Modeling free-electron laser (FEL) oscillators requires calculation of both the light-beam interaction within the undulator and the propagation of the light outside the undulator. We present a paraxial Optical Propagation Code (OPC) based on the Spectral Method and Fresnel Diffraction Integral, which in combination with Genesis 1.3 can be used to perform either steady-state or time-dependent FEL oscillator simulations. A flexible scripting interface is used both to describe the optical resonator and to control the codes for propagation and amplification. OPC enables modeling of complex resonator designs that may include hard-edge elements (apertures) or hole-coupled mirrors with arbitrary\ud
shapes. Some capabilities of OPC are illustrated using the FELIX system as an example

Boller, K.-J.

Karssenberg, J.G.

Slot, P.J.M. van der

Verschuur, J.W.J.

Volokhine, I.V.

English

Modeling free-electron laser (FEL) oscillators requires calculation of both the light-beam interaction within the undulator and the propagation of the light outside the undulator. We present a paraxial Optical Propagation Code (OPC) based on the Spectral Method and Fresnel Diffraction Integral, which in combination with Genesis 1.3 can be used to perform either steady-state or time-dependent FEL oscillator simulations. A flexible scripting interface is used both to describe the optical resonator and to control the codes for propagation and amplification. OPC enables modeling of complex resonator designs that may include hard-edge elements (apertures) or hole-coupled mirrors with arbitrary shapes. Some capabilities of OPC are illustrated using the FELIX system as an example

van der Slot, P.J.M.

Volokhine, I.

University of Twente Research Information

FEL-OSCILLATOR SIMULATIONS WITH GENESIS 1.3J.G. Karssenberg, P.J.M. van der Slot∗, J.W.J. Verschuur, I.V. Volokhine † K.-J. BollerLaser Physics and Non-Linear Optics Group, University of TwentePO Box 217, 7500 AE Enschede, The NetherlandsAbstractModeling free-electron laser (FEL) oscillators requirescalculation of both the light-beam interaction within the un-dulator and the propagation of the light outside the undula-tor. We present a paraxial Optical Propagation Code (OPC)based on the Spectral Method and Fresnel Diffraction Inte-gral, which in combination with Genesis 1.3 can be used toperform either steady-state or time-dependent FEL oscilla-tor simulations. A flexible scripting interface is used bothto describe the optical resonator and to control the codesfor propagation and amplification. OPC enables modelingof complex resonator designs that may include hard-edgeelements (apertures) or hole-coupled mirrors with arbitraryshapes. Some capabilities of OPC are illustrated using theFELIX system as an example.INTRODUCTIONFree-electron laser (FEL) oscillators are complex de-vices. They require simulation of both the amplificationof the radiation field within the undulator and the propaga-tion of the radiation field through the resonator, to correctlypredict the spectral and spatial properties of the output ofthe laser. These properties are very important for the designof user experiments.To date, several codes exist that can simulate an FELamplifier or oscillator (see, e.g., [1]). However, the well-established Genesis 1.3 FEL code [2] is primarily usedfor FEL amplifier or SASE simulations. In this paper wepresent an optical propagation code (OPC) that works to-gether with Genesis 1.3 to simulate FEL resonators. Thefull functionality of Genesis 1.3 is maintained and simula-tion of FEL oscillators can be done both in steady-state andtime-dependent modes.To propagate the radiation field between optical andgain elements, we have implemented three related paraxialmethods: the Spectral Method [3], the Fresnel DiffractionIntegral Method [4] and a modified Fresnel Diffraction In-tegral Method. The latter is based on the normal FresnelDiffraction Integral, however, it includes the ABCD ma-trix of the optical system between input and output plane[5]. The radiation field produced by Genesis 1.3 at the un-dulator’s exit is propagated using one of these methods tothe first optical element. Then the action of that partic-ular optical element is applied to the wave and one of the∗ p.j.m.vanderslot@tnw.utwente.nl† Current address: Philips Research, High Tech Campus 34, 5656 AEEindhoven, The Netherlandspropagation methods is again used to propagate the wave tothe next optical element. This procedure is repeated untilthe undulator’s entrance is reached. Note that propagationthrough a cascaded set of optical elements can be done in asingle step if this set can be represented by a single overallABCD matrix and using the Modified Fresnel DiffractionIntegral for the propagation.We have chosen to separate the optical propagationmodel from the FEL simulation model for two main rea-sons. First, the optical propagation model can be used withdifferent gain models and is, in principle, not limited toFELs. Second, the propagation model can then also beused to propagate the field outside the resonator and deter-mine the field distribution, for example, in the far field orin a user area that can be located at a considerable distancefrom the laser in the case of FELs. The optical propagationcode is available for download [10].In the remainder of this paper, we first describe brieflythe different propagation methods, then the OPC code andend with an example illustrating the capabilities of the com-bination of the OPC code with Genesis 1.3.PARAXIAL OPTICAL PROPAGATIONBy applying a Fourier transform over the transverse co-ordinates, the paraxial wave equation can be written as [5]:(k2x + k2y)ũ − 2ik∂ũ∂z= 0, (1)whereũ(kx, ky, z) =∫ ∞−∞∫ ∞−∞u(x, y, z)e−i(kxx+kyy)dxdy (2)is the Fourier transform of the complex wave amplitudeu(x, y, z). Given a known optical field u0(x, y) at z = 0,the optical field after a propagation over a distance z can beobtained from eq. 1:ũ(kx, ky, z) = ũ0(kx, ky)e−i2zk (k2x+k2y). (3)The propagation method described by eq. 3 is knownas the Spectral Method and consists of applying the spatialFourier transform eq. (2) to the input plane, propagate thefield over a distance z (eq. 3) and applying the InverseFourier Transform to ũ(kx, ky, z). This Inverse FourierTransform is similar to eq. 2 with the roles of ũ, kx andky interchanged with those of u, x and y, respectively, anda factor 1/4π2 added.Proceedings of FEL 2006, BESSY, Berlin, Germany TUPPH037FEL Oscillators and Long Wavelength FELs 407It can be shown that paraxial wave propagation over adistance z = L through a cascaded optical system can berealized in a single step using the elements of the overallABCD matrix in Huygens’ Integral for propagation [5]. Ifwe use the following transform to remove the spherical por-tion of the wave at the input plane [5]:v0(ξ′, η′) ≡√a1a3u0(ξ, η)e−iπ(Ax−Mx)ξ2Bxλ e−i π(Ay−My)η2Byλ (4)and at the output plane:v(x′, y′) ≡√a2a4u(x, y, L)e+iπ(Dx−M−1x )x2Bxλ e+iπ(Dy−M−1y )y2Byλ , (5)where x′ = a1x, ξ′ = a2ξ, y′ = a3x, η′ = a4η, andAx(y) . . . Dx(y) are the ABCD matrix coefficients for thex- and y direction respectively, then Huygens’ Integral canbe written as a modified Fresnel Diffraction Integral:v(x′, y′) = i√Nc,xNc,y×∫ 1−1∫ 1−1K(x′, y′, ξ′, η′)v0(ξ′, η′)dξ′dη′, (6)where the kernel K is given byK(x′, y′, ξ′, η′) = e−iπNc,x(x′−ξ′)2e−iπNc,y(y′−η′)2 , (7)and the equivalent collimated Fresnel numbers Nc,x(y) aregiven byNc,x(y) =Mx(y)a21(3)Bx(y)λ. (8)The arbitrary scaling factors a1..4 in eqs. 4-5 define twomagnification factors Mx(y) = a2(4)/a1(3), if they corre-spond to either the size of a hard aperture or a size suf-ficiently large that the field is just negligible outside thearea covered by that size. Note that for free-space propa-gation A = D =1, C =0 and B = L = z, and usingMx = My =1, reduces eqs. 6-8 to the normal FresnelDiffraction Integral.Both the Spectral and the Modified Fresnel propagationmethods are implemented using Fast Fourier Transformsand therefor their computation time scales as N 2log2(N2)for a N ×N grid. For an equal grid, the Spectral method isthe faster of the two because it requires less operations [6].However, care has to be taken with the Spectral Methodthat the field remains zero at the border of the grid to avoidartificial reflections. The Modified Fresnel Diffraction In-tegral has the advantage that propagation through an opticalsystem, described be a single overall ABCD matrix, is ob-tained in a single step. Another advantage is that the scalingapplied to this method allows a magnification factor for thegrid so that the mesh size in the input plane does not haveto be the same as the mesh size in the output plane.Figure 1: Flowchart of the simulation loop.IMPLEMENTATIONSimulation of an FEL oscillator requires simulation ofboth the gain within the undulator and the propagation ofthe radiation through the remaining part of the resonator,that may contain multiple optical components. Genesis 1.3[2] is used to propagate the optical wave through the undu-lator and calculate the amplification of the wave. The OPCreceives the optical output from Genesis 1.3, and propa-gates it from optical element to optical element using oneof the methods described above, until the undulator’s en-trance is reached and the next round trip can start. If acascaded set of optical elements can be described by a sin-gle ABCD matrix, then propagation through the cascadedset can be done in a single step using the Modified FresnelDiffraction Integral. Output can be produced at various po-sitions, and by the optical elements, which is of advantagefor beam diagnostics and for designing suitable optics for afurther propagation of the output beam.While the actual simulation algorithms are written inFORTRAN, the Perl scripting language is used to both con-trol the program flow and define the resonator geometry.Fig. 1 shows schematically the program flow. The advan-tage of using a script to control the program instead of aconfig file is that it gives the user a lot of freedom to createcomplex models. Table 1 gives an example of a script forsimulation of the FELIX system [7].This example script contains all essential elementsneeded to simulate the FELIX system. The first three linesare headers needed to load a library. Then two objects arecreated, one that corresponds to the genesis program (l.5)and one that corresponds to the optical propagation code(l.6). The genesis object requires the standard configura-tion file for Genesis 1.3 as input. A Perl script function isprovided that gives the user complete control over the pa-rameters in the configuration file. This allows, for example,the use of a Gaussian seed for Genesis 1.3 in the first roundtrip and the resonator feedback as input for Genesis 1.3 inconsecutive round trips.To define the optical propagation through the resonator,the optics object (l.6) accepts a simple script that de-scribes the propagation methods and optical componentsused (l.7-16). The available propagation methods are de-scribed above, and the optical components so far includediaphragms, square apertures, curved mirrors and thinTUPPH037 Proceedings of FEL 2006, BESSY, Berlin, Germany408 FEL Oscillators and Long Wavelength FELs01: #!/usr/bin/perl02: use lib ’./lib’;03: use Physics::OPC;04:05: $genesis = genesis(’./felix.itdp.in’);06: $optics = optics( ”07: fresnel z=1.06208: mirror r=3 R=98% # Downstream mirror09: fresnel z=5.8910: hole r=0.0015 ( # Out-coupling11: dump var=output12: fresnel z=1 # Far field distance13: dump var=far )14: mirror r=4 R=98% # Upstream mirror15: dump var=reflected16: fresnel z=2.358 ” );17:18: for $i (1 .. 75) {19: run $genesis;20: run $optics;21: move $output => "output.$i.dfl";22: move $far => "far.$i.dfl";23: move $reflected => "reflected.$i.dfl";24: move $field => "entrance.$i.dfl";25: }Table 1: Example script for simulating the FELIX FEL os-cillator.lenses. Each optical component is defined by a set ofparameters, for example the downstream mirror (l.8) forFELIX has a curvature (’r’) of 3 m and a reflectivity (’R’)of 98 %. Other possible parameters for a mirror are absorp-tion (’A’), transmittance (’T’), offset (’xoff’ and ’yoff’),and a small tilt (’xr’ and ’yr’). The optical propagationstarts with propagation over a distance z=1.062 m from theundulator’s exit to the downstream mirror using the Fres-nel Diffraction Integral (l.7). Then the downstream mirroris defined (l.8). After the downstream mirror the wave ispropagated to the upstream mirror, again using the Fres-nel Diffraction Integral (l.9). In FELIX, the output beamis extracted through a hole in the upstream mirror. In themodel the ”hole” component (l.10) is processed before theactual mirror component (l.14). The hole component cre-ates two waves, one that is transmitted through the hole,and one that corresponds to the part outside the hole thatremains within the resonator. In this case, the transmit-ted beam corresponds to the radiation extracted from theresonator. The output beam is propagated over a distancesufficiently large to be in the far field (l.11-13). At l.14 thescript continues with the part outside the hole and this isreflected by the upstream mirror. The combination of the”hole” and ”mirror” commands represent the upstream mir-ror with the hole. The resonator is closed by propagationfrom the upstream mirror to the undulator’s entrance (l.16).A-30-20-10 0 10 20 30 x (mm) 2 2.5 3s (mm) 0 5000100001500020000Intensity (a.u.)B-30-20-10 0 10 20 30x (mm) 2 2.5 3s (mm) 0 500010000Intensity (a.u.)Figure 2: Radiation intensity just after reflection on the up-stream mirror (A) and just before the undulator’s entrance(B) as a function of the horizontal cross-section (at y=0)and the position within the optical micro pulse s after 75round trips and for ΔL=-1.5λ.The ”dump” commands indicate positions where output isproduced for further analysis.The last part of the Perl script is the actual simulationloop where we run the two programs consecutively andsave binary dumps (l.21-34) for each round trip at variespositions defined in the optical configuration. The binaryfiles used by the propagation code have the same format asthe field files produced by Genesis 1.3, so existing analysistools can be re-used. Utilities to extract plain text data fromthese files are included with the OPC code. Using plottingtools such as Gnuplot, the user can create all kinds of viewsto analyze the optical pulse.In steady-state mode Genesis 1.3 produces one field dis-tribution, i.e., a single slice, as output, which is then pro-cessed by the propagation code. Genesis 1.3 can also run intime-dependent mode, which includes slippage effects. Inthis mode Genesis 1.3 produces a set of slices representingthe radiation pulse as output. In time-dependent mode, thepropagation code will process all slices produced by Gen-esis 1.3 consecutively. Each slice is propagated using thecenter wavelength of the optical pulse, which is valid onlyfor a narrow bandwidth of the optical pulse. The slowlyvarying amplitude and phase approximation used by Gen-esis 1.3 limits the maximum bandwidth of the pulse. Fur-thermore, an analysis by Dattoli et.al. showed that the rel-ative bandwidth is limit to about 10−3 to 10−2 for typicalparameters of VUV, UV and IR FELs [8].Proceedings of FEL 2006, BESSY, Berlin, Germany TUPPH037FEL Oscillators and Long Wavelength FELs 409EXAMPLEThe script shown in Table 1 is used to illustrate some ofthe capabilities of the OPC code. The script describes theresonator for the FELIX system [7]. The script producesfour binary dump files for each iteration (that is, for eachround trip). The first gives the field distribution just outsidethe outcouple mirror (l.21), l.23 gives the field distributionjust after reflection at the upstream mirror containing thehole for extraction of the radiation, and l.24 gives the dis-tribution at the entrance of the undulator. The far-field dis-tribution (l.22) will not be used here. It should be notedthat the same script is used for both steady-state and time-dependent simulations, the only changes required are madein the genesis configuration file.We performed a time-dependent simulation of theFELIX system. The horizontal cross-section (at y=0) ofthe radiation intensity just after reflection on the upstreammirror and just before the undulator’s entrance are shown inFig. 2 as a function the horizontal position x and the posi-tion s within the optical micro pulse after 75 round trips andfor a cavity detuning ΔL=-1.5λ. This figure clearly showsthe effect of the hole in the mirror (Fig. 2A) on the radi-ation profile and how this profile has changed when prop-agated to the undulator’s entrance (Fig. 2B), that shows anear maximum on-axis intensity. Note that the broad ra-diation profile at the undulator’s entrance may be clippedby the electron beam transport tube. Although not presentin the current script, this can be included by adding a di-aphragm component just after propagation to the undula-tor’s entrance (l.16) in Table 1.If we integrate the intensity over the cross-section of theoptical pulse, we obtain the micro-pulse optical power asa function of the position s within the micro-pulse. Thisis shown in Fig. 3 as a function of the round-trip num-ber. The oscillation in the micro-pulse optical power as afunction of time (=round trip #) is known as the so calledlimit-cycle oscillations [9]. It is more clearly visible in thetotal micro-pulse energy that is shown in Fig. 4 for threedifferent values of the cavity detuning ΔL.CONCLUSIONWe have developed an Optical Propagation Code thatpropagates an arbitrary radiation wave, in the paraxial ap-proximation, through a complex optical system. In com-bination with a code to simulate the gain medium, such asGenesis 1.3 in case of FELs, this optical code can be usedto model the output of a laser oscillator. The use of a flex-ible scripting interface allows a user to create both simpleand complex resonator configurations. The OPC packageis available for download at our website[10].REFERENCES[1] S. Biedron, Y. Chae, R. Dejus, B. Faatz, H. Freund, S. Mil-ton, H.-D. Nuhn, and S. Reiche, Nucl. Instrum. Methods 0 0.5 1 1.5 2 2.5 3s (mm) 10 20 30 40 50 60 70Round trips (#) 0 5e+07 1e+081.5e+08Output power (W)Figure 3: Micro pulse optical power as a function of boththe position s within the pulse and the round trip numberfor a detuning ΔL=-1.5λ. 0 2 4 6 8 10 12 10  20  30  40  50  60  70Micro pulse energy (μJ)Round trips (#)ΔL = -0.5λΔL = -1.0λΔL = -1.5λFigure 4: Total energy of the micro pulse as a function ofround-trip number for a detuning ΔL of -0.5λ, -1λ and -1.5λ respectively.Phys. Res. A, Accel. Spectrom. Detect. Assoc. Equip. A445,110 (2000).[2] S. Reiche, pbpl.physics.ucla.edu/ reiche.[3] E. Sziklas and A. Siegman, Appl. Opt. 14, 1874 (1975).[4] H.A. Hause, “Waves and fields in optoelectronics,” 1984,Prentice-Hall, Englewood Cliffs.[5] A.E. Siegman, “Lasers,” 1986, University Science Books,Mill Valley.[6] I.V. Volokhine, “Design and Numerical Analysis of TUE-FEL II”, Ph.D. thesis, 2003 University of Twente.[7] D. Oepts, A. van der Meer, and P. van Amersfoort, InfraredPhys. Technol. 36, 297 (1995).[8] G. Dattoli, H. Fang, L. Giannessi, M. Richetta, A. Torre,and R. Caloi, Nucl. Instrum. Methods Phys. Res. A, Accel.Spectrom. Detect. Assoc. Equip. A285, 108 (1989).[9] D. Jaroszynski, R.J. Bakker, D. Oepts, A.F.G. van der Meer,and P.W. van Amersfoort, Nucl. Instrum. Methods Phys.Res. A, Accel. Spectrom. Detect. Assoc. Equip. A331, 52(1993).[10] OPC design team, lf.tnw.utwente.nl/opc.html.TUPPH037 Proceedings of FEL 2006, BESSY, Berlin, Germany410 FEL Oscillators and Long Wavelength FELs

FEL-Oscillator simulations with Genesis 1.3

NARCIS 

https://ris.utwente.nl/ws/files/5537378/Karssenberg06FEL.pdf

FEL-Oscillator Simulations with Genesis 1.3

Abstract

Similar works

Full text

Available Versions

University of Twente Research Information

NARCIS

University of Twente Research Information