NBSIR 73-268

Laser Damage in Materials

Albert Feldman, Deane Horowitz, and Roy M. Waxier

Solid State Materials Section

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

August 1973

Semi-Annual Technical Report

Period Covered: February 1, 1973, to July 31, 1973

Prepared for Advanced Research Projects Agency

Arlington, Virginia 22209

ARPA Order No. 2016

NBSIR 73-268

LASER DAMAGE IN MATERIALS

Albert Feldman, Deane Horowitz, and Roy M. Waxier

Solid State Materials Section

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

August 1973

Semi-Annual Technical Report

Period Covered: February 1, 1973, to July 31, 1973

Prepared for Advanced Research Projects Agency

Arlington, Virginia 22209

ARPA Order No. 2016

U, S. DEPARTMENT OF COMMERCE, Frederick B. Dent. Secretary

NATIOIMAL BUREAU OF STANDARDS, Richard W. Roberts. Director

LASER DAMAGE IN MATERIALS

Albert Feldman, Deane Horowitz, and Roy M. Waxier

Solid State Materials Section

Inorganic Materials Division

Institute for Materials Research

ARPA Order No 2016

Program Code Number 3D10

Effective Date of Contract January 1, 1972

Contract Expiration Date December 31, 1973

Principal Investigator Albert Feldman

(301) 921-2840

This is a progress report. The work is incomplete and is

continuing. The views and conclusions contained in this document

are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed

or implied, of the Advanced Research Projects Agency or the U. S.

Government

.

LASER DAMAGE IN MATERIALS

Abstract

Neodymium: glass laser induced damage is observed in

lithivun niobate (LiNbO^) , calcite (CaCO^) , potassiiom dihydrogen

phosphate (KDP) , and deuterated potassixam dihydrogen phosphate

(KD*P) . The damage at the lowest power levels is caused by

inclusions. At higher power levels, filamentary dcimage,

which is indicative of self-focusing, is observed in LiNbO^

.

An analysis of self-focusing data in yttrium aluminum garnet (YAG)

shows that the Kerr effect is the dominant self-focusing

mechanism, with some contribution from the thermal effect.

Bulk and surface damage thresholds in neodymium-doped

thoria : yttrium oxide ceramic are obtained relative to bulk

damage thresholds in several optical materials. For solid

materials, relationships are obtained between the stress-optic

coefficients and the electrostrictive coefficients under

different geometric boundary conditions.

Table of Contents

1. Technical Report Sxunmary 1

1.1 Technical Problem 1

1.2 General Methodology 2

1.3 Technical Results 2

1.4 Department of Defense Implications 3

1.5 Implications for Further Research 4

2. Technical Report 5

2.1 Damage Studies in Lithium Niobate (LiNbO^) , Calcite

(CaCO^) , Potassiiam Dihydrogen Phosphate (KDP) , and

Deuterated Potassiiim Dihydrogen Phosphate (KD*P) . 5

2.1.1 Introduction 5

2.1.2 Experimental Procedure 7

2.1.3 Results and Discussion 7

2.2 Self-Focusing in Yttrium Aliaminxam Garnet 20

2.3 Damage in a Neodymium-Doped Thoria :Yttrixm Oxide

Ceramic Laser Rod 23

2.4 Relations Between Electrostriction and the StressOptic Effect 27

2.4.1 Case I 31

2.4.2 Case II 33

2.4.3 Case III 34

2.5 Acknowledgments 35

2.6 References 37

i

LASER DAMAGE IN MATERIALS

1. Technical Report Siammary

1.1 Technical Problem

The main objective of this program is the measurement of damage

thresholds and the determination of mechanisms associated with selffocusing in materials used in high-energy pulsed laser systems. Selffocusing is the main process which leads to intrinsic bulk damage in

laser materials. The damage appears as filamentary tracks in materials

exposed to high-intensity laser radiation. The study of self-focusing

can be obscured, however, if extrinsic damage processes, such as

inclusion damage, take place.

In this report we examine the bulk damage processes in

crystalline materials, some of which are used in the fabrication of

modulators and Q-switches. We attempt to determine whether inclusions

or self-focusing induced damage are the limiting damage factors in these

materials. When self-focusing is important, we estimate the relative

importance of three mechanisms responsible for self-focusing; the Kerr,

electrostrictive , and thermal effects. Additionally, in a theoretical

analysis we attempt to resolve an apparent discrepancy between different

authors in relations between the electrostrictive coefficients and the

stress-optic coefficients.
1.2. General Methodology

Laboratory experiments were conducted to determine the mechanisms

of damage in calcite (CaCO^) , potassiiam dihydrogen phosphate (KDP) , deuterated potassium dihydrogen phosphate (KD*P) , lithium niobate (LiNbO^)

,

and neodymium-doped thoria : yttrium oxide ceramic. The output of a

Q-switched Nd: glass laser operating in the TEM^^ mode was focused into

the samples with a 181-mm focal length lens. Damage sites in the samples

were examined to ascertain whether inclusion damage or self-focusing

induced damage were the limiting damage processes. In undoped yttrium

aluminiam garnet (YAG) , for which earlier measurements indicated selffocusing damage was present, we calculated a lower bound to the nonlinear

index n^ and the individual contributions of the Kerr effect, electrostriction, and the thermal effect to n^. We have derived relationships between

the electrostrietive coefficients and the stress-optic coefficients

for different geometric boundary conditions using thermodynamic arguments

in order to resolve an apparent discrepancy in the results of different

authors. Thermodynamic free energy relationships were constructed and

sets of Maxwell's equations were obtained by taking appropriate derivatives

of the free energy.

1.3. Technical Results

We find in calcite, KDP, KD*P, and LiNbO^ that inclusions are

the limiting cause of damage (section 2.1). In LiNbO^ we also find

evidence for self-focusing, but this occurs at higher power levels.

We analyze previously reported self-focusing data in YAG

(section 2.2) which was obtained with both linearly and circularly

2.
polarized radiation. We find that the Kerr effect is the dominant

self-focusing mechanism, with some contribution from the thermal effect.

The electrostrictive effect is negligible because YAG has exceedingly

small stress-optic coefficients. Using the theory which we developed in

our previous report, we calculate a lower bound to the nonlinear index n^

and the contributions of the Kerr and thermal effects to n^.

Bulk damage measurements were made in a neodymium-doped

thoria:yttri\am oxide ceramic laser rod (section 2.3) in cooperation with

Charles Greskovich of the General Electric Corporate Research and

Development Laboratory*. The damage threshold relative to several

optical materials was tabulated.

We have obtained relationships between the stress-optic

coefficients and the electrostriction coefficients for three geometric

situations , (2 . 4) . Our derivation is based on earlier theoretical work.

We find that the differing results of several authors correspond to

solutions for different geometric boundary conditions

.

1.4. Department of Defense Implications

The Department of Defense has a need for high-powered solid

state laser systems. Thus it is important (1) to \inderstand the processes

which limit the output power of such systems, (2) to obtain data which

suggest methods for increasing the output power of a given system, and

(3) to verify theories which predict the performance of such systems.

Research sponsored by ARPA.

3.
We have come to the following conclusions: (1) In materials for which

self-focusing is the dominant mechanism leading to damage, such as

glasses, sapphire, and YAG, the Kerr effect is the dominant self-focusing

mechanism. The Kerr effect increases with increasing refractive index;

therefore, for this type of materials, those having low refractive indices

will have high self-focusing thresholds. (2) The thermal effect can be

important for pulse widths ^ 25 ns or for high repetition rate systems.

(3) Inclusion damage limits the performance of many crystals used presently

in modulators and Q-switches, such as calcite, KDP, KD*P, and LiNbO^. In

order to obtain higher damage thresholds, methods must be found to

eliminate the inclusions. When this is done, self-focusing will then

become the limiting process.

1.5. Implications for Further Research

For the remainder of the contract period we intend to study

two problems. (1) We plan to examine in some detail the process of

damage in several neodymium-doped laser glasses, which we discussed

in our previous report, in order to determine the self-focusing mechanisms

in these materials. (2) We plan to examine the self-focusing computer

solutions of Dawes and Marburger so that we can better relate our

damage data to the nonlinear index n^.

The interests of the Department of Defense have been shifting

to longer wavelengths in the infrared (10.6 ym for the CO^ laser and

3.8 ym for the DF laser) . Interest is also expected to shift toward

the ultraviolet region even though there are still no important highpower lasers in this region. Over this wide wavelength range there is
a lack of data on optical materials that could be used for the

construction of optical components. These data include changes of

refractive index with temperature and stress which are important

because components subjected to high-intensity radiation can undergo a

significant change of refractive index due to heat generated by low-level

absorption processes. These changes in index are caused by variations

of temperature and by stresses introduced by temperature gradients.

Therefore, measurements of the refractive index and the change of

refractive index with temperature and stress, in both the infrared and

ultraviolet regions of the spectriam, would be worthwhile. Materials of

cxirrent interest in the infrared region are KCl, ZnSe, sapphire, and

the alkaline-earth fluorides

.

2. Technical Report

2.1. Damage Studies in Lithium Niobate (LiNbO^) , Calcite (CaCO^),

Potassixmi Dihydrogen Phosphate (KDP) , and Deuterated Potassium

Dihydrogen Phosphate (KD*P)

2.1.1. Introduction

The output of high-powered laser systems is limited by the

damage occurring in components exposed to the intense electromagnetic

radiation. Modulators and Q-switches are important components of pulsed

solid-state laser systems. Polarizers and Pockels cells are key elements

in many of these components. Polarizers are constructed from prisms

of calcite or from glass plates stacked at the Brewster angle. Pockels

cells are fabricated from electro-optic crystals; the most common

materials are KDP, KD*P, and LiNbO^.

5.
Several authors have published results of damage studies in

these materials. Poplavskii and Khazov have reported observing

polarization dependent bulk damage in calcite crystals [1] . They were

of the opinion that self-focusing was definitely important in calcite

although they had not observed filamentary tracks typical of selffocusing induced damage.

Damage studies in KDP and LiNbO^ have shown that inclusion

damage is the limiting process [2,3]. Hence, most damage studies in

KDP and LiNbO^ have emphasized surface damage [4-8] and the statistical

nature of this damage process [7,8] . Bass [9] has also examined

filamentary damage in LiNbO^ and has discussed the possibility that selffocusing due to heating is important in producing the damage.

In this report we attempt to identify the principal mechanisms

responsible for bulk damage in optical quality calcite, KDP, KD*P, and.

LiNbO^. We are motivated by the success we have had in identifying the

mechanisms responsible for self-focusing induced damage in several

optical glasses [10-12] . The conclusion we reach is that inclusions

are the principal cause of damage in the above crystals although

, filamentary damage caused by self-focusing is observed in LiNbO^

.

Recent work by Norman Doling [13] shows that inclusions can be a major

factor in surface damage.
2.1.2. Experimental Procedure

The output of a Q-switched Ndtglass laser is focused into the

samples with a 181-mm focal length lens. The characterization of the

experimental apparatus and the laser beam is discussed in a previous

report [10] and is summarized as follows. The laser operates in the

TEMqq mode with a pulse width of 25 ns. The pulse energy is constant

to within + 2%. The focusing lens is located approximately five meters

from the spatial mode selection aperture in the oscillator cavity. The

beam profile at the position of the focusing lens approximates a Gaussian

distribution.

The procedure followed was to fire the laser into the sample

and to observe whether damage was produced. If internal damage occurred,

the sample was moved laterally to the beam and another laser pulse, of

lower energy, was fired into the sample. The laser energy was lowered by

inserting a calibrated neutral-density filter into the beam. If damage

did not occur, the laser energy was raised. In all cases damage was

observed when no filters were present in the beam. When several damage

sites were produced with a single laser pulse, they appeared to be

CO- linear

.

2.1.3. Results and Discussion

Figures 1 to 3 show the damage produced in calcite. The

calcite is of schlieren grade material. Figure 1 shows the damage in

transmitted white light; figure 2 shows the damage with side-light

illiamination. Both photographs were taken through a polarizer to

7.
Fig. 1. Damage sites in calcite as observed in linearly

polarized transmitted white light. The vertical lines define

the planes of focus, the left for the ordinary ray and the

right for the extraordinary ray. The arrow to the right shows

the direction of laser beam propagation, which was perpendicular

to the optic axis. Sites whose ordinates are above the arrow

were produced with the extraordinary ray, those below the arrow

with ordinary ray. The magnification is 9X.

"1 8.
Fig. 2. Damage sites in calcite. These are the same

sites as in Fig. 1 except they are viewed through a polarizer

with side-light illumination. The same conventions are used

as in Fig. 1. The magnification is 6.5X.
Fig. 3. Magnified view of a damage site in calcite as viewed with side-light illximination. The double image, which

is due to the crystal birefringence, occurs because no polarizer was used for obtaining the photograph.

10.
eliminate doiible image formation caused by the crystal birefringence.

The arrows to the side of the figures show the direction of laser beam

propagation, which was perpendicular to the optic axis. Damage sites

whose ordinates are above the arrows were produced with the extraordinary

ray; those below the arrows were produced with the ordinary ray. The

vertical lines on the figures define the planes of focus; the left line

defines for the ordinary ray focus, the right line defines the

extraordinary ray focus.

Observe that the damage sites are randomly distributed about

the focal planes. Filamentary damage, which is associated with selffocusing, is not seen. These observations are indicative of inclusion

damage. In figure 3 we show a magnified view of one damage site. No

polarizer was used in obtaining the photograph so that a double image

appears due to the crystal birefringence. The damage has the appearance

of feather-like cracks radiating from a central region which is the

probable location of the inclusion. Setting a damage threshold for

inclusion damage is difficult because the threshold will depend upon the

diameter of the inclusion. Theoretical calculations [14,15] predict

that, for our experimental parameters, spherical inclusions 0.2 ym

will produce the lowest damage threshold.

Figures 4 to 7 are photographs of damage sites in KDP and

KD*P. The arrows show the direction of the laser beam propagation,

which was along the c-axis of the crystals. The vertical lines define

the focal plane of the laser beam. The random distribution of the

damage sites about the focal plane and the lack of filamentary damage

11.
Fig. 4. Damage sites in KDP as viewed in transmitted

white light. The vertical lines define the plane of focus.

The arrow shows the direction of the laser beam propagation,

which was along the crystallographic c-axis. The magnification is 7X.

12.
Fig. 5. Damage sites in KDP. These are the same sites

as in Fig. 4 except they are viewed with side-light illumination. The conventions used are the same as in Fig. 4. The

magnification is 4.5X.

13.
Fig. 6. Damage sites in KD*P as viewed in transmitted

white light. The conventions used are the same as in Fig. 4.

The magnification is 7.5X.

14.
Fig. 7. Damage sites in KD*P. These are the same sites as in Fig. 6 except they are viewed with side-light illumination.

The conventions used are the same as in Fig. 4.

15.

\
indicate that we are obseirving inclusion type damage. The morphology

of the damage sites in KDP and KD*P is similar, but the sites have a

different appearance than in calcite. The difference can most probably

be attributed to the crystal structures of the materials.

Figures 8 and 9 are photographs of damage observed in LiNbO^.

The arrows indicate the laser beam propagation direction, which was

along the c-axis of the crystal. The vertical lines denote the focal

plane of the laser beam. In figure 8 we can clearly observe damage

caused by four laser shots. The uppermost series of damage sites

(co-linear with the arrow) was produced by an 11.2 mJ pulse with a peak

power of 380 KW. A filamentary damage track was produced, which is

evidence for self-focusing. The filament is clearly seen in an enlargement which is shown in figure 10. In addition to observing self-focusing,

in figure 9 we can also observe damage due to inclusions upstream from

the filament. The three sets of damage sites below the large track of

damage were all produced with pulses of the same energy (3.0 mJ) . While

the main damage occurs in the vicinity of the focal plane, it is clear

that the damage process is non-reproducible. In glasses, we have

observed that the damage process is reproducible. In figure 9 we can

clearly observe damage sites far upstream from the focal plane. We

conclude therefore that the main bulk damage mechanism in LiNbO^ is

due to inclusions. While it is possible to produce self-focusing damage,

it is difficult to draw any conclusions about the self-focusing threshold

and nonlinear index because the occulting effect of inclusions obscures

the determination of the power involved in the self-focusing process.

16.
Fig. 8. Damage sites in LiNbO^ as viewed in transmitted

white light. The conventions used are the same as in Fig. 4.

The magnification is 6.5X. The damage opposite the arrow is

indicative of self-focusing (see Fig. 10)

.

17.
Fig. 9. Damage sites in LiNbO^ . These are the same sites

as in Fig. 8 except they are viewed with side-light illumination. The conventions used are the same as in Fig. 4. The

magnification is 6X.

18.
^

Fig. 10. Filamentary damage in LiNbO^ which is indicative

of self-focusing. The magnification is 42X.

19.
In Table I we present an estimate of energy densities and

maximum power densities of the laser beam in the vicinity of several

damage sites in the crystals under study. We say vicinity because

the numbers are calculated on the beam axis whereas the damage site

may be off the axis. The values cannot be interpreted as thresholds

but can be interpreted as upper bounds to thresholds. The sites chosen

were the farthest upstream. We avoided sites near the focal plane

because this is where aberrations are strongest. The energy and power

densities are calculated from equations that define the propagation

characteristics of Gaussian beams.

;.An examination of the table shows a large variation of energy

density near damage sites in all the materials except calcite. In

calcite the values appear to be quite reproducible even though they

were obtained with laser shots of widely varying energy. The result

suggests that the inclusions in calcite are of uniform size. An examination of the calcite crystal under illumination by a 5 mW helium-neon

laser shows planes with a high density of scatterers. Many of the damage

sites in calcite are located within these planes.

2.2. Self-Focusing in Yttrium Aliuninum Garnet (YAG)

In our previous report we discussed measurements of filamentary

damage tracks produced in YAG with linearly and circularly polarized

radiation. The laser beam propagated along the [111] crystallographic

axis. We found that the self-focusing lengths deduced from these

measurements were in good agreement with theory [10,12,16].

20.
Table I. Energy density and peak power density of

laser beam in the vicinity of damage sites.

Energy Energy Density Power Density

(mJ) (J/cm^) (W/cm^)

Calcite 1 2.4 45 1.55 X 10«

2 14.4 46 1.56 X

3 4.4 46 1.58 X 10*

KDP 1 11.8 76 2.6 X

2 13 130 A C 4 . D X lU

3 10.3 200 7.0 X

4 24.2 460 15.7 X 10«

KD*P 1 12.7 120 4.2 X lO^

2 12.7 280 9.5 X

3 12.7 390 13.4 X 10^

LiNbO^ 1

2

3.03

3.03

24.2

31.3

0.71

0.91

X

X

10^

3 11.2 72.2 2.5 X 10^

21.
In this report we present corrected values for the selffocusing thresholds deduced from the data. These corrections are due

to a recalibration of our thermopile energy meter and to our taking

into account reflections from our sample entrance face. The corrected

thresholds for linearly and circularly polarized radiation are

P = 0.40 MW and P =0.50 MW, respectively. (Throughout this section

c c primed symbols refer to circular polarization; unprimed symbols

refer to linear polarization.) In the theory of Dawes and Marburger,

P^ is the power above which the laser intensity diverges on the

beam axis. Hence, damage will occur at powers less than P^. Thus

P < P^ because P is a measure of the damage threshold for the

c 2 c particular experimental conditions discussed in our previous report.

We are actually measuring a damage threshold.

We have previously discussed a method of calculating the

nonlinear index n^ and the contributions that the Kerr, electrostrictive,

and thermal effects might make to n [10-12] . Here n is defined by

6n = n^ (1)

m 2 o where 6n is the maximum refractive index change occurring during the

m

passage of a laser pulse and is the RMS value of the electric field

at the peak of the pulse. In the paraxial ray approximation, the peak

power of a pulse that leads to critical self-focusing of beams with a

Gaussian profile, in which the spreading of the beam by diffraction

is exactly cancelled by self-focusing, is given by [17-19]

P^ = xV(32TrV) (2) C 2 where X is the wavelength of the laser radiation in air, c is the

22.
velocity of light, and = .273 [16] . A lower bound for n^ can

-13

be derived with eq (2) ; we obtain n^ = 2.7 x 10 esu and

' -13

n^ = 2.1 X 10 esu. The contribution of electrostriction to n^ is

expected to be negligible because YAG has exceedingly small stress-optic

coefficients. For a beam propagating along the [111] axis, the contribuI I tions of the Kerr effect to n^ and have the relation n2(K)/n2(K) = 1.5

if we assume that the Kerr effect is of electronic origin [20] . Using

-13

eqs (20a) and (20b) of reference [10], we find that (K) = 1.7 x 10 esu

-13

and the thermal contribution n^CT) = 0.8 x 10 esu. Thus we find that

the Kerr effect is the dominant self-focusing mechanism. From n^ (T)

-3 -1 we calculate an absorption coefficient a = 5 x 10 cm which is of the

proper magnitude for YAG.

2.3. Damage in Neodymium-Doped Thoria ; Yttrium Oxide Ceramic Laser Rod

In cooperation with Charles Greskovich of the General Electric

Company Corporate Research and Development Center, we conducted damage

studies in a laser rod constructed of neodymium-doped thorium: yttrium

oxide ceramic material. A bulk damage threshold was obtained by focusing

the laser beam into the sample. The damage threshold relative to several

optical materials is given in Table II. Effects due to self-focusing

may be present in the data. Figure 11 is a picture of a bulk damage

site as viewed along the rod axis in transmitted white light. Note

radial cracks that propagate with jogs caused probably by grain

23.
Table II. Relative bulk damage thresholds in several

materials and surface damage threshold of neodymium-doped

a b

thoria : yttrium oxide ceramic (yttralux)

.

Pulse Energy Peak Pulse Power

(mJ) (MW)

BSC 517 22 0.75

Dense Flint SF 55 3.74 0.128

Fused Silica ' 29 1.01

YAG 11 0.36

Yttralox (Bulk) 5.5 0.19

Yttralox (Surface) 1.5 0.051

aObtained by focusing of the output of a Nd: glass laser

(pulsewidth T = 25 ns) into the sample or on the surface

of the sample with a 181-mm focal length lens.

bCommercial materials are identified m this report to

specify the particular substance on which the data were

obtained. In no instance does such identification imply

recommendation or endorsement by the National Bureau of

Standards or that the material identified is necessarily

the best for any application.

24.
Fig. 11. Bulk damage site in a laser rod of neodymiumdoped thoria : yttrium oxide ceramic as viewed along the rod axis

in transmitted white light. The magnification is 75X. Note

radial cracks that propagate with jogs caused probably by grain

boundaries

.

25.
Fig. 12. The same damage site as in Fig. 11 but viewed

between crossed polarizers. Note the highly strained region

in the immediate vicinity of the damage site and also note

the revelation of grain boundaries under stress. The

magnification is 75X.

26.
boundaries. Figure 12 is a picture of the same bulk damage site as

viewed between crossed polarizers. Note the highly strained region in

the immediate vicinity of the damage site and note also the grain

bovmdaries revealed by the stress. The average diameter of the strain

field is approximately 1.5 mm.

In addition to bulk damage, we produced surface damage on

the laser rod by focusing the laser beam onto the sample surface.

Figure 13 is a reflected light micrograph of a surface damage site

located at a three-grain intersection. There is a fine network of

micro-cracks in one grain. All three grains had originally exhibited

this appearance, but, in the process of cleaning the surface for picture

taking, material flaked off the other two grains. Note that the general

surface condition is characterized by polishing scratches.

2.4. Relations Between Electrostriction and the Stress-Optic Effect

The process of electrostriction is of current interest because

it can cause the self-focusing of high-intensity radiation in solids.

Articles have been published that relate the electrostrictive

coefficients to the stress-optic [21] , elasto-optic [22] , or piezodielectric coefficients [23] with differing results. In an early work,

Guggenheim [24] has derived similar relations for liquids which undergo

magnetostriction. He showed that the particular relationship obtained

will depend upon the boundary conditions much as the electric polarization in a solid depends upon the shape of the solid. Guggenheim showed

that solutions are easily obtained only for relatively simple

27.
Fig. 13. Damage site on the surface of thoria :yttriiam

oxide ceramic laser rod. This picture is a reflected-light

micrograph of a surface damage site located at a three-grain

intersection. The magnification is 750X.

28.
configurations

.

In this report we derive relations between the electrostrictive

coefficients and the stress-optic coefficients for dielectric materials

with inversion syitunetry, based on Guggenheim's work. We ignore any

effects due to body rotations. Three cases are treated. The following

notation is used: V is the volume occupied by the electric field;

E

V„ is the volume of the solid in the absence of strain; k. . is an

S ±3

element of the dielectric tensor and K. . ^ is an element of its inverse

tensor; £. . is an element of the strain tensor; O. . is an element of

ID ID

the mechanical stress tensor; <3^j]^£ is a stress-optic coefficient;

^ijkil elasto-optic coefficient; and Y^jj^^ is an electrostriction coefficient; is elastic compliance coefficient. The

phenomenological relationships among these constants are

D. = K. . E .; E. = K. D. (1)

1 ID D 1 ID D

^^ij"^ = ^ijk£ ^k£ = Pijk£ ^k£

^1,0 = T Y. -1,0 E.E . + s, 0 a (3)

kJ6 2 i3k£ 1 J kicmn mn

In eq (2) we ignore the term quadratic in electric field, the

Kerr effect term. Equations (1) and (2) are taken to apply when the

fields are either constant or time varying and stresses and strains are

constant in time. When the fields are time varying, we take a time

average of E^E_. in eq (3) . The tensors K, K q, p and Y will depend

upon the frequency of the field.

29
Following Guggenheim, we can write the free energy of a system

in any one of the following forms , depending on which variables are

meant to be independent:

F, = U - TS - — 1 47r E.D. dV -

1 1

a. .£. . dv

ID ID

(4a)

F^ = U - TS

, ID ID <iV (4b)

F^ = U - TS - —1_

4Tr E.D. dV

1 1 (4c)

F^ = U - TS

4

(4d)

We integrate over a fixed volume of space, which includes V and V . E S

U is the internal energy; T is the temperature; S is the entropy. The

change in internal energy of the system is given by [25]

6U = TdS + —47T E.6d. dV +

1 1

a. .6e. . dv

ID ID

(s,D,e)

where the independent variables are listed to the right. The same

convention is followed below. We then obtain

6f^. =

6F2 =

6F3 =

6f, =

4

-SdT -

477

-SdT + —4lT

-SdT - —4lT

-SdT +

47r

D.6 E. dV -

1 1

E.6 D. dV -

1 1

D.6 E. dV +

1 1

E.6 D. dV +

1 1

e. .6a. . dv

ID ID

e. .6a. . dv

ID ID

a. .6e. . dv

ID ID

a. .6£. . dv

ID ID

(T,E,a)

(T,D,a)

(T,E,e)

(T,D,e)

(5a)

(5b)

(5c)

(5d)

30.
similar expressions can be derived with S as an independent variable.

The above expressions will be used to obtain the desired relationships

between y , q, and K.

2.4.1. Case I

We have a solid of volume v . A uniform electric field is

S

totally enclosed within Vg except for in a thin layer of material at the

boundary of the solid in which the field falls to zero. When the solid

undergoes a strain, we assume that the electric field is still enclosed

within the solid, but V is unchanged. We choose this geometry in

order to neglect the stresses developed because of the constraining

effect of the field free region of the solid. For sufficiently large

volumes, V can be considered equal to V . Using eq (5a), we obtain

31.
For an isothermal process

4TT Fi

V dE.dE.

S 1 :

(7)

and

13 J

T,E

(8)

Taking higher derivatives, we obtain

- 8k. . d^z,.

1 ij kx.

4Tr 8a, n 8e.3e. (9a)

Using eq (5b) , we obtain a similar result

, 3k. . 1 ID

-1 32e

47: do, „ 9d.8d.

kx. 1 J

(9b)

Equations (9a) and (9b) lead to the equality

"^ijkJl 47: "^im '^jn ^nk£ (10)

Equation (10) is used for calculating the electrostrictive strains

induced by focusing a laser beam into the interior of a solid. It

corresponds to the results of references [19] and [21]

.

32.
2.4.2. Case II

Consider a thin slab of material in an external E field

with the large faces of the slab perpendicular to the field.

CASE 31

In this case it is convenient to use eq (5b) because D is continuous

across the slab boundary. End effects are neglected. We obtain

= - SdT + -r- D.dD. + (K. . - 6. .) (1 + e 6 )V D.dD.

2 4tt 1 1 13 13 mn mn Si 3 - £. . da.

.

s 13 1:

(11)

Taking derivatives for the isothermal case , we obtain

= - e (12)

-1 V

3d.5d.

^ 3 J

4Tr 13

V

6. .) (1 + e 6 ) + tI- 6. . 13 mn mn 47T 13

T,a

(13)

33.
Taking higher order derivatives and keeping terms to lowest order, we

obtain

3^£

k£

8d.3d.

1 :

. 9k. . 1 12.

-1

- i- (K. . - 6. .) 6 s TO

4Tr ij ij mn mnkx-

(14)

or

ypqk£ — K .K . 47T pi q3

q. ., n + (K. . - 6 . .) 6 s , n ^i^kil 13 13 mn mnkx, (15)

2.4.3. Case III

Consider a long narrow cylinder whose axis lines up parallel

to a uniform electric field.

CASE HE C D

In this case it is convenient to use eq (5a) because E is continuous

across the cylinder wall. End effects are neglected. We obtain

6F = - SdT - — E dE - (K - 6 ) (1 + e 6 ) V E dE (16)

1 47T 11 2. J xj mn mn Si j - e . . do. . S 13 ij

Taking derivatives for the isothermal case , we obtain

34.
1 = - £

T,E

(17)

8f.

3e.9e.

1 3 T,0

V. V, = - "S. . - — (K. .-6. .) (1 + e 6 ) 477 ij 477 ij ij mn mn

(18)

Taking higher order derivatives and keeping terms to lowest order,

we obtain

e,

0

, 9k. . k£ _ 1 ij_ ^ 1_ 6 ) 6 s 9e.9e. ~ 477 9a, n 477 ij ij mn mnkil

X J k£ J J (19)

But, the susceptibility is

ID

K. . - 6. .

477 (20)

so that

Y — — K K Q + Y 6 S ijk£ 477 im jnTnnk£ '^ijj mn mnk£ (21)

Equation (21) corresponds to the results of Maradudin and Burstein [22]

2 . 5 Acknowledgments

We would like to thank Alvin Rasmussen and Dale West of the

National Bureau of Standards for the calibration of our thermopile.

35.
The photographs and descriptions of the damage sites in the neodyini\aindoped thoria : yttrium oxide ceramic laser rod (figures 11-13) were

supplied by Charles Greskovich.

36.
2.6. References

[1] A. A. Poplavskii and L. D. Khazov, "Anisotropic Resistance

of CaCO^ to Laser Radiation," Zh. Tekh. Fiz. 2_, 407

(1969) [Sov. Phys. —Tech. Phys.].

[2] G. M. Zverev, E. A. Levchuk, and E. K. Maldutis , "Destruction

of KDP, ADP, and LiNbO^ Crystals by Intense Laser Radiation,"

Zh. Eksp. Teor. Fiz. 57_, 730 (1969) [Sov. Phys.—JETP 3£, 400

(1970) ] .

[3] Michael Bass, "NdrYAG Laser Irradiation Induced Damage to

LiNbO^ and KDP," IEEE J. Quantum Electron. QE-7 , 350 (1971).

[4] G. M. Zverev, E. A. Levchuk, V. A. Paskov and Yu. D. Poyadin,

"Damage to the Surface of Lithium Niobate by Light," Zh. Eksp.

Teor. Fiz. 62_, 307 (1972) [Sov. Phys.—JETP 35./ 165 (1972)].

[5] W. D. Fountain, L. M. Osterink, and G. A. Massey, in Damage

in Laser Materials: 1971 , NBS Special Publication 356 (U.S.

Government Printing Office, Washington, D. C, 1971), p. 91.

[6] W. A. Massey and W. R. Hook, "Lithium Niobate Damage Threshold

Measurements Inside a Nd:YAG Q-Switched Laser Cavity," IEEE

J. QuantTjm Electron. QE-7 , 317 (1971) .

[7] M. Bass and H. H. Barrett, in Laser Induced Damage in Optical

Materials: 1972 , NBS Special Publication 372 (U. S. Government Printing Office, Washington, D. C, 1972), p. 58.

[8] M. Bass and H. H. Barrett, in Damage in Laser Materials; 1971 ,

NBS Special Publication 356 (U. S. Government Printing Office,

Washington, D. C, 1971), p. 76.

37.
[9] Michael Bass, in Damage in Laser Materials , NBS Special

Publication 341 (U. S. Government Printing Office, Washington,

D. C. , 1970) , p. 90.

[10] A. Feldman, D. Horowitz, and R. M. Waxier, "Laser Damage in

Materials," National Bureau of Standards Report NBSIR 73-119

(1973).

[11] A. Feldman, D. Horowitz, and R. M. Waxier, "Mechanisms for

Self-Focusing in Optical Glasses," IEEE J. Quantum Electron.,

to be published.

[12] A. Feldman, D. Horowitz, and R. M. Waxier, in Damage in Laser

Materials; 1973 , NBS Special Publication, to be piablished.

[13] N. Boling, in Damage in Laser Materials; 1973 , NBS Special

Piablication, to be published.

[14] H. S. Bennett, in Damage in Laser Materials , NBS Special

Publication 341 (U. S. Government Printing Office, Washington,

D. C. , 1970) , p. 51.

[15] R. W. Hopper, C. Lee, and D. R. Uhlmann, in Damage in Laser

Materials , NBS Special Publication 341 (U. S. Government

Printing Office, Washington, D. C., 1970), p. 55.

[16] E. L. Dawes and J. H. Marburger, "Computer Studies in SelfFocusing," Phys. Rev. 179 , 862 (1969).

[17] W. G. Wagner, H. A. Haus, and J. H. Marburger, "Large-scale

Self-Trapping of Optical Beams in the Paraxial Ray Approximation," Phys. Rev. 175 , 256 (1968).

[18] G. M. Zverev and V. A. Pashkov, "Self-Focusing of Laser

Radiation in Solid Dielectrics," Zh. Eksp. Teor. Fiz. 57 , 1128

38.
(1969) [Sov. Phys.—JETP 30_, 616 (1970)].

[19] E. L. Kerr, "Track Formation in Optical Glass Caused by

Electrostrictive Laser Beam Self-Focusing," Phys . Rev. A 4_,

1195 (1971)

.

[20] F. D. Maker and R. W. Terhune , "Study of Optical Effects Due

to an Induced Polarization Third Order in the Electric Field

Strength," Phys. Rev. 137 , A801 (1965).

[21] A. A. Gundjian, "Electros triction in Germanium," Solid State

Commun. _3' 279 (1965) .

[22] A. A. Maradudin and E. Burstein, "Relation Between Photoelasticity, Electrostriction, and First Order Raman Effect

in Crystals of the Diamond Structure," Phys. Rev. 164 , 1081

(1967) .

[23] H. Osterberg and J. W. Cookson, "The Piezodielectric Effect

and Electrostriction in Anisotropic or Isotropic Media,"

Phys. Rev. 5]^, 1096 (1937).

[24] E. A. Guggenheim, "The Thermodynamics of Magnetization,"

Proc. Roy. Soc. (London) A, 155 , 70 (1936).

[25] E. A. Guggenheim, "On Magnetic and Electrostatic Energy,"

Proc. Roy. Soc. (London) A, 155, 49 (1936).

39.

FORM NBS-IMA (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 73-268

2. Gov't Accession No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Laser Damage in Materials

5. Publication Date Aug US L xy / j

6. Performing Organization Code

NBSIR 73-268

7. AUTHOR(S)

Albert Feldman, Deane Horowitz, and Roy M. Waxier

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/ Work Unit No. 3130440

11. Contract/Grant No.

2016/3D10

12. Sponsoring Organization Name and Address Advanced Research Projects Agency

Arlington, Virginia 22209

13. Type of Report & Period Covered

Semi-Annual Tech.

2-1-73 to 7-31-73

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) Neodymium: glass laser induced damage is observed in lithiiam niobate (LiNbOa) , calcite (CaCOa), potassium dihydrogen phosphate (KDP) , and deuterated potassium

dihydrogen phosphate (KD*P) . The damage at the lowest power levels is caused by

inclusions. At higher power levels, filamentary damage, which is indicative of

self-focusing, is observed in LiNbOs. An analysis of self-focusing data in

yttrivim aluminum garnet shows that the Kerr effect is the dominant self-focusing

mechanism, with some contribution from the thermal effect. Bulk and surface

damage thresholds in neodymixim-doped thoria: yttrium oxide ceramic are obtained

relative to bulk damage thresholds in several optical materials. For solid

materials relationships are obtained between the stress-optic coefficients and

the electrostrictive coefficients under different geometric boundary conditions.

17. KEY WORDS (Alphabetical order, separated by semicolons) Absorption Coefficient; calcite; damage

threshold; deuterated potassium dihydrogen phosphate; electrostriction; electrostricti%

self-focusing; inclusion damage; Kerr effect; laser damage; lithium niobate; nonlinear

indpyn ofnr rpf -rPT-raf-r raft i r»n ;po-t-aggi ;poragg mTnnm riihyrirngpn m nynrngpn phr»c;pha1-<^ phr»c;phai-<^ ; g^l f-foriiQi ng; thermal Self-fCiCUSinr

18. AVAILABILITY STATEMENT thoria : yttrium oxide ceramic;

yttrivmi aluminum garnet.

Cxi UNLIMITED.

for OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

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39

22. Price

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