Control and characterization of long range

focusing of ultrashort pulses for high

intensity beam delivery

Charles G. Durfee

Michael Greco, Amanda Meier, Erica Block

and Jeff A. Squier

Department of Physics, Colorado School of Mines,

Golden, CO 80401

John Gemmer, Shankar Venkataramani, Jerome Moloney

University of Arizona, Tucson, AZ

1

Outline

High intensity beam delivery to a point

Simultaneous spatial and temporal focusing (SSTF) allows

energetic pulse delivery beyond the critical power

Alignment is critical: investigating and characterizing

generalized spatial chirp propagation

High intensity beam delivery to a uniform line

Phase shaping of Bessel-Gauss ring beams for uniform axial

intensity profiles

Direct mapping of E(r) to E(z) allows iterative Gerchberg-

Saxton algorithm

Variational approach with method of stationary phase

provides good initial guess for optimization.

Delivery of pulses to a point with SSTF

Focusing with angular spatial chirp: 4D localization

Inte

nsity

36x smaller

Single beamlet focus x

z t

z

t

x

x

SSTF focus

Pulse front tilt

• XY focus

• Axial focus (Z)

• Temporal focus

C.G. Durfee et al, 20 14244-14259 Optics Express

(2012)

C.G. Durfee et al, invited chapter in “New

Developments in Photon and Materials Research”

(2013)

Nonlinear

interaction

suppressed before

focal plane

Applications of SSTF localization

20

0

m

Micromachining

• Backside ablation

of glass

• No filamentation

Vitek et al, Opt. Express 18, 18086 (2010)

Laser surgery

• Crainiotomy

(mouse)

• Cutting through

5mm of water

• Cutting eye lens (pig) for

cataract surgery

• High axial resolution

avoids collateral damage

Block et al, Bio. Med. Opt. Express (2013)

Suppression of pre-focal plane nonlinearity

Spatial chirp dramatically reduces the effective depth of focus

DOF depends on input beam aspect ratio:

Does not depend on: focal length, pulse duration

Strong reduction of NL propagation prior to space-time focus

I (z) = I0 1+z

zR

æ

èçö

ø÷

2

bBA4

æ

èç

ö

ø÷ 1+

z

zR

æ

èçö

ø÷

2æ

èç

ö

ø÷

é

ë

êê

ù

û

úú

1/2

Relative DOF

Relative B-integral

DOF for full aperture beam

Beam aspect ratio, BA

wy

wx

bBA =wx /wy

Beam aspect ratio, BA

Relative DOF

Relative B-integral

Analytic calculation

of axial intensity:

Scaling for energetic pulse delivery

Example application: LIBS (air, surface target):

threshold pulse intensity for ionization

large area for high S/N ratio

Avoid filamentation and nonlinear focusing for low axial jitter

Test parameters:

800nm, 100fs, 5x1012 W/cm2 on target, 20m range

Conventional focus

• Pcrit ~ 6GW

• Emax ~ 600uJ

• Spot dia ~ 400um

• Lens dia ~ 5cm

SSTF focus βBA=10

• 25x more power for same B-integral

• Pmax ~ 156GW, Emax ~ 16mJ

• Spot dia ~ 4mm for same lens

• 100x more signal (prop to area)

Single-pass grating

compressor has higher

energy throughput

Misalignment: spectral chirp

Poor compression leads to lower peak intensity, less

localization: e.g. grating groove density mismatch

TL pulse duration:

40fs

BA = 4

ϕ3 = 0 ϕ3= 2x105 fs3

z/zR z/zR

+2000 fs2

0 fs2

+4000 fs2

• temporal Airy

pulse

• lower intensity,

axial resolution

Additional ϕ2

Beam expander single-pass compressor

compressed from amp Target

plane

Chirp evolution through focal plane

Tilted evolution of WF curvature geometric chirp

Input chirp moves position of highest intensity

Tpulse: 40fs

BA = 4

ϕ3 = 0

z/zR z/zR

+2000 fs2

0 fs2

+4000 fs2

z/zR

2(z) fs2 -

chirp

+

0 -zR/4 -zR/2 -3zR/4 -zR -2zR -5zR

In-situ dispersion scan (D-scan) pulse

characterization

Geometric dispersion

of SSTF focus: z-

dependent ϕ2

Variation of second

harmonic spectrum

with ϕ2 leads to direct

measurement of

d2ϕ/dω2

No chirp

ϕ2 = 3x103 fs2

ϕ3 = 3x104 fs2

ϕ4 = 1x106 fs4

Geometric additional ϕ2

SH

sp

ectr

um

Geometric D-scan measurements

Vary axial position of SH crystal, collecting integrated

SH spectrum

SH

sp

ec

tru

m

Axial position (= variation of ϕ2)

Third-order limited pulse Fourth-order limited pulse

Fiber

spectrometer

SH

signal Integrating

sphere

Intuitive analysis: double ABCD method

Spatially-chirped beam = superposition of Gaussian beamlets

Linear propagation: independent propagation for each ω beamlet

C.G. Durfee et al, 20 14244-14259 Optics Express (2012)

C.G. Durfee et al, invited chapter in “New Developments in Photon and Materials Research” (2013)

Ray trace beamlet axes with ABCD or non-

paraxial program: gives beamlet position

and angle

Propagate on-axis beamlet with ABCD or Fresnel:

gives local beamlet size and wavefront radius

Incorporate beam angle and displacement into amplitude, phase

to obtain complex spectral domain field: E(x, y, z, ω).

Fourier transform this field to get E(x, y, z, t)

Misalignment: defocus

Changing input divergence to compressor pushes

beamlet waists away from wavelength crossing plane

Diagnostics in progress:

Broadband Sagnac shearing interferometer for real-time

measurement of the divergence and astigmatism of the input

beam

Knife-edge scanning system to directly characterize the focal

quality of each beamlet in parallel

Beam expander single-pass compressor

compressed from amp

Pulse delivery to a line focus:

Shaped remote focusing of Bessel-

Gauss beams

Bessel-Gauss ring beams:

Low intensity away from focal zone

High intensity, small spot at focus

x

z

Bessel Zone

Ring

Zone

Ring

Zone

Use spatial phase shaping to create a beam that has an extended,

constant intensity focal zone at range.

Shaper Phase Input beam

Energetic pulse delivery to flat top

axial profiles

Gaussian axial profile:

Intensity = 2x threshold to ionize over axial FWHM

Energy fluctuations lead to axial length changes

Flat-top or super-Gaussian:

Uniform breakdown, esp impt for machining, waveguides

Line shape less sensitive to energy fluctuations

Sharp boundary at edge of Bessel zone

Ionization not sensitive to phase: use as free parameter for

shaping

Total required energy is proportional to extended length

Axial intensity profile of BG beam

I(z) = I pkw0

2

w(z)2e

-2z2 sin2 g

w(z)2

Axial intensity profile

x

z

Bessel Zone

Ring

Zone

Ring

Zone

Divergence of

beamlet

Overlap of Gaussian

beamlets

With a ring beam, divergence of the beamlets is enough to get good

separation to a ring in far field, Gaussian profile dominates.

I pk =Ein

aeffT p / 4 ln 2( )aeff =

pw0

2

2exp -

k2 sin2 g w0

2

4

é

ëê

ù

ûú I0

k2 sin2 g w0

2

4

é

ëê

ù

ûú

Peak intensity Effective area

Spatial phase shaping:

E(r) to E(z) Fresnel mapping

Analytic transform to map field at lens plane to

longitudinal field (r2 = s)

Integral is exact as shown (Fresnel)

For input ring beam, E(r) = 0 as small r, Fourier

transform:

Allows fast (FFT) iterative optimization of phase-shaped axial

profile.

Confirmed by direct Fresnel propagation.

E r = 0, z( ) = ik0A

2zexp ik0z[ ] E s, z = 0( )exp i

k0s

2z

é

ëêù

ûúds

0

¥

ò

Inspired by Lipson et al, Optical Physics 3rd ed 1995

Ws = k0 / 2z

E Ws( ) = i AWs exp ik0z[ ] E s( )exp iWss[ ]ds-¥

¥

ò

Durfee, Gemmer, Moloney,

Optics Express 21, 15777 (2013)

Gerchberg-Saxton optimization of phase-

only beam shaping

Input beam with phase Target intensity

exp -r - r0( )

2

win2

é

ë

êê

ù

û

úúexp if r( )éë ùûexp i

k0r2

2 f

é

ëê

ù

ûú

I z( ) = exp -(z - f )8

wT

8

é

ëê

ù

ûú

R-space

Z-space

Transform

Force target amplitude

Keep phase

Inverse

transform

Force input amplitude

Keep shaper phase

15th iteration of loop

Initial phase:

- Flat

- Random

- Guess from previous optimization

- “educated guess”

Phase retrieval is different from phase

optimization

- Is there a solution?

- Will we find the best solution?

Stagnation at local minimum:

- input/output algorithm

- Additional random phase

Axial shaping results

Test case:

Axial intensity profile target

optimized

Shaper phase profile Input intensity

Optimized phase

Sh

ap

er

ph

ase (

rad

)

Durfee, Gemmer, Moloney,

Optics Express 21, 15777 (2013)

Gaussian apodization

w0=50μm

r0 = 50mm

2win = 2x5mm

Bessel 1/e2 radius

wB=11μm

Gaussian beamlet confocal

zR=20mm

Bessel axial

length

LB= 1.2 mm

γ0= 2.8o

Shaped Bessel Bessel zone intensity

Fresnel propagate initial amplitude + shaper phase + lens phase

-80

0 8

0

z - f (mm)

-10 0 10

r

(μ

m)

Shaped beam in Bessel zone:

- slightly tapered

- variable approach angle ϒ0(z)

Phase profile dominated by divergence

Dominant quadratic phase: focus or defocus of the beamlets

Beam

direction

Shaper phase profile

0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

downward slope compensates for

greater power in r dr ring

Simple initial guess for rapid convergence

Shaper phase profile has dominant quadratic term

focus or de-focus for beamlets

Use lens before conversion to ring beam to expand beamlet

Reduces required dynamic

range of shaper

Greatly improved/accelerated

G-S algorithm convergence

Shaper phase profile

Input intensity

Optimized phase

Sh

ap

er

ph

ase (

rad

)

Variable target length

Optimized I(z) vs target axial width wT

Log[ fit error ]

wT I(z)

wT

Focal intensity drops off as 1/wT

Variational formulation

For a target (e.g. super-Gaussian) profile ETFT(z), we want to

minimize

Use method of stationary phase to obtain a better estimate of

phase profile. Correct normalization is critical.

Gemmer et al, submitted to

Physica D (2013)

Varying target length E

rro

r

Target width WT (m)

Varying Input Width E

rro

r

Input width W0 (m)

Results for BG beams

1. The method of stationary phase can be used to generate excellent initial

guesses for the GS algorithm in certain asymptotic regimes. Namely

when:

2. There is no hope of using phase shaping when

3. In intermediate asymptotic regimes it is necessary to use GS algorithm.

However, an initial guess made from the method of stationary phase

drastically decreases number of iterations needed.

Summary/ongoing work

Demonstrated delivery of pulses with power greater than critical

power using spatio-temporal focusing.

Experimental beams are not perfect – now expanding theory and

characterization for generalized spatially chirped beams. (CSM)

Developed technique for shaping spatial phase of Bessel-Gauss

beams to produce desired axial profile.

Working on experimental realization of shaped beams, techniques to

scale in energy.

Propagation in turbulent and or weakly nonlinear media (UA, CSM)

Investigate effect of further structuring BG beams with vortex phase and

polarization states.

Investigate high-order harmonic and electron dynamics from

structured beams: SSTF, vortex, radial/azimuthal polarization (CSM,

CU, UA)