1. OPTICAL SYSTEM FOR VARIABLE RESIZING OF ROUND FLAT-TOP DISTRIBUTIONS George Nemeş, Astigmat, Santa Clara, CA, USA gnemes@astigmat-us.com John A. Hoffnagle, IBM Almaden Research Center, San Jose, CA, USA hoffnagl@almaden.ibm.com

2. OUTLINE 1. INTRODUCTION 2. OPTICAL SYSTEMS AND BEAMS – MATRIX TREATMENT 3. VARIABLE SPOT RESIZING OPTICAL SYSTEM (VARISPOT) 4. EXPERIMENTS 5. RESULTS AND DISCUSSION 6. CONCLUSION

3. 1. INTRODUCTION Importance of flat-top beams and spots Obtaining flat-top beams - Directly in some lasers, at least in one transverse direction (transverse multimode lasers; excimer) - From other beams with near-gaussian profiles – beam shapers (transverse single-mode lasers) Obtaining flat-top spots at a certain target plane - Superimposing beamlets on that target plane (homogenizing) - From flat-top beams by imaging and resizing  this work

4. 2. OPTICAL SYSTEMS AND BEAMS: MATRIX TREATMENT Basic concepts: rays, optical systems, beams Ray: R R T = (x(z) y(z) u v)

5. Optical system: S A 11 A 12 B 11 B 12 A B A 21 A 22 B 21 B 22 S = = C D C 11 C 12 D 11 D 12 C 21 C 22 D 21 D 22 Properties: 0 I 1 0 0 0 S J S T = J ; J = ; J 2 = - I; I = ; 0 = - I 0 0 1 0 0 AD T – BC T = I AB T = BA T  det S = 1; S - max. 10 independent elements CD T = DC T A, D elements: numbers B elements: lengths (m) C elements: reciprocal lengths (m -1 ) Ray transfer property of S: R out = S R in

6. Beams in second - order moments: P = beam matrix W M W elements: lengths 2 (m 2 ) P = = = ; M elements: lengths (m) M T U U elements: angles 2 (rad 2 ) Properties: P > 0; P T = P  W T = W; U T = U; M T  M  P - max. 10 independent elements Beam transfer property (beam "propagation" property) of S: P out = S P in S T W = W I Example of a beam (rotationally symmetric, stigmatic) and its "propagation" M = M I U = U I W = W 0 W 2 = AA T W 0 + BB T U 0 In waist: M = 0 ; Output M 2 = AC T W 0 + BD T U 0 U = U 0 plane: U 2 = CC T W 0 + DD T U 0 Beam spatial parameters: D = 4W 1/2 ;  = 4U 1/2 ; M 2 = (  /4)D 0  / ; z R = D 0 /  Beam: P

7. Round spot D (α) Quasi – Image Plane + Cyl. (f, 0) – Cyl. (–f, α ) Incoming beam D 0 y x z Sph. f 0 Block diagram 3 - lens system: + cyl. lens, cyl. axis vertical ( f, 0) - cyl. lens, cyl. axis rotatable about z (-f,  ) + sph. lens (f 0 ) + free-space of length d = f 0 (back-focal plane ) 3. VARIABLE SPOT RESIZING OPTICAL SYSTEM (VariSpot)

8. W 2 = W 2 I; W 2 = A 2 W 0 + B 2 U 0 W 2 (  ) = [(f 0 2 /f 2 ) sin 2 (  )] W 0 + (f 0 2 ) U 0 = W 0 [(f 0 2 /f 2 ) sin 2 (  ) + f 0 2 /z 2 R ] D(  ) = D 0 [(f 0 2 /f 2 ) sin 2 (  ) + f 0 2 /z 2 R ] 1/2 = D m [1 + sin 2 (  )/sin 2 (  R )] 1/2 Compare to free-space propagation: D(z) = D m [1 + z 2 /z 2 R ] 1/2 D m = D 2 (  =  ) = D 0 f 0 / z R =   f 0 D M = D 2 (  =  2) = D 0 [(f 0 2 /f 2 ) + f 0 2 /z 2 R ] 1/2  D 0 f 0 / f (for f/z R <<1) sin(  R ) = f/z R ;  R = “angular Rayleigh range” Perfect imager  B = 0  W 2 = A 2 W 0  beam-independent “Image-mode” of optical system + incoming beam (beam-dependent - Rayleigh range z R ): A 2 W 0 >> B 2 U 0  A 2 >> B 2 /z 2 R f/z R = sin(  R ) << sin(  )  1 VariSpot “image-mode”  D(  )  D 0 (f 0 /f) sin(  ) VariSpot input-output relations

9. 4. EXPERIMENTS Experimental setup

10. Data on experiment Incoming beam data ( = 514 nm) CCD camera data (type Dalsa D7) - Gaussian beam Pixel size: 12  m D 0 = 4.480 mm Detector size: 1024 x 1024 pixels  = 0.154 mrad Dynamic range: 12 bits (4096 levels) z R = 29.1 m Noise level: 2 levels M 2 = 1.05 Attenuator: Al film; OD  3 - Flat-top (Fermi-Dirac) beam D 0 = 6.822 mm  = 0.149 mrad z R = 45.8 m M 2 = 1.55 VariSpot data f Cyl = +/- 500 mm f 0 = 1000 mm  = - 90 0 - 0 0 - 90 0 (manually rotatable mount, +/- 0.25 0 resolution)

11. Fermi-Dirac (F-D) beam profile I(r) = I 0 / {1 + exp [  (r/R 0 - 1)]} R 0 = 3.25 mm  = 16.25 I 0 = 0.0298 mm 2 M 2 (ideal F-D) = 1.50 M 2 (experimental F-D) = 1.55

12. 5. RESULTS AND DISCUSSION Exact  D(  ) = D 0 [(f 0 2 /f 2 ) sin 2 (  ) + f 0 2 /z 2 R ] 1/2 = = D m [1 + sin 2 (  )/sin 2 (  R )] 1/2 Image-mode  D(  )  D 0 (f 0 /f)sin(  ) = D M sin(  ) Gaussian beam D(  ) vs.  D(  ) vs. sin(  E = d min /d max vs. 

13. Exact  D(  ) = D 0 [(f 0 2 /f 2 ) sin 2 (  ) + f 0 2 /z 2 R ] 1/2 = = D m [1 + sin 2 (  )/sin 2 (  R )] 1/2 Image-mode  D(  )  D 0 (f 0 /f)sin(  ) = D M sin(  ) Flat-top (Fermi-Dirac) beam D(  ) vs.  D(  ) vs. sin(  E = d min /d max vs. 

14. Estimating the zoom range in image-mode (“angular far-field”) D(  ) vs.  (small angles) Kurtosis vs.  Blue lines  Image-mode   4 0 D(  )  D 0 (f 0 /f)sin(  ) = D M sin(  )   3 0 - 4 0 Zoom range in image-mode (FD  FD)  13 x - 15 x

15. Examples of spots - Gaussian beam Incoming gaussian beam; D 0 = 4.480 mm Gaussian beam in back-focal plane of f 0 = 1 m spherical lens D f = 0.154 mm

16. Examples of spots - flat-top (Fermi-Dirac) beam Incoming Fermi-Dirac beam; D 0 = 6.822 mm Fermi-Dirac beam in back-focal plane of f 0 = 1 m spherical lens D f = 0.149 mm

17. Examples: VariSpot at working distance Gaussian beam  = 50 0  = 10 0  = 4 0  = 50 0 Fermi-Dirac beam

18. Discussion - Zoom (image-mode) range (D M /D min ) scales with z R /f - Variable spot size scales with f 0 - Cheap off the shelf lenses used, no AR coating - This arrangement already shows (13 - 15) : 1 zoom range for flat-top profiles. D min  1.0 mm; D M  13.6 mm - Estimated (20 - 50) x zoom range for flat-top profiles - Estimated 50  m minimum spot size with flat-top profile - Analysis  smaller spots in “focus-mode”, (“Fourier-transformer-mode”, “angular near-field”) regime (not discussed here)

19. Prototype Zoom = 7 : 1 D min  1 mm; D M  7 mm

20. 6. CONCLUSION New zoom principle demonstrated to resize a flat-top beam at a fixed working distance Zoom factor (dynamic range of flat-top spot sizes): (13 - 15) : 1 D min  1.0 mm; D M  13.6 mm Reasonable good round spots with flat-top profiles Estimated results using this approach (with good incoming flat-top beams and good optics): D min  50  m Zoom factor: (20 - 50) : 1