Universal Polariton Model of Laser-induced Condensed Matter

Damage

V. S. Makin

Institute for Nuclear Energetics, Saint-Petersburg State Polytechnic University,

Sosnovy Bor City, Leningrad region, Russia

Keywords: Laser Radiation, Polarization, Condensed Media, Material Damage, Surface Polaritons, Interference, Micro-

and Nanostructures.

Abstract: The extension of known polariton model of laser-induced condensed matter damage followed by ordered

structures formation for the case of ultrashort pulse durations and condensed matter of different physical

properties was made. In addition to the usual cases of linear or circular polarizations of laser radiation the

case of axi-symmetric polarization was theoretically considered and illustrated by experiments with

dielectrics, semiconductors and metals. The special case of radially polarized laser radiation is distinguished

as one for which laser-produced dynamic resonant the additional energy of excited surface polaritons into

diffraction size focal spot. The advantages of axi-symmetrically polarized laser radiation for materials

treatment in framework of universal polariton model were discussed. In experiments the wide variety of

spatial periods of microstructures were observed which do not described by existing theories. The nonlinear

mathematical model which describes the spatial periods of laser-produced microstructures was suggested.

Model describes the formation of microstructures with periods multiplied by laser wavelength and predicts

the spatial period’s values less than diffraction limit one. The nonlinear theoretical model is illustrated by

published experimental data. The idea to explain the cause of the formation of regular nanostructures with

anomalous orientation on condensed matter surfaces was suggested. It is based on the effect of wedge

(channel) surface plasmon polaritons excitation and their mutual interference under the action of polarized

laser radiation. The new phenomenon of laser-induced anisotropy of metal recrystallization under the action

of nanosecond duration repetitive pulses of linear polarized radiation was experimentally discovered. The

phenomenon was explained as a result of grain-boundary movement by directed flux of skin-layer electrons

dragged by surface plasmon polaritons.

1 INTRODUCTION

In recent years many experimental data were

published causing ordered micro- and nanostructures

formation under interaction of picoseconds and

femtosecond polarized laser radiation with

condensed media having different physical

properties. There is the lack of the models which

universally describes such phenomenon in wide

range of laser (pulse duration, power density, laser

polarization, ...) and material (material properties)

parameters. In present article such model is

proposed with emphasis on values of spatial period

and orientation of formed structures in regimes of

ultrashort laser radiation interaction of linear

polarization. The extension of polariton model for

the case of axi-symmetrical polarization of laser

radiation has been made. The variety of spatial

ordered micro- and nanostructures produced under

the action of laser radiation of long and ultrashort

pulses durations observed in many experiments have

not been explained in framework of known models.

So the problem was exists to create the model which

allows an adequate description of periods and

orientations of formed structures. The nonlinear

mathematical model based on the unimodal logistic

map was suggested for this case. In experiments

with interaction of pulsed laser radiation tiwh metal

it was discovered the anisotropy of recrystallization

process governed by the direction of linear polarized

laser radiation. In the final stage of laser-metal

interaction the quasiperiodic microstructures in the

mode of grooves of thermal etching were formed. So

the problem arises to elucidate the origin of

observed phenomenon. The suggested explanation

180

Makin V..

Universal Polariton Model of Laser-induced Condensed Matter Damage.

DOI: 10.5220/0004846901800186

In Proceedings of 2nd International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS-2014), pages 180-186

ISBN: 978-989-758-008-6

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

of the phenomenon is based on the drag effect of

electrons of metal skin-layer by surface plasmon

polaritons (SPP).

2 UNIVERSAL POLARITON

MODEL

2.1 Extension of Polariton Model

At first polariton model was suggested for

explanation of periodical structures formation under

interaction of pulses of laser radiation of long

durations (>1 ps), Bonch-Bruevich (1982). The

model works in the spectral range of the fulfilment

of inequality:



< -

(1)

for metals, in the conditions of creation of a

semiconductor’s melt (for semiconductors having

semiconductor-metal type phase transition at melting

point); for dielectrics - in the spectral range of the

reststrahlen band (middle IR range). Here 

, 

are

dielectric permittivities for the surface active and

dielectric media, correspondingly.

The extension of polariton model for metals and

fs radiation was suggested by Agranat (1999). But

the question about applicability of polariton model

for semiconductors and dielectrics was open. Few

ideas have been suggested to explain the

microstructures production inside and on the

dielectrics and semiconductors surfaces. For

instance, it was acousto-plasmon model of

Shimotsuma (2003), model based on local surface

plasmons excitation of Bhardwaj (2006), model

accounting for waveguide formation on the

semiconductor’s surface of Martsinovski (2009).

The main idea suggested by us was the

achievement the high concentrations of

nonequilibrium curriers in conduction band such as

during the pulse semiconductor (dielectric) will have

metallic properties. To achieve this, the following

inequality must be fulfilled



< 

sp0

(2)

of excited and usual dielectric, see Fig.1. Each case

the limiting frequency of SPPs 

sp0

is higher than the

nonequilibrium carriers, m and e are mass and

charge of electron, correspondingly. The inequality

(2) is illustrated with the help of dispersion relation

where 

sp0

=

/(1+n

)

1/2

, 

=4n

/m where 

the frequency of laser radiation, n is the refractive

index of dielectric, n

is the concentration of laser

frequency the possibility of existence and excitation

of SPPs exists. Simple estimation shows that critical

concentration of nonequilibrium carriers for

semiconductor silicon is n

~2 10

-3

. This

concentration can be easily achieved in experiments

with femtosecond laser radiation.

Figure 1: Dispersion curve for plane boundary of surface-

active media – semiconductor for surface plasmons with

dissipations. 1 – dispersion curve; 2 – asymptotically

limiting value of dispersion curve for k

; 3 – line of

laser radiation; 4 – light line; k

– wave number of SPP at

laser frequency. Dotted line shows high frequency

dispersion curve behavior for medium with loss.

2.2 Spatial Periods of Structures

The periods of spatial structures predicted by old

polariton model for long pulse durations (normal

incidence of radiation) have two values which

originate from interference of incident laser

radiation with generated by it SPPs, and mutual

interference of SPPs (of opposite propagation

directions): d=/ and d=/2. Here  is the real

part of the refractive index for SPPs of considered

boundary. The old model has not explain more wider

set of periods arises in experiments, for instance,

appearance proportional to  and multiples by

natural numbers periods. Wide variety of periods

appeared in femtosecond laser radiation experiments

with dielectrics, semiconductors and metals was one

of the obstacles to apply the polariton model. So the

question arises to explain this variety of periods.

We suggested the nonlinear mathematical model

to describe the values of periods of laser-induced

structures Makin (2008). The model is founded on

basic models from simple nonlinear one-dimensional

unimodal logistic maps. We follow the close

similarity the expression for total absorbed intensity

I of laser radiation in the case of SPPs excitation:

UniversalPolaritonModelofLaser-inducedCondensedMatterDamage

181

I(x)=I(x)+(II

)

1/2

cos(g

x+)

+(I

)

1/2

cos(g

x+)

(3)

and expression for logistic map (4). Here I

is the

intensity of SPPs, I is the intensity of laser radiation,

,  are initial phases, g

and g

are formed gratings.

Let us introduce new variable x=I

/I. We will

describe our system by one-dimensional unimodal

logistic map:

f(,x)=1+x(1-x), x[0,1], [

, 

] .

(4)

The bifurcation diagram in coordinate (x-) which

corresponds to logistic map (4) is presented on Fig.

2 and describes the set of bifurcations for definite

values of 

(i=1, 2, …). Value d

is the distance

between values х=1/2 and nearest to this point

element of circle of period 2

for = 

*, where 

nearest element is 2

n-1

iteration of point х=1/2:

)

(







(5)



* is the value of parameter , corresponding to

supercircle of period 2

. So the model describes the

set of periods (reversed Feigenbaum’s cascade)

multipled by ½:

…, 4/



, 2/



, /



, /2



, /4



,/8



, …

(6)

The Feigenbaum’s universality is the part of

Sharkovski order. So the nonlinear mathematical

model can be generalized for Sharkovski order.

Figure 2: Bifurcation diagram of reverse Feigenbaum’s

cascade.

Important, that if the period 3 of Sharkovski

order is experimentally realized the appearance of

any period of Sharkovski order is possible. And this

is the rout to chaotic behavior of the system. The

obtained results are well illustrated by published

experimental results including predicted values of

periods far less diffraction limit value Makin (2012).

The obtained theoretical results causing the

nanostructures formation with periods less than the

diffraction limit value are in frame of recently

developed nonlinear Abbe theory of Barsi and

Fleischer (2013).

2.3 Axi-symmetrical Polarization

Let us consider the action of radially polarized laser

radiation (RPR) with selective direction of

polarization =

where  is the angle

characterizing (local) direction of the electric field

vector of the incident wave. It is known that the SPP

frequency is equal to that of the laser radiation,

while its wave vector k

is determined by the law of

conservation of momentum (quasi-momentum):

= k

+mg, (7)

where k

is the tangent component of the wave

vector of laser radiation, g is the vector of resonance

grating that allows conversion of laser radiation to

the surface polaritons (SP), and m is the order of

diffraction. At normal incidence of radiation k

=0,

=g, and



(



(8)

So the induced grating is characterized by

orientation (8) and period of axial ring structures

with a period determined by expression (9), where 

is the real part of the refractive index of the interface

for SPPs. If [0, 2], i.e., takes all possible values,

there appears a grating in the form of circular rings.

d=2









(9)

Figure 3: Distribution of (a, c) intensity and (b, d)

corresponding microstructures formed under the influence

of a train of femtosecond laser pulses with (a) radial and

(2012).

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Hence, the resonance grating corresponding to

the interference of the radially polarized laser

radiation with driven by it SPs represents axi-

symmetric ring structures with a period determined

by (9). The experimental results obtained by Lou

(2012) and Hnatovski (2011) support the above

consideration; namely, it was shown that radially

polarized femtosecond laser radiation interacting

with the surface of silicon induced an axisymmertic

ring structure with period d=/690 nm and depth

up to 300 nm (=806 nm, = 35 fs, Q=0.26 J/cm

Fig. 3b.

Figure 4: Ordered microstructures formed on the surface

of fused silica by a train femtosecond laser pulses (=775

nm, =200 fs): (a, c) radially polarized radiation; (b, d)

azimuthally polarized radiation. The power densities in

Fig. 4c and 4d are 1.6 times higher than in Figs. 4a and 4b,

Hnatovski (2011).

A similar structure was observed on the surface

of fused silica (d=/n260 nm, =775 nm, =200

fs) Fig 4a. Here, n is the refractive index of fused

silica n=. The high power density of the laser

radiation used by Lou (2012) and Hnatovski (2011)

has induced nonthermal phase transition, as a result

of which the medium was acquiring the properties of

surface active agent.

Based on similar arguments, it can readily be

shown that the azimuthally polarized laser radiation

causes formation of radial structures of damage (see

Figs 3c, 3d, 4b, 4d). Note, that formation of

structures was interpreted by Hnatowski (2011) as a

“print” of the field under tight focusing, while no

interpretation was offered by Lou (2012).

The interaction of axi-symmetrically polarized

laser radiation with metal (stainless steel) also

causes the formation of corresponding

axisymmetrical gratings Allegre (2013), according

to theoretical considerations, see Fig.5.

The character of the formed structures can be

explained based on other considerations. It is known

that RPR interacting with axisymmetric ring

resonance grating excite SPPs propagating in the

radial direction, Zhan (2006). The opposite

statement is also correct: interference of the RPR

with radially propagating SPPs produces an axi-

symmetric grating.

If SPPs propagate towards the center of the

structure, additional absorption related to their

excitaton and energy transfer in the direction of the

structure center (Fig. 6) will cause formation of a

peak of intensity due to the waves being in phase in

the center of the structure. Indeed, in the area where

the fields of the laser radiation and the excited SPPs

add together, the total field has the form:

E=E

, (10)

where

=A(z- r i



(1)

r)exp(ik

r)exp(-it).

(11)

A is the amplitude, r is the radius vector, r and z are

the unit vetors, i=k

, H

(1)

is the first-order Hankel

function of first kind,  is the frequency, and k

the SPP wave number. Expression (11) is valid if

r>>1, i.e., for r>>/2.

The problem of additional power P(r)

redistribution due to SPPs excitation and their

energy transfer towards the center of irradiated zone

of radius r

was formulated and solved by Makin

(2013). As a result, assuming that r

, L=

-1

~

P(r)/S



r=r



[1- 1/(P

)]P

8L/

1/~

-1

(12)

where S

r=r

is area of SPPs focusing with r=r

, r

diffraction limited radius, P

is constant, L is the

SPP’s propagation length along the considered

boundary. Hence, the RPR (in the universal

polariton model) ensure formation of “hot” areas

under the conditions of additional (other than

Fresnel) absorption of energy of the laser radiation

and its focusing through the SPP energy transfer

towards the structure center. The effect of additional

absorption of energy of laser radiation by resonance

dynamic microrelief gratings, Bazhenov (1986),

should take place also in the case of processing of

metals by radially polarized radiation of high-power

continuous-wave technological CO

lasers. The

experimentally observed increase in the laser being

processed (at the front and the walls) has a

processing efficiency by radially polarized radiation

by a factor of 1.5 – 2 (data obtained by Trumpf,

Niziev (1999)) was attributed to tighter focusing

geometry, Dorn (2003), and the fact that the

coefficient of absorption of radiation by the surface

maximum possible value corresponding to

UniversalPolaritonModelofLaser-inducedCondensedMatterDamage

183

absorption of a p-polarized wave.

Figure 5: Optical microphotos illustrated polarization

vector of laser radiation (=800 nm, =15 fs, Q=1.5 J/cm

)

distribution and formaion at thefocal plane at stainless

steel surface ordered microstructures. The arrows show the

direction of the structure sg, g  E. Modal convertor

(SLM) allows to get beams of laser radiation with linear,

ircular, radial and azimuthal polariation and

corresponding areas of laser-indued surfae dmage: (a), (b),

(c), (d), Allegre (2013).

Figure 6: Schematic illustration of formation of an

axisymmetric ring resonant grating on the surface of a

condensed media by radially polarized laser radiation; the

grating converts incident radiation into surface polaritons

that are focused onto the structure center. The inset shows

polarization of laser radiation in the beam cross section.

The magnification of the electronic image of the

surface of silicon (Fig.3b) allowed seeing small-

scale structures covering ridges of the main

resonance structures of the ring type with period

d=230 nm. The grating vector of the small-scale

structure was oriented approximately along the

tangent line to the ring structures. There structures

are the result of excitation of channels SPPs

propagating along the ridges of the ring structures of

the main resonance relief pattern and their

contribution to interference (see section 2.3), Makin

(2012).

Ring structures with period d=/4=/4n125

nm could also be seen on the surface of fused silica

under irradiation with a radially polarized radiation

(Fig. 4a); simultaneously, the initial stage of

formation of structures with d=/8n64 nm (upper

left corner) could be seen. Here, n(=1.45) is the

refractive index of fused silica. The image also

reveals nanostructures with d40 nm the grating

vector of which is parallel to the tangent line to the

axi-symmetric grating. The characteristic scale of

structures in Fig.4c obtained at higher power

densities of laser radiation was also d130 nm. The

fact that the theoretical expression governing the

period of the resonance structures for fused silica

contains index n indicates that the structures are

localized in glass volume in the near surface layer

Makin (2012). Formation of ring-type

nanostructures with a period of a multiple of 2 is

related to Feigenbaum’s universality Makin (2008)

(see the section 2.2). Formation of ordered

nanostructures with such small periods in glass (k=4,

8) is observed and interpreted for the first time (see,

e.g., Makin (2013)).

Axi-symmetric ring-type gratings were

experimentally observed in the regime of interaction

of long CO

laser pulses (1s) with fused silica

and were caused by interference of unpolarized

radiation with surface phonon polaritons driven by

it.

Note that the interpreted effect of formation of

ordered damage structures by nontraditionally

polarized radiation can be used for creation of

devices controlling the polarization of laser radiation

via reconstruction of polarization from the character

of spatial distribution of produced ordered structures

(solution of the inverse problem).

2.4 Anomalous Grating Orientation

In the interaction of polarized femtosecond laser

radiation the formation of nanostructures with

anomalous orientation gE

(normal incidence) was

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184

observed for condensed matter with physically

different properties: metals, semiconductors,

dielectrics by Makin (2009). The period d of such

structures was small in comparison with wavelength

of incident radiation, usually d<<. Recently we

have observed the formation of such structures in

regime of long pulse durations (titanium, germanium

Makin (2011)). The origin of their appearance was

not clear. We proposed the model based on the

mutual interference of so called wedge (or channel)

surface plasmon polaritons propagating along the

ridges (hollows) of main resonant structures (with

opposite propagation directions). The dispersion

relation of such wedge (channel) SPPs can

sufficiently differ from one for planar boundary

(their dispersion curve lies under the curve for plane

boundary SPPs curve). So the periods of structures

which corresponds to such interference can be as

small as ~40-60 nm for metals, ~50-60 nm for

compound semiconductors and ~40 nm for quartz

glass and corresponding gratings have anomalous

orientation (see section 2.2). Note that excitation of

wedge (channel) SPPs is one more way for laser

radiation energy transfer into heat dissipated inside

irradiated materials.

2.5 Quasi-periodic Gratings on Metal

In our experiments the effect of formation of

quasiperiodic gratings on surface of polished

titanium under the action of 10 ns pulses of linear

polarized laser radiation (=1,06 m, =10 ns) was

discovered. The process of anisotropic metal

recrystallization governed by polarization of laser

radiation was experimentally observed. The period

of produced structures in mode of grooves of

thermal etching having orientation gE

was 5 m

and rises to 6 m with laser power density. Later

analogous experimental data were achieved for

titanium and amorphous titanium alloys for

femtosecond laser pulse durations. To explain the

observed phenomenon the physical model was

suggested by Makin (2013). The model involves the

excitation of SPPs by the grain boundaries in the

preferential direction of electric field vector of

incident laser radiation, drags of electrons of skin-

layer by SPPs and movement of grain boundaries,

step by step from pulse to pulse, via transfer of

electron’s momentum to them.

3 CONCLUSIONS

The extension of polariton model for the case of

ultrashort laser pulse durations and materials with

different physical properties: metals,

semiconductors, and dielectrics was made. The

nonlinear mathematical model to describe the spatial

periods of laser-induced structures in the process of

laser-matter interaction involving excitation of

surface polaritons was suggested and experimentally

verified for different materials: dielectrics, metals,

ceramics and semiconductors. The nonlinear model

describes the formation of structures with periods

proportional to  and sufficiently lower values of

periods in comparison with diffraction limit value.

The resonant axi-symmetrical micro- and

nanostructures structures which arise under the

interaction of axi-symmetric polarization of laser

radiation with matter were theoretically predicted

and experimentally illustrated for metals,

semiconductors and dielectrics.

The origin of formation of the nanostructures

with anomalous orientation gE

for long and

ultrashort laser pulse durations were cleared as a

result of excited wedge or channel SPPs

participation in interference process.

The effect of anisotropy of metal

recrystallization under the action of long and

ultrashort laser pulse durations of linear polarized

laser radiation was discovered. The theoretical

explanation of observed effect was suggested based

on the effect of grain boundary movement by

electrons of skin-layer dragged by surface plasmon

polaritons.

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