Analysis of Statistical Correlations and Intensity Spiking in the Self-Amplified Spontaneous-Emission Free-Electron Laser

The narrow band chaotic output of the self-amplified spontaneous-emission free-electron laser (SASE FEL) exhibits intensity spikes. In the linear regime before saturation, we use an approach developed by Rice to determine probability distributions for the peak values of intensity in both the time and frequency domains. We also find the average number of spikes per unit time or frequency. In addition, we derive joint probabilities for the intensity in the output pulse to have values 1 I and 2 I at times 1 t and 2 t , and for the spectral intensity to have values ~ 1 I and ~ 2 I at frequencies 1 ω and 2 ω . PACS: 41.60.Cr, 02.50.-r Submitted to: Physical Review ST-AB 1 Permanent address: Brookhaven National Laboratory, Upton, NY 11973 * Work supported by Department of Energy contracts DE–AC03–76SF00515 and DEAC02-98CH10886.


I. INTRODUCTION
The theory of high-gain single-pass free-electron lasers has been developing since the late 1970's [1][2][3][4][5].In the absence of an external seed laser, the SASE FEL starts up from the shot noise in the electron beam.Because SASE starts from the shot noise, a proper theory requires a statistical treatment of the output radiation [6][7][8][9][10][11][12][13][14][15][16][17].Average properties of the output were studied in [6][7][8][9][10][11][12] and fluctuations were considered in [13][14][15][16][17].To describe the shot noise, one considers the arrival time of the individual electrons at the undulator entrance to be independent random variables, and one determines the statistical properties of the output radiation by averaging over the stochastic ensemble of arrival times.In the linear regime before saturation, it follows from the Central Limit Theorem [18] that the probability distribution describing the spectral intensity ( ) ω

I
, or the time-domain intensity ( ) t I , is the negative exponential distribution [14] and the intensity fluctuation is 100%.
The output intensity as a function of time exhibits spiking [13] (see Fig. 1), and the width of the intensity peaks is characterized by the coherence time [14,15], , where ω σ is the SASE gain bandwidth.The spectral intensity also exhibits spikes (Fig. 2), and the width of the spectral peaks is inversely proportional to the electron bunch duration b T .
At a fixed position z along the undulator, consider the energy in a single SASE pulse, where b T is the duration of an electron bunch having uniform average density.For z fixed, one can think of dividing the pulse into M statistically independent time-intervals of width T coh .The energy fluctuation within a single coherent region is 100%, but the fluctuation W W / σ of the energy in the entire pulse is reduced and given by [14,15] ( ) Here, M is defined [14,19] to be the number of modes in the radiation pulse.The energy per pulse is described by the gamma distribution [14,18,19], (1.4) Let ( ) ( ) ( ) be the radiated SASE electric field, where is the undulator resonant frequency.In the description (at fixed z) of the statistical properties of the SASE output, two important quantities [14] are the field correlation function and the intensity correlation function In the linear region before saturation, they are related by (1.7) The energy fluctuation W σ in a pulse can be expressed in the form . (1.9) In this paper, we advance the statistical description of SASE by applying the mathematical analysis of random noise developed by Rice [18].Some of the results of our analysis were presented in ref. [17].In the earlier work [14] reviewed above, two important distributions of a "global" nature were derived: the exponential distribution [Eq.1.1)] applies to an arbitrary time or frequency, and the gamma distribution applies to the total radiated energy.In this paper, we study what might be termed "local" statistical properties.In the time-domain, we determine the joint probability [Eq.(3.14)] that at a fixed position z along the undulator axis, the normalized intensity in the radiation pulse has the values Q 1 and Q 2 at times t 1 and t 2 .We also find the probability per unit time [Eq.(4.10)] of observing a spike with maximum normalized intensity Q.In the frequency domain, we derive the joint probability [Eq.(3.21)] that the normalized spectral intensity has the values ~1 Q and ~2 Q at frequencies 1 ω and 2 ω .We also find the probability per unit frequency interval [Eq.(4.18)] of observing a peak in the frequency spectrum with maximum normalized intensity Q .
In Eq. (1.3), the number of modes M in the output pulse was defined in terms of the pulse energy fluctuation.Another quantity of interest is the number of intensity spikes s N (Fig. 1).Following Rice [18], we characterize the temporal spikes by the existence of local maxima of the intensity.From Eq. (4.14), we find that It is interesting to note that all known statistical properties of SASE derived in previous work [14] and in this paper can be derived from Rice's study [18] of shot noise in telephone systems, see also ref. [19].
Our paper is organized as follows: In Section II, we review the calculations of field and intensity correlations for the SASE radiation.In Section III, we use the Central limit theorem to determine the joint probability distributions mentioned above.In Section IV, we find the probability per unit time of observing a spike with maximum normalized intensity Q.From this result, we determine the average number of temporal spikes per unit time.We also find the probability per unit frequency interval of observing a peak in the frequency spectrum with maximum normalized intensity Q .From this result, we determine the average number of spectral spikes per unit frequency.In Section V, we present an approximate calculation of the probability per unit time of observing a spike with maximum normalized intensity Q>>1.This gives one an intuitive (albeit mathematical) picture of the spikes making up the SASE output.A summary of our results is given in Section VI.

II. CORRELATIONS
We work within the classical, one-dimensional approximation, in the linear regime before saturation.The uniform density electron bunch contains N e electrons and has length, L b = cT b , long compared to the coherence length of the output radiation.We restrict our attention to the radiated field inside the electron bunch sufficiently far from the back end so that coherent effects [12,20,21] can be neglected.The radiation field has the form, with amplitude A given by where k w is the undulator wavenumber, 0 2 γ mc is the initial electron energy and a w is the undulator strength parameter.The SASE green's function is denoted by ( ) t z h , and the arrival time of the j th electron at the undulator entrance (z=0) by j τ .The independent random variables j τ (j=1,…,N e ) are considered to be uniformly distributed over the interval We use brackets to represent an average over the arrival times.For the uniform distribution, . We also introduce the shorthand notation: ) , ( 1 . The correlation of the amplitudes at a given position z along the undulator, at two different times is given by: In Eq. (2.3) and in the following, we do not explicitly show the z-dependence of the functions, since we are working at fixed z.While deriving Eq. (2.3), we retain only the dominant contributions characterized by the absence of rapid phase variation.These correspond to keeping pair-wise equal summation indices from the A and A * terms.
It is often useful to introduce the Fourier transform (2.4) The correlation of Eq. ( 2.3) can now be expressed as The correlation of the intensities at two different times can be calculated as follows: This is the result of Eq. (1.7).Again we have kept only those terms without rapid phase variation.
Deep into the exponential gain regime, a saddle point analysis yields the Gaussian approximation to the green's function where a n d ρ is the Pierce parameter [3].We can write this asymptotic approximation in the form (2.9) Taking the Fourier transform of (2.8), we find ) Here, ω σ is the SASE gain bandwidth (see Fig. 2).Note that has a dependence on s ω that is not explicitly shown.Using the Gaussian approximation of Eq. (2.8) in Eq.
(2.3) we derive (2.12) We define the normalized amplitude, and use the notation a 1 =a(z,t 1 ) and a 2 =a(z,t 2 ).Under the approximations we are employing, β .We note that in terms of this notation, the field correlation introduced in Eq. (1.5) is given by ( ) and the intensity correlation introduced in Eq. (1.6) is ( ) . In the high-gain regime before saturation, it follows from Eqs.

III. JOINT PROBABILITY DISTRIBUTIONS
In Eq. (2.2), the amplitude A is represented as a sum of independent random terms; it follows that probability distributions describing the output radiation are determined from the Central Limit Theorem [18].The central limit theorem (see Appendix A) states that of N independent random vectors approaches the normal law as ∞ → N . For simplicity consider 0 = j r ; then as where ) ,..., ( 1 is a K-dimensional row-vector (the superscript T indicates transpose) and V the corresponding column vector.The symmetric matrix M is comprised of the second moments: M -1 is the inverse of the matrix M. Note that when the central limit theorem applies, the distribution is Gaussian and hence is determined by the second moments.Under these conditions, one need not compute all the higher moments to determine the distribution-a great simplification.
As an illustration, consider the special case when .Again, we take 0 = = j j y x so independence In this case, the central limit theorem implies that the probability distribution P(X,Y) for The probability P(Q,φ )dQdφ for finding Q between Q and Q+dQ, and φ between φ and φ φ d + , is given by As shown by Rice [18], the Central Limit Theorem also enables us to determine the joint probability distribution describing the field amplitudes at two different times.
The required correlations are determined from Eqs. (2.14), and we find:   [18]: I 0 is the Bessel function of imaginary argument of order zero.Note that when . The distribution of Eq. (3.14) has been used to describe narrow band chaotic light [22].Some mathematical properties of the distribution have been studied in [23], where the following expansion in terms of Laguerre polynomials was derived It is now of interest to determine the conditional average The corresponding fluctuation is We note from Eq. (3.17) that , and is greater than Q 1 when Q 1 <1.This is the statistical basis for the appearance of spikes in the radiation output.
The analysis just presented in the time-domain can be extended into the frequency-domain.Let ) , ( ~ω ∆ z A be the Fourier transform of A(z,t), where where and sinc(x)=(sin x)/x.As in the time-domain, it is possible to write Eq. (3.21) in terms of Laguerre polynomials.

IV. INTENSITY PEAKS
We can provide a statistical description of the spikes in the output, first in the time-domain and then in the frequency-domain.To prepare for the discussion of the intensity peaks, we first review the required mathematics developed by Rice [18].
Consider the stochastic function ) ; ,..., , ( We can now determine the probability p(y 1 )dtdy that F has a maximum with value between y 1 and y 1 +dy in the time interval t 1 <t<t 1 +dt.At a maximum, the derivative of F is zero and its second derivative is negative.From the result of Eq.( 4.3), we see that ( ) , the matrix M of second moments, Eq. (3.2), is independent of t.We use the variable Q=R 2 , and we choose the normalization K(z) such The probability dtdQ Q p t ) ( that a maximum of Q with value between Q and Q+dQ is found in time interval dt is determined by [18], For Q>>1, the sum over n in (4.10) clearly peaks near n=Q.One can treat n as a continuous variable, convert the sum over n to an integral and use the method of steepest descent to obtain Another method for evaluating the sum over n in (4.10) for large Q is to separate it into two pieces Summing over alternate values of n and recognizing that the lower limit n=0 in the sums can be modified slightly without affecting the overall limit as , one obtains the asymptotic approximation of Eq. (4.11) by noting that  The number of peaks per unit time is The probability of finding a maximum with normalized intensity between Q and Q+dQ is p t (Q)/N t dQ, which is plotted in Fig. 3.The average peak height is 56 . 1 = Q and the rms peak height fluctuation is 1.27.In Eq. (4.14) and Fig. 3, ( ) is determined from Eq. (4.10).
In the frequency domain, (4.17)Following Rice's analysis [18], we find that the probability independent of ω , with (4.20) The number of peaks per unit frequency is The probability of finding a maximum with normalized spectral intensity between , which is plotted in Fig. 4. The average spectral peak height is 66 . 1

V. APPROXIMATE DESCRIPTION OF INTENSITY SPIKES
Motivated by the result for the conditional average, Eq. (3.17), it is reasonable to assume that the output is comprised of a series of peaks, and in the region near each maximum, the intensity profile can be approximated by where Q p is the maximum intensity of the peak centered about t=t p .
Recall from Section IV that p t (Q)dQdt is the probability of finding in the radiation output an intensity peak of magnitude between Q and Q+dQ, in time interval dt.Let us consider an intuitive argument for determining the large Q behavior of p t (Q).For large Q 1 , the probability of the intensity Q being greater than Q 1 can be evaluated in two ways.
First, it is equal to Alternatively, it can be written as , where ) , ( represents the region in t, under a peak having maximum value Q, for which the intensity is greater than Q 1 .From Eq. (5.1), we find . Equating these two expressions for the probability that the intensity Q be greater than Q 1 , we derive the following integral equation for p t (Q): For large Q 1 , the exponential behavior on the right hand side of Eq. ( . If one changes the variable of integration to y=Q-Q 1 , then Eq. ( 5.2) becomes Clearly, the important values of y are of the order of 1.Thus, for large Q 1 , one has y<<Q 1 and ) The solution of (5.4) for large Q . Thus the solution of Eq.
(5.2) for large Q is in reasonable agreement with the exact value, 1, of this average.Similarly, let us consider the output intensity spectrum to be comprised of a sum of peaks also in reasonable agreement with the exact value, 1.

VI. SUMMARY OF RESULTS
In this paper, we have used the approach of Rice [18] to extend our knowledge of SASE statistics.In earlier work [14], it was shown that the probability distribution for the intensity at a given time or frequency is the exponential distribution [Eq.(1. In the time-domain (Fig. 1), we have derived the probability p t (Q)/N t dQ [Eq.(4.10) and Fig. 3] of finding a maximum with normalized intensity between Q and Q+dQ.The number of peaks per unit time was found to be  Many of the statistical properties of the radiation reviewed in the introduction to this paper have been verified experimentally [24].In particular, it was observed that the fluctuations of the SASE radiation energy followed the gamma distribution [Eq.(1.4)], and the output of a narrow-band monochromator behaved as expected from theory.
Spikes in the spectral distribution of intensity have been measured experimentally [25][26][27].If experimental conditions are sufficiently controlled so that fluctuations are dominated by shot noise and not variations in experimental parameters such as charge, bunch length or emittance, then the distribution of peak heights can be measured and compared with the prediction of Eq. (4.18).Also, one can compare the number of peaks per unit frequency with the prediction of Eq. (6.4) In terms of the spectral density w(f) defined by Rice [18], this correlation of the electric fields can be expressed as

Figure 1 . 1 Figure. 2 .
Figure 1.Intensity spiking in the time-domain (arbitrary units).The width of the peaks is characterized by the SASE coherence time

7 )
Returning to the FEL problem, let us express the normalized field amplitude defined in Eq. (2.13) as .Limit Theorem, it is seen that the probability P(x,y)dxdy for finding x between x and x+dx, and y between y and y+dy, is given by follows from Eqs. (3.1) and (3.12) that

the normalized spectral intensities at frequencies ω 1 and ω 2
is determined in a manner similar to the derivation in the time-domain leading to Eq.(3.14).We find,

Figure. 3 .
Figure. 3. p t (Q)/N t dQ is the probability of a peak in the time-domain having normalized intensity between Q and Q+dQ (dimensionless variables).

( 4 .
18) and Fig.4] of finding a maximum with normalized spectral intensity between In this paper, we have derived the joint probability P(Q 1 ,Q 2 )dQ 1 dQ 2 for finding normalized intensity between Q 1 and Q 1 +dQ 1 at time t 1 , and Q 2 and Q 2 +dQ 2 at time t 2 : . In the future, if one develops methods to directly measure spikes in the time domain, then the predictions of Eqs.(4.10) and (6.3) can be compared with experiment.i