Fig. 1. 4f relay transmission system. Relay imaging with two lenses to reproduce the phase and intensity distribution of the laser beam on the thin clisk^[6]

Fig. 2. Graph of beams with different spot radii and wavefront curvatures propagating in a 4f system (The diopter of thin-disk is 0, and the diopter refers to the reciprocal of the focal length)

Fig. 3. Transmission curves of beams within 5 tandem 4f systems when the diopter of the thin-disk is different

Fig. 4. (a) Optical path of a 4f system consisting of one lens and two prism pairs; (b) position of the beam passing through the lens^[6]

Fig. 5. Thin-disk multi-pass amplifier with parabolic mirrors and prisms^[6]

Fig. 6. 12-pass thin-disk amplifier based on 4f relay imaging^[7]

Fig. 7. 14-pass thin-disk amplifier based on 4f relay imaging

Fig. 8. 18-pass amplifier based on a dual thin-disk 4f system^{[8, 11]}

Fig. 9. 14 pass amplifier based on 4f relay imaging system. (a) Top view of the single thin-disk dual-pass amplifier; (b) schematic diagram of non-folded optical path transmission; (c) physical diagram of the thin-disk multi-pass amplifier^[12]

Fig. 10. Top view of the thin-disk multi-pass amplifier with compensation mirror based on relay imaging^[14]

Fig. 11. Improved relay imaging optical path diagram (Using a compensating mirror instead of a parabolic mirror with a compensating mirror)^[16]

Fig. 12. Thin-disk 12-pass amplifier based on a 4f relay imaging + kaleidoscope system^[20]

Fig. 13. Thin-disk 64 pass amplifier based on a dual 4f relay imaging system^[13]

Fig. 14. Schematic diagram of the optical path of the 24-pass amplifier. (a) The optical path passes continuously through 1-disk-2-K₂-3-disk-4-K₁-5-disk-6-K₂-7, where 1-7 represents the mirror numbers in Figure 14 (b), K₁ and K₂ represent the concave mirror K₁ and convex mirror K₂, respectively. K₁-K₂ defines the optical stable cavity. (b) The reflector array number and the lateral projection position of other elements

Fig. 15. Variation in (a) output spot and (b) wavefront curvature inverse with a diopter of thin-disk for the 16-pass 4f amplifier and optical Fourier transmission multi-pass amplifier. The red dashed line represents the 4f multi-pass amplifier, the blue solid line represents the optical Fourier transmission multi-pass amplifier, and the gray solid line represents the optical Fourier transmission multi-pass amplifier in ideal circumstances^[24]

Fig. 16. Beam propagation of an 8-pass amplifier based on optical Fourier transmission. (The black line represents the diopter of the thin-disk at 0. The red and blue lines represent the diopter of the thin-disk are ±1/(40f), and f is the focal length of the 4f system)^[24]

Fig. 17. Beam propagation of a practical optical Fourier transform 8-pass amplifier that shortens the transmission distance. (The black line represents the diopter of the thin-disk at 0. The red and blue lines represent the diopter of the thin-disk = ±1/(40f), and f is the focal length of the 4f system)^[24]

Fig. 18. (a) Top view^[25], (b) stereo optical path diagrams, and (c) physical view of the lens array^[25] of the 20-pass amplifier

Fig. 19. (a) Physical drawing of the vertical retro-reflector^[26]; (b) comparison of the optical path between the vertical retro-reflector and the plane reflector

Fig. 20. Fourier transmission multi-pass amplifier with an active stabilization system^[26]

Fig. 21. The relationship between the small signal gain of three eight-pass amplifiers and the measured deflection angle of the thin-disk. The red symbols represent conventional Fourier transmission multi-pass amplifiers, the blue symbols are taken from the same amplifiers but with the M2 lens being replaced by a vertical rearward reflector, and the green symbols represent the Fourier transmission multi-pass amplifiers equipped with an active stabilization system^[26]

Fig. 22. Optical path diagram of the near collimated beam propagation multi-pass amplifier^[27]

Fig. 23. (a) Overall optical path layout of picosecond multi-pass amplifier, (b) optical path of a multipass cell and (c) picture of a single array of mirrors^[40]

Fig. 24. Spot radius distribution in the optical path of the picosecond multi-pass amplifier^[40]

Fig. 25. Overall optical path of the 720 mJ picosecond laser^[41]

Fig. 26. Optical path diagram of the thin-disk large-aperture ring amplifier^[43]

Fig. 27. The output laser parameters of some reported multi-pass amplifiers. (a) Pulse repetition frequency vs pulse energy, (b) pulse width vs peak power, and (c) average output power vs peak power