Wave Optics of Synchrotron Radiation

Synchrotron radiation is a unique source of infrared radiation being highly polarized, pulsed, with the broad emission band and about thousand times brighter than standard thermal source. All just mentioned synchrotron radiation adventures apply to a large choice of experimental techniques and investigations. Among them are high-pressure studies, earth science and biology, microspectroscopy, reflectance and absorption spectroscopy for surface study, time-resolved spectroscopy and ellipsometry. Interest in infrared synchrotron radiation goes back to the 1980s (Duncan & Williams, 1983). At present numerous infrared beamlines have been developed at synchrotron radiation facilities throughout the world, see, e. g., (Bocci et al., 2008; Carr & Dumas, 1999; Guidi et al., 2005; H. Kimura et al., 2001; S. Kimura et al., 2001; S. Kimura et al., 2006; Roy et al., 2006; Williams & Dumas, 1997). Efforts to improve the radiation beam characteristics lead to elaboration more and more sophisticated beamline optics. To achieve this goal, we need to know all characteristics of emitted radiation, its intensity distribution, polarization and phase distribution. In particular, the synchrotron radiation wave properties play a major role in conventional diagnostics of electron beams in storage rings (Andersson et al., 2006; Elleaume et al., 1995; Fang et al., 1996; Flanagan et al., 1999; Hs & Huang, 1993; Weitkamp et al., 2001). In this case, the image of the electron beam is formed by an optical lens. The synchrotron radiation diffraction on the lens iris aperture restricts the resolution of the beam profile measurements. Infrared synchrotron radiation was used for longitudinal beam diagnostics at FLASH free electron laser (Grimm et al., 2008; Paech et al., 2006, 2007). Recent trends show an increased usage of synchrotron radiation interferometers for high precision measurements of the electron beam sizes (Artemiev et al., 1996; Chubar, 1995; Hiramatsu et al., 1999; Katoh & Mitsuhashi, 1999; Naito & Mitsuhashi, 2010). For proper interpretation of observed data, we also need to know synchrotron radiation phase distributions.


Introduction
Synchrotron radiation is a unique source of infrared radiation being highly polarized, pulsed, with the broad emission band and about thousand times brighter than standard thermal source. All just mentioned synchrotron radiation adventures apply to a large choice of experimental techniques and investigations. Among them are high-pressure studies, earth science and biology, microspectroscopy, reflectance and absorption spectroscopy for surface study, time-resolved spectroscopy and ellipsometry. Interest in infrared synchrotron radiation goes back to the 1980s (Duncan & Williams, 1983). At present numerous infrared beamlines have been developed at synchrotron radiation facilities throughout the world, see, e. g., (Bocci et al., 2008;Carr & Dumas, 1999;Guidi et al., 2005; H. S. Kimura et al., 2006;Roy et al., 2006;Williams & Dumas, 1997). Efforts to improve the radiation beam characteristics lead to elaboration more and more sophisticated beamline optics. To achieve this goal, we need to know all characteristics of emitted radiation, its intensity distribution, polarization and phase distribution. In particular, the synchrotron radiation wave properties play a major role in conventional diagnostics of electron beams in storage rings (Andersson et al., 2006;Elleaume et al., 1995;Fang et al., 1996;Flanagan et al., 1999;Hs & Huang, 1993;Weitkamp et al., 2001). In this case, the image of the electron beam is formed by an optical lens. The synchrotron radiation diffraction on the lens iris aperture restricts the resolution of the beam profile measurements. Infrared synchrotron radiation was used for longitudinal beam diagnostics at FLASH free electron laser (Grimm et al., 2008;Paech et al., 2006Paech et al., , 2007. Recent trends show an increased usage of synchrotron radiation interferometers for high precision measurements of the electron beam sizes (Artemiev et al., 1996;Chubar, 1995;Katoh & Mitsuhashi, 1999;Naito & Mitsuhashi, 2010). For proper interpretation of observed data, we also need to know synchrotron radiation phase distributions.
As far as we know, the paper (Sabersky, 1973) was one of the first papers to pioneer the investigation of geometrical-optical properties of synchrotron radiation coming from the curving relativistic electron beam. The phase-space techniques, commonly applied to charged beam optics, was used for analysis. The focusing of synchrotron radiation by a convex lens within the framework of geometrical optics was considered in (Green, 1976;Ogata, 1987). The qualitative agreement with the experimentally observed data was found, although the quantitative discussion needs taking into account of the diffraction effects. The synchrotron radiation treatment as a laser-like Gaussian beam with a small opening angle was performed in (Coisson & Marchesini, 1997;Kim, 1986;Ogata, 1991;Takayama et al., 1998Takayama et al., , 1999. The important benefit to the use of this approximation is that the Gaussian beams have been much studied. It has been shown however that the Gaussian approximation has some limitations when calculating the synchrotron radiation coherence. At the same time the Gaussian beam approximation can reasonably be applied to the horizontally polarized component of synchrotron radiation only. It has been also found that the Gaussian beam approximation is poor for undulator radiation (Kim, 1986). In addition, as it was noted (Miyahara & Kagoshima, 1995), the brightness function defined by Kim can be negative (that has no physical meaning) and should therefore be modified.
The most-used scheme of diffraction phenomena implementation into synchrotron radiation theory was suggested by A. Hofmann and F. Meot (Hofmann & Meot, 1982). This technique involves the following steps. First, the synchrotron radiation from a single electron has supposedly the phase distribution of a point source that is spherically symmetrical one, while its amplitude distributions (for both horizontal and vertical polarizations) should be identical to that of an electron in a homogeneous magnetic field. Under these assumptions and supposing small observation angles, for such source we can calculate the Fraunhofer diffraction pattern. Then, assuming that we have a transverse Gaussian distribution of such point-like sources, and that they are not coherent (it corresponds to incoherent radiation of different electrons in the beam), we can calculate the image distribution by a convolution of the single-electron radiation diffraction pattern with the point sources distribution. Assume further that, instead of moving relativistic beam, we have a uniform longitudinal distribution of such immobile point sources along the circular trajectory of the beam, which are again incoherent. Then we can calculate the resultant image distribution from the electron beam by straightforward integration along longitudinal coordinate. This scheme is relatively simple, easy-to-interpret physically and allows easily estimating the optical resolution for the beam cross-section measurements. It was employed by a number of groups for optimization of the electron beam profile monitor systems and experimental date interpretations, e.g. (Andersson & Tagger, 1995;Arp, 2001;Clarke, 1994;Hs & Huang, 1993). The main weakness of this method is that some steps in this scheme are not deduced from the fundamental principles of radiation theory that is to say from the Maxwell equations. Close inspection of this model shows that some assumptions seem not sufficiently self-consistent. Thus, considering the synchrotron radiation as the radiation from the immobile point sources distributed uniformly along electron's trajectory, we in fact presume that the radiation, generated by the electron at neighboring sections of its path, are incoherent, but this is not the case. The simplest counter example is undulator radiation, where the radiation, emitted by the electron at different periods (5-10 and more cm in length), is coherent. Accordingly, there is no reason to expect the opposite for synchrotron radiation. Rather, we should expect that the radiation from different parts of the electron trajectory is coherent in the case of synchrotron radiation as well.
In terms of wave optics a consistent definition of the problem is as follows. Let a physical device consists of multiple optical parts. Each part has characteristics known beforehand. It is sufficient to know the radiation wave amplitude and phase at every point of the device entrance window in order to take advantage of the Helmholtz -Kirchhoff integral theorem (Born & Wolf, 1986). If these values are known, one can calculate the distribution of the radiation intensity on the device output screen.
The analysis of synchrotron radiation phase distribution based on the fundamental radiation principles was presented at the Eleventh Russian Conference on the usage of Synchrotron Radiation (Novosibirsk, 1996) (Smolyakov, 1998a(Smolyakov, , 1998b. Using exact solutions of the Maxwell equations, the expansion of the synchrotron radiation phase in powers of small observation angles was obtained. A leading quadratic term of this expansion shows that synchrotron radiation by its phase distribution nature is much closer to the radiation emitted by an immobile point source rather than to the longitudinally extended source according to the geometrical-optical approach mentioned above. At the same time the correction cubic term produces a self-aberration effect of synchrotron radiation. The synchrotron radiation exhibits a searchlight effect which manifests itself in the amplitude distribution non-homogeneity in the vertical direction. Using the known amplitude and phase distributions, the intensity distribution of the focused synchrotron radiation from one electron (single electron image) was computed (Smolyakov, 1998c). The effect of the lens aperture on the optical resolution of the electron beam profile measurement system by means of synchrotron radiation was also analyzed. Later the similar results were obtained in the paper (Bosch, 1999), in the part which deals with the synchrotron radiation focusing by a reflecting sphere with an aperture.
An accurate simulation of synchrotron radiation propagation through the beamline optical system inevitably needs the application of specialized computer codes. Nowadays two codes are mostly in use: the code SRW (Chubar & Elleaume, 1998) and the code PHASE (Bahrdt, 1997(Bahrdt, , 2007. Comparison studies of these codes may be found in (Bowler et al., 2008). Some distributions of focused synchrotron radiation with and without phase corrections were simulated with the help of SRW .
The codes, which are based on the wave propagation, are precise but time consuming. It should be mentioned that simpler and hence faster approaches are also used for analysis of beamline optical system properties (Ferrero et al., 2008).
In this chapter we will consider solely the case of standard synchrotron radiation, namely, the radiation generated by highly relativistic electrons while they pass through the uniform magnetic field of bending magnets in storage rings. Intensity distributions of synchrotron radiation, as well as its polarization properties, have been studied quite intensively and are widely covered in the literature. The amplitudes of horizontally (  ) and vertically (  ) polarized wave components of synchrotron radiation are expressed in terms of modified Bessel functions. Although the first formulas for the wave amplitudes were derived more than 60 years ago, regular study of the synchrotron radiation phase distributions started relatively recently, about dozen years ago and not yet analyzed in full measure. Here an exact expression for the phase distribution in synchrotron radiation wave will be derived. Surprisingly, the resultant exact formula is written in terms of elementary functions only, though this formula is rather cumbersome. The lens aperture effect on the focused radiation intensity distribution is analyzed.
We do not consider edge radiation in this chapter. A comprehensive list of papers on edge radiation can be found in (Geloni et al. 2009a(Geloni et al. , 2009bKorchuganov & Smolyakov, 2009;Smolyakov & Hiraya, 2005).

Wave optics of convex lens
To gain greater insight into physics of synchrotron radiation wave optics, let us consider a standard case of the point source radiation focusing by a refractive lens, see Fig. 1. Within www.intechopen.com the framework of geometrical optics, the set of rays ( 1 R and 2 R ) from the point source P , located at a distance 0  D ahead of the lens, is focused to an image point S aft of the lens at the distance L so that: , where f is the focal length of the lens. In terms of wave optics, a point source emits a divergent spherical wave with the spherically symmetric amplitude and phase distributions. Here . In its passing through the ideal lens, the wave does not change its amplitude distribution (no absorption) while the phase distribution is transformed from the divergent-type one into the convergent-type wave as it is shown in Fig. 1. This is due to the fact that the lens, with its spherical shape and correspondently with smoothly varied thickness, imparts the proper different phase delays to different parts of the incident wave. Supposing that the lens sizes are far less than the distance to the source D , we have for the incident wave at 0  z : After passing the ideal infinitely thin lens with the focal length f , the correspondent distributions of the outgoing wave ) 0 , , will be equal to: Here the phase factor describes the action of the lens on the radiation passing through it. With a knowledge of the radiation distributions ) 0 , , aft of the lens and applying the Kirchhoff integral theorem (Born & Wolf, 1986), we can calculate the radiation intensity at any point of the observation screen Assume for simplicity that the lens is rectangular in shape: x l x  , y l y  , where x l 2 and y l 2 are the lens horizontal and vertical sizes respectively. Considering the integral (4) in the where Notice that the focused spot has horizontal and vertical sizes . Second, the radiation intensity at the . It is physically clear that with the lens horizontal aperture x l 2 increasing, the photon flux through the lens varies proportionally and the horizontal size of the spot is correspondently diminished. So, the density is quadratic in x l 2 . Similar to the vertical aperture y l 2.

Time-domain analysis of synchrotron radiation
Here, we will consider synchrotron radiation in the time domain and describe it in terms of electric field of the emitted wave and its arrival time to an observer. A relativistic electron generates synchrotron radiation in the uniform magnetic field of a storage ring bending magnet. Let us consider a physical experiment with a geometry shown in Fig. 2. The electron rotates anticlockwise.
www.intechopen.com For simplicity, we will restrict here our qualitative analysis to the case of radiation in the storage ring median plane XOZ . Briefly, the idea of analysis is as following.
In uniform magnetic field, an electron generates synchrotron radiation homogeneously along its orbit. In the median plane on which the electron moves, the wave shape of the emitted radiation (the electric field temporal structure, curves 2, 3 and 4 in Fig.2) is the same at any observer positions in this plane (e.g., points A , D and B ) provided the distances from the emitting points a , O and b respectively to the observers are the same. In the strict sense the distances aA , OD and bB are slightly different from each other. Here, however, this difference can be neglected being negligibly small in comparison with the distance itself. The difference in the field between observers A , D and B is in the relative arrival time A  , D  and B  of the radiation field. The distinctions   between arrival time A  , D  and B  cannot be neglected since the quantity    c is not negligibly small as compared with the radiation wavelength, where c is the speed of light. This arrival time difference gives rise to the phase difference in the frequency domain.
Let a relativistic electron moves along its circular orbit in a horizontal plane XOZ (the magnetic field is aligned with the vertical Y-axis). It is convenient to choose a frame of reference in such a way that at the initial moment 0  t the electron was at the origin of the coordinates with its velocity vector directed along the Z-axis. Then the equations of the electron motion are: Here, R is the electron orbit radius,  is the angular velocity of the electron , c is the speed of light.
Let us consider a points at the observation screen (plane A-D-B, device entrance window), which is transversal to the Z-axis. The observer, who is at the point } , 0 , 0 { D , will detect the peak in the electric field distribution of the radiation wave (synchrotron radiation field pulse) at the moment c D   . This is obvious because, at time 0  t the electron was at the origin of coordinates, and the vector of its velocity pointed to the observer with coordinates If there are other observers located at some different points A or B of the observation screen, they will detect a similar synchrotron radiation pulse. Nevertheless, these observers will detect the signal peak at different instants. Obviously, the time detected by each observer consists of two components. The first part ) ( s e x t is the time, when the electron was at such point of its orbit ) ( e t r  from where the velocity vector pointed to the observer with coordinates (points a or b in Fig.2). The other part is the time of radiation propagation from position of the electron )) ( ( to the point at the observation At some moment t the electron's position was ) (t r  and its velocity was pointed to the observer with the coordinates . It can easily be shown from Fig. 2 that: It follows from Eq. (6) and Eq. (8) that: The case 0 ) is of interest for our analysis. By using the relation , we can get from Eq. (9): This equation is quadratic in terms of ) ( t tg  with the solution: www.intechopen.com This is precisely the solution of the two solutions of the quadratic equation (10) It is convenient to define the function ) , see Fig. 2.
As a result, we will get the following expression for the arriving time ) Looking ahead, we note that the value of is a phase of the radiation with the wavelength  .
The s x -dependence of the function is shown in Fig. 3a. The parameters of Siberia-2 storage ring were used for this simulation: electron beam energy 2.5 GeV (electron reduced energy is equal to  =4892), electron orbit radius R =4905.4 mm, synchrotron radiation critical energy 7.2 keV (Korchuganov et al., 2005;Korchuganov & Smolyakov, 2009). The distance of the observation plane from the emission point was taken to be D = 20000 mm.

The quadratic behaviour of retardation
suggests that the synchrotron radiation phase distribution is very close to that of an immobile point source. Closer examination of the derived above expressions shows that the correspondent equivalent point source, which produces the similar phase distribution, should be placed at the point with coordinates . Practically the value 2 2 R  is very small; in our case it is equal to 4 10  mm. The front position for this point source is described by the following expression: is shown in Fig. 3b. It should be pointed out that this difference is small, less to 3 10  mm at the horizontal aperture boundaries 200  mm in our example. It means that the synchrotron radiation phase can be considered at zero approximation as the spherically symmetric phase of the correspondent immobile point source. Expanding Eqs. (13) -(15) in powers of s x we get the following approximation for the difference in front positions of synchrotron radiation and point source front (Fig. 3b): It describes the curve in Fig. 3b with a very good accuracy. Though this term is cubic in s x and is very small at first glance, it plays an important part in synchrotron radiation imaging, as we will see below.
Finally we will give another well-known example, which on closer examination also shows the phase distribution of synchrotron radiation. Let an electron moves anticlockwise along the circle trajectory ) (t r  , see Fig. 4. The radiation pulse (the maximum of the generated electric field) moves along the velocity vector . Let us substitute Eqs. (6) and (7) into this relation, fix an observation point in time  and consider the set of points in time t when the radiation was emitted: . We will get the simultaneous distribution of the radiation pulses in the space, the well-known spiral radiation pattern, see Fig. 4. Some comments to this pattern can be found in (Jackson, 1999). The numbers of computed diagrams of electric field lines of radiating electron are published by (Tsien, 1972), see also (Shintake, 2003).  . What is important for us in this picture is the fact that the spiral pattern is tangent to the screen and locally has the behaviour of quadratic function relative to horizontal (X) coordinates as it is shown at the right side of Fig.4. So, this spiral pattern also displays the phase distribution similar to the point source phase behaviour.

Synchrotron radiation phase distribution
In this section we will derive the exact expressions for synchrotron radiation phase. Let us consider an electron moving along trajectory ) (t r  with reduced energy 1   . Electric field in the emitted wave which is observed at the point } , where c is the speed of light, e is the electron charge,   is its reduced velocity, , and t is emission time: The radiation spectrum with wavelength  is defined by the Fourier transform: The unit vectors of polarization  q  and  q  are directed transversally to the radiation propagation and describe horizontally and vertically polarized radiation correspondently. The number of photons per unit time per unit area per unit relative spectral interval emitted by an electron beam with current I is equal to: It can easily be shown by the direct differentiation of Eq. (18) that: Using Eq. (21) we can write: Here T is an arbitrary quantity to be defined below. It is important to keep in mind that T does not depend on time t .
Substituting Eqs. (17) and (18) Once the radiation intensity is calculated, the Eq. (24) should be substituted into Eq. (20) and the phase ) , ( X T   is of no concern. But the applications of optical elements generate a need for the radiation phase ) , It is a matter of direct verification to prove the following exact relation: Substituting Eq. (27) into Eq. (26) and integrating by parts with using Eq. (23), we get: It should be noted that so far the exact formulas (17) -(28) were used and no approximations were employed. Particular attention should be paid to the phase as its scale of variation is very small, the phase changes by  2 along the radiation wavelength. Furthermore, we will derive the exact expression for synchrotron radiation phase distribution which is given by Eq. (25). At the same time we will apply the standard far-field approximation in the radiation amplitudes calculations, see Eq. (28).
We detect the radiation at the point } , Similarly, the expansion for It can be easily derived from Eq. (30) and Eq. (31) that: Then we obtain from Eq. (28): Usually, calculating synchrotron radiation intensity, the observation point is chosen in such a way that 0 x n  thus substantially simplifying the resultant calculations. It is possible for the amplitudes calculations because they are axially uniform and depends on vertical angle only. In this case we obtain

as a purely real function and
as a purely imaginary function. But this simplification is not applicable when calculating the phase distribution of synchrotron radiation and we need to apply more accurate calculating procedure.

Substituting this relation into Eq. (34) and denoting
, we obtain: Using Eq. (36), we can easily derive from Eq. (33) the well-known expressions for synchrotron radiation fields: is the synchrotron radiation critical wavelength,

 x
Ai and  x i A  are the Airy function and its derivative.
We have obtained standard formulas, describing the synchrotron radiation amplitudes  , see Eq. (25): Notice that ) (x t e is the moment when the electron, having the position )) ( ( x t r e  , points to the observer with the coordinates . Then the phase distribution of synchrotron radiation is described by the following exact expression: It is worth noting that we have obtained, as a result, the exact analytic expression involving only elementary functions. It is important to outline also that the phase difference, rather than the phase value ) (X   , has a physical sense. In other words, the phase is defined up to a constant value identical for all the points of the optical device entrance window.

Synchrotron radiation regarded as a point-source radiation
Consider now the phase up to the third power inclusive. This expansion has the form: If both angles x  and y  are sufficiently small, one can neglect the term , which is of the third order of smallness. The size of the corresponding region can be evaluated from condition  As a result, one finds that the term ) In this case ) ( ) ( 2 X X      and the distribution of the synchrotron radiation phases coincides with the phase distribution of an immobile point source located at the point R usually is very small and can be neglected. Then it is seen from Eq. (44) that the equivalent immobile point source is located at the origin of the coordinates. In this case the synchrotron radiation properties are rather well understood. The spatial distributions of the synchrotron radiation amplitudes are described by Eqs. (37) and (38). The synchrotron radiation phase distribution inside the angle limited by Eqs. (46) coincides with that of an equivalent spherical wave propagating from the origin of the coordinates and is described by Eq. (44). For the Siberia-2 storage ring (  =4892, c  =0.175 nm,  =6000 nm) the region given by Eq. (46) is rather large: mrad. Any decrease in the radiation wavelength reduces the size of this region.
All the above stated concerned a single relativistic electron moving along the circular trajectory with zero initial conditions see Eqs. (6) and (7) and Fig. 5. This electron passed through the origin of the coordinates, where its velocity was directed along the Z -axis. The immobile point source, which is equivalent in phase distribution to the synchrotron radiation generated by this electron, is at the origin of the coordinates. However, in practice the electron beam has always some spread in both positions and angles. It is clear that for electrons with different orbits the locations of the equivalent immobile point sources would www.intechopen.com also be different. The explicit calculation of the position of the equivalent point source for an arbitrary electron involves rather cumbersome formulas and is beyond the scope of this chapter. However, for an electron with a trajectory lying in the storage ring median plane, the solution is reasonably simple. , which was outside our analysis here, plays an important role in synchrotron radiation optics, determining its self-aberration effect.

Focusing of synchrotron radiation
Let us consider the standard experimental layout for synchrotron radiation imaging shown in Fig. 6. For simplicity sake the geometry with an ideal refractive lens is considered. The distance between the origin of coordinates and lens is equal to D , the distance from the lens to the observation screen is equal to L . Fig. 6. Synchrotron radiation imaging experiment geometry: 1 -electron trajectory; 2 -lens, 3 -observation screen.
The ideal lens with focal length f does not change the synchrotron radiation amplitudes but adds an extra shift in the radiation phase distribution: Here is the synchrotron radiation field. Its amplitudes 0 E  are given by Eqs. (37) and (38) and the phase ) is given by Eqs. (42) -(45). Notice that the cubic term ) will also be taken here into account.  (49), we can find with Eq. (20) the intensity distribution of the focused synchrotron radiation.
The computed images of horizontally and vertically polarized synchrotron radiation are shown in Fig. 7. The simulations were performed at the following conditions: orbit bending radius 4905.4 mm; electron beam energy of 2.5 GeV; beam current 0.1 A; beam emittance is zero (one-electron approximation); lens focal length 5 m; distance from tangential source point to the lens 10 m; distance from the lens to the observation screen 10 m; horizontal and vertical sizes of the lens 600 mm (angular aperture 60 mrad); radiation wavelength 6000 nm. The lens focuses the synchrotron radiation emitted into all vertical angles, see Fig. 8. Intensity distributions are given in photons/s/mm 2 /(0.1%bandwidth). It is significant that the intensity of focused vertically polarized synchrotron radiation is equal to zero in the electron orbit plane ( 0  y ), see Fig. 7b. This is a result of the antisymmetry of the vertically polarized electric field with respect to the electron orbit plane, see Fig. 8  , will cancel each other. At large horizontal aperture of the lens (600 mm) the cubic term ) in the radiation phase manifests itself as additional asymmetric fringes in the radiation image (selfaberration effect). Real electron beam emittance will smooth out these fringes. Nevertheless, such asymmetry was observed in 100 -20000 cm -1 spectral range at BL43IR infrared beamline of Spring-8 (Ikemoto et al., 2003), see Fig. 9. Let us consider now the influence of the lens horizontal aperture on the synchrotron radiation imaging. It follows from Eq. (46) that the natural scale for the angular aperture is 3 3 . But at large horizontal aperture of the lens, the self-aberration property of synchrotron radiation changes dramatically the radiation image characteristics. For the case of point source radiation its intensity at the focused spot center varies with the square of the lens horizontal aperture, see the end of the section 2.1. This is also true for synchrotron radiation focusing by relatively small lens; at small lens  its value Such kind of problems, which are caused by the self-aberration effect of synchrotron radiation (the term ) ( 3 X   ), can be neutralized by using of the so-called "magic mirror".
The idea how to bring a segment of circular orbit to an isochronous focus and profile of such mirror in the orbit plane was presented in (Lopez-Delgado & Szwarc, 1976). The expansion of formula for "magic mirror" to include the vertical direction was made in (S. . The "magic mirror" for synchrotron radiation is similar to the ideal lens for a point source radiation since it compensates the aberrations caused by the term ) ( 3 X   in design. It is employed at some infrared beamlines (H. S. Kimura et al., 2006).

Conclusion
The analysis of wave optical properties of synchrotron radiation given in this chapter shows their unconventionality. On the one hand, in the zero-order approximation the phases of the synchrotron radiation emitted by each of the electrons coincide with the phases of the radiation emitted by the corresponding immobile point sources. That is why standard optical equipment works well with synchrotron radiation paraxial beams. On the other hand, synchrotron radiation has the property of self-aberration. To improve beamline performance, the specialized optical elements, such as the "magic mirror", should be used in the beamline optical system. The use of the exact formula for the phase distribution of synchrotron radiation provides a way of developing new optical elements which are optimized for synchrotron radiation utilization.