Gaussian beams 9.1 9. Gaussian beam propagation 9.3 Gaussian beam imaging 9.4 tilted spherical mirror 9.5 f-θ scan lens 9.6 Gaussian beam with general astigmatism 9.7 Laser Cavity Design 9.1 OSLO Kogelink Li Kogelink Li Applied Optics and Optical Engineering, 1979 Appl. Opt. 5, 1550-1567(1966) 9. Gaussian beam propagation Helmholtz ( x, y, z) + k u( x, y, z) = 0 u 9.1 π u(x,y, z) k k= λ 9.1 u spherical As exp( ± ik x + y + z ) ( x, y, z) = 9. x + y + z z u ( x y, z) = A exp( ikz), 9.3 plaue p ± 9-1
9. 9.3 A s A p z u ( x y, z) = ( x, y, z) exp( ikz), ϕ 9.4 (x,y,z) 9.4 9.1 (x,y,z) ϕ ϕ ϕ ϕ + + ik = 0 x y z z 9.5 z (x,y,z) z paraxial equation parabolic ϕ ϕ ϕ + ik = 0 x y z 9.6 9.6 x y u(r,z) u w 0 1, = exp i( kz Φ( z) ) r w( z) w ( r z) + ik R ( z) ( z) 9.7 r + = x y 9.8 Φ λz 1 ( z) = tan πw0 9.9 9.7 w R z w 0 9.7 w R wavefront z w 1/e r=w 1/e 9.1 w 9-
9.1 9.7 z=0 w(z) R(z) w ( z) = w 0 λ z 1 + π w0 9.10 R ( z) π w0 = z 1 + λ z 9.11 9.10 z=0 w (z) w 0 w 0 beam waist 9.11 z 9.11 z R(z) 9. z R OSLO OSLO OSLO OSLO OSLO Z OSLO z R 9-3
9. w, w 0, z R 9.10 9.11 1 z far field 1/e θ = tan λ λ πw 0 πw0 1 ~ 9.1 Reyleigh Rayleigh range Z R π w z 0 R = 9.13 λ Rayleigh w Rayleigh collimation length confecal parameter b πw0 b = = λ z R 9.14 q 1 1 λ = i 9.15 q R πw 1 exp ikr q 9-4
q z πw ( ) 0 q z = i + z 9.16 λ 3 orthogonal x y 90 q A q B q n + ' = n' C q + D n 9.17 q' n' q n A,B,C,D 9.17 A,B,C,D X 9.17 ABCD R q q q 9.15 q λ 0 λ 0 =nλ ABCD 1 qˆ n n λ = = i 9.18 q R πw Aqˆ + B qˆ = 9.19 Cqˆ + D astigmatic x y xz yz q x q y 1 k ( ) ( ) x y ϕ x, y, z = q + xq y exp i 9.0 qx q y 9-5
ABCD xz yz q y 9.0 z q x ϕ sin φ q y ( ) ( ) + + x, y, z = q q exp i + x y + sin( φ ) 9.1 x y 1 k cos φ qx sin φ qx cos q y φ 1 q y 1 q x xy 9.6 9.1 simple astigmatism orientantim, 9.1 general astigmatism aligned 90 Arnaud Kogelink Appl. Opt. 8, 1687 (1969) 9.1 q x,q y 9.17 ABCD OSLO ABCD spreadsheet ABCD hard edged 9.7 Siegman M Siegman Proc. SPIE14, (1990) M near field moments 9.1 λ w 0θ = 9. π w0 9-6
H λ w 0H = M w0θ = M 9.3 π M M M λ M ABCD M λ w 9.3 Gaussian beam imaging ABCD m 0 M = 1 1 f m 9.4 m f ABCD n=n'=1 q' q ABCD mq q' = 9.5 1 1 q + f m q 9.5 w' R' w ' = m w 9.6 m Rf R' = 9.7 f mr m object plane R= 9.7 R R'=-mf positive lens f m R' focal shift R'= R=f/m slow beam Fresnel a R Fresnel a /λr 9-7
ABCD Catalog database 500mm -1 0.638 m Melles-Griot 01LDX48 9.3 9.3 0.5mm 0 0 0 ABCD 9.4 9-8
9.4 ABCD 9.5 9.5 ABCD 1-1.1017 10 3 mm 0.84804mm 0 Gaussian apodization 1 9-9
40 mm 9.6 9.6 x y 0.501mm 9.7 9.7 9-10
z=-138.583mm 0.76 9.7 9.7 w0 0.13 = = 0. 75 9.8 w 0.5 9.8 ( z=0 z=-138.583415) z=-138.583 0.131mm 9.9 9.9 z=-138.583 real circular diaphragm 1/e 9.10 9-11
9.10 x,y 9.11 9.1 Mahajan JOSA A3, 470 (1986) OSLO PSF 9.11 9-1
9.1 9.4 tilted spherical mirror DeJager Noethen Appl. Opt. 31, 199(199), Appl. Opt. 31, 660(199) 45-50mm -1/3 DeJager Noethen 9.13 9.13 9-13
1mm 9.14 9.14 DeJager Noethen Sags YFS XFS 9.5 f-θ scan lens 9-14
f-θ scan lens Tateoka Minoura 4179183 DeJager Noethen 300mm 8.65 14.35 9.15 ( OSLO 5.4 LT (configurations) ) 9-15
9.15 0 1.5mm 9.16 >> trr 0.0 >> tgb ful all 1.5 1.5 0.0 0.0 0.0 9.16 9-16
9.17 9.16 9.17 9.18 >>cfg >>trr 0 >> tgb ful all 1.5 1.5 0.0 0.0 0.0 9-17
9.18 OSLO 8 YSPT SIZE Bt OSLO z 17.97 0.047835 cos 17.97 =0.0455 Bt 9.6 Gaussian beam with general astigmatism Arnaud Kogelnik 45 (stigmatic) 50mm 00mm 500mm 0.638 500mm 9.19 9.0 9-18
9-19
9.19 9.0 5 plane surface z=0 6 11 100mm 6 50 m 0.15mm 9.1 >>trr 0 >> tgb ful all 0.15 0.15 0.0 0.0 0.0 9-0
9.1 0.15mm 6 5 BEAM AZMTH PHASE AZMTH 6 11 Y SPT SIZE, X SPT SIZE BEAM AZMTH 11. 11. 9.7 Laser Cavity Design mode self-consistent 9-1
Fabry-Porot 5mm 0.5mm Nd:YAG 10mm fused silica Melles Griot,01LQF005 OSLO 6.1 MG LQF005 YAG 0.5mm YAG 5mm 9.3 9.3 9.4 9-
9.4 YAG 0 OSLO 0 0 Conic 0 Conic 0 0.01 10 m Conic 9.5 9.5 SCP 0 Conic 5 5 0 YAG YAG /3 SCP *yagmode *yagmode 9.6 ( OSLO5.4 LT 9.6 ) 9-3
9.6 *yagmode 5 YAG 1 0.5 = 0.1667 9.7 3 9.7 9-4
9-5
9-6