* Corresponding author. Tel.: #1-202-767-0034; fax: #1- 202-734-1280.  Work supported by the United States Department of En- ergy, O$ce of Basic Energy Sciences, under Contract No. W- 31-109-ENG-38. E-mail address: freund@mmace.nrl.navy.mil (H.P. Freund). Nuclear Instruments and Methods in Physics Research A 445 (2000) 53}58 Nonlinear harmonic generation and proposed experimental veri"cation in SASE FELs  H.P. Freund*, S.G. Biedron, S.V. Milton Science Applications International Corporation, 1710 Goodridge Drive, McLean, VA 22102, USA Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA Abstract Recently, a 3D, polychromatic, nonlinear simulation code was developed to study the growth of nonlinear harmonics in self-ampli"ed spontaneous emission (SASE) free-electron lasers (FELs). The simulation was applied to the parameters for each stage of the Advanced Photon Source (APS) SASE FEL, intended for operation in the visible, UV, and short UV wavelength regimes, respectively, to study the presence of nonlinear harmonic generation. Signi"cant nonlinear harmonic growth is seen. Here, a discussion of the code development, the APS SASE FEL, the simulations and results, and, "nally, the proposed experimental procedure for veri"cation of such nonlinear harmonic generation at the APS SASE FEL will be given.  2000 Published by Elsevier Science B.V. All rights reserved. 1. Introduction An important goal of the synchrotron radiation community is to reach shorter wavelengths than 1A s with higher coherent powers. There are several techniques under consideration to reach these goals: self-ampli"ed spontaneous emission (SASE) at shorter wavelengths [1], which requires higher beam energies and/or shorter wiggler periods; the two-wiggler FEL ampli"er scheme [2]; and high-gain harmonic generation [3]. In SASE FELs, there are naturally occurring nonlinear harmonics that could be used to reach these shorter, desired wavelengths. In this paper, a discussion of the non- linear mechanism is provided. In addition, the ap- plication of the 3D polychromatic code MEDUSA to several planned experiments on the SASE FEL under construction at the Advanced Photon Source (APS) is described. 2. The multi-purpose simulation code: MEDUSA MEDUSA is based on a Gauss-Hermite mode expansion [4,5], in conjunction with a source- dependent expansion [6] for the dynamic evolution of the mode spot sizes and the curvature of the phase fronts. As a result, MEDUSA constitutes a slow-time-scale formulation of Maxwell's equa- tions where the amplitudes and phases of each mode, as well as the spot size and phase front 0168-9002/00/$ - see front matter  2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 1 1 3 - 3 SECTION I.