第九章高斯光束（Gaussian beams）

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1 Gaussian beams 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, (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

2 A s A p z u ( x y, z) = ( x, y, z) exp( ikz), ϕ 9.4 (x,y,z) (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 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 πw w R z w w R wavefront z w 1/e r=w 1/e 9.1 w 9-

3 z=0 w(z) R(z) w ( z) = w 0 λ z 1 + π w R ( z) π w0 = z 1 + λ z 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

4 9. w, w 0, z R 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

5 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

6 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 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

7 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' = 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

8 ABCD Catalog database 500mm m Melles-Griot 01LDX mm ABCD

9 9.4 ABCD ABCD mm mm 0 Gaussian apodization 1 9-9

10 40 mm x y 0.501mm

11 z= mm w = = w ( z=0 z= ) z= mm z= real circular diaphragm 1/e

12 9.10 x,y Mahajan JOSA A3, 470 (1986) OSLO PSF

13 tilted spherical mirror DeJager Noethen Appl. Opt. 31, 199(199), Appl. Opt. 31, 660(199) 45-50mm -1/3 DeJager Noethen

14 1mm DeJager Noethen Sags YFS XFS 9.5 f-θ scan lens 9-14

15 f-θ scan lens Tateoka Minoura DeJager Noethen 300mm ( OSLO 5.4 LT (configurations) ) 9-15

16 mm 9.16 >> trr 0.0 >> tgb ful all

17 >>cfg >>trr 0 >> tgb ful all

18 9.18 OSLO 8 YSPT SIZE Bt OSLO z cos = Bt 9.6 Gaussian beam with general astigmatism Arnaud Kogelnik 45 (stigmatic) 50mm 00mm 500mm mm

19 9-19

20 plane surface z= mm 6 50 m 0.15mm 9.1 >>trr 0 >> tgb ful all

21 mm 6 5 BEAM AZMTH PHASE AZMTH 6 11 Y SPT SIZE, X SPT SIZE BEAM AZMTH Laser Cavity Design mode self-consistent 9-1

22 Fabry-Porot 5mm 0.5mm Nd:YAG 10mm fused silica Melles Griot,01LQF005 OSLO 6.1 MG LQF005 YAG 0.5mm YAG 5mm

23 9.4 YAG 0 OSLO 0 0 Conic 0 Conic m Conic SCP 0 Conic YAG YAG /3 SCP *yagmode *yagmode 9.6 ( OSLO5.4 LT 9.6 ) 9-3

24 9.6 *yagmode 5 YAG =

25 9-5

26 9-6

第六章像質品估（Image Evaluation）

第六章像質品估（Image Evaluation） 6.1 Image Evaluation 6. 6.3 6.3.1 6.3. Spot size Analysis 6.3.3 wavefront Analysis 6.4 Point spread functions 6.4.1 6.4. 6.4.3 6.5 fiber coupling 6.6 Energy Distribution 6.7 Transfer functions 6.7.1 6.7.

第九章 高斯光束（Gaussian beams）

第九章高斯光束（Gaussian beams）