Synopsis of Yang/Wang’s Analysis and Optimization on Single-Zone Binary Flat-Top Beam Shaper Blake...

Synopsis of Yang/Wang’s Analysis and Optimization on Single-Zone Binary Flat-Top

Beam Shaper

Synopsis of Yang/Wang’s Analysis and Optimization on Single-Zone Binary Flat-Top

Beam Shaper

Blake Anderton

Mon, Dec. 8 2008

The BIG problem …The BIG problem …

• Overview: gain medium illuminated as shown.

Veryexpensive

lasergain

mediumf

Fourier-conjugate planes

The BIG problem …The BIG problem …

• Common input: circularly symmetric Gaussian

Veryexpensive

lasergain

mediumf

Gaussian input

Gaussian output

The BIG problem …The BIG problem …

• Result: Strange and

not-so-wonderful things happen.

Veryexpensive

lasergain

mediumf

Gaussian input

Gaussian output

Too much light at center(scoring, crack)

(also not good)

Too little light at edge

(lost opportunity)

(not good)

But what if…But what if…

…you modify the input beam’s phase to produce a uniform (“flat-top”, “circ”) pattern at gain medium?

Veryexpensive

lasergain

medium

Gaussian input

UNIFORMoutput beam (!)

Benefits of uniform beam at gain medium • no scoring (too much light)• no missed gain opportunities at edges (too little light)

?

Phase plate(ei scaling factor)

The “ideal” … The “ideal” …

• Phase plate produces “Bessinc” entering lens• Output: “perfect” circ in Fourier plane

Gaussian input

?

Phase plate

“Bessinc” “Circ”

The “ideal” …& why it costs too much

The “ideal” …& why it costs too much

• Phase plate produces “Bessinc” entering lens• Output: “perfect” circ in Fourier plane• Requires: continuous-phase plate (high

precision etching)

Gaussian input

$$$

CONTINUOUS-phase plate

“Bessinc” “Circ”

How to mimic a BessincHow to mimic a Bessinc

• Compare plate input and ideal output– Input:

Gaussian– Output: Bessinc

• Key difference: y-values (pos/neg)

BEFORE phase plate

AFTER phase plate (ideal)

?Gauss Bessinc

How to mimic a BessincHow to mimic a Bessinc

• Where do first negative Bessinc values occur (radially)?

• We designate that ring’s width by w.

ww

How to mimic a BessincHow to mimic a Bessinc

• Where do first negative Bessinc values occur (radially)?

• We designate that ring’s width by w.

• Add remaining bands with approx. same width w

ww

w

w

w

w

How to mimic a BessincHow to mimic a Bessinc

• Make Gaussian values negative within these bands.

• Result is negative where Bessinc is negative.

ww

w

w

w

w

How to mimic a BessincHow to mimic a Bessinc

• Net result (right):– Not quite Bessinc – better than Gaussian.

• With this going into the lens, how do we build a phase plate to make this?

?

Goes here

How do we make this phase plate?How do we make this phase plate?

• We seek to specify in ei.• What values must take?

• We seek to specify in ei.• What values must take?

– Everything btwn 0 and 2

How do we make this phase plate?How do we make this phase plate?

• We seek to specify in ei.• What values must take?

– Everything btwn 0 and 2– Just 0 or (inside/outside the

bands).

How do we make this phase plate?How do we make this phase plate?

• We seek to specify in ei.• What values must take?

– Everything btwn 0 and 2– Just 0 or (insdie/outside the

bands).

• How to produce these ’s?– Etch a substrate (index n)

height profile h(r) such that

= 0 or

How do we make this phase plate?How do we make this phase plate?

How many bands do we really need?How many bands do we really need?

• At right: effect of 1st, 2nd bands

• Bands 2+ have minimal effect.

Effect of 1st band

(BIG)

Effect of 2nd band

(less than big)

What if phase plate ONLY had 1 band (no 2nd, 3rd, …)?

• Plot: comparing 1-, 2-, 3-band plates.

• Result: Not too different!

• Bottom line: only 1 band is needed.

Don’tneed

Don’tneed

Don’tneed

Don’tneed

Don’tneed

Don’tneed

How many bands do we really need?How many bands do we really need?

How far are we from ideal? 3 Figures of Merit (FOMs)

How far are we from ideal? 3 Figures of Merit (FOMs)

• How do you compare the Fourier-plane field produced by a binary-phase plate to the “ideal” circ? With 3 figures-of-merit:

– Uniformity: “ringing” in central zone (less ringing is better)

Plot of typical Fourier-plane

intensity produced by single-banded

binary-phase plate

Central zone is defined as having

intensity ≥ 90% peak

How far are we from ideal? 3 Figures of Merit (FOMs)

How far are we from ideal? 3 Figures of Merit (FOMs)

• How do you compare the Fourier-plane field produced by a binary-phase plate to the “ideal” circ? With 3 figures-of-merit:

– Uniformity: “ringing” in central zone (less ringing is better)

– Steepness: slope of central-zone boundary (steeper is better)

Plot of typical Fourier-plane

intensity produced by single-banded

binary-phase plate

Central zone is defined as having

intensity ≥ 90% peak

How far are we from ideal? 3 Figures of Merit (FOMs)

How far are we from ideal? 3 Figures of Merit (FOMs)

• How do you compare the Fourier-plane field produced by a binary-phase plate to the “ideal” circ? With 3 figures-of-merit:

– Uniformity: “ringing” in central zone (less ringing is better)

– Steepness: slope of central-zone boundary (steeper is better)

– Efficiency: energy spread beyond central-zone (less loss is better)

Plot of typical Fourier-plane

intensity produced by single-banded

binary-phase plate

Central zone is defined as having

intensity ≥ 90% peak

Check with experimentCheck with experiment

• System characteristics– = 633 nm (wavelength)– = 420 m (Gaussian e-1 radius and

inner-radius r0 of single phase-plate zone)

– f = 200 mm (lens focal length)– d = 184 mm (obs. plane location)– = 0.8768 (phase on a h = 0.52

m, n = 1.534 single-zone, binary-phase plate)

Value of d was changed to compensate the departure in etch depth h (designed for 0.47 um, actually got 0.52 um)

Check with experimentCheck with experiment

• Comparing predicted/actual results:

– Predicted• Uniformity: U = 2.2%• Steepness: K = 0.61• Efficiency: = 75%

– Actual• Uniformity: U < 3%• Steepness: K = 0.59• Efficiency: = 72.3%

Experi-mental

results: Intensity along x-

direction

Experi-mental

results: Intensity along y-

direction Good agreement!

SummarySummary• Driving problem: non-uniform beam at gain medium.

• Costly ideal: continuous-phase plates.

• Affordable alternative: binary-phase modification of a

Gaussian’s similarity to “ideal” (circ-producing) Bessinc

• Manufacturing: relating phase level to etch depth h in a

substrate of index n

• Simplifying: marginal benefits of more than one zone

• Figures-of-merit: uniformity, steepness, efficiency

• Verification: through experiment

ConclusionConclusion

Single-zone, binary-phase plates provide an affordable, mechanically-feasible option for

producing a uniform field in an optical system’s focal plane.

Relevance to OptomechanicsRelevance to Optomechanics

• Provides mechanically-feasible “phase grating” implementation.

• Exemplifies a system with mechanical compensation capability (d,).

• Yang/Wang also include tolerancing examples: zone width, etch depth sensitivities at +/- 10% of design values.

• Exemplifies cost benefits of designing for manufacturability (1-zone vs. 2+).

Thank you!Thank you!

How far are we from ideal? 3 Figures of Merit (FOMs)

How far are we from ideal? 3 Figures of Merit (FOMs)

• How do you compare the Fourier-plane field produced by a binary-phase plate to the “ideal” circ? With 3 figures-of-merit:

Plot of typical Fourier-plane

intensity produced by single-banded

binary-phase plate

Central zone is defined as having

intensity ≥ 90% peak

Selecting observation distance d and phase for optimum FOMs

Selecting observation distance d and phase for optimum FOMs

• Now, best d and depend on FOMs: U, K, .– They’ll differ from (d = f, ).

• If we set d = f,

we need

• If we set ,

we need d = 0.81f

d = f

0.79

25

d =0.81 f

Bes

t F

OM

s

Bes

t F

OM

s

Check with experiment

• System characteristics– = 633 nm (wavelength)– = 420 m (Gaussian e-1 radius and

inner-radius r0 of single phase-plate zone)

– f = 200 mm (lens focal length)– d = 184 mm (obs. plane location)– = 0.8768 (phase on a h = 0.52

m, n = 1.534 single-zone, binary-phase plate)

Value of d was changed to compensate the departure in etch depth h (designed for 0.47 um, actually got 0.52 um)