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

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

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

2. OPTICAL SYSTEMS AND BEAMS: MATRIX TREATMENT

Basic concepts: rays, optical systems, beams

Ray: R

RT = (x(z) y(z) u v)

Optical system: S

A11 A12 B11 B12

A B A21 A22 B21 B22

S = = C D C11 C12 D11 D12

C21 C22 D21 D22

Properties: 0 I 1 0 0 0S J ST = J ; J = ; J2 = - I; I = ; 0 = - I 0 0 1 0 0

ADT – BCT = IABT = BAT det S = 1; S - max. 10 independent elementsCDT = DCT

A, D elements: numbers B elements: lengths (m)

C elements: reciprocal lengths (m-1)

Ray transfer property of S: Rout = S Rin

Beams in second - order moments: P = beam matrix

<x2> <xy> <xu> <xv> <xy> <y2> <yu> <yv> W M W elements: lengths2 (m2)P = <RRT> = = ; M elements: lengths (m) <xu> <yu> <u2> <uv> MT U U elements: angles2 (rad2) <xv> <yv> <uv> <v2>

Properties: P > 0; PT = P WT = W; UT = U; MT M P - max. 10 independent elements Beam transfer property (beam "propagation" property) of S: Pout = S Pin ST

W = W IExample of a beam (rotationally symmetric, stigmatic) and its "propagation" M = M I U = U I W = W0 W2 = AAT W0 + BBTU0

In waist: M = 0 ; Output M2 = ACT W0 + BDTU0 U = U0 plane: U2 = CCT W0 + DDTU0

Beam spatial parameters: D = 4W1/2; = 4U1/2; M2 = (/4)D0/; zR = D0/

Beam: P

Round spot D(α)

Quasi – Image Plane+ Cyl. (f, 0)

– Cyl. (–f, α)

Incoming beam

D0

y

x

z

Sph. f0

f0

Block diagram 3 - lens system: + cyl. lens, cyl. axis vertical ( f, 0) - cyl. lens, cyl. axis rotatable about z (-f, ) + sph. lens (f0) + free-space of length d = f0 (back-focal plane)

3. VARIABLE SPOT RESIZING OPTICAL SYSTEM(VariSpot)

W2 = W2 I; W2 = A2 W0 + B2 U0

W2() = [(f02/f2) sin2()] W0 + (f0

2) U0 = W0 [(f02/f2) sin2() + f0

2/z2R]

D() = D0 [(f02/f2) sin2() + f0

2/z2R]1/2 = Dm[1 + sin2()/sin2(R)]1/2

Compare to free-space propagation: D(z) = Dm[1 + z2/z2R]1/2

Dm = D2( = ) = D0 f0 / zR = f0

DM = D2( = 2) = D0 [(f02/f2) + f0

2/z2R]1/2 D0 f0/ f (for f/zR <<1)

sin(R) = f/zR; R = “angular Rayleigh range”

Perfect imager B = 0 W2 = A2 W0 beam-independent

“Image-mode” of optical system + incoming beam(beam-dependent - Rayleigh range zR): A2 W0 >> B2 U0 A2 >> B2/z2

R f/zR = sin(R) << sin() 1

VariSpot “image-mode” D() D0 (f0/f) sin()

VariSpot input-output relations

4. EXPERIMENTS

Experimental setup

Data on experiment

Incoming beam data ( = 514 nm) CCD camera data (type Dalsa D7)

- Gaussian beam Pixel size: 12 m

D0 = 4.480 mm Detector size: 1024 x 1024 pixels

= 0.154 mrad Dynamic range: 12 bits (4096 levels)

zR = 29.1 m Noise level: 2 levels

M2 = 1.05 Attenuator: Al film; OD 3

- Flat-top (Fermi-Dirac) beam

D0 = 6.822 mm

= 0.149 mrad

zR = 45.8 m

M2 = 1.55

VariSpot data

fCyl = +/- 500 mm

f0 = 1000 mm

= - 900 - 00 - 900

(manually rotatable mount, +/- 0.250 resolution)

Fermi-Dirac (F-D) beam profile

I(r) = I0 / {1 + exp [ (r/R0 - 1)]}

R0 = 3.25 mm

= 16.25

I0 = 0.0298 mm2

M2 (ideal F-D) = 1.50

M2 (experimental F-D) = 1.55

5. RESULTS AND DISCUSSION

Exact D() = D0 [(f02/f2) sin2() + f0

2/z2R]1/2 =

= Dm[1 + sin2()/sin2(R)]1/2

Image-mode D() D0(f0/f)sin() = DMsin()

Gaussian beam

D() vs. D() vs. sin( E = dmin/dmax vs.

Exact D() = D0 [(f02/f2) sin2() + f0

2/z2R]1/2 =

= Dm[1 + sin2()/sin2(R)]1/2

Image-mode D() D0(f0/f)sin() = DMsin()

Flat-top (Fermi-Dirac) beam

D() vs. D() vs. sin(

E = dmin/dmax vs.

Estimating the zoom range in image-mode (“angular far-field”)

D() vs. (small angles) Kurtosis vs.

Blue lines Image-mode 40

D() D0(f0/f)sin() = DMsin()

30 - 40 Zoom range in image-mode (FD FD) 13 x - 15 x

Examples of spots - Gaussian beam

Incoming gaussian beam; D0 = 4.480 mm

Gaussian beam in back-focal plane of f0 = 1 m spherical lens

Df = 0.154 mm

Examples of spots - flat-top (Fermi-Dirac) beam

Incoming Fermi-Dirac beam; D0 = 6.822 mm

Fermi-Dirac beam in back-focal plane of f0 = 1 m spherical lens

Df = 0.149 mm

Examples: VariSpot at working distance

Gaussian beam

= 500 = 100 = 40

= 500

Fermi-Dirac beam

Discussion

- Zoom (image-mode) range (DM/Dmin) scales with zR/f

- Variable spot size scales with f0

- Cheap off the shelf lenses used, no AR coating

- This arrangement already shows (13 - 15) : 1 zoom range for flat-top

profiles. Dmin 1.0 mm; DM 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)

Prototype

Zoom = 7 : 1

Dmin 1 mm; DM 7 mm

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

• Dmin 1.0 mm; DM 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):

Dmin 50 m

Zoom factor: (20 - 50) : 1