Exact solution for second harmonic generation in XFELs

Optics Communications 271 (2007) 207–218 www.elsevier.com/locate/optcom

Exact solution for second harmonic generation in XFELs Gianluca Geloni *, Evgeni Saldin, Evgeni Schneidmiller, Mikhail Yurkov Deutsches Elektronen-Synchrotron (DESY), HASYLAB, Notkestrasse 85, 22607 Hamburg, Germany Received 11 April 2006; received in revised form 11 October 2006; accepted 13 October 2006

Abstract The generation of harmonic radiation via a non-linear mechanism, driven by electron bunching at the fundamental frequency, is an important option in the operation of high-gain Free-Electron Lasers (FELs). By utilizing harmonic generation at a large scale facility, the production of intense radiation at shorter wavelengths for the same electron beam energy is feasible. This paper describes a theory of second harmonic generation in planar undulators with particular attention to X-Ray FELs (XFELs). Our study is based on an exact analytical solution of Maxwell’s equations, derived with the help of a Green’s function method. Up-to-date theoretical understanding of second harmonic generation is limited to the estimation of the total radiation power, which is based on a comparison of the right hand side of the wave equation for the first harmonic with the right hand side of the equation for the second harmonic, the latter being incorrectly modified. The exact solution should be obtained by solving the wave equation itself. Our work yields correct parametric dependencies and specific predictions of additional properties such as polarization, angular distribution of the radiation intensity and total power. The most surprising prediction is the presence of a vertically polarized part of the second harmonic radiation, whereas current knowledge predicts a horizontally polarized field.  2006 Elsevier B.V. All rights reserved. PACS: 41.60.Cr; 52.35.g; 41.75.i Keywords: Free-electron Laser (FEL); X-rays; Even harmonic generation

1. Introduction In a Free-Electron Laser the electromagnetic field at the fundamental harmonic interacts with the electron beam. As a result, the beam is bunched in a non-linear (sinusoidal) ponderomotive potential. When the bunching is strong enough, the beam current exhibits non-negligible Fourier components at harmonics of the fundamental. In the case of Self Amplified Spontaneous Emission (SASE) only the transverse ground mode of the fundamental harmonic survives, due to the transverse mode selection mechanism [1] in the high-gain regime, and is responsible for the bunching mechanism. As a result, the non-linear Fourier components radiate coherently. This phenomenon is referred to as (nonlinear) harmonic generation of coherent radiation. *

Corresponding author. Tel.: +49 4089984506; fax: +49 4089944475. E-mail address: [email protected] (G. Geloni).

0030-4018/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.019

The process of harmonic generation of coherent radiation can be considered a purely electrodynamical one. The harmonics of the electron beam density are driven by the electromagnetic field at the fundamental frequency, but the bunching contribution due to the interaction of the electron beam with the radiation at higher harmonics can be neglected. This leads to important simplifications. In fact, in order to perform a numerical analysis of the characteristics of higher harmonic radiation, one has to solve the self-consistent problem for the fundamental harmonic only. FEL self-consistent codes present numerical methods to solve this problem. Then, the harmonic contents of the beam current can be calculated. These contents enter as known sources in the electrodynamical process. The solution of Maxwell’s equations accounting for these sources gives the desired characteristics of higher harmonics radiation. Note that it does not matter what mechanism is responsible for a given harmonic content of the beam

208

G. Geloni et al. / Optics Communications 271 (2007) 207–218

current. It may be the bunching mechanism due to the nonlinear ponderomotive potential but also, for example, external modulation. The overall result is that a coherent source radiates at a given harmonic of the fundamental. As a result, simulation codes dealing with harmonic generation are not self-consistent. They simply have to compute the solution of Maxwell’s equations, while the sources are considered as given input data (for example, obtained by means of self-consistent FEL-codes). In this paper we restrict our attention to even harmonics in planar undulators. Non-linear generation of the second harmonic radiation, in particular, is important for extending the attainable frequency range of an XFEL facility. The high peak-brilliance increase of XFELs with respect to third generation light sources (up to eight orders of magnitude) makes the second harmonic contents of the XFEL radiation very attractive from a practical viewpoint. Moreover, it is also important in connection with experiments that make use of the fundamental harmonic only. In fact, one must be able to correctly estimate the higher harmonics effects to distinguish between non-linear phenomena induced by the fundamental and linear phenomena due to the second harmonic. Non-linear harmonic generation has been the subject of theoretical studies in high-gain SASE FEL both for odd [2] and even harmonics [3,4,8], where the electrodynamical problem was dealt with. The practical interest of these studies is well underlined by the fact that they were followed by both numerical analysis [5] and experiments, that have been carried out in the infra-red and in the visible range of the electromagnetic spectrum [6,7]. Experimental results are compared with numerical analysis which in turn relies on analytical studies. This emphasizes the importance of a correct theoretical understanding of the subject. Remarkably, understanding of harmonic generation of coherent radiation does not require the introduction of radically new physical concepts. As has been said before, the problem is reduced to the solution of Maxwell’s equations for coherent sources independent of the mechanism that is responsible for the presence of such sources. In the space–frequency domain a single particle is described in terms of harmonics of the charge and current densities. Then, an exact expression for a certain harmonic radiation from a single particle, in particular the second, can be calculated solving Maxwell’s equations. It was, in fact, presented a long time ago and can be found in Synchrotron Radiation textbooks (e.g. [9,10]). To describe harmonic generation of coherent radiation one has to consider the fact that many electrons are involved in the radiative process. Electrons have given offsets and phases, and radiate coherently, as a whole or in part, due to the longitudinal modulation of the beam current at the considered harmonic. In order to account for this, the expression for a given harmonic from a single particle has first to be generalized to include the arbitrary offset and angle of the electron trajectory with respect to the undulator axis. Then, this generalized expression has to be convolved with the

coherent particle distribution. Note that in the limiting case for a zero beam emittance (i.e. a filament beam) the final result just coincides with the expression for the harmonic from a single particle. We conclude that the key ideas involved in the harmonic generation mechanism are not much different from those regarding harmonic generation from a single particle. For a given frequency component the electromagnetic wave equation dictates both a characteristic longitudinal length (that is the radiation formation length) and a characteristic transverse length. As we will see, when the beam transverse size is smaller than the characteristic transverse length, the entire electron beam behaves like a single electron. The harmonics of the beam current are simply interpretable in terms of the harmonic contents of a single particle current. In this case, all the particles act coherently and the radiated intensity scales with the square number of the electrons in the beam. When the transverse size of the beam increases and becomes much larger than the characteristic transverse length, less electrons contribute collectively to the field and if the beam current remains constant the total radiated power is decreased. The characteristic transverse length is specified in a natural way after a dimensional analysis of the problem. The first treatment [3] of non-linear generation of even harmonics does not account for the presence of such a parameter, which was followed by others [4,8]. We find that, in these works, an important part of the source terms is neglected and an incorrect expansion of the particle trajectory is performed. Estimations of the second harmonic power are then based on comparison of the right hand side of the wave equation for the second harmonic with the right hand side of the wave equation for the first harmonic. In contrast to this, exact calculations should be based on a solution of Maxwell’s equations. We find that these works predict an incorrect dependence of the second harmonic field on the parameters of the problem. Results of [4] are extended in [8] to the case of an electron beam moving off-axis through the undulator. One of the conclusions in [8] is that the second harmonic power increases when an angle between the beam and the undulator axis is present. We find that the power of the second harmonic radiation should never increase when such an angle is present. As we will see, it is independent of the angle in optimal situations when the microbunching wavefront is matched with the beam propagation. In this paper, that was inspired by a method [11] developed to deal with Synchrotron Radiation from complex setups, we present a theory of second harmonic generation in high-gain FELs. First we give, in Section 2, an exact analytical solution of the wave equation for the second harmonic generation problem. The procedure employed to derive such a solution shows the advantages of a Green’s function method. In Section 3, our result is used to calculate for a particular case the specific properties of the second harmonic radiation such as polarization, directivity diagram and total power. The most surprising prediction

G. Geloni et al. / Optics Communications 271 (2007) 207–218

of our theory is that the electric field is not polarized entirely in the horizontal plane, as it is usually believed, but exhibits, though remaining linearly polarized, a vertically polarized component too. Following the presentation of our theory, in Section 4 we comment on the differences between our approach and the present understanding of the second harmonic generation mechanism. In Section 5, we draw conclusions about our present work. 2. Complete analysis of second harmonic generation mechanism As has been said in Section 1, the process of (second) harmonic generation of coherent radiation is a purely electrodynamical one. First, proper initial conditions are given as input to an FEL self-consistent code, which calculates the electron beam bunching from the interaction of the beam with the first harmonic radiation. Then, the results from the self-consistent code are used as electromagnetic sources to solve the problem of second harmonic generation. For simplicity, in the following we will consider a beam modulated at a single frequency x as the source. To fix the ideas, we may consider the modulated beam entering a setup consisting of an upstream bend, a planar undulator, and a downstream bend. However, as we will stress later on, an application of the resonance approximation will allow us to neglect field contributions from non-resonant structures, i.e. from the bends, so that our calculations are not restricted to this particular setup. One may always write the longitudinal current density jz along the undulator as a sum of an unperturbed part independent of the modulation and of the time, jo, and a term responsible for the beam modulation, ~jz , at frequency x (perturbation) r? ; tÞ ¼ jo ðz;~ r? Þ þ ~jz ðz;~ r? ; tÞ: jz ðz;~

ð1Þ

x þ ry~ y identifies the position Here t is the time, and ~ r? ¼ rx~ of a point on a transverse plane located at the longitudinal coordinate z. We assume that we can write the unperturbed part jo as if all the particles were moving coherently, that is ðcÞ

r? Þ ¼ jo ð~ r? ~ r? ðzÞÞ; jo ðz;~ ðcÞ

ð2Þ

where ~ r? ðzÞ describes the coherent motion. This assumption is always verified, for instance, in the case of a single particle, when jo is simply a d-Dirac function, or in the case of a monochromatic beam. In order for Eq. (2) to be valid when some energy spread is present we should assume that the transverse size of the electron beam is not smaller than the typical wiggling motion of the electrons. In this case, the validity of Eq. (2) has an accuracy given by the relative deviation of the particles energy from the average value dc/c, where c is the relativistic Lorentz factor. Since for the FEL process dc/c is, at most, of the order of the efficiency parameter, we have dc/c  1 and Eq. (2) is valid with the same accuracy of the FEL theory. However, it should be noted here that the average energy of the beam

209

is to be considered, in general, a function of the coordinate z, i.e. c = c(z). This function has to be given as a result of start-to-end simulations and considered as an input for our electromagnetic problem. In this paper we will work under the paraxial approximation, i.e. we assume, with no restriction, that the square of the average Lorentz factor in the longitudinal direction, c2z , is much larger than unity. This means that the average longitudinal velocity vz can be substituted everywhere with the speed of light in vacuum, c, with the exception of resonant phases, where small deviations from c are important. The perturbation ~jz can then be written as     ðcÞ ðcÞ ~jz ðz;~ r? ; tÞ ¼ jo ~ r? ðzÞ ~a2 z;~ r? ~ r? ðzÞ r? ~  Z z   dz0  exp ix  ixt þ C:C: : ð3Þ 0 0 vz ðz Þ Rz In order to correctly calculate the phase x 0 dz0 =vz ðz0 Þ  xt in Eq. (3) one has to account for the dependence of vz, on the position z. The function vz(z) can be recovered from the ðcÞ knowledge of ~ r? ðzÞ and of the average energy of the beam c = c(z). The function ~a2 , instead, is to be considered a result of the FEL self-consistent code used to calculate the initial condition for the electrodynamical problem. Its dependence on z describes the evolution of the modulation through the beamline and accounts for emittance and energy spread effects. Values assumed by ~a2 are not necessarily real. In fact, there canR be a z-dependent phase shift z with respect to the phase x 0 dz0 =vz ðz0 Þ  xt. If the beam is deflected at angles gx and gy in the horizontal and in the vertical direction with respect to the z axis, the velocity of the coherent motion depends also on the deflection angles. Renaming position and velocity of the coherent motion with no deflection with the subscript ‘‘(nd)’’ one obtains ! g2x þ g2y vz ðz; gÞ ¼ vzðndÞ ðzÞ 1  ; 2 ð4Þ ~ v?ðndÞ ðzÞ þ vzðndÞ ðzÞ~ g; v? ðz; gÞ ¼ ~ and ðcÞ

ðcÞ

~ gÞ ¼ ~ r?ðndÞ ðzÞ þ ~ gz: r? ðz;~

ð5Þ

Also ~a2 will depend on ~ g. The exact dependence is fixed by the way the beam is prepared and should be regarded as a condition for the orientation of the microbunching wavefront. In all generality we can write   ðcÞ ~a2 ¼ ~a2 z;~ r? ~ r? ðz;~ gÞ : ð6Þ The total current density can be written as     ~ vðz;~ gÞ  ðcÞ ðcÞ ~ jðz; t;~ gÞ ¼ jo ~ r? ðz;~ gÞ 1 þ ~a2 z;~ r? ~ r? ðz;~ gÞ r? ~ vz  Z z   dz0  exp ix  ixt þ C:C: : ð7Þ 0 gÞ 0 vz ðz ;~

210

G. Geloni et al. / Optics Communications 271 (2007) 207–218

In the limit for c2z  1, i.e. in paraxial approximation, one can express the charge density as j j ð8Þ q¼ z ¼ z: vz c Eqs. (7) and (8) give us the expressions to be used as sources for the inhomogeneous wave equation for the transverse field ~ E? E?  c2 r2 ~

~ E? o2 ~ ~? q þ 4p oj? : ¼ 4pc2 r 2 ot ot

ð9Þ

Eq. (9) is a partial differential equation of hyperbolic type. We look for solutions for ~ E? in the form ~ ~ e ? exp½ixðz=c  tÞ þ C:C: E? ¼ E ð10Þ The description of the field given in Eq. (10) is quite gen~ e ? varies slowly along z with eral, but is useful only when E respect to the length k = 2pc/x. In that case, in fact, Eq. (10) amounts to a factorization of ~ E? as the product of a fast and a slowly varying function of z. Substitution of Eqs. (7), (8) and (10) into Eq. (9) yields  2ix o o2 ~ 2 e? þ E r? þ c oz oz2   h 4p z i ix ~ ~ exp i Us  x ¼ ð11Þ a2 : v ?  r ? jo ~ c c c2 In Eq. (11) we put Z z dz0 Us ðz;~ ; gÞ ¼ x 0 gÞ 0 vz ðz ;~

ð12Þ

while r2? is the Laplacian operator acting on transverse coordinates only. In our case of study it is possible to simplify Eq. (11) with the help of the paraxial approximation. In fact, the longitudinal direction z is such that c2z  1. As a result, the radiation formation length along the z axis, typical of our process, is much longer than the wavelength k. This allows one to neglect the second order derivative of ~ e varies the field with respect to z in Eq. (11), because E slowly along z with respect to the wavelength k. Therefore ~ e ? as we may write the Maxwell equation describing E    h i ix 2ix o ~ ~ e ? ¼ 4p exp i Us  x z ~ v a2 : r2? þ  r E ? ? jo ~ c oz c c c2 ð13Þ 1

With the aid of the appropriate Green’s function we find an exact solution of Eq. (13) without any other assumption about the parameters of the problem. A Green’s function for Eq. (13), namely the solution corresponding to the unit point source, satisfies the equation  2ix o 2 r? þ Gðzo  z; r~ ?o  r~ ? Þ ¼ dðr~ ?o  r~ ? Þdðzo  zÞ; c oz ð14Þ 1 Note the similarity of this problem with the problem of solving Schroedinger equation for a free particle in two dimensions.

and, in an unbounded region, can be written explicitly as ( ) 0 j2 ~ ~ 1 j r  r ? ?o ~0 exp ix ; Gðzo  z0 ; r~ ?o  r ? Þ ¼  4pðzo  z0 Þ 2cðzo  z0 Þ ð15Þ assuming zo  z 0 > 0. When zo  z 0 < 0 the paraxial approximation does not hold, and the paraxial wave equation (13) should be substituted, in the space–frequency domain, by a more general Helmholtz equation. However, the radiation formation length for zo  z 0 < 0 is very short with respect to the case zo  z 0 > 0, i.e. there is no radiation for observer positions zo  z 0 < 0. As a result, in this paper we will consider only zo  z 0 > 0. It follows that the observer is located downstream of the sources. This leads to the solution   Z Z 1 1 0 1 ix ~ 0 0 0 ~ e ? ðzo ;~ ~ ~ v r?o Þ ¼  dz ðz ;~ gÞ  r E d r ? ? ? c 1 zo  z0 c2     ðcÞ ðcÞ r? ðz0 ;~ gÞ ~ a2 z0 ; ~ r0 ? ~ r? ðz0 ;~ gÞ  jo ~ r0 ? ~ #  ( " ) j~ r?o  ~ r 0 ? j2 z0 0 þ i Us ðz ;~ gÞ  x  exp ix ; 2cðzo  z0 Þ c ð16Þ

~0 r ?

where represents the gradient operator (acting on transverse coordinates) with respect to the source point, while ðzo ;~ r?o Þ indicates the observation point. Note that the integration is taken from 1 to 1 because the sources are understood to be localized in a finite region of space (i.e. ~a2 is different from zero in a finite region of space). This is in agreement with zo  z 0 > 0 and within the applicability criteria of the paraxial approximation. Integration by parts of the gradient terms leads to ! Z Z ~ ix 1 0 1 gÞ ~ r0 ? r?o  ~ v? ðz0 ;~ ~ 0 e ~ E? ¼  2  dr ? dz c 1 zo  z0 c zo  z0     ðcÞ ðcÞ  jo ~ r0 ? ~ r? ðz0 ;~ gÞ ~a2 z0 ; ~ r0 ? ~ r? ðz0 ;~ gÞ

  exp iUT ðz0 ; ~ r0 ? ;~ gÞ ; ð17Þ where the total phase UT is given by " #   2 z0 j~ r?o  ~ r0 ? j UT ¼ Us  x þx : c 2cðzo  z0 Þ

ð18Þ

We now make use of a new integration variable ðcÞ ~ r? ðz0 ;~ gÞ so that l¼~ r0 ? ~ Z 1 1 ~ e ? ¼  ix E dz0 c2 1 zo  z0 ! Z ðcÞ 0    0 ~ ~ ~ r ðz ;~ gÞ ~ r ðz ;~ gÞ  l v ? ?o ? ~ a2 z0 ;~   d~ l l j o l ~ c zo  z0  exp½iUT ðz0 ;~ l;~ gÞ;

ð19Þ

and

" #   ðcÞ 2 z0 j~ r?o ~ r? ðz0 ;~ gÞ  ~ lj UT ¼ Us  x þx : c 2cðzo  z0 Þ

ð20Þ

G. Geloni et al. / Optics Communications 271 (2007) 207–218

We will consider the case of a planar undulator and we will be interested in the total power of the second harmonic emission and in the directivity diagram of the radiation in the far zone. Accounting for the beam deflection angles gx and gy we model the electron transverse motion as  

 cK 0 0 ~ sin ðk w z Þ þ gx vz ~ v? ðz ;~ gÞ ¼  x þ gy v z ~ y ð21Þ c and

 K ðcÞ ~ r? ðz0 ;~ gÞ þ ~ l¼ ðcosðk w z0 Þ  1Þ þ gx z0 þ lx ~ x ck w

 þ gy z0 þ ly ~ y:

kw eH w ; 2pme c2

ð22Þ

ð23Þ

where (e) is the (negative) electron charge, me is the electron mass, and Hw is the maximum of the magnetic field produced by the undulator on the z axis. Moreover, one has ! g2x þ g2y 0 Kgx cUs 4c2 K2 ’  þ  sinð2k w z0 Þ z x 2 k w c 8c2 k w 4c2  K 2 þ

Kgx cosðk w z0 Þ: ck w

p¼1

where Jp indicates the Bessel function of the first kind of order p, while a and w are real numbers. We will be interested in frequencies around the second harmonic ð28Þ

where

Here kw = 2p/kw, kw being the undulator period. K is the undulator parameter K¼

We will make use of the well-known expansion (see [12]) 1 X J p ðaÞ exp½ipw; ð27Þ exp½ia sinðwÞ ¼

x2o ¼ 4k w cc2z ;



211

c2 : 1 þ K 2 =2

c2z ¼

ð29Þ

~ e ?2 the second harmonic contribution calIndicating with E culated at frequencies around x2o one obtains Z 1 Z 1 Z Lw =2    ~ 0 ~ a2 z0 ;~ e ?2 ¼ ix2o E dl dl dz j l exp½iU0  x y o l ~ c2 zo 1 1 Lw =2   1 1 X X inp  J m ðuÞJ n ðvÞ exp 2 m¼1 n¼1  K  i ðexpfi½Rx þ 1k w z0 g  expfi½Rx  1k w z0 gÞ 2c   0 0 x þ ½ðhy  gy Þ expfiRx k w z g~ y ; þðhx  gx Þ expfiRx k w z g ~ ð30Þ

ð24Þ where

We will now introduce the far zone approximation. Substitution of Eqs. (24), (22) and (21) in Eq. (17) yields the following field contribution calculated along the undulator Z Z Lw =2 ~ ~ e ? ¼ ix E d l dz0 jo ð~ lÞ~ a2 ðz0 ;~ lÞ c 2 zo Lw =2   K 0 sinðk w z Þ þ ðhx  gx Þ ~  exp½iUT  x þ ðhy  gy Þ~ y ; c ð25Þ

Rx ¼

  z0 K2 2 2 2 1þ þ c ððhx  gx Þ þ ðhy  gy Þ Þ UT ¼ x 2c2 c 2  K2 Kðhx  gx Þ 0 0 sinð2k w z Þ  cosðk w z Þ  2 ck w c 8c k w c   K 1 2 2 zo þx ðhx  gx Þ  ðhx lx þ hy ly Þ þ ðhx þ hy Þ : k w cc c 2c ð26Þ Here hx and hy indicate the observation angles xo/zo and yo/zo. Moreover, the integration in Eq. (25) is performed in dz 0 over the interval [Lw/2, Lw/2], where Lw = Nwkw is the undulator length, and Nw is the number of undulator periods. The reason why the integration is limited to the region inside the undulator is that, working under the resonance approximation in the limit for Nw  1, one can neglect contributions to the field due to non-resonant elements outside the undulator [11].

ð31Þ

with x1 1

  1 K2 2 2 2 þ c ½ðhx  gx Þ þ ðhy  gy Þ  : ¼ 1þ 2k w cc2 2

ð32Þ

Moreover u¼

x2o K 2 ½1  K 2 =ð4c2 Þ n o; x1 4 1 þ K 2 þ c2 ½ðhx  g Þ2 þ ðhy  g Þ2  x y 2

ð33Þ

v¼

x2o 2Kc½1  K 2 =ð4c2 Þðhx  gx Þ x1 1 þ K22 þ c2 ½ðhx  gx Þ2 þ ðhy  gy Þ2 

ð34Þ

where 

x  n  2m; x1

and U0 ¼ x2o



 K 1 zo ðhx  gx Þ  ðhx lx þ hy ly Þ þ ðh2x þ h2y Þ : k w cc c 2c ð35Þ

In the limit for Nw  1, assuming that ~a2 does not vary much in z 0 over a period of the undulator kw, the exponential functions in the integrand of Eq. (30) can exhibit fast oscillations in z 0 . If all exponential functions oscillate rapidly, the overall result of the integration is suppressed. However, there are values Rx that correspond to situations when at least one of the exponential functions is unity. For these values, which are Rx = 0, Rx = 1 or Rx = 1, the result of the integration is not suppressed. If x = x2, x2 being defined by

212

G. Geloni et al. / Optics Communications 271 (2007) 207–218

x2 ¼ 2x1 ;

ð36Þ

these values of Rx correspond to n = 2  2m, n = 3  2m and n = 1  2m respectively. Neglecting all other terms and imposing x = x2 + Dx2 we obtain ~ e ?2 ¼ ix2o E c 2 zo

Z

Z

1

1



Z

1

dlx

Lw =2

dly 1

dz0 jo ð~ lÞ~ a2 ðz0 ;~ lÞ exp½iU0 

Lw =2 1  X

 Dx2 K 0  exp i kwz i ðJ m ðuÞJ 32m ðvÞ 2c x1 m¼1     i½3  2mp i½1  2mp  exp J m ðuÞJ 12m ðvÞexp 2 2   i½2  2mp ~ þðhx  gx ÞJ m ðuÞJ 22m ðvÞ exp x 2     i½2  2mp ~ y : þ ðhy  gy ÞJ m ðuÞJ 22m ðvÞexp 2 ð37Þ

For any value of K and hx  gx much smaller than 1/cz, v is a small parameter and only the smallest indexes in the Bessel functions Jq(v)  vq in Eq. (37) give a non-negligible contribution. As a result, Eq. (37) can be drastically simplified. One can write (compare, for instance, with [11]) Z 1 Z 1 ix2o ~ e E ?2 ¼ 2 ½Aðhx  gx Þ~ x þ Bðhy  gy Þ~ y dlx dly c zo 1 1 Z Lw =2  dz0 exp½iU0 jo ð~ lÞ~ a2 ðz0 ;~ lÞ exp½iCz0  Lw =2 n o 2 2  exp i2c2z ½ðhx  gx Þ þ ðhy  gy Þ k w z0 ; ð38Þ where we defined      2K 2 K2 K2 K2 J  J þ J ; 0 2 1 2 þ K2 2 þ K2 2 þ K2 2 þ K2 ð39Þ  2 K ; ð40Þ B ¼ J1 2 þ K2

A¼

In Eq. (38) we used the fact that Dx2 ¼ 2c2z ½ðhx  gx Þ2 þ ðhy  gy Þ2  þ C; x1

ð41Þ

with C¼

x  x2o : x1o

ð42Þ

In Eq. (42), x1o is defined as x1o ¼ 2k w cc2z :

ð43Þ

Moreover, under the resonant approximation, we have i x2o h zo ðhx lx þ hy ly Þ þ ðh2x þ h2y Þ : ð44Þ U0 ¼ c 2

Fig. 1. Illustration of the behavior of the ratio Q(K) between the second harmonic field contribution due to the gradient of the density part of the source and the contribution due to the current part of the source.

The detuning parameter C should indeed be considered as a function of z, i.e. C = C(z) which can be retrieved from the knowledge of c = c(z). It is important to see that the terms in J1 in Eqs. (39) and (40) are due to the presence of the gradient term in ~? ðjo ~a2 Þ in Eq. (13), which has been omitted in [3] and r later on in [4,8]. We find that, without the gradient term, one recovers results quantitatively incorrect for the xpolarization component. In Fig. 1 we plot the ratio of the contribution of the radiation field due to the gradient of the density part of the source, and the contribution due to the current part of the source for the x-polarization component. This is a function Q(K) of the K parameter only and it can be written as Q¼

~ ?2g E ~ ?2c E

     2 þ K2 K2 K2 K2 ¼ J1 J0  J2 ; 2K 2 2 þ K2 2 þ K2 2 þ K2 ð45Þ where the subscript ‘‘g’’ stands for ‘‘gradient’’ and ‘‘c’’ stands for ‘‘current’’. As it can be seen from Fig. 1, the gradient term always contributes for more than one fourth of the total field, independently of the values assumed by K. Also, if the gradient term is omitted, the entire contribution to the field polarized in the y direction would go overlooked. The inclusion of the gradient term in the source part of the wave equation should not be considered as a peculiarity of the second harmonic generation mechanism. In Synchrotron Radiation theory from bending magnets, for instance, the presence of such a source term is customary and it is responsible, as here, for part of the horizontally polarized field and for the entire vertically polarized

G. Geloni et al. / Optics Communications 271 (2007) 207–218

field. Moreover, the gradient term is always associated with an integration by part. Therefore, it is always accompanied with the gradient of the Green’s function, which is responsible for a term proportional to the observation angle hx,y. Eq. (38) can also be written as h i ~ e ?2 ¼ ix2o exp i x2o zo ðh2 þ h2 Þ ½Aðhx  gx Þ~ E x x y c 2 zo 2c Z 1 Z 1 Z 1 þ Bðhy  gy Þ~ y dlx dly dz0 1 1 1 h x i 2o ðhx lx þ hy ly Þ  exp i n xc o 2o ~ð2Þ ðz0 ;~  exp i ½ðhx  gx Þ2 þ ðhy  gy Þ2 z0 q l; CÞ; 2c ð46Þ ~ð2Þ as where we have defined q ~ð2Þ ðz0 ;~ l; CÞ ¼ jo ð~ lÞ~ a2 ðz0 ;~ lÞ exp½iCz0 H Lw ðz0 Þ; q

ð47Þ

0

H Lw ðz Þ being a function equal to unity over the interval [Lw/2, Lw/2] and zero everywhere else. Its introduction simply amounts to a notational change. Namely it accounts for the fact that the integral in dz 0 is performed over the undulator length in Eq. (38), while it is performed from 1 to 1 in Eq. (46). It should be noted that, usually, computer codes do not present the functions ~ a2 and exp[iCz 0 ] separately as we did. Rather they combine them in a single product, usually known as the complex amplitude of the electron beam modulation with respect to the phase ~ð2Þ as a given funcw = 2kwz 0 + (x/c)z 0  xt. Regarding q tion allows one not to bother about a particular presentation of the beam modulation. Eq. (38) or, equivalently, Eq. (46) are our most general result, and are valid independently of the model chosen for the current density and the modulation. It is interesting to note that, when one writes Eq. (38) in the form of Eq. (46), one obtains an expression which is formally similar ~ð2Þ ðz0 ;~ to the spatial Fourier transform of q l; CÞ with respect 0 ~ ~ð2Þ is a to z and l. There are two problems though. First, q function of ~ g, which appears in the conjugate variable to z 0 and, second, if c = c(z) one has x2o = x2o(z). 3. Analysis of a simple model Let us treat a particular case. Namely, let us consider the case when we can consider cðzÞ ¼ c ¼ const, when C(z) = 0 and when  h x i 2o ~ð2Þ ðz;~ q gx lx þ gy ly H Lw ðzÞ; lÞ ¼ jo ~ ð48Þ l a2o exp i c with a2o = const and

! l2x þ l2y Io ~ jo ðlÞ ¼ exp  : 2pr2 2r2

ð49Þ

Here Io and r are the bunch current and transverse size respectively.

213

This particular case corresponds to a modulation wavefront perpendicular to the beam direction of motion. In this case Eq. (46) can be written as h i ~ e ?2 ¼ ia2o x2o exp i x2o zo ðh2 þ h2 Þ E x y c 2 zo 2c Z Z 1 Z 1

 1 x þ Bðhy  gy Þ~ y dlx dly dz0  Aðhx  gx Þ~ 1 1 1 n x o 2o ½ðhx  gx Þlx þ ðhy  gy Þly   exp i nx c o 2o 2 2  exp i ½ðhx  gx Þ þ ðhy  gy Þ z0 jo ð~ lÞH Lw ðz0 Þ 2c ð50Þ and amounts, indeed, to the spatial Fourier transform of lÞH Lw ðz0 Þ. We obtain straightforwardly jo ð~ h i ~ e ?2 ¼ iI o a2o x2o Lw exp i x2o zo ðh2 þ h2 Þ ½Aðhx  gx Þ~ E x x y c2 zo 2c   Lw x2o 2 2 þ Bðhy  gy Þ~ ½ðhx  gx Þ þ ðhy  gy Þ  ysinc 4c  2 2  r x2o 2 2  exp  ½ðhx  gx Þ þ ðhy  gy Þ  : ð51Þ 2c2 If the beam is prepared in a different way so that, for instance, the modulation wavefront is not orthogonal to the direction of propagation of the beam, Eq. (46) retains its validity. However, in general, Eq. (46) is not a Fourier ~ð2Þ includes a phase transform. It is if cðzÞ ¼ c ¼const, and q factor of the form exp½ia~ g ~ l. The case a = 1 has just been treated. The case a < 1 corresponds, instead, to a modulation wavefront that is orthogonal neither to the z axis nor to the direction of propagation. Going back to our particular case in Eq. (51), a subject of particular interest is the angular distribution of the radiation intensity along the ~ x and ~ y polarization directions which will be denoted with I2(x,y). Upon introduction of normalized quantities rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi x2o Lw h ¼ 8pN w cz h; h^ ¼ c rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi x2o Lw ð52Þ ^g ¼ g ¼ 8pN w cz g; c rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ^lx;y ¼ x2o lx;y ¼ 8pN w cz lx;y cLw Lw and of the Fresnel number N¼

x2o r2 ; cLw

ð53Þ

one obtains   I 2ðx;yÞ ^hx  ^gx ; ^hy  ^gy    2 1 2 2 ¼ const ^gx;y  ^hx;y sinc2 ½ð^hx  ^gx Þ þð^hy  ^ gy Þ  4 n o 2 2  exp N ½ð^hx  ^gx Þ þ ð^hy  ^gy Þ  : ð54Þ

214

G. Geloni et al. / Optics Communications 271 (2007) 207–218

Note that in the limit for N  1, Eq. (54) restitutes the directivity diagram for the second harmonic radiation from a single particle. In agreement with Synchrotron Radiation textbooks [9,10] neither of the polarization components of I2(x,y) has azimuthal symmetry. This is not the case for the first harmonic, where only the x polarization component is present and has azimuthal symmetry. As an example, the directivity diagram in Eq. (54) is plotted in Fig. 2 for different values of N as a function of ^ hx  ^gx at ^ hy  ^ gy ¼ 0 for the horizontal polarization component. The next step is the calculation of the second harmonic power. The power for the x- and y-polarization components of the second harmonic radiation are given by Z 1 Z 1 c 2 W 2ðx;yÞ ¼ dxo dy o jE?x;y ðzo ; xo ; y o ; tÞj 4p 1 1 Z 1 Z 1 c ~ ?x;y ðzo ; xo ; y o Þj2 ; ¼ dxo dy o jE ð55Þ 2p 1 1 where ð. . .Þ denotes averaging over a cycle of oscillation of the carrier wave. We will still consider the model specified by Eqs. (48) and (49) with C = 0. It is convenient to present the expressions for W2x and W2y in a dimensionless form. After appropriate normalization they both are a function of one dimensionless parameter only, that is  b 2x ¼ W b 2y ¼ F 2 ðN Þ ¼ ln 1 þ 1 : W ð56Þ 4N 2 b 2x ¼ W 2x =W ð2Þ and W b 2y ¼ W 2y =W ð2Þ are the normalHere W ox oy ð2Þ ized powers, while the normalization constants W ox and ð2Þ W oy are given by ! ! ð2Þ W ox A2 2a22o I 2o : ð57Þ ¼ ð2Þ c W oy B2

For practical purposes it is convenient to express Eq. (57) in the form ! !   W ð2Þ A2 Io ox 2 ½2a  ¼ W ; ð58Þ b 2o 2 cI W ð2Þ B A oy where Wb = mec2cIo/e is the total power of the electron beam and IA = mec3/e ’ 17 kA is the Alfven current. The function F2(N) is plotted in Fig. 3. The logarithmic divergence in F2(N) in the limit for N  1 imposes a limit on the meaningful values of N. On the one hand, the characteristic angle ^hmax associated with the intensity distribution is given by ^h2max  1=N . On the other hand, the expansion of the Bessel function in Eq. (38) is valid only as ^h2 K N w . As a result we find that Eq. (56) is valid only up to values of 1 N such that N J N 1 w . However, in the case N < N w we deal with a situation where the dimensionless problem parameter N is smaller than the accuracy of the resonance approximation  N 1 w . In this situation our electrodynamic description does not distinguish anymore between a beam with finite transverse size and a point-like particle and, for estimations, we should replace ln(N) with lnðN 1 w Þ. We will now compare our results for the second harmonic with already known results for the first. The case treated in [13] corresponds to a modulation wavefront orthogonal to the direction of propagation, exactly as specified here for the second harmonic (i.e. perfect resonance with Eqs. (48) and (49) valid) and allows direct comparison of results. The outcomes of [13] have been presented, similarly to what has been done here for the second harmonic, in dimensionless form. After appropriate normalization, one finds b 1 ðN Þ ¼ W 1 ¼ F 1 ðN Þ W ð1Þ W0     2 1 N N2 arctan ¼ þ ln ; p N 2 N2 þ 1 ð1Þ

where the normalization factor W 0 is given by

Fig. 2. Plot of the directivity diagram for the radiation intensity as a function of ^ hx  ^ gx at ^ hy  ^gy ¼ 0 for the horizontal polarization component, for different values of N.

Fig. 3. Illustration of the behavior of F2(N).

ð59Þ

G. Geloni et al. / Optics Communications 271 (2007) 207–218

   Io K2 ð1Þ W 0 ¼ W b ½2p2 a21o  N w A2JJ ; cI A 2 þ K 2 AJJ being defined as   K2 K2 AJJ ¼ J 0  J1 : 4 þ 2K 2 4 þ 2K 2

ð60Þ

ð61Þ

Here a1o is the analogous of a2o for the first harmonic. For notational reasons, a1o is one half of the original modulation level ain in Eq. (27) of [13]. It should also be noted that all N in Eq. (59) are multiplied by a factor 1/2 with respect to what is reported in [13]. This is because we are referring all results to the Fresnel number for the second harmonic. The function F1(N) is plotted in Fig. 4. We can compare more quantitatively the normalized power for the second and for the first harmonic   2 3 1 ln 1 þ 2 F 2 ðN Þ p 4 4N  5; ¼ ð62Þ F 1 ðN Þ 2 arctan 1  þ N ln N 2 N

2

N 2 þ1

while, from a practical viewpoint, the comparison between the real powers is equal to ð2Þ ð2Þ W 2 W ox þ W oy F 2 ðN Þ ¼ ð1Þ F 1 ðN Þ W1 W0

2 þ K 2 a22o A2 þ B2 F 2 ðN Þ : ¼ F 1 ðN Þ A2JJ ð2pÞ2 N w K 2 a21o 2

at N  1:

ð63Þ

ð64Þ

In Fig. 5 we plot the behavior of F2/F1 as a function of N and its asymptotic, p/(4N), for N  1. Finally, it is possible to study the ratio between the second harmonic power due to the y vertical and the x horizontal polarization components, that is only a function of

Fig. 4. Illustration of the behavior of F1(N).

Fig. 5. Solid line: illustration of the behavior of F2/F1 as a function of N. Dashed line: its asymptotic, p/(4N), for N  1.

the K parameter and is simply given by the ratio R = W2y/W2x RðKÞ ¼

It is interesting to calculate Eq. (62) in the limit N  1. We have F 2 ðN Þ p ! F 1 ðN Þ 4N

215

W 2y B2 ðKÞ ¼ : W 2x A2 ðKÞ

ð65Þ

A plot of R(K) is given in Fig. 6. As one can see, the relative magnitude scales from 4% in the case K  1 up to about 6% in the limit K  1. The vertical polarization component of the radiation depends quite weakly on the K parameter. The knowledge of the polarization contents of the radiation, even if relatively small as in this case, can be important from an experimental viewpoint. For example, in the VUV wavelength range, the reflection coefficients of many materials (e.g. SiC, that is widely used for mirrors) exhibit a

Fig. 6. Illustration of the behavior of the ratio between the second harmonic power due to the y vertical and the x horizontal polarization components, R(K).

216

G. Geloni et al. / Optics Communications 271 (2007) 207–218

complicated behavior, and there may be even an order of magnitude difference depending on the polarization of the radiation. It should be noted that R(K) is independent of the particular model chosen for the beam modulation as can be seen by inspecting Eq. (38). It is also important to remark that the second harmonic radiation from a planar undulator is linearly polarized, since vertical and horizontal polarization components are characterized by the same phase factor. This fact is well-known in Synchrotron Radiation theory for a single particle and it is true for any observation angle and any harmonic of the radiation from a planar undulator [10]. A different behavior takes place in the case of bending magnet radiation, when vertical and horizontal polarization components exhibit a relative p/2 phase shift, indicating circular polarization. At this point, a comment on what has been done before is needed. We calculated the electric field, the angular intensity distribution and the power for the second harmonic making a particular assumption about the electron beam modulation in Eq. (48). This amounts to an analysis of the modulation wavefront orthogonal to the direction of propagation of the beam. The same assumption has been implicitly made calculating the first harmonic power (the expression in [13] has been used, which does not account for deflection angles). In this particular case we have seen that the total power of the second harmonic radiation does not depend on the deflection angles gx and gy. In the more general situation we find that the second harmonic power can be independent of the beam deflection angle (like in the situation treated by us) or can decrease due to the presence of extra oscillating factors in ~ l in Eq. (46). In contrast with our findings, an increase of the total power is reported in [8], due to the presence of deflection angles.

gradient term in the source part of the wave equation is dropped. Second, the particles trajectory in the transverse x direction is expanded (we will comment on this later on). Because of this the Fresnel number N is not identified as the main physical parameter of the problem. Third, after these operations, the wave equation is not solved. The second harmonic power is estimated, instead, in the following way: (a) the square of the source parts of the wave equation for the second harmonic is calculated; (b) the square of the source parts for the first [8] or the third [3,4] harmonic is calculated; (c) the ratio between the square of the source parts for the second and either the first [8] or the third [3,4] is taken. As a result, incorrect magnitude of the second harmonic power and polarization characteristics are predicted. Let us briefly comment on these points by analyzing more in detail the approach followed in [3,4,8]. First, as we have seen in Section 2 the gradient term in the wave equation is responsible for a contribution to the total intensity for the second harmonic both for the horizontal and for the vertical polarization components, and it cannot be neglected. Doing so results in an overall incomplete result. Namely, one would obtain only part of the horizontally polarized component of the field. Second, going further with the derivation in [3,4,8], the motion of the electrons in the x-direction is written as a sum of a fast oscillation due to the undulating motion and a slow motion due to the betatron functions. In the y-direction instead, only the slow motion due to betatron functions is present. The beam distribution is then considered as a collection of individual point-particles, i.e. a sum of d-Dirac functions. For the ith electron one may write

4. Discussion

and

Following the presentation of our theory, in this Section we discuss differences between our approach and the currently accepted treatment of second harmonic generation. It is worth beginning with a summary of the steps which led us to our main results. First, we started from the wave equation assuming that the electromagnetic sources are given externally by some code calculating the electron beam bunching at the second harmonic. Second, after applying the ultrarelativistic approximation (c2  1) and the resonance approximation (Nw  1), both non-restrictive ones, we exactly solved the wave equation using a Green’s function method. Third, we calculated the angular distribution of intensity assuming a given beam modulation and we derived an expression for the total power radiated at the second harmonic by integrating the expression for the angular distribution of intensity. Finally, we compared the expression for the second harmonic power with the analogous expression for the first harmonic. In [3], and later on in [4,8], the ultrarelativistic and the resonance approximation were used too, but there are three points in their treatments that we find incorrect. First, the

y 0i ðzÞ ¼ y i ðzÞ;

x0i ðzÞ ¼ xi ðzÞ þ Dxi ðzÞ

ð66Þ

ð67Þ

where Dxi(z) describes the fast oscillation, while xi ðzÞ and y i ðzÞ describe the slow motion. All the d-Dirac in the x coordinate on the right hand side of the wave equation are subsequently expanded as dðxi  xi ðzÞ  Dxi ðzÞÞ ’ dðxi  xi ðzÞÞ  Dxi ðzÞd0 ðxi  xi ðzÞÞ; ð68Þ based on the only assumption that the transverse beam dimension is much larger than the wiggling amplitude of the electron motion. It should be noted that, based on this assumption, the ratio between the wiggling amplitude and the transverse beam dimensions, K/(ckwr)  1, is identified as the main physical parameter of the theory and is denoted as the coupling strength of the second harmonic emission. In contrast with this we have found that the correct solution of the wave equation depends on the Fresnel number, but not on the coupling strength (see, for instance, Eq. (54)). In this regard it is suggestive to write 1/N = Lwc/(r2x) as

G. Geloni et al. / Optics Communications 271 (2007) 207–218

1/N = [K/(ckwr)]2 · pNw(2 + K2)/(4K2). As K/(ckwr) assumes a fixed value (for instance, much smaller than unity according to the assumption above), our Fresnel number can assume any value, depending on the number of undulator periods Nw. On the one hand, if N  1 we have a behavior F 2 ðN Þ  1=ð4N 2 Þ / N 2w so that F2(N)/ F1(N) / Nw as has been seen in Eq. (64). Then, the ratio W2/W1 is independent of Nw as one can see from Eq. (63). On the other hand, if N  1 we obtain F2(N)  ln[1/(4N2)]  const. so that F2(N)/F1(N) is independent of Nw while W 2 =W 1 / N 1 w . In contrast with this, being based on the coupling strength parameter only, current understanding of the second harmonic mechanism predicts that W2/W1 is always independent of Nw. Finally we find that, from a mathematical viewpoint, the expansion in Eq. (68) constitutes an incorrect operation done on the right hand side of the wave equation. To show this, we need to consider the mathematical structure of the wave equation. For any polarization component we are dealing with a differential equation X e x;y ðx; y; zÞ ¼ fx;y ðzÞ LE dðxi  xi ðzÞ  Dxi ðzÞÞdðy i  y i ðzÞÞ; i

ð69Þ with



L¼

r2? þ

2ix o ; c oz

ð70Þ

where fx,y(z) is a function containing the appropriate phase factor. This is essentially equivalent to our starting equation, Eq. (13). The only differences are that the sources are presented in a different way and that, at that stage, we had already assumed that the transverse beam dimensions are not smaller than the wiggling amplitude of the electron motion. The problem with the expansion in Eq. (68) is that Dxi = Dxi(z) is a function of the longitudinal coordinate and that the Green’s function for the wave equation depends on both longitudinal and transverse coordinates. Let us see this point in more detail. If we call G(zo  z 0 , x0  x 0 , y0  y 0 ) the Green’s function of the operator L we have ~ x;y ðzo ;xo ;y o Þ E Z 1 Z ¼ dx0 1

1

dy 0

Z

1

1

dz0 Gðzo  z0 ;xo  x0 ; y o  y 0 Þfx;y ðz0 Þdðx0  xðz0 Þ

1

 Dxðz0 ÞÞdðy 0  y ðzÞÞ;

ð71Þ

that is ~ x;y ðzo ; xo ; y o Þ ¼ E

Z

1

dz0 Gðzo  z0 ; xo  xðz0 Þ

1

 Dxðz0 Þ; y o  y ðz0 ÞÞfx;y ðz0 Þ:

ð72Þ

It follows that the expansion of the d-Dirac in Eq. (68) is mathematically equivalent to the expansion of the Green’s function G in Dxi(z 0 ) around xo  xi ðz0 Þ. However, under the only assumptions xi ðz0 Þ  jDxi ðz0 Þj and y i ðz0 Þ 

217

jDxi ðz0 Þj, we cannot expand the Green’s function in Dxi(z 0 ) around xo  xi ðz0 Þ. In fact we have Gðzo  z0 ; xo  xi ðz0 Þ  Dxi ðz0 Þ; y o  y ðz0 ÞÞ 6¼ Gðzo  z; xo  xi ðz0 Þ; y o y i ðz0 ÞÞ  dGðzo  z0 ; n; y o  y i ðz0 ÞÞ  Dxi ðz0 Þ  dn

;

ð73Þ

n¼xo xi ðz0 Þ

because G is simultaneously a function of z not only through Dxi(z 0 ), xi ðz0 Þ and y i ðz0 Þ but also through zo  z 0 . 5. Conclusions In this paper we addressed the mechanism of second harmonic generation in Free-Electron Lasers in the case of planar undulators. We found that an early treatment of this phenomenon [3] includes incorrect operations made on the source term of the wave equation, which describes the electrodynamical part of the problem. Namely, an important part of the source term is dropped and, second, an expansion of the particles trajectory in the transverse horizontal direction is performed. We find that there is no ground for such a step. Moreover, this leads to the identification of the ratio of the amplitude of the electron wiggling motion and the electron beam transverse size with the main physical parameter of the problem. Such parameter does not play any role in our theory. The same steps were also followed in [4,8]. After the above-mentioned operations, the wave equation is not solved. Rather, an estimation of the second harmonic power is given. First, the square of the source parts in the wave equation for the second and either for the first [8] or the third [3,4] harmonic is calculated. Second, the ratio of the square of the second harmonic source part and either the square of the first or the third is taken. Finally, authors of [8] introduced the notion that the second harmonic power increases when a deflection angle between the beam trajectory and the undulator z direction is present. In contrast to this, we find that such power can only decrease or, at most, be independent of the deflection angle, depending on how the beam modulation is prepared. By solving analytically the wave equation with the help of a Green’s function technique we derived an exact expression for the field of the second harmonic emission. We limited ourselves to the steady-state case, which is close to practice in High-Gain Harmonic Generation (HGHG) schemes. For the rest, we did not make restrictive approximation. Our solution of the wave equation may therefore be used as a basis for the development of numerical codes dealing with second harmonic emission, which should be taking, as input data, the electron beam bunching for the second harmonic calculated by self-consistent FEL codes. We found that, in general, the second harmonic field presents both horizontal and vertical polarization components and that the electric field is linearly polarized. Moreover, the magnitude of the power associated to the vertical polarization component relative to that associated to the

218

G. Geloni et al. / Optics Communications 271 (2007) 207–218

horizontal polarization component is a function of the undulator deflection parameter only. Using our result, we analytically calculated the directivity diagram and the power associated with the second harmonic radiation assuming a particular beam modulation case. We expect that these expressions may be useful for cross-checking of numerical results. In this paper we presented a theory of the second harmonic generation mechanism in XFELs and pointed out several notions which we consider incorrect. In this regard, it should be noted that some of them appear to go beyond the subject of harmonic generation itself. We have seen that, according to these notions there should be an increase of the second harmonic power when an angle between the beam direction and the undulator axis is present. In our understanding they seem to imply, very much like a general property following from Maxwell’s equations, that the solution of Maxwell’s equations for any electron beam in an undulator can show a non-trivial dependence on the angle between the trajectory and the undulator axis. If this was the case, this conclusion should be valid, in particular, for a single particle. We find that this is not correct. For a single particle in an undulator, the dependence of the electric field on the angle between the average direction of the particle and the undulator axis, gx,y is related to the chosen reference system, usually one with the z axis aligned with the undulator axis. A simple rotation of an angle gx,y to a system with the z axis aligned with the electron average velocity would give a result independent of such an angle. This means that the basic characteristics of undulator radiation, in particular the intensity distribution at fixed frequency and the spectrum at fixed observation angle hx,y, depend on the combination (hx,y  gx,y) only. In other words, the presence of an angle gx,y between the electron direction and the undulator axis has the only effect of intro-

ducing a rotation in the expression of the electric field, leaving otherwise unvaried all its characteristics, including its resonance frequency. Acknowledgements The authors wish to thank Martin Dohlus (DESY) for useful discussions and Josef Feldhaus (DESY) for his interest in this work. Particular thanks go to Paul Radcliffe (DESY) for his help during the development of the final version of this work. References [1] E. Saldin, E. Schneidmiller, M. Yurkov, The Physics of Free Eelectron Lasers, Springer, 2000. [2] Z. Huang, K. Kim, Phys. Rev. E 62 (2000) 5. [3] M. Schmitt, C. Elliot, Phys. Rev. A 34 (1986) 6. [4] Z. Huang, K. Kim, Nucl. Instr. and Meth. Phys. Res. A 475 (2001) 112. [5] H. Freund, S. Biedron, S. Milton, Nucl. Instr. and Meth. Phys. Res. A 445 (2000) 53. [6] A. Tremaine et al., Phys. Rev. Lett. 88 (2002) 204801. [7] S. Biedron et al., Nucl. Instr. and Meth. A 483 (2002) 94. [8] Z. Huang, S. Reiche, Generation of GW-level, sub-Angstrom radiation in the LCLS using a second-harmonic radiator, in: R. Bakker et al. (Ed.), Proceedings of the FEL 2004 Conference, Trieste, Italy, 2004, p. 201. [9] H. Wiedemann, Synchrotron Radiation, Springer-Verlag, Germany, 2003. [10] H. Onuki, P. Elleaume (Eds.), Undulators, Wigglers and their applications, Taylor & Francis, 2003. [11] G. Geloni, E. Saldin, E. Schneidmiller, M. Yurkov, Paraxial Green’s Functions in Synchrotron Radiation Theory, DESY 05-032, ISSN 0418-9833 (2005). [12] D. Alferov, Y. Bashmakov, E. Bessonov, Sov. Phys. – Tech. Phys. 18 (1974) 1336. [13] E. Saldin, E. Schneidmiller, M. Yurkov, Nucl. Instr. and Meth. A 539 (2005) 499.