- Email: [email protected]
Calculation of intracavity laser modes in the case of a partial waveguiding in the vacuum chamber of the “CLIO” FEL
Nuclear Instruments and Methods in Physics Research A 475 (2001) 524–530
Calculation of intracavity laser modes in the case of a partial waveguiding in the vacuum chamber of the ‘‘CLIO’’ FEL Rui Prazeres*, Vincent Serriere1 LURE, 209d, Universite! de Paris Sud, BP 34, 91898 Orsay cedex, France
Abstract A new method to calculate the propagation of the laser field in a waveguide is presented here. This method uses the ‘‘Fast Fourier Transform’’, without mode decomposition, and it is compatible with the propagation both in a waveguide and in the free space. This is interesting in the case of the FEL, when the optical cavity is not completely guided, and when the guiding occurs softly or only in a limited part of the waveguide; for example, when the input wave is not guided at the entrance and becomes guided at the end. A numerical code, using this method for the wave propagation, is described here. It gives the total losses of the optical cavity, and the transverse profile of the laser at any point of the cavity. This code allowed us to calculate the best parameters of the ‘‘CLIO’’ infrared FEL, corresponding to the minimum of losses. r 2001 Published by Elsevier Science B.V. PACS: 41.60.Cr; 42.79.Gn; 84.40.Az Keywords: Free electron laser; Waveguide
1. Introduction For the Free Electron Lasers (FEL) operating in the far infrared, the diffraction of the light inside the optical cavity, necessitates the use of a waveguide in place of the vacuum chamber of the undulator. In some cases, this waveguide does not match the total length of the optical cavity, because of the free space required for the magnetic dipoles which are guiding the electron beam. As a consequence, the propagation of the laser field in
*Corresponding author. Tel.: +33-1-64-46-80-91; fax: +331-64-46-41-48. E-mail address: [email protected] (R. Prazeres). 1 Present address: ESRF, BP 220, F-38043 Grenoble Cedex, France.
the optical cavity involves both waveguide propagation and free space propagation. On the other hand, the undulator vacuum chamber (i.e. the waveguide) must have a flat cross-section in order to allow a minimum value for the undulator gap. We can consider, in first approximation, that the waveguide is of horizontal plane type. As a consequence, the guiding of the wave only occurs in the vertical plane. The numerical operator of wave propagation, which is defined here, is a reduction to a plane waveguide of a more general method used for rectangular waveguides [1]. The absorption in the walls of the waveguide is not considered here. The absorption is more important for the high order waveguide modes, but these modes are principally lost by diffraction, which is due to clipping of the laser profile by the mirror or
0168-9002/01/$ - see front matter r 2001 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 5 9 9 - 6
R. Prazeres, V. Serriere / Nuclear Instruments and Methods in Physics Research A 475 (2001) 524–530
by the waveguide entrance. The other numerical methods which have been used until now are mostly based on a decomposition in the waveguide proper modes [2,3]; they are more cumbersome and imprecise because they require a large number of modes.
verify [1] that this method is still valid for the propagation of the proper modes in a waveguide. The convolution of a proper mode with G(x,y) gives ~pq ðx; y; 0Þ Gðx; yÞ ¼ E ~pq ðx; y; 0ÞexpðikLÞexpðidjÞ
E
ð4Þ
where
plL p2 q2 dj ¼ þ 4 a2 b2
2. Wave propagation 2.1. In free space The theory of diffraction at short distance gives the evolution of a field distribution Cðx; y; zÞ when propagating along the z-axis [4]. Considering the complex amplitude Cðx; y; 0Þ at z=0, the field C0 ðx0 ; y0 ; LÞ at distance z=L is given by a convolution of C with the ‘‘Green function’’ G(x,y): C0 ðx; y; LÞ ¼ Cðx; y; 0Þ Gðx; yÞ
ð1Þ
where Gðx; yÞ ¼
525
ik ikL ikðx2 þy2 Þ=2L e e : 2pL
The convolution with the Green function can be calculated numerically as a simple product, in Fourier space, by TF[G](u,v), where TF[G](u,v) is the Fourier Transform of G(x,y): ’ þv Þ TF½G ðu; vÞ ¼ eikL eiplLðu : 2
2
ð2Þ
In the design of a numerical code [5], we can define a propagation operator [P] which is a combination of three operators: ~ 0 ¼ ½P C ~ ¼ ½FFT1 ½ TFðGÞ ½FFT C ~ C
ð3Þ
where [FFT] is the 2D ‘‘Fast Fourier Transform’’, [ TF(G)] is an operator which makes a product, element by element, with TF[G](u,v), and [FFT1] is the inverse FFT. 2.2. In a waveguide The proper modes TEpq and TMpq of a rectangular waveguide are well known [6]. As shown in Section 2.1, the convolution Cðx; yÞ Gðx; yÞ gives the propagation in the free space, i.e. a priori not in a waveguide. Nevertheless, we can
corresponds to the phase shift of the proper mode TEpq or TMpq for the waveguide propagation when f bfc where fc is the cut-off frequency, a and b are the transverse dimensions of the waveguide. As a consequence, this propagation operator can also calculate the longitudinal evolution in a waveguide of a linear combination of proper modes, i.e. to any profile Cðx; yÞ: This method can be extended [1] to the general case where f also may be close or smaller than fc; using in Eq. (4) the exact expression for the phase shift of the modes {p,q}: " 2 2 #1=2 oL 2pL pl ql 1þ ¼ þ : Dj ¼ vj l 2a 2b The advantage of this method of calculation is the full compatibility with the case of a guided wave (waveguide propagation), an unguided wave (free space propagation) and any intermediate regime where the guiding occurs softly or only in a limited part of the waveguide. The numerical operator [P] is a numerical approximation of the convolution with G(x,y). The operator [P] uses the FFT routine, applied to a wave Cðx; yÞ which is sampled in the domain [[Dx; Dy]]. In normal conditions, as in Section 2.1, the amplitude of C must be equal to zero on the boundary of the domain [[Dx; Dy]]; otherwise, the result of the FFT is false. For the FFT of a proper ~pq ðx; yÞ is extending in the whole mode, the field E domain [[Dx; Dy]], and the final result of the FFT should be false. Nevertheless, if the size of the FFT domain [[Dx; Dy]] is equal to the transverse periodicity of the proper modes, then the result of the FFT is correct. This corresponds to the condition Dx ¼ 2a and Dy ¼ 2b; where a and b are
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the transverse dimensions of the waveguide. When this condition is respected, the transverse periodicity of the proper modes is equal to the periodicity of the FFT, and we can replace the convolution by the operator [P]: ~0 ðx; y; LÞ ¼ ½P E ~pq ðx; y; 0Þ ¼ E pq ~pq ðx; y; 0Þ Gðx; yÞ:
E
ð5Þ
In the following part of the paper, we will always consider that the operator [P] is defined in the area [[2a,2b]]. ~ ðx; yÞ Let us consider now a given wave front C at z=0, which can be decomposed in a sum ~ m ðx; yÞ of proper modes TEpq and TMpq C ~ ðx; yÞ: ð6Þ ~ m ðx; yÞ ¼ Sðapq :E ~TMpq ÞaC ~TEpq þ bpq E C The profile C is generally nonperiodic, whereas ~ m ðx; yÞ comes Cm is periodic. The periodicity of C from the periodicity of the proper modes. The ~ m is only equal to C in the waveguide area wave C [[a,b]]. In summary, the propagation in free space can be calculated using the operator [P] applied to ~ ðx; yÞ; whereas the the nonperiodic profile C propagation in a waveguide must be calculated using the same opertator [P] applied to the ~ m ðx; yÞ: This profile C ~ m can be periodic profile C ~ obtained from C using an operator of mirror symmetries, called ‘‘Mosaic transform’’, which is described below. The mode decomposition is not required here. This method of calculation of propagation is very rapid because it only uses one FFT and one FFT1.
2.3. The propagation operator ~ m ðx; yÞ is deduced from The periodic profile C ~ Cðx; yÞ by the application of an operator [M] that we call ‘‘Mosaic transform’’. This operator involves mirror symmetries which are equal to the symmetries of the proper modes TEpq and TMpq. The case of a plane waveguide is more simple because it only uses the symmetry in the vertical axis. The relevant diagram of symmetry for the ‘‘Mosaic transform’’ of the polarization Ex(x,y) is shown in Fig. 1. The left part of the figure displays ~ ðx; yÞ in the FFT domain the input wave C [[Dx; Dy]], with Dy ¼ 2b: There is no condition for Dx because it is a plane waveguide: no symmetry is required in the horizontal axis, and ~ is only the propagation is free. The profile C defined in the waveguide area, for b/2oyob/2. The white part must be filled with the function ~ ðx; yÞ by mirror symmetries according to Fig. 1. C The symmetry properties of TEp and TMp are the same. However, the polarizations Ex and Ey of the same proper mode have opposite signs in the symmetries. Therefore, there are two mosaic operators, [Mx] and [My], respectively, for the two polarizations Cx and Cy : Fig. 1 shows the operator [Mx], which corresponds to a mirror symmetry with a sign inversion of C: This inversion can be observed in the representation of the amplitude of the TE1 mode in Fig. 1. The operator [My] corresponds to the same representation, in Fig. 1, but with a mirror symmetry without sign inversion of C: These two ‘‘Mosaic operators’’
Fig. 1. Mosaic transform [Mx] for the polarization Ex(x, y): the domain on the left is Cðx; yÞ in [[Dx,Dy]], with Dx=2b. The domain on the right is Cm ðx; yÞ: The curve on the right-hand side represents the TE1 mode.
R. Prazeres, V. Serriere / Nuclear Instruments and Methods in Physics Research A 475 (2001) 524–530
~ ðx; yÞ in order can be applied to any input wave C ~ to obtain the mosaic profile Cm ðx; yÞ which is used in the waveguide propagation: Cxm ¼ ½Mx Cx and Cym ¼ ½My Cy : The complete operator of waveguide propagation includes some other operators: a diaphragm operator [D] which sets to zero the field outside of the waveguide, and a couple of resampling operators [S+] and [S]. This resampling is used for switching between the FFT domains [[Dx,Dy]]: one domain for the free space areas of the optical cavity, and one restricted domain with the condition Dy=b for the waveguide area. In summary, the propagation subroutine will use the following series of operators, ~ ðx; yÞ: applied to the profile C ~ x ðx; yÞ ¼ ½S ½D ½P ½Mx ½D ½Sþ Cx ðx; yÞ C C0y ðx; yÞ ¼ ½S ½D ½P ½My ½D ½Sþ Cy ðx; yÞ: ð7Þ
3. Simulations for ‘‘CLIO’’ 3.1. The numerical code A numerical simulation, involving the propagation operator described in Section 2.3, has been
527
done in order to calculate the losses of the optical cavity of the FEL when a waveguide is used in place of the undulator vacuum chamber. The numerical code [5] uses several operators: for the reflection on the cavity mirrors, for the transversal gain distribution G(x,y) and for the propagation both in free space or in the waveguide. Fig. 2 shows a layout of the optical cavity. Each mirror is defined by the reflection coefficient for the amplitude, the radius of curvature and the diameter. The gain transverse distribution is a Gaussian function G(x,y), centered on axis, in the center of the undulator. The field of validity of the propagation operator is the same for free space and waveguide propagation: it is limited to the paraxial approximation (small angles). The program calculates the propagation in the cavity of a wave front Cðx; yÞ of complex amplitude, representing the optical field. This wave front is sampled in the transverse plane by 128 128 points, representing a total area of 10 10 cm. The process starts from a Gaussian TEM00 field distribution C0 ðx; yÞ in the center of the cavity. This initial field distribution is normalized to jjC0 jj ¼ 1: The iterative procedure of propagation in the cavity leads to a convergent solution Cn ðx; yÞ; where n is the number of cavity round
Fig. 2. Layout of the optical cavity used in the numerical code.
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R. Prazeres, V. Serriere / Nuclear Instruments and Methods in Physics Research A 475 (2001) 524–530
trips. Between n=20 and 40 iterations are necessary to obtain a convergent solution, corresponding to the saturation limit. The value of the gain at saturation Gs is not known in advance and depends on the losses of the final mode. This gain value is determined during the iterations, in parallel with the mode, until it reaches the saturation limit. After convergence, the total losses of the cavity can be deduced from the gain value at saturation Gs. The size 10 10 cm of the sampling area is determined in order to fit the maximum width of the laser profile, i.e. on the mirrors. The laser profile cannot be much wider than the mirrors, for which the diameter is F ¼ 38 mm in the case of CLIO. On the other hand, the number of points for the sampling depends on the computer capability, but it must be sufficient to allow a reasonable resolution for the sampling in
the middle of the waveguide, where the gain profile fits the electron beam profile: i.e. about s ¼ 2 mm for ‘‘deep’’ infrared operation of CLIO (l>20 mm). 3.2. Numerical results The aim of these simulations is to obtain the parameters of ‘‘CLIO’’ which are giving a minimum of losses in the optical cavity. In the first set of simulations, the cavity losses have been calculated for various mirror radius of curvature Ry1 and Ry2 for the vertical axis on both mirrors. The horizontal radius of curvature remains constant with Rx1=Rx2=3 m, because the wave is not guided along x-axis. The density plots in Fig. 3 represent the losses versus Ry1 and Ry2. Four laser wavelengths have been used: l=25, 50, 100
Fig. 3. Density plot of cavity losses versus the mirror radius of curvature Ry1 and Ry2 on vertical axis for both mirrors.
R. Prazeres, V. Serriere / Nuclear Instruments and Methods in Physics Research A 475 (2001) 524–530 100
A : losses for a non waveguided vacuum chamber total cavity LOSSES (%)
B : losses without vacuum chamber 80
C : losses with the plane waveguide
60
A 40
20
C
B mirror losses : 8.7%
0 0
20
40
60
80
100
Wavelength ( µm)
Fig. 4. Cavity losses calculated for the standard set of mirrors R1=R2=3 m.
529
and 200 mm. The small losses correspond to the dark points. The point with the label ‘‘CLIO’’ corresponds to the standard cavity of CLIO with Rx=Ry=3 m on both mirrors. The transverse dotted line corresponds to the theoretical limit of stability of the optical cavity 0o(1L/Ry1)(1L/ Ry2)o1 in vertical plane: only the points above this line are in the stable area. The white points with a cross correspond to a nonconvergent simulation : the laser profile C is not stable along the consecutive cavity round trips. All these points are below the limit of stability of the cavity. At a shorter wavelength, l ¼ 25 mm, the minumum of losses corresponds to the standard cavity with symmetric mirrors: Ry1=Ry2=3 m. However, for the larger wavelength l ¼ 200 mm, the smaller
Fig. 5. Transverse profiles Cðx; yÞ at both ends of the waveguide and on mirror 2.
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losses are only obtained with toroidal mirrors: with Ry1=1.1 m and Ry2=1.7 m. In this configuration, each radius of curvature Ry is equal to the distance between the mirror and the closer entrance of the waveguide (see Fig. 2): Ry1=d1 and Ry2=d2. This point stays below the limit of cavity stability, but this limit only corresponds to a free cavity which is not the case here. Note that for any wavelength larger than 100 mm, the standard cavity of CLIO with Ry1=Ry2=3 m gives very large losses which do not allow lasing. This result is in agreement with some calculations already published [3] which lead to the same conclusion by empirical methods. However, this simulation gives the absolute value of the cavity losses, and such an extended view of the FEL losses versus the cavity parameters could not be obtained with empirical methods. We can also observe in Fig. 3 that many points below the limit of stability, at l=25 and 50 mm, are still converging points with low losses (dark points). Only a few of them are marked with a white cross (nonconverging simulation). This means that, in principle, the lasing can be obtained in this area, but the laser may be strongly unstable. As shown in Fig. 3, working at wavelengths larger than 100 mm requires toroidal mirrors. Therefore, it is interesting for us to know where the wavelength limit of laser operation is with the standard set of mirrors R1=R2=3 m. Fig. 4 shows the cavity losses obtained with the simulation for R1=R2=3 m. The flat curve B corresponds to a simulation without the vacuum chamber: the losses of 8.7% correspond to the mirror reflectivity r1=0.99 and r2=0.93. The curve A corresponds to the cavity losses with an unguided vacuum chamber, which is represented
by one iris on both ends with 15 mm in vertical and infinite in horizontal. As expected, the losses are increasing strongly with the wavelength because of the diffraction. The losses with a waveguide are represented by the curve C. They include the mirror losses (8.7%), and the waveguide losses by cutting of the mode at the entrance. Note that there is a minimum of losses for l ¼ 60 mm. Fig. 5 displays the transverse profiles at both ends of the waveguide and on the mirror 2. These profiles correspond to the positive direction of propagation of the wave along the z-axis. For l ¼ 100 mm, the profile is truncated at the waveguide entrance. At l ¼ 60 mm, the laser profiles exhibit a smooth structure, close to a Gaussian profile, which leads to minimum losses. At l ¼ 55 mm, the profile exhibits a more complicated structure, which gives larger diffraction losses as for l ¼ 60 mm.
References [1] R. Prazeres, A method of calculating the propagation of electromagnetic fields both in waveguides and in free space, using the Fast Fourier Transform, Eur. Phys. J. Appl. Phys., in press. [2] K.W. Berrymann, T.I. Smith, Nucl. Instr. and Meth. A 318 (1992) 885; R.L. Elias, et al., Phys. Rev. Lett. 57 (4) (1986) 424. [3] Li Yi Lin, A.F.G. Van der Meer, Rev. Sci. Instrum. 68 (12) (1997) 4342. [4] S. Lowenthal, Y. Belvaux, Rev. Opt. 46 (1) (1967) 1. [5] R. Prazeres, M. Billardon, Nucl. Instr. and Meth. A 318 (1992) 889. [6] G. Dubost, Propagation libre et guid!ee des ondes e! lectromagn!etiques, Masson, Paris 1995, ISBN:2-22584792-4.
Calculation of intracavity laser modes in the case of a partial waveguiding in the vacuum chamber of the ‘‘CLIO’’ FEL Rui Prazeres*, Vincent Serriere1 LURE, 209d, Universite! de Paris Sud, BP 34, 91898 Orsay cedex, France
Abstract A new method to calculate the propagation of the laser field in a waveguide is presented here. This method uses the ‘‘Fast Fourier Transform’’, without mode decomposition, and it is compatible with the propagation both in a waveguide and in the free space. This is interesting in the case of the FEL, when the optical cavity is not completely guided, and when the guiding occurs softly or only in a limited part of the waveguide; for example, when the input wave is not guided at the entrance and becomes guided at the end. A numerical code, using this method for the wave propagation, is described here. It gives the total losses of the optical cavity, and the transverse profile of the laser at any point of the cavity. This code allowed us to calculate the best parameters of the ‘‘CLIO’’ infrared FEL, corresponding to the minimum of losses. r 2001 Published by Elsevier Science B.V. PACS: 41.60.Cr; 42.79.Gn; 84.40.Az Keywords: Free electron laser; Waveguide
1. Introduction For the Free Electron Lasers (FEL) operating in the far infrared, the diffraction of the light inside the optical cavity, necessitates the use of a waveguide in place of the vacuum chamber of the undulator. In some cases, this waveguide does not match the total length of the optical cavity, because of the free space required for the magnetic dipoles which are guiding the electron beam. As a consequence, the propagation of the laser field in
*Corresponding author. Tel.: +33-1-64-46-80-91; fax: +331-64-46-41-48. E-mail address: [email protected] (R. Prazeres). 1 Present address: ESRF, BP 220, F-38043 Grenoble Cedex, France.
the optical cavity involves both waveguide propagation and free space propagation. On the other hand, the undulator vacuum chamber (i.e. the waveguide) must have a flat cross-section in order to allow a minimum value for the undulator gap. We can consider, in first approximation, that the waveguide is of horizontal plane type. As a consequence, the guiding of the wave only occurs in the vertical plane. The numerical operator of wave propagation, which is defined here, is a reduction to a plane waveguide of a more general method used for rectangular waveguides [1]. The absorption in the walls of the waveguide is not considered here. The absorption is more important for the high order waveguide modes, but these modes are principally lost by diffraction, which is due to clipping of the laser profile by the mirror or
0168-9002/01/$ - see front matter r 2001 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 5 9 9 - 6
R. Prazeres, V. Serriere / Nuclear Instruments and Methods in Physics Research A 475 (2001) 524–530
by the waveguide entrance. The other numerical methods which have been used until now are mostly based on a decomposition in the waveguide proper modes [2,3]; they are more cumbersome and imprecise because they require a large number of modes.
verify [1] that this method is still valid for the propagation of the proper modes in a waveguide. The convolution of a proper mode with G(x,y) gives ~pq ðx; y; 0Þ Gðx; yÞ ¼ E ~pq ðx; y; 0ÞexpðikLÞexpðidjÞ
E
ð4Þ
where
plL p2 q2 dj ¼ þ 4 a2 b2
2. Wave propagation 2.1. In free space The theory of diffraction at short distance gives the evolution of a field distribution Cðx; y; zÞ when propagating along the z-axis [4]. Considering the complex amplitude Cðx; y; 0Þ at z=0, the field C0 ðx0 ; y0 ; LÞ at distance z=L is given by a convolution of C with the ‘‘Green function’’ G(x,y): C0 ðx; y; LÞ ¼ Cðx; y; 0Þ Gðx; yÞ
ð1Þ
where Gðx; yÞ ¼
525
ik ikL ikðx2 þy2 Þ=2L e e : 2pL
The convolution with the Green function can be calculated numerically as a simple product, in Fourier space, by TF[G](u,v), where TF[G](u,v) is the Fourier Transform of G(x,y): ’ þv Þ TF½G ðu; vÞ ¼ eikL eiplLðu : 2
2
ð2Þ
In the design of a numerical code [5], we can define a propagation operator [P] which is a combination of three operators: ~ 0 ¼ ½P C ~ ¼ ½FFT1 ½ TFðGÞ ½FFT C ~ C
ð3Þ
where [FFT] is the 2D ‘‘Fast Fourier Transform’’, [ TF(G)] is an operator which makes a product, element by element, with TF[G](u,v), and [FFT1] is the inverse FFT. 2.2. In a waveguide The proper modes TEpq and TMpq of a rectangular waveguide are well known [6]. As shown in Section 2.1, the convolution Cðx; yÞ Gðx; yÞ gives the propagation in the free space, i.e. a priori not in a waveguide. Nevertheless, we can
corresponds to the phase shift of the proper mode TEpq or TMpq for the waveguide propagation when f bfc where fc is the cut-off frequency, a and b are the transverse dimensions of the waveguide. As a consequence, this propagation operator can also calculate the longitudinal evolution in a waveguide of a linear combination of proper modes, i.e. to any profile Cðx; yÞ: This method can be extended [1] to the general case where f also may be close or smaller than fc; using in Eq. (4) the exact expression for the phase shift of the modes {p,q}: " 2 2 #1=2 oL 2pL pl ql 1þ ¼ þ : Dj ¼ vj l 2a 2b The advantage of this method of calculation is the full compatibility with the case of a guided wave (waveguide propagation), an unguided wave (free space propagation) and any intermediate regime where the guiding occurs softly or only in a limited part of the waveguide. The numerical operator [P] is a numerical approximation of the convolution with G(x,y). The operator [P] uses the FFT routine, applied to a wave Cðx; yÞ which is sampled in the domain [[Dx; Dy]]. In normal conditions, as in Section 2.1, the amplitude of C must be equal to zero on the boundary of the domain [[Dx; Dy]]; otherwise, the result of the FFT is false. For the FFT of a proper ~pq ðx; yÞ is extending in the whole mode, the field E domain [[Dx; Dy]], and the final result of the FFT should be false. Nevertheless, if the size of the FFT domain [[Dx; Dy]] is equal to the transverse periodicity of the proper modes, then the result of the FFT is correct. This corresponds to the condition Dx ¼ 2a and Dy ¼ 2b; where a and b are
526
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the transverse dimensions of the waveguide. When this condition is respected, the transverse periodicity of the proper modes is equal to the periodicity of the FFT, and we can replace the convolution by the operator [P]: ~0 ðx; y; LÞ ¼ ½P E ~pq ðx; y; 0Þ ¼ E pq ~pq ðx; y; 0Þ Gðx; yÞ:
E
ð5Þ
In the following part of the paper, we will always consider that the operator [P] is defined in the area [[2a,2b]]. ~ ðx; yÞ Let us consider now a given wave front C at z=0, which can be decomposed in a sum ~ m ðx; yÞ of proper modes TEpq and TMpq C ~ ðx; yÞ: ð6Þ ~ m ðx; yÞ ¼ Sðapq :E ~TMpq ÞaC ~TEpq þ bpq E C The profile C is generally nonperiodic, whereas ~ m ðx; yÞ comes Cm is periodic. The periodicity of C from the periodicity of the proper modes. The ~ m is only equal to C in the waveguide area wave C [[a,b]]. In summary, the propagation in free space can be calculated using the operator [P] applied to ~ ðx; yÞ; whereas the the nonperiodic profile C propagation in a waveguide must be calculated using the same opertator [P] applied to the ~ m ðx; yÞ: This profile C ~ m can be periodic profile C ~ obtained from C using an operator of mirror symmetries, called ‘‘Mosaic transform’’, which is described below. The mode decomposition is not required here. This method of calculation of propagation is very rapid because it only uses one FFT and one FFT1.
2.3. The propagation operator ~ m ðx; yÞ is deduced from The periodic profile C ~ Cðx; yÞ by the application of an operator [M] that we call ‘‘Mosaic transform’’. This operator involves mirror symmetries which are equal to the symmetries of the proper modes TEpq and TMpq. The case of a plane waveguide is more simple because it only uses the symmetry in the vertical axis. The relevant diagram of symmetry for the ‘‘Mosaic transform’’ of the polarization Ex(x,y) is shown in Fig. 1. The left part of the figure displays ~ ðx; yÞ in the FFT domain the input wave C [[Dx; Dy]], with Dy ¼ 2b: There is no condition for Dx because it is a plane waveguide: no symmetry is required in the horizontal axis, and ~ is only the propagation is free. The profile C defined in the waveguide area, for b/2oyob/2. The white part must be filled with the function ~ ðx; yÞ by mirror symmetries according to Fig. 1. C The symmetry properties of TEp and TMp are the same. However, the polarizations Ex and Ey of the same proper mode have opposite signs in the symmetries. Therefore, there are two mosaic operators, [Mx] and [My], respectively, for the two polarizations Cx and Cy : Fig. 1 shows the operator [Mx], which corresponds to a mirror symmetry with a sign inversion of C: This inversion can be observed in the representation of the amplitude of the TE1 mode in Fig. 1. The operator [My] corresponds to the same representation, in Fig. 1, but with a mirror symmetry without sign inversion of C: These two ‘‘Mosaic operators’’
Fig. 1. Mosaic transform [Mx] for the polarization Ex(x, y): the domain on the left is Cðx; yÞ in [[Dx,Dy]], with Dx=2b. The domain on the right is Cm ðx; yÞ: The curve on the right-hand side represents the TE1 mode.
R. Prazeres, V. Serriere / Nuclear Instruments and Methods in Physics Research A 475 (2001) 524–530
~ ðx; yÞ in order can be applied to any input wave C ~ to obtain the mosaic profile Cm ðx; yÞ which is used in the waveguide propagation: Cxm ¼ ½Mx Cx and Cym ¼ ½My Cy : The complete operator of waveguide propagation includes some other operators: a diaphragm operator [D] which sets to zero the field outside of the waveguide, and a couple of resampling operators [S+] and [S]. This resampling is used for switching between the FFT domains [[Dx,Dy]]: one domain for the free space areas of the optical cavity, and one restricted domain with the condition Dy=b for the waveguide area. In summary, the propagation subroutine will use the following series of operators, ~ ðx; yÞ: applied to the profile C ~ x ðx; yÞ ¼ ½S ½D ½P ½Mx ½D ½Sþ Cx ðx; yÞ C C0y ðx; yÞ ¼ ½S ½D ½P ½My ½D ½Sþ Cy ðx; yÞ: ð7Þ
3. Simulations for ‘‘CLIO’’ 3.1. The numerical code A numerical simulation, involving the propagation operator described in Section 2.3, has been
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done in order to calculate the losses of the optical cavity of the FEL when a waveguide is used in place of the undulator vacuum chamber. The numerical code [5] uses several operators: for the reflection on the cavity mirrors, for the transversal gain distribution G(x,y) and for the propagation both in free space or in the waveguide. Fig. 2 shows a layout of the optical cavity. Each mirror is defined by the reflection coefficient for the amplitude, the radius of curvature and the diameter. The gain transverse distribution is a Gaussian function G(x,y), centered on axis, in the center of the undulator. The field of validity of the propagation operator is the same for free space and waveguide propagation: it is limited to the paraxial approximation (small angles). The program calculates the propagation in the cavity of a wave front Cðx; yÞ of complex amplitude, representing the optical field. This wave front is sampled in the transverse plane by 128 128 points, representing a total area of 10 10 cm. The process starts from a Gaussian TEM00 field distribution C0 ðx; yÞ in the center of the cavity. This initial field distribution is normalized to jjC0 jj ¼ 1: The iterative procedure of propagation in the cavity leads to a convergent solution Cn ðx; yÞ; where n is the number of cavity round
Fig. 2. Layout of the optical cavity used in the numerical code.
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trips. Between n=20 and 40 iterations are necessary to obtain a convergent solution, corresponding to the saturation limit. The value of the gain at saturation Gs is not known in advance and depends on the losses of the final mode. This gain value is determined during the iterations, in parallel with the mode, until it reaches the saturation limit. After convergence, the total losses of the cavity can be deduced from the gain value at saturation Gs. The size 10 10 cm of the sampling area is determined in order to fit the maximum width of the laser profile, i.e. on the mirrors. The laser profile cannot be much wider than the mirrors, for which the diameter is F ¼ 38 mm in the case of CLIO. On the other hand, the number of points for the sampling depends on the computer capability, but it must be sufficient to allow a reasonable resolution for the sampling in
the middle of the waveguide, where the gain profile fits the electron beam profile: i.e. about s ¼ 2 mm for ‘‘deep’’ infrared operation of CLIO (l>20 mm). 3.2. Numerical results The aim of these simulations is to obtain the parameters of ‘‘CLIO’’ which are giving a minimum of losses in the optical cavity. In the first set of simulations, the cavity losses have been calculated for various mirror radius of curvature Ry1 and Ry2 for the vertical axis on both mirrors. The horizontal radius of curvature remains constant with Rx1=Rx2=3 m, because the wave is not guided along x-axis. The density plots in Fig. 3 represent the losses versus Ry1 and Ry2. Four laser wavelengths have been used: l=25, 50, 100
Fig. 3. Density plot of cavity losses versus the mirror radius of curvature Ry1 and Ry2 on vertical axis for both mirrors.
R. Prazeres, V. Serriere / Nuclear Instruments and Methods in Physics Research A 475 (2001) 524–530 100
A : losses for a non waveguided vacuum chamber total cavity LOSSES (%)
B : losses without vacuum chamber 80
C : losses with the plane waveguide
60
A 40
20
C
B mirror losses : 8.7%
0 0
20
40
60
80
100
Wavelength ( µm)
Fig. 4. Cavity losses calculated for the standard set of mirrors R1=R2=3 m.
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and 200 mm. The small losses correspond to the dark points. The point with the label ‘‘CLIO’’ corresponds to the standard cavity of CLIO with Rx=Ry=3 m on both mirrors. The transverse dotted line corresponds to the theoretical limit of stability of the optical cavity 0o(1L/Ry1)(1L/ Ry2)o1 in vertical plane: only the points above this line are in the stable area. The white points with a cross correspond to a nonconvergent simulation : the laser profile C is not stable along the consecutive cavity round trips. All these points are below the limit of stability of the cavity. At a shorter wavelength, l ¼ 25 mm, the minumum of losses corresponds to the standard cavity with symmetric mirrors: Ry1=Ry2=3 m. However, for the larger wavelength l ¼ 200 mm, the smaller
Fig. 5. Transverse profiles Cðx; yÞ at both ends of the waveguide and on mirror 2.
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losses are only obtained with toroidal mirrors: with Ry1=1.1 m and Ry2=1.7 m. In this configuration, each radius of curvature Ry is equal to the distance between the mirror and the closer entrance of the waveguide (see Fig. 2): Ry1=d1 and Ry2=d2. This point stays below the limit of cavity stability, but this limit only corresponds to a free cavity which is not the case here. Note that for any wavelength larger than 100 mm, the standard cavity of CLIO with Ry1=Ry2=3 m gives very large losses which do not allow lasing. This result is in agreement with some calculations already published [3] which lead to the same conclusion by empirical methods. However, this simulation gives the absolute value of the cavity losses, and such an extended view of the FEL losses versus the cavity parameters could not be obtained with empirical methods. We can also observe in Fig. 3 that many points below the limit of stability, at l=25 and 50 mm, are still converging points with low losses (dark points). Only a few of them are marked with a white cross (nonconverging simulation). This means that, in principle, the lasing can be obtained in this area, but the laser may be strongly unstable. As shown in Fig. 3, working at wavelengths larger than 100 mm requires toroidal mirrors. Therefore, it is interesting for us to know where the wavelength limit of laser operation is with the standard set of mirrors R1=R2=3 m. Fig. 4 shows the cavity losses obtained with the simulation for R1=R2=3 m. The flat curve B corresponds to a simulation without the vacuum chamber: the losses of 8.7% correspond to the mirror reflectivity r1=0.99 and r2=0.93. The curve A corresponds to the cavity losses with an unguided vacuum chamber, which is represented
by one iris on both ends with 15 mm in vertical and infinite in horizontal. As expected, the losses are increasing strongly with the wavelength because of the diffraction. The losses with a waveguide are represented by the curve C. They include the mirror losses (8.7%), and the waveguide losses by cutting of the mode at the entrance. Note that there is a minimum of losses for l ¼ 60 mm. Fig. 5 displays the transverse profiles at both ends of the waveguide and on the mirror 2. These profiles correspond to the positive direction of propagation of the wave along the z-axis. For l ¼ 100 mm, the profile is truncated at the waveguide entrance. At l ¼ 60 mm, the laser profiles exhibit a smooth structure, close to a Gaussian profile, which leads to minimum losses. At l ¼ 55 mm, the profile exhibits a more complicated structure, which gives larger diffraction losses as for l ¼ 60 mm.
References [1] R. Prazeres, A method of calculating the propagation of electromagnetic fields both in waveguides and in free space, using the Fast Fourier Transform, Eur. Phys. J. Appl. Phys., in press. [2] K.W. Berrymann, T.I. Smith, Nucl. Instr. and Meth. A 318 (1992) 885; R.L. Elias, et al., Phys. Rev. Lett. 57 (4) (1986) 424. [3] Li Yi Lin, A.F.G. Van der Meer, Rev. Sci. Instrum. 68 (12) (1997) 4342. [4] S. Lowenthal, Y. Belvaux, Rev. Opt. 46 (1) (1967) 1. [5] R. Prazeres, M. Billardon, Nucl. Instr. and Meth. A 318 (1992) 889. [6] G. Dubost, Propagation libre et guid!ee des ondes e! lectromagn!etiques, Masson, Paris 1995, ISBN:2-22584792-4.