grating equation that is a consequence of interference at a periodic structure, while Eq. (27) comes from Snell’s law, which is a consequence of material dispersion. When expanded to first order, both devices lead to the same effect: the linear dependence of the angle of diffraction (refraction) on the angle of incidence and the wavelength. We mention that it is possible to derive a general framework that features the common working principle of different spectroscopic devices . With this approach, the dispersion produced by a prism can be treated by the same formalism as the dispersion introduced by a grating.8), one readily obtains
Let us have a first glimpse at the amount of tilt induced by a prism and a grating. Let light at illuminate a commercially available grating with . Let the input angle be , which results in an output angle of the first diffraction order of . In this case, the angular dispersion is , and the tilt angle, given by Eq. (15), is . On the other hand, for a prism  with a refractive index and at a wavelength of , the tilt angle is . The resulting values of the tilt angle give a hint that one can obtain less angular dispersion with currently used prisms than with conventional gratings. Notwithstanding, losses can be much higher in setups with gratings than in those with prisms, unless an optimized grating is designed. We should notice that a larger amount of pulse-front tilt can be obtained by combining sequences of prisms  or by using combinations of gratings and prisms .
2.4. Pulse-Front Tilt versus Angular Dispersion
Until now, we have seen that angular dispersion generates pulse-front tilt. In the cases considered, we have used both concepts indiscriminately to describe the effect of a prism and of a grating. But is the introduction of angular dispersion the only option to generate pulse-front tilt? This question was addressed in , where the authors showed that spatial chirp may also lead to the generation of pulses with pulse-front tilt. Although there may be alternative procedures, for the sake of simplicity we will restrict ourselves to this case.
Previously we analyzed the concept of the tilt angle in the spatial domain and in the spatiotemporal domain . Since the spatial chirp can be conveniently described  in the spatial and frequency degrees of freedom , in what follows we will focus our analysis on these variables. First, let us consider the amplitude of a pulse on diffraction off a grating in the variables. This is obtained by Fourier transforming Eq. (12) from the time to the frequency domain and for . To obtain analytical results, we assume the input beam to have a Gaussian shape in time and space, characterized by the widths and , respectively. With these assumptions, the complex envelope of the beam on diffraction is written as
Let us consider light beams that apart from angular dispersion display linear spatial chirp and GVD characterized by the parameters ξ and δ, respectively. The amplitude of the beam is now written as33) gives us the spatiotemporal shape of the pulse. After some straightforward calculation, one obtains that the intensity of the beam is 
Pulses diffracted by two gratings, as described in the previous sections, are a good example of pulses that show pulse-front tilt without angular dispersion. Between the two gratings, the beam propagates in free space. Performing the Fourier transform of of Eq. (21) from time to frequency, we obtain
Comparing with Eq. (33), we can see that the diffraction effects arising from the propagation from one grating to the other introduce spatial chirp and an effective second-order dispersion parameter .
One can have pulses that show pulse-front tilt and no angular dispersion. This can be achieved by introducing, for instance, spatial chirp and group-velocity disperion (GVD). The combined effect of two dispersive elements and diffraction produced by the propagation of the pulse from one element to the other can effectively generate the spatial chirp and GVD required.
3. Angular Dispersion in Nonlinear Optics
Since the invention of the laser, nonlinear optics has become one of the most fruitful areas of optics. Nonlinear effects are employed in many important applications of optical technologies, such as optical fiber communications or high-resolution imaging and detection of biological tissue. Two important concepts that are at the core of nonlinear optics are considered here: the generation of optical waves at new wavelengths, analyzed in Subsections 3.2, 3.4, and the existence of an optical entity that can exist only in the realm of nonlinear optics: the soliton. Solitons are analyzed in Subsection 3.3.
Paradoxically, the observation of most effects in nonlinear optics depends on the specific linear properties of the materials used. It is here where angular dispersion, a linear effect, plays an important role, making it possible to modify the dispersive properties of materials to allow the observation of certain nonlinear effects .
3.1. Pulse Compression and Pulse Stretching
The first two important applications that we analyze in this tutorial that use the angular dispersion introduced by a series of gratings or prisms are pulse compression and chirped pulse amplification (CPA). In both cases, the combined use of nonlinear effects and dispersive effects aims at shortening (pulse compression) or broadening (pulse stretching) pulses in the temporal domain in order to amplify ultrashort pulses.
3.1a. Pulse Compression Techniques
The goal of pulse compression techniques considered in this subsection is to modify the properties in frequency and time of a transform-limited pulse of time duration in order to generate a new pulse with a shorter duration . This can be achieved with the scheme shown in Fig. 7. The goal of the first stage is to broaden the spectrum of the pulse. For this purpose, we can use, for instance, the nonlinear effect of self-phase modulation (SPM) in an optical fiber . In SPM, the temporal width of the pulse is not changed, but because of the appearance of a quadratic temporal phase chirp, the bandwidth is enhanced. Afterward, one needs to translate the frequency broadening into pulse compression, erasing any frequency chirp introduced in the first stage. This will render the new pulse transform limited again, but this time with a shorter time duration thanks to the increased bandwidth.
Let us assume that the input pulse is a transform-limited Gaussian pulse that can be written in the temporal and frequency domains as
After Fourier transforming this equation, we get42), we can see that the frequency bandwidth of the pulse after propagation in the fiber increases and is given by43) predicts a theoretical fourfold enhancement of the bandwidth.
Equation (42) shows that SPM also introduces a positive quadratic frequency chirp of the form with
The angular dispersion introduced by gratings or prisms can be used to compensate such a quadratic chirp. As was shown in Subsection 2.2, a pair of gratings separated by a distance L in a vacuum introduces a quadratic negative frequency chirp . Therefore, to achieve compensation it is required that .
The combined effect of self-phase modulation, which broadens the bandwidth of a pulse while it propagates down an optical fiber, and of the angular dispersion introduced by dispersive elements (gratings and/or prisms), which renders the resulting pulse transform limited, allows the generation of a new, compressed pulse with a shorter time duration.
As an example, in  input pulses at the wavelength of were focused into a polarization-preserving fiber long. The authors observed a factor of 3 increase in the frequency spectrum, from to about . A pair of gratings with and a slant angle of 30° set apart were used to compress the pulse after propagation in the fiber, measuring the output pulses with a time duration of .
For the sake of simplicity, we have considered only the effects of the presence of quadratic phase terms in the frequency domain. Notwithstanding, apart from SPM, other dispersive and nonlinear effects might cause the appearance of nonquadratic frequency chirp terms, especially when we are dealing with ultrashort pulses. The effects to be included are higher-order dispersion, cross-phase modulation, self-steepening, and the self-induced Raman effect . On the other hand, pairs of gratings also introduce nonquadratic phase terms that should also be taken into account. With this information in hand, one can use appropriately engineered combinations of prisms and gratings to erase the frequency chirp generated in the nonlinear fiber. In , the authors used combinations of prisms and gratings to compensate not only for the quadratic but also for the cubic phase of ultrashort optical pulses. They obtained compressed pulses as short as .
In general, as one moves to shorter pulses and higher peak powers, the phase terms, induced by nonlinear and dispersive effects during propagation in the fiber, present more complicated shapes in an increasingly larger bandwidth. In order to compensate for these phases for generating ultrashort pulses, one needs to consider appropriately tailored arrangements of prisms and gratings that are correspondingly more sophisticated .
3.1b. Chirped Pulse Amplification
Until now, we have analyzed a two-stage pulse compression scheme: the first stage broadens the spectrum, and the second stage renders the pulse transform limited. Certain applications require not only pulse compression at a certain stage, but also pulse stretching in another stage. This is the case for chirped pulse amplification (CPA), a scheme to amplify pulses that avoids the serious damage that high peak powers of several gigawatts per square centimeter can cause to gain media due to the effect of self-focusing . To avoid this damage, one should reduce the peak power of the pulse before the pulse enters the amplifying stage.
The general scheme of CPA is shown in Fig. 8(a). The first stage is intended to reduce the peak power of the input pulse by introducing a quadratic phase term. Although this can be achieved by propagation in an optical fiber, the high powers used can nonetheless generate other undesirable effects that can be avoided by employing angular dispersion with pairs of gratings. If we again consider an input pulse given by Eq. (39) with peak power , the pulse at the end of the pulse stretching stage is written as
The second stage of CPA is the amplification of the pulse. Assuming that the gain bandwidth is broad enough to amplify the whole frequency spectrum of the pulse and that it does not introduce any frequency chirp, the final stage should compensate for the angular dispersion introduced in the first stage. Therefore, it should introduce an effective quadratic chirp of value .
The role of angular dispersion in chirped pulse amplification (CPA) is to stretch a pulse before it is amplified in order to reduce its peak power and to recompress it afterwards.
We have seen in Subsection 2.2 that a pair of gratings separated by a distance L can introduce only an effective anomalous dispersion (negative dispersion). To generate effective GVD values with a positive sign, we have to introduce elements that modify the propagation of the pulse between the two gratings. One example is shown in Fig. 8(b). Two lenses are separated a distance d from each grating. The separation between the two lenses is , where f is the focal lens of each lens. After some straightforward calculation , which takes into account the paraxial propagation of the pulse from one grating to the other, one finds that the transfer function of the system can be written as20), one can see that a quadratic phase term in frequency of the form appears as 30] several compression and stretching stages were used to generate transform-limited pulses with of energy and peak power, using mismatched grating stretcher–compressors.
3.2. Achromatic Phase Matching
3.2a. Broadband Second-Harmonic Generation
Second-harmonic generation (SHG) is a nonlinear process in which light at a certain frequency (fundamental frequency, FF) interacts with a nonlinear crystal and generates a nonlinear polarization in the medium at double frequency that causes the generation of optical radiation at the same frequency (the second harmonic, SH). At a more fundamental level, SHG is a process in which the atoms or molecules that make up the nonlinear material absorb two photons at frequency and re-emit a single photon at frequency . To do so, the interacting waves have to satisfy the energy and momentum conservation laws.
In the field of nonlinear optics, momentum conservation is usually called the phase-matching condition, which comes from the requirement to match the phases of the interacting waves [29, 31, 32]. The electric field of the fundamental and second-harmonic waves can be written as , , and the consequent phase-matching condition is50), would be difficult to satisfy.
On the other hand, for one wavelength it is relatively easy to satisfy the condition given by Eq. (50) in birefringent media where the polarization of the fundamental and second-harmonic waves are chosen to be mutually orthogonal and their refractive indices follow different dispersion curves. It is, however, more complicated to satisfy the phase-matching condition for a broader range of frequencies, which is necessary for the conversion of short light pulses that exhibit a correspondingly broader spectrum.
The k vectors of the interacting fields can be expanded in a Taylor series about the central frequency, i.e., , where are the central frequencies and are frequency deviations. The fulfilment of the phase-matching condition given by Eq. (50) over a broader range of frequencies can then be satisfied when the group velocities of the interacting waves are equal,
The GVM could be neglected when the crystal length L is much smaller than the temporal walk-off length , , which is defined as33, 34]
It is possible to increase the bandwidth of spectral acceptance by using a shorter nonlinear crystal, because the spectral acceptance is inversely proportional to . A major disadvantage is a serious reduction of the conversion efficiency, because the signal energy at the central frequency is proportional to the square of the crystal length. The decrease of the crystal length would require a corresponding increase of the input power to compensate for the reduction of conversion efficiency. By extension, the reduction of the spectral acceptance of the nonlinear crystal effectively sets a minimum duration of the SH pulse that can be achieved. For example, in , the spectral acceptance of a long BBO crystal was measured at a FF of . The obtained acceptance of imposes an approximate minimum pulse duration of , assuming a Gaussian pulse shape.
It is here where the angular dispersion comes to help. It was shown in Subsection 2.2 that with the help of angular dispersion it is possible to control the group velocities and higher-order dispersion terms. Simply changing the angle of incidence of light at a diffraction grating, we can tune the dispersive properties to the desired values.
Let us try to get a further insight into the effects of the use of pulses with angular dispersion in the process of SHG [36, 37]. Let us consider the full frequency dependence of the FF and SH wave vectors and , respectively. In order to enhance the efficiency of SHG over a larger frequency range, the phase-matching condition must be fulfilled over a broader range of frequencies. If the beam passes through a prism or a grating that introduces angular dispersion, each frequency of the outgoing diverging beam propagates in a different direction. One can choose such angular dispersion so that each frequency enters the nonlinear medium at such an angle that the phase-matching condition is satisfied for every frequency along its direction of propagation. The phase-matching condition, given by Eq. (50), requires now that
We let and denote the central wavelengths of the FF and SH waves in vacuum, respectively, and the angle of the direction of propagation of the central frequencies. The refractive indices of both waves are denoted and . If we expand Eq. (54) to first order and use the expression for the total derivative , and similarly for the SH wave, we obtain
The use of pulse-front tilt allows us to fulfill the condition by effectively increasing the temporal walk-off length . Its maximum value is obtained when the effective group velocities are equal, . Without loss of generality, let us consider a type-I SHG process, where both the FF and the SH waves might be extraordinary waves. Let us designate the spatial walk-off angles as and . The effective group velocities of the interacting waves are (see Subsection 2.2)8), one can show that the group-velocity matching requires that
Let us consider SHG in a uniaxial birefringent crystal. The walk-off angle can be written as 59) into Eq. (57), we again obtain the required angular dispersion for the group-velocity matching  given by Eq. (55).
We should notice that Eq. (55) does not explicitly show the need to use a configuration with a nonzero spatial walk-off for the FF and/or SH waves. However, using Eq. (59), we conclude that a configuration where the FF or SH waves experience spatial walk-off is required. It should be noted that this is true in collinear configurations only, where the FF and SH waves propagate in the same direction inside the nonlinear crystal. Group-velocity matching can also be obtained without spatial walk-off in noncollinear geometries, where the FF and SH waves propagate in different directions, as described in [25, 38]. In most of these configurations, even if the spatial walk-off is nonzero, it plays a minor role, because the main correction to the pulse group velocity results from the angle of the noncollinear interaction.
The previous analysis gives us a clear picture of how the use of beams with pulse-front tilt allows us to increase the spectral acceptance bandwidth of the SHG process. Through selection of the appropriate amount of angular dispersion, one generates new effective group velocities of the interacting waves, making the group velocities of the FF and SH waves equal. This is equivalent to selecting directions of propagations for each frequency such that each frequency fulfills the condition of phase-matching in its own direction.
Figure 9(a) shows the evolution of a SH pulse with GVM compensation. Thanks to the perfect phase matching over a broader range of frequencies, the SH quickly and efficiently builds up. By comparison, in Fig. 9(b) without GVM compensation, we can see the SH wave hardly appears, and after a few millimeters of propagation, it disappears completely owing to backconversion that transfers the energy back into the fundamental wave. The evolution of the slowly varying envelopes depicted in Fig. 9 is calculated by use of the evolution equations described in Subsection 3.2b.
Achromatic phase matching uses angular dispersion to modify the group velocity of the fundamental and the second-harmonic (SH) waves in the process of second-harmonic generation (SHG). When the effective group velocities are made equal, the spectral acceptance of the frequency-doubling crystal increases, which allows the generation of SH pulses with enhanced bandwidths and thus shorter time durations.
One of the advantages of the method described here is the possibility to control the group velocities of the interacting waves in any frequency region of interest and in any nonlinear crystal, which significantly extends the range of frequencies and materials that can be used for the generation of ultrashort pulses through the SHG of short input pulses. It is especially important when no materials are available that can directly be used at specific wavelengths of interest . For instance, broadband SH pulses have been generated in type-I BBO at in a long crystal , at in a long crystal , or around and at in a type-II long BBO crystal . In all of these cases, if the nonlinear crystals were used without pulse-front tilt, the bandwidth and the efficiency of the SH wave would have been severely reduced.
Another example of the capability of the pulse-front tilt technique to enhance the bandwidth of SHG was demonstrated in . The authors used wide FF pulses at that were to be upconverted in a long periodically poled crystal (PPLN). A schematic of their configuration is shown in Fig. 10. In their case, the vector phase-matching condition required an angle of between the FF beam and the grating wave vector of the poling, as shown in Fig. 10(b). The beam was focused elliptically to achieve a spot wide in the x direction in which the pulse-front tilt was introduced, and wide in the y direction. Figure 11 shows the main results. wide SH pulses at were generated that correspond to a pulse duration of if measured by autocorrelation, considering a Gaussian pulse. For the sake of comparison, the spectrum of the upconverted wave in collinear SHG with a crystal of identical length without GVM compensation was also measured. The measured spectrum was , which corresponds to in time. A noncollinear quasi-phase-matched configuration in combination with angular dispersion demonstrated a 14-fold increase of the spectral acceptance.
3.2b. Evolution Equations
Let us consider type-I SHG pumped by pulsed FF light. The electric field of the FF and SH waves can be written as with and h.c. standing for Hermitian conjugate. The evolution of the slowly varying amplitudes can be described by two coupled equations 
If the input FF signal has a large beam waist and is tilted, it can be shown that the temporal evolution of the FF and SH waves can be described by 61), we have to assume that the pulse-front tilt of the FF wave is mirrored in the generated SH wave with the same value of the tilt angle . The validity of this assumption can be verified by solving Eqs. (60) with an input FF beam with pulse-front tilt. In Fig. 12 we show the SH intensity as a function of the crystal length under different conditions. Two conclusion can be readily drawn. First, the reduced Eqs. (61) are a valid approximation of the more general Eqs. (60). Second, GVM severely diminishes SH efficiency conversion.
3.3. Solitons in Media
An initially short pulse propagating along the longitudinal direction z in a medium will broaden in time owing to the chromatic dispersion (GVD) of the material. The pulse will also broaden in the transverse spatial dimensions because of diffraction. The nonlinear interaction of the light beam with the atoms of the material modifies the main features of light propagation and can even allow, under appropriate circumstances, the temporal and spatial broadening caused by dispersion and diffraction to be overcome. The interplay among dispersion, diffraction, and nonlinearity can produce localized wave packets whose features in space or time (or both) do not change during propagation. These localized objects are called spatial or temporal solitons, respectively. When solitons present a bell-shaped intensity profile, they are called bright solitons. The localized structures that correspond to a null of the optical signal are called dark solitons. The existence of bright or dark solitons depends on the specific dispersive properties of the nonlinear media and on the type of nonlinearity that produce them.
In certain cases, when wave propagation can be described by very specific equations, such as the Korteweg–de Vries equation or the nonlinear Schrödinger equation, one can give a very precise mathematical definition of what a soliton is [43, 44]. In more real physical scenarios, many effects should be considered that make the evolution equations more complicated. Nevertheless, the equations that describe the evolution of the relevant physical parameters still predict the existence of localized waves with solitonlike behavior. To encompass such solitary waves, one only needs to generalize the definition of what a soliton is [45, 46]. In general, a soliton is a localized structure that propagates undistorted over long distances because of the balance between dispersion and diffraction on the one side and the nonlinearity of the material on the other side.
As has already been said, optical solitons can be considered in the temporal or spatial domains. Consider a light beam that propagates in a medium. If in some way the transverse spatial shape of the optical field is confined (for example in a waveguide), the degrees of freedom of interest that describe the light propagation are the direction z and the temporal variable t. In this case, the solitons are referred to as temporal. On the other hand, if the temporal variable can be considered constant, for example by using long pulses or continuous-wave (cw) lasers, the spatial variables will govern the wave propagation and the solitons are referred to as spatial.
The nonlinear effect responsible for the generation of optical solitons depends on the particular scheme being used. For example, parametric interactions in media allow the generation of multicolor solitons formed by waves with different frequencies. The formation of these so-called quadratic solitons is mediated by the interaction of the FF and the SH waves in a SHG geometry. On the other hand, self-focusing due to cubic Kerr nonlinearity allows the observation of temporal solitons when single-mode optical fibers are used , and the photorefractive effect in electro-optic materials can create a saturable nonlinear refractive index, where photorefractive solitons can be observed .
In this subsection of the tutorial, we will concentrate on quadratic solitons, i.e., the solitons produced in materials in a SHG geometry. The key point to understand the enabling role of angular dispersion in the generation of solitons can be seen by defining the effective lengths that determine the relevant quantities that enter into play and by writing the equations that describe the evolution of the FF and SH waves as a function of these new variables . If we consider a typical temporal width , beam waists and , and a value of the peak amplitude N, it is possible to define a normalized field amplitude, , and spatial and temporal dimensionless variables , , , . With these definitions, and using the subindex 1 for the FF wave and the subindex 2 for the SH, the evolution equations (60) become
- Dispersion length (FF),
- Dispersion length (SH),
- Diffraction length (FF),
- Diffraction length (SH),
- Spatial walk-off length,
- Temporal walk-off length,
- Absorption length (FF),
- Absorption length (SH),
- Nonlinear length (FF),
- Nonlinear length (SH),
- Coherence length,
For the sake of simplicity, notice that the diffraction length is defined with the value of the waist , corresponding to the size in the dimension in which the soliton will be formed. Angular dispersion is introduced in the transverse dimension x. On the other hand, the spatial walk-off length is defined with the same value of the waist, because the walk-off angle lies in the plane of angular dispersion, i.e., in the direction x. For both and , the quantity of interest is .
The balance between the linear effects included in Eqs. (62) and the nonlinear parametric interaction between the FF and SH waves is mediated by the ratio of the characteristic lengths corresponding to the dispersion and diffraction lengths. The observation of different types of solitons implies that the relationship between all of these characteristic lengths is tailored accordingly.
In order to observe one-dimensional and two-dimensional spatial quadratic solitons, there are three main requisites that have to be satisfied: (a) material dispersion must be small enough to render the temporal effects on the propagation negligible, (b) the crystal length must be larger than, or at least comparable with the diffraction length in order make the effects of nonlinear induced focusing observable, and (c) the spatial walk-off length must be larger than, or at most comparable with the diffraction length so that solitons can be excited with the currently available peak powers [50, 51].
Spatial quadratic solitons have been observed in one dimension . Materials with appropriate dispersive properties were chosen such that the use of cw, or even picosecond pulses, renders negligible all the temporal effects. For the case of one dimension, a planar waveguide of was used to confine the beam in one transverse dimension. A cw beam with a waist in the nonconfined dimension of yields a diffraction length of and . In addition, the spatial walk-off is negligible, and hence the conditions for the generation of spatial solitons are satisfied.
Spatial quadratic solitons have also been observed in two dimensions . They were observed in a type-II KTP bulk sample pumped by a laser pulse at . The beam waist of the input FF beam was , leading to , and the walk-off angles were and , giving . With these values it is easy to see that it is possible to observe spatial solitons, since and .
For observing temporal solitons, restrictions analogous to the ones described for spatial solitons have to be satisfied: (a) spatial effects on the beam propagation must be minimized, for example, by using beams with a large waist size, (b) the length of the nonlinear crystal must be larger than, or at least comparable with the dispersion length, (c) the temporal walk-off length must be larger than, or at most comparable with [50, 51] the dispersion length , and (d) the dispersion must have the appropriate sign of anomalous dispersion (i.e., ) to support the existence of bright solitons. The conditions to obtain spatial and temporal solitons are summarized in Table 1.
From the conditions described above for the observation of temporal solitons, it is clear that the possibility of controlling the group velocities and and GVDs and is crucial for the observation of temporal solitons. As shown in Subsection 2.2, this is precisely what angular dispersion allows us to do, and it thus gives us a tool for the generation of temporal quadratic solitons.
For a quantitative explanation of the use of angular dispersion for the generation of solitons, let us consider the generation of spatiotemporal solitons in a type-I crystal pumped by a laser pulse centered at . For , the refractive index at the phase-matching angle is ; the GVM between the fundamental and the SH waves is ; the GVD for the FF wave is ; the GVD for the SH wave is ; and, finally, the FF and SH walk-off angles that lie in the x direction are and , respectively. For the generation of spatiotemporal solitons, the spatial pump profile is elliptical. In the spatial dimension in which the soliton will be formed, let us say y, the waist is chosen such that the diffraction length is smaller than the length of the crystal. For this type of experiment, typical crystal lengths are of the order of a few millimeters; so the waist in the y direction is set to be .
All of these values allow us to calculate the effective length parameters and determine the feasibility of observing quadratic spatiotemporal solitons. The numerical values for the effective lengths relevant to Table 1 are
With these numbers, it is possible to see that for a long crystal, and condition (b) for spatial solitons is satisfied. On the other hand, in the temporal domain, , , and the dispersion is normal . It is not then possible to observe quadratic temporal solitons, since the conditions listed above for the observation of temporal solitons are not satisfied.
To observe temporal solitons in the discussed example, it is necessary to reverse the sign of the GVD parameters, reduce the effects of the GVM, and make the dispersion length smaller than the length of the material. As was mentioned before, and from the discussion in Subsection 2.2, one way to accomplish this is by using angular dispersion. Consider the case in which the crystal is flanked by two gratings of . When the pump beam impinges on the grating at an angle (the output angle of the first-order diffraction would be ), angular dispersion is introduced in the x dimension, and the front of the pulse becomes tilted by a tilt angle . In order for Eqs. (23, 24) to be applicable, the pump beam has an elliptical spatial distribution: is a few millimeters, and .
According to the second term of Eq. (24), the tilt angle introduces an additional anomalous dispersion of . The new effective GVM is . And the new GVDs at the FF and SH waves are and , respectively. The new effective characteristic lengths obtained by inserting the gratings are
The values previously obtained with no pulse-front tilt are shown again for the sake of comparison. It can now be seen that the conditions for the observation of spatiotemporal solitons are satisfied: , the magnitude of the GVM between the FF and SH waves has been highly reduced so that , and the dispersion has become anomalous, thus enabling the excitation of bright solitons.
In the experimental implementations of spatiotemporal quadratic solitons, BBO and are some of the materials most widely used. The first temporal solitons were observed in a type-I BBO crystal, where pulses of duration at were injected . Shortly after , spatiotemporal quadratic solitons were observed in a type-I crystal with highly elliptical pulses at . In later work , the generation of spatiotemporal solitons in BBO and was studied in detail.
A typical experimental setup used to generate spatiotemporal solitons with the help of angular dispersion  is shown in Fig. 13. In this setup, the gratings that introduce angular dispersion and modify the dispersive properties of the nonlinear crystal are clearly seen. Cylindrical lenses to control the spatial shape of the beam and guarantee the necessary ellipticity that allows the tailoring of group velocities and GVD are also depicted.
The presence of spatiotemporal solitons generated with the setup of Fig. 13 was demonstrated by measuring the temporal pulse duration and the beam waist of the soliton wave in the y transverse dimension at different propagation distances . Figure 14 shows the experimental results: the dashed curves represent the temporal and spatial broadening that the optical wave would suffer if the nonlinearity were not active, i.e., working at a low pump power. On the other hand, the experimental black points were measured for a high pump power and reflect the fact that the temporal and spatial widths of the soliton wave do not change during propagation thanks to the interplay among the nonlinear effect, dispersion, and diffraction. This is precisely the fact that reveals the presence of spatiotemporal solitons.
3.4. Generation of Terahertz Waves
The generation of electromagnetic pulses at THz frequencies is of great interest in various fields. One of the main areas of application of THz waves is the probing and detection of materials, since the characteristic energies of many interactions in molecules occur in this region . These waves with submillimeter wavelengths at the crossing of the far-infrared and microwaves are, in general, not that easy to produce because their wavelength is too long for optical devices and too short for electronic circuits.
One of the ways to produce THz waves is to use a special case of frequency difference generation, called “optical rectification.” In this process, a photon at frequency is absorbed by an atom of a nonlinear medium, and two new photons are generated: one at the optical frequency , and another one at a much lower frequency (THz). For the process of optical rectification to be efficient, the phase-matching conditions between all the interacting waves have to be satisfied.
Let us first consider a case in which the optical and the THz waves propagate in the same direction z (collinear configuration). The pump beam is an intense optical pulse with central frequency , and a large beam area (plane wave). The electric field of the pump writes , where is the wavenumber of the optical pulse, is the refractive index at the central frequency, and h.c. stands for Hermitian conjugate. The slowly varying amplitude can be written as
When the conditions for optical rectification are satisfied, the pump beam that interacts with a nonlinear crystal generates a nonlinear polarization at the THz frequency Ω, which is written as 64) reveals that the phase of the nonlinear polarization goes as . So for the newly generated THz wave to build up, the phase-matching condition65) in a Taylor series to first order, we obtain 
In materials with a large nonlinear coefficient, such as , is much larger than in the frequency band of interest . For instance, for a pump at , and ; so collinear phase matching is not possible in this case. However, if we could modify the optical group index it would then be possible to satisfy the phase-matching condition given by Eq. (67). This is precisely what can be achieved by introducing angular dispersion in the pump beam as described in Subsection 2.2: if a laser pulse acquires pulse-front tilt, the group index is changed. To clarify this idea we can rewrite Eq. (23) in terms of the group index. Noticing that , we obtain
In a more general case, THz generation occurs in a noncollinear geometry; i.e., the optical and THz waves do not propagate along the same direction. Figure 15 depicts the general configuration considered where the THz wave is generated at an angle with respect to the pump beam direction of propagation. Different optical frequencies, ω and , propagate inside the nonlinear crystal in different directions, and , determined by the amount of angular dispersion ν introduced in the pump. In the noncollinear case, the phase-matching condition must be written in the vector format 60]
Equation (71) is the key expression that highlights the role of angular dispersion in the generation of THz waves by means of optical rectification. By changing with the help of dispersive elements, we can tune the noncollinear angle and satisfy the phase-matching conditions. Furthermore, the introduction of different amounts of angular dispersion allows the tuning of the frequency of the generated THz wave .
In a dispersive medium, the tilt angle Φ is given by Eq. (8), and the relationship between the angles ν and Φ is given by Eq. (14), which if rewritten in terms of the refractive index and the group index of the optical wave yield 8, 72), we see that ; i.e., the angle of propagation of the THz wave is perpendicular to the pulse front that is tilted at an angle ν in the plane (dashed lines in Fig. 15).
The efficiency of the generation of THz waves by means of the process of optical rectification of femtosecond pulses depends on the fulfillment of the phase-matching condition between the optical and THz waves. By using optical pulses with pulse-front tilt, not only do we make it possible to comply with the phase-matching condition, but also we can tune the THz frequency at which this happens, which adds tunability to the scheme.
The full potential of using tilted pulses to tune the frequency of the generated THz waves was demonstrated in . This is shown in Figs. 16, 17. The THz output was tuned between 1 and in at a temperature of by changing the tilt angle ν between 59° and 64°. Figure 16 shows the spectra of the THz waves when the tilt angle ν is changed. Figure 17 plots the change of the central frequency of the THz waves as a function of the tilt angle. The measured energy is also shown. The capability of generating even higher frequencies would require the use of pulses with pulse durations below in less absorbing materials.
We remark that the use of pulse-front techniques allows the use of pumps in different frequency bands that may be more optimal. For instance, pulse energies of up to with a spectral bandwidth of up to were obtained by optical rectification of laser pulses with energy and pulse duration , achieving a conversion efficiency of , an order of magnitude higher than the one measured when using other materials in an optimized geometry.
4. Angular Dispersion as an Enabling Tool in Quantum Optics
Until now, we have seen the role of angular dispersion in applications in the field of nonlinear optics. We may regard these applications as belonging to classical optics in the sense that we are dealing with large amounts of photons whose properties can accurately be described by the classical Maxwell equations. In this section, on the other hand, we will see that angular dispersion is also useful in quantum optics when, for example, pairs of photons are created.
We will focus on the generation of pairs of photons by means of the process of spontaneous parametric downconversion (SPDC). SPDC is a nonlinear optical process in which an intense pump impinges on a nonlinear material and occasionally creates a pair of photons of lower frequencies. The correlations of the photon pairs produced by SPDC are of particular interest from both the fundamental and the practical points of view. Fundamentally, SPDC photon pairs are present at the core of many experiments to test the validity of the foundations of quantum mechanics. And, practically, new technologies based on their correlations promise improvements over their classical counterparts, among them quantum communications and information processing or clock synchronization.
In this tutorial, we will concentrate on the frequency properties of the SPDC photons, namely, on the bandwidth and the type of frequency correlations between the two photons. The appropriate frequency content of paired photons depends on the particular application under consideration [64, 65, 66, 67]. Some applications require frequency-correlated photons, some require frequency-anticorrelated photons, and some require frequency-uncorrelated photons. For this reason in recent years various methods to tailor at will the spectral properties of paired photons have been developed [68, 69, 70, 71].
In this section, we will see how the use of light beams with pulse-front tilt in SPDC allows us to control the bandwidth and the type of frequency correlation. These two frequency properties can be tuned by a proper tailoring of the group velocities and GVD parameters of all the waves that interact in the nonlinear process [72, 73]. As we have seen throughout the tutorial, this tailoring is precisely what angular dispersion enables. Indeed, it will be the pulse-front tilt angle that will play the role of a control parameter to tune, at will, the bandwidth and the type of frequency correlations of the SPDC photons. Remarkably, when using pulse-front tilt, there is no need for any particular engineering of the SPDC source. Moreover, the method is independent of the material and can be used in any frequency range where other methods do not work.
As we will see in what follows, angular dispersion allows us to increase the bandwidth of the joint spectrum of paired photons, an important point for the generation of very narrow temporal biphotons. In addition, the use of pump beams with pulse-front tilt makes possible the generation of frequency-correlated, frequency-anticorrelated, and even uncorrelated photon pairs. The latter case offers a very attractive applicability of angular dispersion for the generation of heralded indistinguishable and pure single photons with a tunable frequency bandwidth .
4.1. Angular Dispersion in Spontaneous Parametric Downconversion
Let us consider the generation of SPDC photons in a collinear configuration depicted in Fig. 18 when the two downconverted photons propagate in the same direction. Unlike the typical SPDC setups, the nonlinear medium is placed between two diffraction gratings that introduce angular dispersion. It can be shown that after the two gratings, the quantum state of the downconverted photons can be written as 74), we can see that the frequency content of the downconverted light is determined by the length of the crystal L, by the spectral characteristics of the pump beam , and by the dispersive properties of the nonlinear material expressed by the phase-mismatch term along the longitudinal direction, , with , where is the refractive index and .
The role of angular dispersion is to modify due to the dependence of on q. Inspection of Eq. (20) shows us that angular dispersion can thus modify the frequency shape of the phase-matching function . After the second grating, downconverted photons are detected by using strong spatial filters, e.g., single-mode fibers, so that . To get a further physical insight, let us expand up to second order. Considering walk-off for the pump, the signal, and the idler,23, 24): and . We assume that there is perfect phase matching for . From these expressions, it is clear that, as described in Subsection 2.2, angular dispersion allows us to modify the effective inverse group velocity and effective GVD. Employing pump pulses tilted by an angle Φ in SPDC configurations, it is possible to control the frequency properties of paired photons.
4.2. Tunable Control of Frequency Correlations of Paired Photons
To see how angular dispersion can be used to control the type of frequency correlation of SPDC photons, let us consider the first-order terms of Eq. (75). One obtains frequency-anticorrelated photons when and frequency-correlated photons when . Frequency-uncorrelated photons are obtained if the effective group velocity of the pump is equal to that of the signal or the idler, and/or .
Figure 18 shows the experimental arrangement used to measure the joint spectrum of the entangled paired photons. Before being detected, each of the photons forming a pair passes through its respective monochromator, which is scanned to measure the joint spectrum . Figure 19 shows the experimental results that demonstrate the feasibility to fully control the frequency correlations in SPDC via angular dispersion. In particular, the measurements were performed by using a type-II BBO crystal cut for collinear degenerate phase matching.
In Fig. 19, different types of frequency correlation are observed by varying the tilt angle Φ. The first row of Fig. 19 shows the case with no tilt. As expected for a pulsed pump and type-II phase-matching, the spectra of the signal and idler photons are different, one being narrower than the other, which is a consequence of their different group velocities. The following rows correspond to different values of the tilt angle. The second and fourth rows of Fig. 19 depict the cases of highly frequency-anticorrelated and highly frequency-correlated photons. The third row shows the interesting case of frequency-uncorrelated pairs.
The frequency uncorrelation observed in the third row of Fig. 19 is indeed a signature of the presence of a separable quantum state. To demonstrate full separability of the two-photon state, it is also required that there be no phase entanglement . Theoretically, we can calculate the Schmidt decomposition of the state given by Eq. (73) when the separability condition of the group velocities is fulfilled. In that case, the entropy of entanglement is nearly 1, and the Schmidt decomposition contains only one mode, revealing the separability of the quantum state . In this way, the scheme offers a possibility to generate paired photons in a separable state, a quantum state so desired in quantum information processing applications. Frequency-uncorrelated photons can well serve as a source of heralded single photons: the detection of one of the photons heralds the presence of its twin photon in the setup without in any way changing its state. Experimentally, a tomographic analysis or a four-photon experiment such as the one described in  would be needed to fully demonstrate the separability and therefore the purity of the generated single photons.
The frequency correlations of entangled paired photons generated in the process of spontaneous parametric downconversion can be tuned independently of the frequency band and of the nonlinear crystal used. The photons can exhibit frequency anticorrelation, frequency correlation, and even frequency uncorrelation only through control of the amount of angular dispersion.
4.3. Controlling Bandwidth of Paired Photons
As we have mentioned before, Eq. (74) tells us that the bandwidth of the downconverted light is determined by the length and the dispersive properties of the nonlinear material, by the geometry of the SPDC configuration (collinear or noncollinear, type I or type II) and by the spectral characteristics of the pump beam. In the following, we will see that the use of SPDC with a pump beam with angular dispersion allows us to control the SPDC bandwidth as well .
To further clarify the ideas, we consider the case of a narrowband pump, for example, a picosecond pulsed laser. In this case, energy conservation dictates , and Eq. (75) reduces to72]. If the first-order terms of are removed by using angular dispersion, the dependence of the bandwidth on the length will go as . In addition, the higher dispersion terms are much weaker, which further broadens the spectrum. In a type-I SPDC process, if the tilt Φ is such that , the first nonzero terms in are of fourth order , and the dependence of the bandwidth on the length of the crystal goes as .
The values of the tilt angle that maximize the bandwidth for type-II and type-I processes are 
The effect of introducing angular dispersion to increase the bandwidth of the SPDC photons was demonstrated experimentally by using the setup of Fig. 18 with a BBO crystal cut for degenerate type-II collinear phase matching. Figure 20(a) depicts the joint spectrum for the case without angular dispersion, and Fig. 20(b) shows the joint spectrum when a tilt was introduced.
To get a quantitative value of the increase in the bandwidth, let us examine Fig. 21. The solid curves represent the theoretical predictions, and the dots are the experimental results. Figures 21(a), 21(b) correspond to the spectra of single counts of the signal and idler photons. The curve on the left corresponds to no tilt, , and the one on the right to the case of . For the case with no gratings, a FWHM bandwidth of is obtained, while for the case with gratings, the FWHM bandwidth is , leading to a sevenfold increase of the bandwidth. Figures 21(c), 21(d) depict the coincidence counts. The widths along the antidiagonal line (at ) of the joint spectra, Figs. 20(a), 20(b), are plotted. The coincidence width of broadens to .
The effect of spectrum broadening can be employed to generate a very short temporal biphoton that is given by the Fourier transform of the joint spectral amplitude . It was shown in  that when the pulse-front tilt is used, the spectral phase profile remains smooth and very flat, which translates into ultrashort nearly transform-limited biphotons with a temporal correlation of a few femtoseconds. The detection of one of the photons of the pair determines the detection of the other photon located in a distant place within a time window given by the biphoton’s temporal width. This phenomenon could expediently be used, e.g., for clock synchronization. The pulse-front-tilt technique contrasts with other methods in which the increase in bandwidth is not directly accompanied by a decrease of the correlation time. That can be caused by a particular shape of the spectral phase or when the phase relationship between individual frequencies, and thus coherence, is lost. This is, for example, the case of white-light continuum generated by Kerr self-phase modulation.
Up to this point we have considered the effects of angular dispersion when a collinear geometry is used. However, it is worth mentioning that the combination of the pulse-tilt techniques described above with the use of noncollinear geometries further expands the possibilities to control the joint spectrum of paired photons . In noncollinear geometries it is possible to map the spatial characteristics of the pump beam into the spectra (spatial-to-spectral mapping) [78, 79], providing another way to manipulate the joint spectral amplitude of the biphoton.
The bandwidth of paired photons generated in the process of spontaneous parametric downconversion can be enhanced or reduced, independently of the frequency band and of the nonlinear crystal used. It is possible to generate spectral bandwidths of up to hundreds of nanometers in the visible optical range that translate into temporal correlations between the two photons of just a few femtoseconds.
The angular dispersion of light is an old physical phenomenon that was already discussed at the beginning of the 18th century by Isaac Newton in his book Opticks , where he describes how white light decomposes into colors and diverges after passing through a prism.
In this tutorial we have shown how, with the appearance of the laser, angular dispersion has become an important enabling tool in different areas of nonlinear and quantum optics. The key tool is the possibility to modify the dispersive properties of materials by using light pulses with suitable amounts of angular dispersion. The use of these pulses in many applications has been described with such an unifying view.
Most times, this common perspective is absent in scientific papers and technical reports, or at least is not clearly seen, because of the use of different notation in each field or because emphasis is put on diverse aspects each time.
Section 2 of this tutorial hopefully offers such a view, and each application considered in Sections 3, 4 is analyzed under the general unifying framework developed in Section 2. The first two applications, pulse compression and CPA, described in Subsection 3.1, are nowadays routinely used in commercial systems for compressing and stretching optical pulses. The next application is achromatic phase matching, considered in Subsection 3.2, which enables us to enhance the capability of frequency doublers for efficiently doubling ultrashort pulses.
The excitation of quadratic temporal solitons is described in Subsection 3.3, where it is shown that the introduction of angular dispersion permits the observation of temporal solitons. In Subsection 3.4, we have described how the use of pulses with pulse-front tilt allows us to satisfy the condition of phase matching, a requisite not easily achievable, but notwithstanding necessary, for the efficient generation of THz waves in the process of optical rectification of femtosecond laser pulses. Even more, the method allows us to tune the frequency of the generated THz wave, allowing the implementation of tunable generators of THz waves. Finally, the generation of entangled paired photons with tunable bandwidth and frequency correlations is analyzed in Section 4.
Subsection 3.3 and Section 4 are two outstanding cases that exemplify the role of light beams with angular dispersion. By altering the unfavorable conditions offered by most natural materials, the use of angular dispersion makes it possible to observe physical effects that would not be possible otherwise.
The necessary conditions for the observation of temporal solitons in quadratic nonlinear media are not met in commonly used nonlinear crystals. This is also the case for the observation of frequency-entangled photons that show frequency correlation or frequency uncorrelation. In both cases, pulses with angular dispersion allow us to modify the dispersive properties of media, effectively engineering new materials that meet the necessary requirements in terms of new effective group velocity and GVD parameters.
Summarizing, light beams with angular dispersion, or pulse-front tilt, allow us to perform tasks in nonlinear and quantum optics not possible otherwise, highlighting the role of angular dispersion as enabling tool.
We thank F. Wise, A. Schober, and J. Hebling for providing some of the figures that appear in this tutorial, helping to greatly improve its clarity. We give special thanks to L. Torner and S. Carrasco for long and helpful discussions during many years about the role of angular dispersion in nonlinear optics, especially concerning the generation of quadratic spatiotemporal solitons and the control of the frequency correlations of entangled photons. We thank X. Liu, F. Wise, and P. Di Trapani for occasional, but illuminating, discussions about solitons. We thank P. Loza and D. Artigas for helpful discussions concerning the process of SHG. Finally, we also thank R. Trebino and S. Akturk for discussions about the relationship between angular dispersion and pulse-front tilt. This work was supported by the European Commission (Qubit Applications, contract 015848), by the Government of Spain (Consolider Ingenio CSD2006-00019, FIS2007-60179), and was supported in part by FONCICYT project 94142.
Tables and Figures
1. I. Newton, Opticks: or a Treatise of the Reflections, Refractions, Inflections and Colours of Light, 4th ed. (William Innys, 1730), http://books.google.com/books?id=XXu4AkRVBBoC.
2. Y. R. Shen, Principles of Nonlinear Optics (Wiley-Interscience, 2002).
3. P. N. Butcher and D. Cotter, Elements of Nonlinear Optics (Cambridge Univ. Press, 1991).
4. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
5. Richardson Gratings, Diffraction Grating Handbook (Newport, 2000), http://gratings.newport.com/information/handbook/handbook.asp.
6. H. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1983).
7. O. E. Martínez, “Pulse distortions in tilted pulse schemes
for ultrashort pulses,” Opt. Commun. 59, 229–232
8. O. E. Martínez, “Grating and prism compressors in the case of
finite beam size,” J. Opt. Soc. Am. B 3, 929–934
9. J. Hebling, “Derivation of the pulse front tilt caused by
angular dispersion,” Opt. Quantum
Electron. 28, 1759–1763
10. J. Hebling, K. L. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by
tilted-pulse-front excitation and their application
possibilities,” J. Opt. Soc. Am. B 25, B6–B19
11. V. G. Dimitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Crystals
(Springer, 1997). [CrossRef]
12. J. P. Torres, S. Carrasco, L. Torner, and E. W. Van Stryland, “Frequency doubling of femtosecond pulses in
Opt. Lett. 25, 1735–1737
13. R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial
walk-off in collinear three-wave interaction with tilted
pulses,” Opt. Lett. 21, 973–975
14. R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric
generator/amplifier of visible femtosecond pulses,”
IEEE J. Quantum Electron. 34, 459–464
15. E. B. Treacy, “Optical pulse compression with diffraction
gratings,” IEEE J. Quantum Electron. 5, 454–458
17. O. E. Martínez, J. P. Gordon, and R. L. Fork, “Negative group-velocity dispersion using
refraction,” J. Opt. Soc. Am. A 1, 1003–1006
19. J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, 1995).
20. J. Hebling and Z. Márton, “Theory of spectroscopic
devices,” J. Opt. Soc. Am. B 23, 966–972
22. O. E. Martínez, “Hybrid prism-grating ultrashort pulse
compressor,” Opt. Commun. 83, 117–122
24. X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast
optics,” Opt. Commun. 242, 599–604
25. J. A. Fullop and J. Hebling, “Applications of tilted-pulse-front excitation,” in Recent Optical and Photonic Technologies, K. Y. Kim, ed. (Inteh, 2010), chap. 11, http://sciyo.com/books/show/title/recent-optical-and-photonic-technologies?PHPSESSID=0imuiv8409jhevei6sb4toh0d5
26. C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, “Compression of femtosecond optical
pulses,” Appl. Phys. Lett. 40, 761 (1982). [CrossRef]
27. R. L. Fork, C. H. Brito Cruz, P. C. Becker, and C. V. Shank, “Compression of optical pulses to six
femtoseconds by using cubic phase compensation of femtosecond optical
pulses,” Opt. Lett. 12, 483–485
28. D. Strickland and G. Mourou, “Compression of amplified chirped optical
pulses,” Opt. Commun. 56, 219–221
29. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics
(Wiley, 1991). [CrossRef]
30. Y. Zaouter, J. Boullet, E. Mottay, and E. Cormier, “Transform-limited pulses from a nonlinear-fiber chirped-pulse amplifier using a
mismatched grating stretcher-compressor,” Opt.
Lett. 33, 1527–1529
31. A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, 1997).
32. R. W. Boyd, Nonlinear Optics (Academic, 2008).
33. A. Yariv, Quantum Electronics (Wiley, 1975).
34. P. W. Milonni and J. H. Eberly, Lasers (Academic, 1988).
35. G. Szabó and Zs. Bor, “Frequency conversion of ultrashort
pulses,” Appl. Phys. B 58, 237–241
36. O. E. Martínez, “Achromatic phase matching for second
harmonic generation of femtosecond pulses,” IEEE
J. Quantum Electron. 25, 2464–2468
37. V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strizhevskii, “Method for compensating the phase-matching
dispersion in nonlinear optics,” Sov. J. Quantum
Electron. 4, 1090–1098
40. G. Szabó and Zs. Bor, “Broadband frequency doubler for femtosecond
pulses,” Appl. Phys. B 50, 51–54
41. B. Richman, S. Bisson, R. Trebino, E. Sidick, and A. Jacobson, “Efficient broadband second-harmonic
generation by dispersive achromatic nonlinear conversion using only
prisms,” Opt. Lett. 23, 497–499
42. A. Schober, M. Charbonneau-Lefort, and M. Fejer, “Broadband quasi-phase-matched
second-harmonic generation of ultrashort optical pulses with spectral angular
dispersion,” J. Opt. Soc. Am. B 22, 1699–1713
43. M. J. Ablowitz, Solitons and the Inverse Scattering Transform
(Society for Applied and Industrial Mathematics
(SIAM), 1981). [CrossRef]
44. P. G. Drazin and R. S. Johnson, Solitons: an Introduction
(Cambridge Univ. Press,
45. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, 1997).
46. A. Hasewaga and Y. Kodama, Solitons in Optical Communications (Clarendon, 1995).
47. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse
narrowing and solitons in optical fibers,” Phys.
Rev. Lett. 45, 1095–1098
49. C. R. Menyuk, R. Schieck, and L. Torner, “Solitary waves due to cascading,” J. Opt. Soc. Am.
B 11, 2434–2443
51. S. Carrasco, J. P. Torres, L. Torner, and F. Wise, “Walk-off acceptance for quadratic soliton
generation,” Opt. Commun. 191, 363–370
52. R. Schieck, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due
to cascaded second-order nonlinearities in planar
waveguides,” Phys. Rev. E 53, 1138–1141
53. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial
solitary waves in a quadratic medium,” Phys. Rev.
Lett. 74, 5036–5039
54. X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal
solitons,” Phys. Rev. Lett. 82, 4631–4634
55. P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second
harmonic generation with tilted pulses,” Phys.
Rev. Lett. 81, 570–573
56. X. Liu, K. Beckwitt, and F. W. Wise, “Two-dimensional optical spatiotemporal
solitons in quadratic media,” Phys. Rev.
E 62, 1328–1340
57. F. Wise and P. Di Trapani, “The hunt for light bullets–spatiotemporal solitons,” Opt. Photon. News 13(2), 29–32 (February 2002).
58. J. Hebling, K.-L. Yeh, K. A. Nelson, and M. C. Hoffmann, “High-power THz generation, THz nonlinear
optics, and THz nonlinear spectroscopy,” IEEE J.
Sel. Top. Quantum Electron. 14, 345–353 (2008). [CrossRef]
59. K. L. Vodopyanov, “Optical generation of narrow-band terahertz
packets in periodically-inverted electro-optics crystals: conversion efficiency
and optimal laser pulse format,” Opt.
Express 14, 2263–2276
61. A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almasi, and J. Hebling, “Scaling up the energy of THz pulses created
by optical rectification,” Opt. Express 13, 5762–5768
62. J. Hebling, A. G. Stepanov, G. Almasi, B. Bartal, and J. Kuhn, “Tunable THz pulse generation by optical
rectification of ultrashort pulses with tilted pulses,”
Appl. Phys. B 78, 593–599
66. A. Valencia, G. Scarcelli, and Y. H. Shih, “Distant clock synchronization using
entangled photon pairs,” Appl. Phys.
Lett. 85, 2655–2657
67. T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single
photons prepared by means of conditional measurements on a biphoton
state,” Eur. Phys. J. D 18, 237–245
68. J. P. Torres, F. Macià, S. Carrasco, and L. Torner, “Engineering the frequency correlations of
entangled two-photon states by achromatic phase matching,”
Opt. Lett. 30, 314–316
69. O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, “Two-photon coincident-frequency entanglement
via extended phase matching,” Phys. Rev.
Lett. 94, 083601 (2005). [CrossRef]
71. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single
photons in pure quantum states,” Phys. Rev.
Lett. 100, 133601 (2008). [CrossRef]
72. T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for
spontaneous parametric down-conversion driven by a narrow pump
pulse,” Phys. Rev. A 56, 1534–1541
73. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time
correlations in multiphoton states,” Phys. Rev.
A 64, 063815 (2001). [CrossRef]
75. J. P. Torres, M. W. Mitchell, and M. Hendrych, “Indistinguishability of entangled photons
generated with achromatic phase matching,” Phys.
Rev. A 71, 022320 (2005). [CrossRef]
76. K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert
space,” Phys. Rev. A 75, 050101(R) (2007). [CrossRef]
77. M. Hendrych, X. Shi, A. Valencia, and J. P. Torres, “Broadening the bandwidth of entangled
photons: a step towards the generation of extremely short
biphotons,” Phys. Rev. A 79, 023817 (2009). [CrossRef]
78. S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous
parametric down-conversion,” Phys. Rev. A 70, 043817 (2004). [CrossRef]
79. A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled
photons,” Phys. Rev. Lett. 99, 243601 (2007). [CrossRef]
Juan P. Torres leads a research group in Optics at the ICFO-Institute of Photonic Sciences in Barcelona, Spain, since 2003. He is also associate professor at the Universitat Politècnica de Catalunya since 1996, where he teaches courses on electromagnetic theory and several aspects of Photonics. He did his undergraduate studies in physics at the Universitat de Barcelona from 1982 to 1987, and received a Ph.D. in Sciences from the Universitat Politècnica de Catalunya in 1994. Juan P. Torres' main area of interest is Nonlinear and Quantum optics. In particular, he is interested in exploring theoretically and experimentally the unique features of new types of optical waves, such as solitons and vortex beams, and basic concepts of quantum theory, such as entanglement and decoherence. Juan P. Torres has co-authored numerous scientific papers published in international peer-reviewed journals and has established collaborations with research groups in many different countries. Nowadays, he is very interested in applying concepts and techniques born and developed in nonlinear and quantum optics to the life sciences.
Martin Hendrych received his Ph.D. degree in quantum optics from Palacky University, Olomouc, Czech Republic, in 2003. During his study, he experimentally implemented quantum key distribution, quantum identification, and quantum secret-sharing schemes. In 2000, he was awarded a NATO Advanced Science Fellowship to fund his stay in the Quantum Imaging Laboratory at Boston University, Boston. Upon completion of his degree, he worked as a Research Scientist at the Joint Laboratory of Optics of Palacky University and the Institute of Physics of the Czech Academy of Sciences, Olomouc, Czech Republic. Since 2005, he is a Research Fellow at ICFO—Institute of Photonic Sciences, Barcelona, Spain. Among his main research areas are quantum and nonlinear optics, design and implementation of sources of entangled photons in the fields of quantum information and quantum communications, frequency entanglement, dispersion control, and Bragg-reflector waveguides.
Alejandra Valencia works at ICFO—The Institute of Photonic Sciences in Barcelona, Spain, since October 2005. She did her undergraduate studies in physics at Universidad de los Andes, Bogotá, Colombia (1994–1999). In 2002, A. Valencia received her Master in Sciences degree and in 2005 her Ph.D., both from the University of Maryland Baltimore County (UMBC), USA. The topic of her dissertation was the study of protocols for clock synchronization based on the characteristics of entangled photon pairs. As a postdoctoral researcher, her interest has been mainly oriented towards the engineering of the frequency correlations of entangled photon pairs and the generation of pure single photons. A. Valencia has coauthored various scientific papers published in international peer-reviewed journals and has established collaborations with research groups in different countries. Nowadays, she works in the knowledge and technology transfer unit (KTT) of ICFO developing all the outreach and scientific divulgation activities of the institute.