NONLINEAR OPTICAL RESPONSE OF CYANOBIPHENYL LIQUID ...€¦ · The key characteristics of liquid...

Journal of Nonlinear Optical Physics & MaterialsVol. 9, No. 3 (2000) 365–411c©World Scientific Publishing Company

NONLINEAR OPTICAL RESPONSE OF CYANOBIPHENYL

LIQUID CRYSTALS TO HIGH-POWER,

NANOSECOND LASER RADIATION

SVETLANA G. LUKISHOVA

Liquid Crystal Institute, Kent State University, Kent, OH, 44242, USAAlso Institute of Radioengineering and Electronics of the Russian Academy of Sciences,

11 Mokhovaya, 103907, Moscow, RussiaPresent address: IRE-SL, 172 Works Rd., Honeoye Falls, NY 14472, USA

Received 7 June 2000

Results from investigations are summarized into: (1) transient refractive and absorptive(two-photon) nonlinearities at 0.532 µm by the Z-scan method, and (2) reflective non-linearity in the near-IR, of linearly nonabsorbing cyanobiphenyl liquid crystals undernanosecond laser irradiation.(1) For isotropic liquid crystals at the several-nanosecond time scale and severaltens-micrometers beam-waist-diameter, transient molecular-reorientation and ther-mal/density refractive nonlinearities compete in changing the sign of the total transientrefractive nonlinearity. For the different, given pulse durations, the influence of cou-pled thermal and density effects on nonlinear refraction depends, through buildup time,on the beam-waist diameter. Nonlinear absorption coefficients depend on the incidentintensity. For planar nematic layers, cumulative effects in heating (and in refractivenonlinearity) were observed even at low, 2–10 Hz pulse repetition rate. These results areuseful for optical power limiting applications, and for intensity and beam-quality sensorsof pulsed, high-power lasers.(2) Reflective nonlinearity of chiral-nematic (cholesteric) mirrors near selective reflec-tion conditions for circular polarized light at λ = 1.064 µm was studied both under freespace irradiation and inside a laser resonator. Specially chosen experimental irradiation

conditions make it possible to attribute the observed changing of reflectivity to athermalhelix unwinding by the optical field. The results can find applications in laser-resonatormirrors, Q-switches and soft apertures for beam-profile formation, and also in showingthe limits of use cholesteric optical elements in high-power laser beams.

1. Introduction

Liquid crystals (LC)1,2 have become essential in display technology.3 Much less

well known is the use of LCs in, and the potential for improvement of, coherent

optical devices, such as lasers. For example, linear optical and electrooptical laser

elements4–8 made of LCs (mirrors, filters, retardation plates, modulators (light

valves)) are successfully used for more than 20 years. Discovery in 1980 of a giant

refractive optical nonlinearity 9,10 in LCs and its athermal, ten- to hundredfold

365

366 S. G. Lukishova

enhancement by dissolving a small amount (∼ 0.1%) of dye (chromophore) in the

LC-fluid,11 prompted further studies of the nonlinear optical response in these

materials, see, e.g. Refs. 4, 10 and 12–20. It is safe to say that, to date, no nonlinear

optical LC devices, in particular of χ(3) type, have progressed past the prototype

stage. In no small measure, this is caused by the uncertainties in values, i.e. lack of

reliable values, of nonlinear hyperpolarizabilities found in the literature.

The key characteristics of liquid crystals as a fluid material are the long-range

correlation between anisotropic molecules and the cooperative nature of their inter-

action with different, external fields.1–3 Figure 1 shows schematically the molecular

order of the so-called nematic (a) and isotropic (b) phases of LCs made of “rod-

like” molecules. The nematic phase with largely unidirectionally aligned molecules,

director (see arrows in Fig. 1), exists only within a restricted temperature interval

below the nematic/isotropic-liquid phase transition. While in this phase, the long

axis of the LC molecules can be uniformly aligned both perpendicular (homeotropic)

and/or parallel (planar, see Fig. 1(c)) to the fluid container’s walls, by special sur-

face treatment. Alignment is readily achieved in thin (less than∼ 200 µm thickness)

fluid layers.

(a) (b) (c)

Fig. 1. Nematic LC (a); isotropic LC (b); planar-aligned layer of nematic LC (c).

The attractiveness of LCs as a material includes their relatively low cost, ease

of use, including the possibility of filling various volumes (e.g. optical fibers21 and

porous glasses22), their very low absorption over a wide spectral interval including

the visible and IR,13 high damage threshold to laser radiation,4,5 and large electro-

optical and thermooptical coefficients.12,13 Molecular design provides wide latitude

in modifying structure/propertie relations. At least 80 000 LCs are now synthe-

sized. The present paper considers only simple model systems, cyanobiphenyl LCs3

(widely used in display technology), with a stable room-temperature nematic phase.

In addition to nematic phases, there exists the possibility to create spatially peri-

odic structures (chiral nematic, smectic phases).1–3 Furthermore LCs may serve as

solvent for different chromophores11 and/or dispersed as micrometer-size particles

in a polymeric matrix,23 adding to the variability of properties of possible liquid

crystalline devices.

The purpose of this paper is to update the reader on reliable values for LCs

nonlinear-optical materials parameters for several-nanosecond pulse duration. This

is a timely issue as comparing reported values is fraught by a multitude of potential

pitfalls that were not considered yet in the literature (to my knowledge). For in-

stance, the irradiation geometry and various time constants play a momentous role

Nonlinear Optical Response of Cyanobiphenyl Liquid Crystals 367

in what value, and even sign, the transient nonlinear parameters may take. The

consideration of these problems will be useful for constructing liquid-crystalline

nonlinear optical devices, e.g. optical power limiters8 (using both pure and chro-

mophore-doped LCs12,13) and/or intensity and laser beam quality meters,24 for very

often used commercial green-irradiation Nd:YAG lasers, with several nanosecond

pulse duration and 2–10-Hz pulse repetition rate. As to the switching application,

an athermal reflective nonlinearity in chiral nematic mirrors will be considered in

connection with spatial and temporal profile formation of laser radiation.

The structure of this paper is as follows. In the present section, Sec. 1 (Intro-

duction), a short review of two principal mechanisms of refractive nonlinearities

in nematic LCs (orientational (Sec. 1.1) and/or thermal and density (Sec. 1.2))

is presented next for both the oriented-nematic and isotropic case. In Sec. 1.1,

the electronic refractive nonlinearity will be introduced, characterized much smaller

value for these materials. Mechanisms of nonlinear absorption will be reviewed in

Sec. 1.3. Reflective nonlinearity of chiral nematic mixtures based on helix unwind-

ing will be reported in Sec. 1.4. Section 1.5 contains short preview of problems

considered in present paper. Section 2 describes characterization of materials and

LC-cell fabrication (Sec. 2.1 — nematic and isotropic cells, Sec. 2.2 — chiral ne-

matic (cholesteric) mirrors). Section 3 presents two frequently used experimental

set ups (Z-scan (Sec. 3.1) and for measurements of reflective nonlinearity (Sec. 3.2)).

Section 3.1.1 describes determining the absorptive and refractive nonlinearities from

Z-scan data. The experimental results of author’s Z-scan measurements of transient

nonlinear refraction and nonlinear absorption of cyanobiphenyl LCs under several

nanosecond, 0.532-µm laser irradiation will be considered in Sec. 4 for isotropic LC

layers and in Sec. 5 for planar nematic layers. In Sec. 6, various transverse effects

in laser beam will be presented including cumulative spatial self-phase modulation

elliptical ring formation (Sec. 6.1), “dark” spot formation, asymmetric scattering

and cross-structure formation (Sec. 6.2). Numerical modeling of heat diffusion will

be discussed in Sec. 7. Section 8 contains experimental results on athermal reflective

nonlinearity of cholesteric liquid crystal mirrors under selective-reflection conditions

for free space irradiation (Sec. 8.1) and inside laser resonator (Sec. 8.2). Section 9

concludes the paper and provides an outlook for further possible experiments.

1.1. Orientational Refractive Nonlinearity

This section deals with the difference between molecular orientation in an optical

field (and its contribution to the nonlinear refraction) of LCs,25,26 and of ordinary

organic liquids with dipole molecules which also possess strong orientational non-

linearity, e.g. CS2.

As in ordinary organic liquids, individual (or pseudoindividual) molecular re-

sponse (nuclear reorientation) in LCs has been detected in the picosecond range.27,28

The peculiarity of a liquid crystalline material is the collective and correlated molec-

ular reorientation in the optical field (the so-called optically induced Freedericksz

368 S. G. Lukishova

transition). It is much slower than individual molecular response but contributes

enormously stronger to the nonlinear susceptibility χ(3)(−ω, ω, ω,−ω) and nonlinear

refractive index n2 (see Sec. 3.1.1) than optically driven reorientation by individual

molecules. The giant optical nonlinearity9,10 observed in nematic layers for CW-

laser-operation has nine orders of magnitude higher χ(3)(−ω, ω, ω,−ω) and n2 than

CS2 (n2(CS2) = 1.2.10−11 esu29), but its response time is slow (submilliseconds to

seconds).17

1.1.1. Nematic Layers

Once experimental progress permitted, it was realized that for intense fields,

fast collective and correlated orientational response of nematic molecules is

possible16,17,30,31 even for nanosecond irradiation. The buildup time and the mag-

nitude of the refractive nonlinearity are determined by the incident intensity.30 For

instance, for 6-ns excitation, the orientational-relaxation time of aligned nematic

5CB was observed to be of the order ≤ 1–100-µs.30 In a separate work,31 20-ns

pulses were shown to trigger formation of elliptical diffraction rings (similar to the

CW-optical-Freedericksz-transition9,10,32,33) as the result of orientational spatial

self-phase modulation.

For picosecond irradiation of oriented nematics the collective orientational effects

are weaker. Z-scan measurements for 0.532-µm, 33-ps pulse duration, yield values

for 5CB LC34 of n2par = 0.87n2(CS2) and n2perp = 0.58n2(CS2).

At high laser intensities the nonlinear response to picosecond and nanosecond

laser irradiation can be accompanied by intense hydrodynamic flow which cou-

ples to the reorientational process, the so-called flow-reorientational (alignment)

effects.35,36 This effect is not driven by density or temperature modulations but by

photoelastic stresses.

1.1.2. Isotropic LCs

Isotropic LCs are very attractive because of their excellent transparency even for

beam paths as long as tens of cm, opening an opportunity for use as nonlinear

materials in fibers.21 (Nematic LCs do not scatter light and are transparent only

in oriented layers). First experiments on short-pulse laser irradiation of isotropic

LCs have been reported in the early 1970–80s in connection with the optical Kerr-

effect and steady-state and transient self-focusing studies in organic liquids.37–42

The peculiarity of isotropic LCs (in difference with ordinary organic liquids) is

that its response time can be varied by temperature over a wide range, because its

temperature dependence scales as (T − Tc)−1, where Tc is the nematic/isotropic

phase transition temperature.18,37–44

For isotropic LCs, the value of steady-state n2 was measured to be more than

∼ 100 times in excess of CS2 with a dependence on the temperature buildup time

tbup of the order of several tens to hundreds of nanoseconds.18,37–44 For 6–9-ns pulse


duration, the orientational nonlinear refraction of isotropic LCs is transient41 and

transient n2 is expected to have a smaller value than the steady-state n2.18,37–44 It is

very important to note that the value of steady-state n2 depends on the temperature

as (T − Tc)−1 as does the response time, but the transient value of n2 is nearly

independent of the temperature.41 As an example, the transient values for MBBA

were reported to be of one order of magnitude larger than that of CS2.38,39

For picosecond irradiation of isotropic LCs, values of n2 range from 0.05 to

0.13 n2(CS2) for 1.064-µm, 30-ps pulse duration, for both Schiff base and ester

compounds.45

1.1.3. Electronic Nonlinearity

By comparison with the above mentioned orientational effects, electronic contribu-

tions to n2 that are fast (10−15s) for well-known LCs can be considered negligible.

Experimental value of electronic χ(3) (CARS) were found to be only ∼ 5 times

higher than that for glass (χ(3)(glass) ∼ 10−2 χ(3)(CS2)), for MBBA LC (both in

the nematic and isotropic state).46,47 Measurements of electronic contribution of

naphthyl core compounds6,48 both in the ordered and isotropic state (nearly degen-

erate four-wave mixing) at 1.053 µm showed values of χ(3) ∼ (0.1− 0.8) χ(3) (CS2).

These small electronic contributions are not straightforward to explain, especially as

similar conjugated structures of some polymers49 have thousand times higher value

of electronic n2. There is an opinion,50 that molecules with high hyperpolarizabil-

ities do not possess LC phases or have LC-phase in very narrow and unpractical

temperature region.

1.2. Thermal and Density Refractive Nonlinearity

As in any absorbing medium, thermal mechanism causes refractive nonlinearity

in nematic and isotropic LCs. Thermal nonlinearity coupled with density

changes12,13,21,36 competes with the orientational nonlinearity in changing the sign

of the total refractive nonlinearity. Another mechanism of density changes,

electrostriction,51 with buildup time in the nanosecond range, is comparable with

thermal/density contributions if the medium’s absorption coefficients are low. As

will be seen later (Sec. 1.3), at 0.532 µm a strong heating mechanism exists for the

LC model compounds considered here. That is why electrostriction is neglected

from now on.

Refractive index changes by heating are the sum of two contributions:

∆n = (∂n/∂ρ)T∆ρ+ (∂n/∂T )ρ∆T , (1)

where ρ is the density, T is the temperature. It is very important in the current

context that each term in (1) has its own characteristic turn-on time, and will

contribute differently under transient and steady-state regimes.52,53

370 S. G. Lukishova

1.2.1. Thermal-Density Nonlinearity

For several nanosecond pulse duration and several tens of micrometers beam waist,

the buildup time tac of the thermal-density-nonlinearity (∆n = (∂n/∂ρ)T∆ρ) can

be close to the laser pulse duration to.

tac = ro/Vs , (2)

where ro is the beam radius and Vs is the velocity of sound (Vs (LCs) ∼ 1500 m/s).

For tac > to the thermal-density-nonlinearity will not develop during the pulse.

Experimental values of n2 for this transient regime were reported in Ref. 52 for

absorbing organic liquid. It was found experimentally, that the transient absolute

value of n2 for to = 5 ns diminishes with ro increasing from 9 to 32-µm.

Acoustic-grating generation in LCs have been intensively studied both under

subpico-, pico- and nanosecond excitations.12,13,17,27,54

1.2.2. Thermooptical Effects

The thermooptic coefficient dn/dT in nematics is extraordinarily large, ranking

among the largest of all known materials.16 Not unexpected for these highly aniso-

tropic molecules, it depends on the orientation of the molecules. E.g., dnpar/dT ∼−2.5 · 10−3grad−1 for 5CB in the temperature interval 26–31◦C for incident polar-

ization parallel to the molecular orientation direction.55 For “perpendicular” po-

larization, dnperp/dT is of opposite sign and equals ∼ −1/2dnpar/dT .16 In the

proximity of the phase transition to the isotropic state, the slope of both dnpar/dT

and dnperp/dT steepens.55

For nematic LCs it was well-known that the thermooptical effect experiences

a lengthening of its buildup time (tens–hundreds of nanoseconds), rendering it in-

significant on a several-nanosecond time scale.12,13,36,56 Similar buildup times are

reported in Ref. 52 for absorbing organic fluid. Thermooptical effects for ro = 32 µm

were found to be more significant for pulses longer than ∼ 100 ns.52

Because of the slow buildup time, thermooptical (thermal-lens) spatial self-

phase-modulation rings in beam cross-section were observed only for CW laser

radiation.57–59 LC heating was in this case due to absorbing coatings on cell glass-

substrates, dissolving absorbing dyes in LCs and/or irradiation by high-average

power (∼ 1 W for 100-µm spot-size57) beams.

For pulsed lasers there exist, to my knowledge, no reports on thermal spatial self-

phase modulation rings, even at repetition rate regimes, although in Ref. 60 cumu-

lative effects (thermal memory) at repetition rate higher than 5 Hz for nanosecond

irradiation of planar-aligned nematic LC layers are mentioned.

Also important for the current paper is the thermal diffusion-relaxation time14

τdif ∼ D−1(1/r2o + π2/L2

o)−1 , (3)

where D is the thermal diffusion coefficient, Lo is the thickness of the cell. Coeffi-

cient D of nematic LCs is anisotropic. Typical τdif ∼ 10−2 − 10−6s14 for nematics

layers.


1.3. Nonlinear Absorption Mechanisms and

Laser-induced Damage

Heating of cyanobiphenyl LCs by short-pulse laser radiation in the visible range

that drives photoacoustical and thermooptical effects, is caused by two-photon

absorption,27,61 concurrent or subsequent excited-state absorption,17,27,54 and

the efficient decay of the excited states through radiationless-recombination

channels.17,27 Strong nonlinear absorption was observed in optical power limiting

studies of LCs12,13,20,45,62–65 and Z-scan measurements.34,66–70

Linear absorption coefficients α of cyanobiphenyl LCs are incapable, however,

to produce heating effects at low incident average powers. Table 1 lists data13,71

for linear absorption coefficients α for 5CB and E7 (see Sec. 2).

Table 1. Linear absorption coefficients (α) of 5CB and E7.

Liquid crystal λ = 0.532 µm λ = 1.064 µm

5CB α ∼ 0.02 cm−1,13 α ∼ 0.04 cm−1 71 α ∼ 0.08 cm−1 13,71

E7 α ∼ 0.05 cm−1 13 —

1.3.1. Nonlinear Absorption Coefficients of Nematic and Isotropic LCs

For the nematic phase, the two-photon absorption coefficient has a several-times-

higher value for incident polarization parallel to the molecular dipole direction than

for perpendicular polarization.61 Already first experiments on optical power lim-

iting, acoustic grating generation and Z-scan measurements in LCs provided the

experimental evidence for two-photon absorption dichroism.34,54,63

Absolute values of nonlinear absorption coefficient β at several nanosecond pulse

duration are found to be one–two orders of magnitude higher than under several

tens of picosecond irradiation (see Table 2).17,20,34,54,65–70,72,73 In explanation of

such a difference, there is some speculation in the literature about excited state

absorption that strongly contributes to nonlinear absorption under nanosecond ir-

radiation, and especially long-living-triplet-state absorption that can be significant

over nanosecond time spans. The correct answer is not clear yet, in my view. Spec-

troscopic investigations of diluted solutions of cyanobiphenyl LCs74–76 show that

at 0.532-µm triplet absorption is much weaker than singlet observed, though, for

several-tens of picoseconds excitation.

1.3.2. Laser-induced Damage of LCs

Laser-damage issues are brought up here for two reasons, one obvious and one

potentially biasing measurements discussed so far. Laser damage obviously limits

LC devices’ useful power range and useful life. More important, in liquid crys-

tals it may start out as a transient effect prior to becoming a permanent material

372 S. G. Lukishova

Table 2. Nonlinear absorption coefficients (β) of liquid crystals at λ = 0.532 µm.

LC material Layer Layer β Pulse I Refer.

geometry thickness cm/GW duration GW/cm2

Nanosecondresults

5CB Planar, paral. 25 µm 180–650 7 ns 0.45–1.1 34

5CB Planar, paral. 25 µm 182 6.1 ns 1.17 69

5CB Planar, perp. 25, 120 µm 36 6.5–7 ns — 34, 67

5CB Isotropic 25, 120 µm 114 7 ns — 34

CB15 Isotropic 25, 120 µm 38 7 ns — 34

5CB Isotropic 1–2 mm 7–20 7 ns 0.3–0.9 present

5CB Isotropic 105 µm 89 7 ns 0.8 present

5CB Planar, paral. 105 µm 126 7 ns 0.36 present

CB15 Isotropic 1–2 mm 4.5–30 7 ns 0.2–0.9 139, pres.

CB15 Isotropic 25 µm 138 7 ns 0.75 present

ILC (mixture

of ester Isotropic 5 mm 60 20 ns 0.07–0.7 73

LCs)

Picosecond

results

5CB Planar, paral. 40 µm 0.3+0.5 I 28 ps 4–15 68

5CB Planar, paral. 120 µm 2.26 33 ps 8.33 34, 67

5CB Planar, perp. 120 µm 0.78 33 ps 23.3 34, 67

Schiff base

and ester Isotropic 5 mm 0.6 30–67 ps 1–25 45, 62

LCs

Cyanobi- Isotropic 2 mm 0.2–0.23 1.5 ps, 33 65

phenyl LCs 0.587 µm

ILC Isotropic 5 mm 2.3 66 ps 4–80 72, 73

modification. In this transient form, i.e. as a microscopic bubble77,78 that within

fractions of a second may redissolve back into the bulk, it may constitute one possi-

ble reason for the high nonlinear absorption coefficients measured under nanosecond

irradiation. In such experiments higher fluences (J/cm2) per pulse were used than

under picosecond irradiation. Bubbles scatter incident light77 and disappear on a

much longer time scale than nanoseconds. In Ref. 78 the importance was shown of

purifying “as received” LCs through filtration to remove condensation centers for

bubble formation on the surface of the LC cell under the irradiation by power laser

radiation even in the picosecond time span.

It should be pointed out that high-power laser applications of LCs are not the

principal sector among all LC applications. Commercial vendors of LCs are there-

fore unlikely to provide materials optimized for high-power-laser use. That is why

two types of possible contributors to laser damage of LC materials must be avoided.


They are undissolved extrinsic impurities79 that can be removed using particle

filters and intrinsic impurities (dissolved gaseous components that can be removed

by degassing as well as synthesis byproducts or catalyst traces that can be removed

by chromatography).

Although laser-induced damage in LCs was intensively studied in connection

with optical elements for high-power, thermonuclear-scale lasers,4,5 there exist few

attempts to characterize the damage in the presence of two-photon absorption. I

will describe here the main results of laser damage studies in LCs that seem useful

in the current context.

In Refs. 4 and 5 it was shown that after purification of the “as received” LC

samples the laser-damage threshold for massive material modification (visible burst

of stable bubbles and glass chips in the cell surface and/or a web of scatter lines)

improved sharply. For purified (using 0.2-µm particle filter) and degassed, isotropic

5CB, thresholds for permanent photolysis of ∼ 9.6 J/cm2 and ∼ 4.4 J/cm2 were ob-

served under single and multiple shots respectively (1 ns pulse duration, 1-mm beam

diameter, λ = 1.053 µm). Under similar conditions, transient-bubble “damage” ob-

servable only under 110× darkfield magnification in “as received” LCs occurred

at ∼ 0.76 J/cm2 for isotropic 5CB and ∼ 0.89 J/cm2 for isotropic E7 (single shot

threshold).80

In the presence of two-photon absorption (λ = 0.532 µm) laser-induced damage

for isotropic, “as received” LC was studied at much longer, 20-ns pulse duration.12,13

Damage was characterized as appearance of visible bubbles. The “so-called” damage

threshold strongly increased from ∼ 5 J/cm2 to 80 J/cm2 with diminishing beam

diameter from ∼ 200 µm to 28 µm, a typical indicator for the presence of extrinsic,

localized factors in the material.

More detailed studies of other liquids (mostly water) were reported, e.g. in

Refs. 81–83. For organic liquids79 it was shown that under high-temperature py-

rolysis, caused by the optical breakdown and the following recombination of its

products, free carbon in soot form83,84 can be generated. UV-laser photolysis of

liquid aromatic compounds for production of graphite and polymeric carbon was

reported.85 In the case of LCs, carbon particles can also be created by the spikes

in energy that sporadically take place in short-pulse commercial lasers. These car-

bon particles may accumulate in the LC fluid over many shots and cause damage

to it.

1.4. Nonlinear Reflection of Thin Layers of Chiral Nematics

Near Selective Reflection Conditions

This section will deal with nonlinear optical properties of planar nematic LC layers

with chiral additive in the nematic fluid that posses the so-called selective reflec-

tion. The phenomenon will be discussed of nonlinear changing of reflectivity of

such chiral LC-mirrors by light-field only, both theoretical prediction and some ex-

periments including its application in self-adaptive switching laser-resonator-device.

374 S. G. Lukishova

The experiments were carried out at λ = 1.064 µm with no linear and nonlinear

absorption of the liquid crystalline material.

1.4.1. Helicoidal Structure of Chiral Nematic (Cholesteric) LCs and

Selective Reflection of Light

One of the most important properties of LCs containing chiral molecules (molecules

having structures such that mirror images are not superposable, like right and left

human hands) is a tendency to develop helicoidal structures.86 This chiral nematic

(cholesteric) LC helical structure leads to the important optical property of selective

(Bragg-like) reflection1,87,88 in wavelength and circular polarization. One full 360◦

rotation of the molecular layers is defined as one pitch length, Po, see Fig. 2. The

selective reflection wavelength λo = Ponav,1,87 where the average refractive index

of cholesteric LC, nav = (npar +nperp)/2. Pitch length can be adjusted for selective

reflection in visible and IR region using the proper concentration of chiral additive

in a nematic mixture. For a given mixture composition, it may be also controlled

by temperature or an external fields. Cholesteric mirrors with planar-aligned LC

layer (Fig. 2) were used in lasers since 1978.89–93 Their conventional operational

mechanism as other cholesteric-liquid-crystal optical elements (e.g. circular polariz-

ers, notch filters, soft (apodized) apertures) is based on linear propagation of light

through the medium.4,5,7,8,94,95

Half of pitch

Circular polarized laser radiation

Glass substrates

LC

Spacers

Fig. 2. Schematics of cholesteric LC mirror.

1.4.2. Nonlinear Selective Reflection

The effect of helix-pitch changes induced by an elliptically polarized, intense light

wave was predicted theoretically in 1982.96,97 In Ref. 96, it was calculated in detail

how this effect would manifest itself. At a threshold intensity of ∼ 106 W/cm2,

a drop in selective reflection coefficient would set in, leading to bistability in the

reflected beam. Such a bistability and helix-pitch dilation are different from ther-

mal effects caused by dye-photoabsorption in the cholesteric-LC-solvent. Thermal

nonlinearities in dye-doped cholesteric LC-layers have been investigated by many

authors, e.g. Refs. 98–101, but are of no interest here.


1.4.3. Experimental Results on Athermal Changing the Cholesteric Pitch

by the Light Field only

The challenge in observing helix pitch dilation and its unwinding by optical fields

arises from the larger than E ∼ 104 V/cm electric-field strength requirement on the

optical wave that needs to act on the cholesteric LC for time periods commensurate

with the characteristic unwinding time of the helix th ∼ several milliseconds. There-

fore, neither short-pulse intensities exceeding by nearly an order of magnitude those

estimated needed102 nor rather lower intensities applied CW in Refs. 103–106, suc-

ceeded in triggering nonlinear changes in the reflection coefficient of nonabsorbing

cholesteric LCs.

The first successful experiments revealing small pitch dilation (without signifi-

cant drop in cholesteric LC reflectivity) were carried out in 1990. The pitch change

revealed itself in the form of retro-self-focusing and pinholing (soft aperture) of cir-

cularly polarized, CW, Gaussian beams reflected by the cholesteric LC output cou-

pler of a Nd:YAG laser.103–106 Change in coupler “curvature” allowed the resonator

to operate stably in TEM00 mode without intracavity aperture. The authors ex-

plained their results with an intensity-dependent helix pitch dilation in a spatially

non-uniform Gaussian beam, that caused phase shifts in the reflection along the

beam radius.106

The first athermal drop in cholesteric LC reflectivity in response to strong, circu-

larly polarized laser radiation was simultaneously observed by my group107–111 and

authors of Refs. 112–113. In my group, this effect was ascertained in four cholesteric

LC mirrors with different substrate-surface-treatment methods for planar-alignment

of LC-molecules. A special laser operational mode was used that assured several

short pulses (∼ 100 ns) with E ∼ 2.2–7 times threshold value to act on the medium

over the time interval th (∼ms). A 4.5 kHz pulse repetition rate laser mode was

used to accumulate pitch changes from pulse to pulse up to and including helix

unwinding. Pitch-dilation-induced reflectivity drop was detected both under free-

space-propagation conditions of the driving field as well as inside resonators in which

the cholesteric LC acted as end mirror. Inside the laser resonator, the reflectivity

drop of the cholesteric LC terminated lasing action in pulsed mode. Lasing recov-

ered again when the laser mode was switched to CW-operation (low peak power

density). The observed effect was independent of the average power density and,

in the presumed absence of two-photon absorption (λ = 1.064 µm), this made it

possible to attribute it to an athermal increase in the cholesteric LC pitch in the

optical wave field. Further details of these experiments will be considered in Secs. 2

and 8.

The authors of Refs. 112 and 113 used the property of nematic LCs to achieve

faster orientational response time to high-intensity beams (see Sec. 1.1). At incident,

single-pulse (in difference with my experiments with high-repetition rate pulses)

intensities of ∼ 100 MW/cm2, the cholesteric-LC-mirror reflectivity dropped from

90 to 25% even for single, nanosecond pulses. A cholesteric LC mirror has been

376 S. G. Lukishova

further used in an electrooptically Q-switched laser in place of the conventionally

used combination of dielectric mirror plus polarizer. Pulses of 10-ns duration with

peak intensities up to 1 GW/cm2 have been obtained. It was also shown that the

light-induced reflectivity drop leads to passive cavity-dumping and an improved

transverse beam profile.

1.5. Preview of Present Paper

The results of several sets of measurements of nonlinear refraction, absorption, and

reflection of cyanobiphenyl LCs with different cell thickness and various orienta-

tional order (isotropic, nematic, chiral nematic) will be considered in this paper.

1.5.1. Nonlinear Refraction Measurements at λ = 0.532 µm

• Several-nanosecond time-scale, the Z-scan measurements of transient value of

n2 of isotropic 5CB and CB15 LCs with beam-waist diameter 35–50 µm show

positive value of n2, 2.5–3.7 times in excess of n2(CS2), in difference with

7-µm-diameter measurements of Refs. 34 and 66 where negative-sign values were

reported for isotropic 5CB. This difference is explained by dependence of the value

of transient thermal-density nonlinearity with negative sign on beam-waist diam-

eter (see Sec. 1.2). For 7-µm diameter, tac is less than pulse duration time and

nonlinear refraction as a result of density changes prevails under the positive-sign,

transient, orientational nonlinearity.

• It will be shown that cumulative effects in thermal refractive nonlinearity are

important for planar-aligned nematic LCs even for low, 2–10-Hz pulse-repetition-

rate irradiation which is commonly used in many commercial lasers. As a result,

the response of either an LC optical-power-limiting device or of a Z-scan sample

will differ between the first irradiation and after several seconds of irradiation.

These cumulative effects are also the origin for the spatial self-phase-modulation

transverse effects characterized by a slow buildup time. Z-scan measurements

show dependence of the value of nonlinear refraction on the incident intensity,

e.g. at I = 0.36 GW/cm2, n2 (5CB) ≈ −20n2(CS2). Numerical modeling of

the heat diffusion at 10-Hz pulse-repetition-rate confirms sufficient cumulative

heating of LCs by several degree K at the center of a Gaussian beam. Such local

heating is a plausible mechanism for the observed spatial self-phase modulation

rings.

1.5.2. Nonlinear Absorption Measurements at λ = 0.532 µm

• Z-scan measurements of nonlinear absorption under several nanosecond irra-

diation of 1–2-mm layer-thickness, isotropic 5CB and CB15 show an order of

magnitude smaller values in comparison with thin (25–125-µm thickness) layers.

This layer-thickness dependence points toward a key role surfaces (and possible

“damage” of LC-material near the surface (see Sec. 1.3)) play in measurements.


• Measured magnitudes of nonlinear absorption coefficient β depend on the incident

intensity, e.g. β ∼ 4.5–30 cm/GW at I ∼ 0.2–0.9 GW/cm2 for isotropic 5CB and

CB15. That is a more than an order of magnitude higher value than for picosecond

measurements (see Table 2 and Sec. 1.3).

1.5.3. Nonlinear Selective Reflection at λ = 1.064 µm

More detailed results than in Sec. 1.4 will be presented on characterization and

experiments with four cholesteric LC mirrors for laser resonators. Laser irradiation

conditions will be considered for changing the transmission of these mirrors in high-

power laser beam with possible explanation of the effect by athermal helical pitch

dilation and unwinding by the field of a light wave. This remarkable phenomenon of

changing the chirality by a light field only can be used in laser-resonator devices for

temporal (Q-switching) and spatial (soft aperture) profile formation under certain

laser operational regimes.

2. Liquid Crystal Material and Cell Characterization

For Z-scan measurements the well known cyanobiphenyl LCs, 5CB (K15) 4-cyano-

4′-n-pentylbiphenyl,114 and CB15114 (4′-(2-methylbutyl)-4-biphenylcarbonitrile

were used. These materials have identical chemical composition, but CB15, in

difference to 5CB, contains a chiral carbon (Fig. 3) in its pentyl chain.

The E7 mixture of cyanobiphenyl compounds114,115 (Fig. 4(a)) together with

the chiral additive CB15 was used for cholesteric LC mirror preparation.

Fig. 3. Space-filling model of nematic 5CB and chiral additive CB15.

378 S. G. Lukishova

Fig. 4. Molecular structures of the constituents making up the LC E7 (a) and chromophorebis[di-n-butylamino]stilbene (b).

2.1. Nematic and Isotropic LC Layer Preparation

LCs were supplied by BDH. Degassed 5CB served as sample material having been

filtered through a 5-µm filter in a clean room, prior to filling of the cell under regular

laboratory conditions. Samples of 5CB were kept above the nematic/isotropic phase

transition Tc = 35.3◦C114 under control of an Instec heater control unit. CB15 is

isotropic at room temperature.

The two-photon absorbing chromophore bis[di-n-butylamino]stilbene116

(Fig. 4(b)) was added in some experiments of Sec. 6 (7.4 weight %) to a 1:4 mixture

by weight of 5CB and 7CB (Fig. 4(a)).a

For studies of “thick” samples of LCs in the isotropic state, spectroscopic quartz

cells of 0.5, 1 and 2 mm beam pathlength (NSG Precision Cells, Inc.) were used,

as well as for calibration with CS2 (99.9% HPLC grade, Sigma-Aldrich).

“Thin” cells were homemade from glass plates with size ∼ 2.5 cm × 2.5 cm.

Mylar spacers of different thickness (25, 50, 105 and 125-µm) were used between

the two glass plates. For “thin” cells of LCs in the isotropic state Fisher Finest

Premium microscopic slides of 1-mm thickness were used. For oriented nematic

layers of 5CB, planar alignment (Fig. 1(c)) was carried out at room temperature.

To align LC molecules in layer thickness up to 125-µm uniformly in one direction,

spin-coated, rubbed polyimid glass plates (sodalime or borosilicate) were prepared

with the same dimensions as Fisher slides. Coating and rubbing were carried out

in a clean room, but cell assembly and filling with LCs under ordinary laboratory

conditions. Devcon 5-Minute epoxy was used for sealing the “thin” cells.

aChromophore and 7CB and 5CB mixture composition were provided by Prof. J. Perry (Universityof Arizona).


2.2. Chiral Nematic (Cholesteric) LC Mirrors

The liquid-crystal mirrors (Fig. 5) were prepared by the Moscow Organic Inter-

mediates and Dye Institute NIOPIK from photochemically stable mixtures of the

nematic E7 and the chiral twisting agent CB15 (both from E. Merck) with mix

ratio adjusted for maximum reflectivity around the wavelength λ = 1.064 µm (21.4

wt% CB15 + 78.6 wt% E7).91 Materials were prepared, filtered, analyzed, and

processed in a clean room at the Laboratory for Laser Energetics (University of

Rochester). The average refractive index of the mixture was nav = 1.557. The

results of the spectrophotometric measurements of the transmission coefficient of

each of the three, sealed mirrors as a function of wavelength λ for nearly completely

circular-polarized radiation are presented in Fig. 6.

Planar alignment of the cells was achieved by the following methods. It is very

important that at one inner side of the LC-mirror-cell liquid crystalline molecules

are strongly fixed along one direction on the substrate surface (so-called strong an-

choring). On the opposite inner side of the mirror, LC molecules should not be fixed

(weak anchoring). For all mirrors the side of the glass cell with the strong-anchored

director was polyimide coated and then rubbed. The weak-anchoring side was left

unbuffed but covered with (1) polyimide — mirror 1; (2) polyvinylcinnamate and

Fig. 5. Photograph of cholesteric LC mirrors using either mylar spacers (visible) or spacer spheres(invisible).

Fig. 6. Spectral transmittance of three cholesteric LC mirrors.

380 S. G. Lukishova

subsequent photoorientation — mirror 2; (3) polyvinylcinnamate and subsequent

photoorientation and additional shearing of the cell — mirror 3; (4) polyimide

only — mirror 4. The liquid-crystal layer thickness in the hermetically sealed cells

(1–3) were 12.8, 13.8, and 6.5-µm, respectively, and in unsealed (open contact with

air) cell (4) — 20-µm.

3. Experimental Set Ups

3.1. Z-scan Measurements of Nonlinear Absorption and

Nonlinear Refraction

Among various methods for measuring n2, e.g. Refs. 51 and 117–120, the Z-scan

method29,121–123 is attractive owing to its simplicity. In this method for measuring

nonlinear refraction and absorption (transmission),29,121–123 a sample is scanned

along the optical, Z-axis through the focus of a lens, while the far-field transmission

is recorded as a function of sample position (Fig. 7). The nonlinear absorption

coefficient is derived by collecting light from the whole beam (open aperture Z-

scan). For determination of nonlinear refraction, a “small” aperture is used in front

of the detector, admitting only rays close to the Z-axis (closed aperture Z-scan).

The ratio of closed-aperture measurement values divided by open-aperture values

yields the “Z-scan curve” for determination of nonlinear refraction.

It should be noted that for pulsed, solid-state lasers with beam quality and

divergence far from the diffraction limit, the Z-scan method, simple in principle,

requires great care in ensuring reliable results.122 Pulse-to-pulse fluctuations in laser

radiation (both in energy, pulse duration and/or in beam diameter) tend to make

the Z-scan method, straightforward as it is for CW, power-stabilized laser sources,

a nontrivial task for pulsed, especially commercial nanosecond lasers with beam

quality far from ideal.

The experimental set up used is shown in Fig. 8. A frequency-doubled,

Q-switched (6–9 ns pulse duration at Gaussian FWHM) Nd:YAG laser (Quantel In-

ternational (Continuum) model YG-682S-10), using a feedback-stabilized, injection-

seeded, single-longitudinal-mode, unstable resonator with Gaussian cavity end

mirror124 and two, single-pass amplifiers, provided, at 2, 5 and 10-Hz repetition

rate, an output beam-spot diameter of ∼ 8 mm at 0.532 µm.

To optimize beam quality, a 1.5-mm-diameter Teflon aperture was placed in the

central part of the beam,125 such that an Airy spot at a reasonable distance from the

Teflon aperture (using mirrors M and beamspliter BS) was formed. A converging,

Sample

Detector

Z

ApertureLens

Fig. 7. Schematic of Z-scan measurements.


Fig. 8. Experimental set up for measurements of nonlinear refraction and absorption by the Z-scanmethod.

9.5 or 10.5-cm focal-length lens L was placed at 4 m from the Teflon aperture.

To remove far-field side lobes caused by the Teflon aperture, a 2.75-mm-diameter,

metal aperture MA was placed in front of the lens. The same aperture MA was

placed at the reference channel. For maintaining a linear incident polarization and

changing the input-pulse energy to the sample, a Glan-prism-polarizer and a quartz

λ/2 waveplate combination was placed in the beam.

Signal D1 and reference D2 pyroelectric joulemeters (Gentec ED-100A with

amplifier EDX-1, vendor recalibrated) or fast photodetectors (Hamamatsu S1722-

01 and Motorola MRD510, respectively) were connected to a digitizing oscilloscope

(Textronix TDS 640). An AT-GPIB/TNT board from National Instruments was

used for data acquisition and processing. Laser pulse shape was monitored by a

sub-ns-risetime Motorola MRD510 photodiode, connected to a Hewlett Packard

HP 54111D digitizing oscilloscope. For closed-aperture Z-scan measurements, a

“small” aperture was placed in front of D1. Sometimes the entire photodetector

was used as the “aperture”. For open-aperture measurements, an additional lens in

front of D1 was used to collect all laser radiation leaving the sample. The samples

were placed on an Aerotech mechanical translation stage controlled by an Unidex

1 motion controller. To measure the average126 beam diameter near the waist

produced by the focusing lens of the Z-scan set up, an 8-bit, WinCam CCD-beam

analyzer by Merchantek (pixel size 8.3 µm × 8.6 µm) was used in a separate, low-

intensity, equivalent channel formed with beamspliter BS and additional lens with

the same focal length as the lens in the Z-scan channel. Beam-waist diameter was

also measured with knife-edge technique and using Z-scan of CS2 with well-known

nonlinear refractive index n2.

The same CCD-camera was also employed for acquisition and analysis of beam

profile on the lens and far-field patterns during the Z-scan. For adjusting the size

of the pattern to the size of the sensor, an additional lens was used between the

sample and the camera. Some far-field patterns were recorded from a screen by

382 S. G. Lukishova

Fig. 9. Beam profile at the lens along X-axis (a) and Y -axis (b); beam diameter in the waistregion (c): • — for X-axis, � — for Y -axis; and pulse shape (d).

a standard video camera/digitizer combination, permitting the recording of each

pulse.

Figure 9 shows near Gaussian spatial distribution of the laser-beam intensity at

the lens in the Z-scan set up (a), (b), typical beam caustic near the waist (c) and

near Gaussian temporal pulse shape (d). Figure 9(d) is a result of storage of ∼ 100

shots.

It must be noted next that both sensitivity and accuracy of the absolute Z-scan

measurements critically depend on the incident beam profile and laser beam quality.

In Ref. 127, numerical modeling showed that, even with a slight deviation from a

perfect Gaussian beam distribution, the Z-scan curve calculation using the standard

formula for the Gaussian beams can yield erroneous, nonlinear optical parameters.

Influence of laser beam quality was studied numerically,122,128 where even for beam

quality factor M2 > 1.2, errors were found in the absolute Z-scan measurements

(M2 (see Ref. 129) expresses in terms of beam divergence as M2 = θ/θ00, where θ

and θ00 are the real beam divergence and TEM00 mode divergence). In difference

to absolute measurements, in relative measurements by Z-scan method even low-

quality beams are found to be useful.130


In order to remove errors connected with deviation of the spatial profile from

the Gaussian shape,127 relative Z-scan measurements are presented here through

the use of CS2 as a calibration material before and after each measurements of an

LC-sample.

For every translation-stage step (1 mm) 30 irradiation pulses were accumulated,

and, after 5% “windowing”,122 these data were averaged. The accuracy of measure-

ments of n2 and β is estimated to be ∼ ±20% dominated by the incident-intensity

fluctuations.

3.1.1. Evaluation of n2 and β from the Z-scan Data

In light of different definitions of n2 in the literature,51,117,118,131 the value of the

nonlinear refractive index n2 and the nonlinear refractive coefficient γ are defined

in this paper by n = no + 1/2n2|E|2 and n = no + γI. Here n is the total refractive

index of the material, and no is the linear refractive index, E is the (complex) optical

electric field amplitude inside the material, I is the optical intensity. Conversion

between n2 and γ is made via n2[esu] = (cno/40π)[m2/W ], where c is the light

speed (m/s) in vacuum.

For isotropic materials and a linearly polarized laser beam, n2 and the nonlinear

susceptibility χ(3)(−ω, ω, ω,−ω)49–51,117,118,131–134 are related simply by51

n2 = (12π/no)χ(3)1111(−ω, ω, ω,−ω) , (4)

where χ(3)1111 is one of three nonzero elements of χ(3).

For the Gaussian profiles the values of γ and n2 can be calculated from the

Z-scan curves using the following formula29

∆T tr = 0.406|∆Φ|(1− S0.25) , (5)

where ∆T tr is the difference between the maximum value of the normalized trans-

mittance T tr and its minimum value, ∆Φ is the nonlinear phase shift on axis during

Z-scan, S is the linear transmittance of the “small” aperture for a Gaussian input

beam (S = 1 − exp(−2r2a/R

2o)), where ra is the “small” aperture radius, Ro is the

beam radius at the place of “small” aperture in the linear regime).

The nonlinear phase shift ∆Φ = 2πλ−1γIL, where I is the peak axial intensity

at the waist, assuming a temporally and spatially Gaussian-shaped pulse, L is the

sample length.

In this paper intensity is defined as

I(r, t) = 2E(τπ3/2r2o)−1 exp(−2r2/r2

o − t2/τ2) . (6)

Here E is the total energy, the pulse-length full width at half maximum (FWHM)

to = 2τ(ln 2)1/2, ro is the beam waist-radius at 1/e2 level. Using these pulse and

beam parameters,

Ipeak = 4E(ln 2)1/2(toπ3/2r2

o)−1 ≈ 2E/(toπr2o) . (6′)

384 S. G. Lukishova

As to nonlinear absorption, here an assumption is made that two-photon absorp-

tion135 is the principal loss mechanism. The fundamental equation describing this

intensity loss with depth z is

dI/dz = −βI2 . (7)

The solution to (7) at z = L is

I(L) = I(0)/(1 + βI(0)L) = I(0)/(1 +Q) . (7′)

Sample transmittance, T tr(L) = (1 + Q)−1 = 1 −∆T trabs, where ∆T tr

abs is obtained

from the open-aperture Z-scan curve. For a Gaussian spatial and temporal profile

T tr(L) = 2Q−1π−1/2

∫ ∞0

ln(1 +Qe−x2)dx . (8)

Expansion of the logarithmic function yields a linear relationship (valid for small

Q) between T tr(L) and I(0) where the slope is proportional to the nonlinear co-

efficient β:

T tr(L) ≈ 1 + 2−3/2βI(0)L , and ∆T trabs ≈ −23/2βI(0)L . (8′)

3.2. Measurements of Nonlinear Transmittance of

Cholesteric Mirrors

Nonlinear irradiation of cholesteric LC mirrors was carried out by two approaches:

cholesteric LC samples were either irradiated in the focus of a freely propagating

laser beam (for the experimental layout see Fig. 10) or inside a laser resonator

(Fig. 11) in which the cholesteric LC structure served as output coupler.

Nd:YAG laser

Quartz polarizer

Photodiode

λ/4

Glass filters

Lens

CLC mirror

Power meter

Fig. 10. Experimental set up for free-space experiments with cholesteric LC mirrors.

Fig. 11. Schematic of the cholesteric-LC-mirror-equipped laser oscillator.


A Nd:YAG laser operated in two-regimes: (1) in CW mode and (2) in an acousto-

optically Q-switched mode, emitting high-repetition-rate (4.5 kHz) pulse trains,

each pulse being 500-ns long. In either mode, the average power reaching the

cholesteric LC samples was between 0.3 and 1 W.

The incident and transmitted Nd:YAG powers were detected by a calibrated

photodiode and a power meter. Pulse duration and repetition rate were measured

by an avalanche photodiode and a photomultiplier. To increase the incident power

density, a lens with a focal length of about 5-cm was placed in front of the cholesteric

LC mirror in free-space experiments.

Beam cross sections were monitored by a CID 2220A video camera (CID Tech-

nologies Inc.) and video processing set up. From the recorded beam profiles,

1/e2 intensity beam-waist diameters of between 50 and 220-µm were derived for

different sets of measurements. Peak-power densities (for 500-ns pulses) of ∼106 − 107 W/cm2 were used in pulse-repetition rate mode.

For the laser-resonator studies, a plane-parallel cavity was formed by a dielectric

nontransmitting mirror and a cholesteric-LC output coupler (Fig. 11). An active

Nd:YAG element was pumped by a continuously operating arc flashlamp filled with

krypton (the voltage across this lamp was 100 V). Pumping of the active element

generated a thermal lens whose optical power was +1 dioptre kW−1. Switching to

high pulse-repetition rate operation was performed by an acousto-optical switch.

A quartz λ/4 plate was placed inside the cavity between the cholesteric LC

mirror and the active element: in the absence of this plate there was no lasing.

The diameter of the optical beam at the laser output was ∼ 0.8 mm at an average

output power of 1 W.

4. Results of the Z-scan Measurements of Isotropic Liquid Crystals

4.1. Nonlinear Refraction

Results from the Z-scan measurements of 2-mm 5CB samples136 in the isotropic

state at 45◦C for I = 0.66 GW/cm2 are presented in Fig. 12(a)–(c). For comparison,

Z-scan curves for CS2 with 1 and 2-mm cell thickness are shown also for the same ex-

perimental conditions (Fig. 12(d)) for I1 = 1.4 GW/cm2 (•) and I2 = 1.2 GW/cm2

(∆). From data in Fig. 12(c) we derive γ(5CB) = 3.7 γ(CS2) and n2(5CB) = 3.7

n2(CS2). Beam-waist diameter is ∼ 35 µm in these measurements, and to = 7 ns.

Over the temperature interval 35.6–50◦C, the value of n2 (5CB) changed little:

it varied from 2.5 to 3.7 n2 (CS2).136,137

Typical Z-scan curves for 2-mm, CB15 cells136,137 at room temperature are

presented in Fig. 13(a)–(c) for I = 0.74 GW/cm2. For comparison, a Z-scan curve

for CS2 with 2-mm thickness is presented as well (Fig. 13(d)) for I = 0.73 GW/cm2.

The results from these curves yield the following values: γ(CB15) = 2.6 γ(CS2) and

n2(CB15) = 2.5n2(CS2). Beam-waist diameter is ∼ 36 µm, and to = 6.8 ns.

In both cases, isotropic 5CB and CB15, self-focusing (converging lens) was ob-

served. Aperture factor was S ∼ 0.1 in these measurements.

386 S. G. Lukishova

0.50.60.70.80.9

11.1

-10 -5 0 5 10Z, mm

Nor

mal

ized

tran

smitt

ance

(a)

0.50.60.70.80.9

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(b)

0.60.70.80.9

11.11.21.31.41.51.6

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tran

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0.9

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1.1

1.2

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Fig. 12. Z-scan curves for isotropic CB15: open aperture (a); closed aperture (b); closed/openaperture (c); and closed aperture for CS2 (d).

Fig. 13. Z-scan curves for isotropic 5CB: open aperture (a); closed aperture (b); closed/openaperture (c); and closed aperture for CS2 (d).


4.2. Nonlinear Absorption

The main results of β-measurements are as follows:

(1) a remarkably large value of β is found for “thin” layers (e.g. for isotropic CB15,

β ∼ 100–150 cm/GW for 25–125-µm cells at I ∼ 0.7–0.8 GW/cm2) in compar-

ison with that from “thick” samples (β ∼ 4.5–30 cm/GW for 1–2-mm cells of

isotropic CB15 and 5CB at 45◦C at I ∼ 0.2–0.9 GW/cm2). Typical curves for

β versus layer thickness136 for isotropic CB15 are presented in Fig. 14(a) at the

same incident intensity I ∼ 0.75–0.78 GW/cm2.

(2) a dependence of β (cm/GW) on incident intensity (Fig. 14(b)) for isotropic 5CB

at 45◦C, “thick” samples.

(3) for isotropic 5CB near the nematic/isotropic phase transition, β values vary by

up to a factor of ∼ 6. Five degrees above the phase transition up to 50◦C, β

does not vary beyond experimental error.

020406080

100120140160

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Layer thickness, mm

, cm

/GW

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

Intensity, GW/cm2

, cm

/GW

(a)

(b)Fig. 14. Dependence of nonlinear absorption coefficient on layer thickness (a) and the incidentintensity (b).

An explanation of the dependence of β on layer thickness may not rest on the

argument of a Rayleigh length shorter than the sample pathlength. Measurements

of the beam caustic near focus (Fig. 9(c)) show that both “thin” and “thick” cells

388 S. G. Lukishova

used in this experiment were shorter than the Rayleigh length. In addition, for a

sample larger than the Rayleigh length, an open aperture Z-scan curve must be

flat-bottomed.122

5. Results of the Z-scan Measurements of Thin Planar

Nematic Layers

To take into account the well-known nonlinear-absorption anisotropy in the nematic

phase, i.e. nonlinear absorption being larger for incident polarization parallel to the

nematic director and weaker for perpendicular polarization,61 orientation-dependent

measurements were carried out.136–138 Figure 15 shows the results from open (a)

and closed (b) aperture Z-scans for a 5CB layer of 105-µm thickness, irradiated by

a parallel-polarization (I = 0.36 GW/cm2) pulse. Results for the ratio of closed-

aperture data by open-aperture data are presented in Fig. 15(c). For comparison,

0.80.85

0.90.95

11.05

1.1

-5-4 -3-2 -1 0 1 2 3 4

Z, mm

Nor

mal

ized

tran

smitt

ance

(a)

0.80.85

0.90.95

11.05

-6-5-4-3-2-1 0 1 2 3 4

Z, mm

Nor

mal

ized

tran

smitt

ance

(b)

0.940.960.98

11.021.041.06

-6-5-4-3-2-1 0 1 2 3 4Z, mmN

orm

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ed tr

ansm

ittan

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orm

aliz

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ansm

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(d)

LC-dipolesE

k

Fig. 15. Z-scan curves for planar nematic 5CB: open aperture (a); closed aperture (b); closed/openaperture (c); and closed aperture for CS2 (d).


a closed-aperture Z-scan for CS2 is shown in Fig. 15(d) (I = 1.2 GW/cm2). In all

cases of Fig. 15, S = 0.5.

The results of calculations are as follows: γ = −18.5 γ(CS2). Evaluation of

β yields a value of ∼ 126 cm/GW. The large values of γ (and n2) by themselves

are remarkable, but so is the sign of the nonlinear refraction coefficient (negative).

These data are in good agreement with those of Refs. 34, 66, 67 and 70 where also

a negative nonlinear refraction coefficient was observed for an oriented thin layer of

5CB and parallel-polarized laser light.

The magnitude of β obtained in this measurement for ∼ 105-µm cell probably

cannot be considered as a “true” two-photon absorption coefficient (see Sec. 1.3 and

Sec. 4.2 (Fig. 14(a)).

6. Cumulative Spatial Self-phase Modulation Elliptical

Ring Formation in the Beam and Other Transverse Effects

The LC molecules with their tendency to align with the electric-field direction of

an applied optical wave do in general not experience torque when the LC molecular

dipole is parallel to the linear polarization (nematic cell surface is perpendicular to

the beam propagation direction) of the incident pulse (see Fig. 15, drawing at the

bottom).

Contrary to this expectation, during periodic illumination of planar-nematic

cells in such geometry over time spans of ∼ 0.5–ten seconds the linearly polarized

beam is shown here to experience an imprint of a far-field pattern of high-contrast,

concentric, elliptical diffraction rings whose major axis orients itself perpendicular

to the incident polarization direction.138,139 Once formed, each set of rings may

remain stable for up to several minutes. Upon continued pulsed irradiation over

several minutes, the number of these rings varies systematically with laser intensity

(sometimes between 1–2 and (up to) 20 rings). This ring pattern differs from that

typically obtained from orientational spatial self-phase modulation. In our case,

the ring pattern evolves under polarization conditions for which, in principle, no

orientational effects may occur (polarization of laser light is parallel to the director

orientation). Under CW-irradiation at twice the power density than the average

power density under pulsed illumination, the effect fails to exist.

The key characteristics of the effect are as follows:

• This effect and its development exhibit threshold behavior depending on both the

incident peak intensity and the geometry of irradiation, but not on the average

power density.

• The effect’s evolution depends on the irradiation geometry (focal beam diameter)

and, for a given geometry, on the cell thickness.

• The effect exists at both 2-Hz and 10-Hz pulse repetition rate, however, in the

2-Hz case the threshold increases (cumulative effect).

• Adding the chromophore does not change the character of the effect but only low-

ers its threshold by ∼ 2–2.5 times (for the tested dopant type and concentration).

390 S. G. Lukishova

• When the LC cell was rotated around the light-propagation direction, the ellipti-

cal rings’ major and minor axes rotated in the same handedness, but the number

of rings diminished as the angle between LC director and incident linear polar-

ization increased. However, when the LC director was oriented perpendicular to

the light polarization, the effect vanished entirely at incident intensities below

∼ 1 GW/cm2.

6.1. Large-beam Diameter (Slow Focus)

At 160-µm beam diameter, cells of thickness 25–50-µm did not permit evolution of

the effect (only chaotic break-up of the beam was observed), while larger thickness

(105–125-µm) cells did. The elliptical ring pattern developed from the original beam

spot through several “rays” of scattered light (Fig. 16). Images in Fig. 16 (105-µm

layer thickness of 5CB and 7CB mixture with chromophore, I = 0.5 GW/cm2) were

obtained by recording from the screen elliptical ring patterns onto magnetic tape

at full repetition rate, and digitizing each frame afterwards.

Figure 17 shows the evolution of the ring-image with diminishing incident in-

tensity for layer thickness 125-µm. As well as in Fig. 16, for lowering the effect

Fig. 16. Slow focus irradiation: time evolution of the elliptical ring pattern in the far-field.

Fig. 17. Slow focus irradiation: set of elliptical ring patterns with diminishing incident intensityfrom (a) to (g).


threshold doping of the mixture of 5CB and 7CB LCs with the two-photon ab-

sorbing chromophore (see Sec. 2.1 and Fig. 4(b)) was used. The same patterns are

observed in pure 5CB (or mixture), but at incident intensity levels twice larger than

used here.

6.2. Small-beam Diameter (Tight Focusing).

Asymmetric Scattering, “Dark” Spot and

Cross-structure Formation

For tight focusing (35–50-µm beam-waist diameter), the effect of elliptical ring for-

mation was observed for all used cell-thicknesses (25–30, 50, 105, 125-µm). In all

tight-focusing experiments no chromophores were used. In this case, the evolution

of the far-field pattern from the initial beam spot into the elliptical rings passed

through the appearance of what is referred to here as “dark ” spot (Fig. 18) or

wide dark ring (“donut” shape) around the initial beam spot and a stage of asym-

metric scattering (Fig. 19).138,139 Figure 18 (for 5CB sample with 125-µm layer

thickness) shows changes of the far-field pattern with increasing intensity from

I < 0.3 GW/cm2 (high-contrast, narrow-diffraction-ring on beam diameter is a

result of spatial self-phase modulation) to I ∼ 0.6 GW/cm2 (“dark spot” instead of

a bright beam, “negative” of the original beam spot). This “dark” spot effect was

observed only for small-beam diameter (tight focusing). It can be explained by the

additional widening of the far-field diffraction angle in the self-defocusing medium

in the divergent beam. It should be noted that the same “dark ” spot was observed

not only under “parallel” polarization, but under “perpendicular” as well and even

for the LC in the isotropic state. In the latter two cases, the threshold of the effect

doubled over that for “parallel” polarization and neither asymmetric scattering nor

elliptical rings were observed. The “dark”-spot effect was also reported in Ref. 140

for thermal self-defocusing of CW-radiation in an absorbing isotropic liquid. Theo-

retical analysis140 has shown that the “dark”-spot formation is a common feature of

a divergent Gaussian beam transmitted through an arbitrary defocusing medium.

The scattering asymmetry (Fig. 19) follows a simple rule: the scattered intensity

is always highest in the direction orthogonal to the incident polarization direction.

The buildup time of such scattering near its threshold was measured (using magnetic

tape recording and digitizing each frame afterward) of the order of ∼ 0.3–1 second.

Figure 19 (5CB sample with 30-µm layer thickness, I = 0.43 GW/cm2) shows

(a) (b)

Fig. 18. Tight focusing: far-field patterns at low incident intensity (a); “dark spot” (b).

392 S. G. Lukishova

Fig. 19. Tight focusing: far-field asymmetric scattering CCD-image (a); 3D-rendering of spatialbeam profile (b); X-axis spatial intensity distribution (c); Y -axis spatial intensity distribution (d).

Fig. 20. Tight focusing: elliptical rings in the far-field with increasing incident intensity from leftto right.

details of asymmetric scattering and the “donut ”-shape “dark ” spot at the beam

initial spot. The central bright spot and the “donut” black ring are circles. At the

1/e2 level, the asymmetric scattering spot diameter at the X-direction (parallel to

the incident polarization) is ∼ 1.5 times larger than that in the Y -direction.

At higher intensity (I ∼ 0.5 GW/cm2), asymmetric scattering becomes stronger,

and the first ring appears (Fig. 20, left for 25-µm pathlength of 5CB). At still

higher intensities (∼ 0.7 GW/cm2), asymmetric scattering evolves into the multiple-

elliptical ring pattern whose major axis is oriented perpendicular to the incident po-

larization (Fig. 20, right for 105-µm pathlength of 5CB). Under certain conditions,

the divergence angle of the elliptical fringes abruptly increases, exceeding the spa-

tial capture range of the CCD-camera’s detector. This necessitates capturing the

fringe pattern on a white paper screen, and imaging the screen on to the detector

by a conventional objective. Figure 20, right is the result of such imaging. For scale

comparison, the absolute size of the initial beam spot observed in Fig. 20 left and

right does not change throughout the evolution of the pattern.


Fig. 21. Tight focusing: far-field elliptical ring pattern (left) and result of 45◦ rotation (right).

Fig. 22. Different types of the cross-structure in the far-field.

When the LC cell was rotated around the light-propagation direction, the ellip-

tical rings’ major and minor axes rotated in the same handedness, but the number

of rings diminished as the angle between LC director and incident, linear polar-

ization increased (Fig. 21, 50-µm cell thickness: left I = 0.77 GW/cm2, right

I = 0.88 GW/cm2). By the time the LC director is oriented fully perpendicular to

the light polarization, the effect vanishes entirely.

At the highest employed intensity (I ∼ 0.8–1 GW/cm2 for tight focusing), the

elliptical ring pattern may develop into a cross structure139 for both tight focusing

and slow focus. Figure 22 shows different versions of such cross structures (left —

for tight focusing (5CB, 25-µm layer thickness), right — for slow focus (5CB and

7CB mixture with chromophore, 125-µm layer thickness)), depending on specific

laser-pulse or irradiation-geometry parameters. Each specific form is reproducible

for a defined laser operational mode. Equally important, the polarization state

of the far-field laser light depicted in Fig. 22 remains the same as in the incident

beam, in difference with cross patterns resulting from the sample being held between

crossed-polarizers, reported, e.g. in Ref. 141. Under CW-excitation of equivalent

power, we did not observe any of the above described effects.

7. Numerical Modeling of Cumulative Effects and Explanation

of Spatial Self-phase Modulation Effects

For an explanation of the slow buildup time spatial self-phase modulation as well

as of its absence under CW irradiation of comparable power density, the

394 S. G. Lukishova

heating of LC was further scrutinized. In a pure LC-material, the major contrib-

utor to heating is radiationless decay from excited states addressed by two-photon

absorption and excited-state absorption17,27 (see Sec. 1.3). Assuming that the glass

substrates do not absorb at all, numerical solutions have been sought for the heat

balance equation. The finite-element solutions were obtained using the commercial

ANSYS/Thermal code),b including thermal conduction in the LC-material and the

two glass-walls of the LC-cells, convection from the glass walls to air, and enthalpy

effects at the phase-transition temperature. Two-dimensional transient thermal

analysis was carried out for the initial condition of a Gaussian spatial temperature

distribution written by the laser pulse into the material. Temperature-dependent

material parameters for 5CB (thermal conductivity, specific heat) were provided in

file form by the authors of Ref. 142.

For lack of literature data on radiationless energy coupling efficiencies from two-

photon and excited-state absorption, here the assumption is made of a Gaussian

temperature profile equivalent to the in-focus laser-beam profile (1/e2 diameter =

160 µm) with maximum temperature Tmax. Other device parameters used in

the simulation were: LC layer thickness L = 125 µm, glass substrate thickness

Lglass = 375 µm, room temperature Troom = 26◦C. In order to comply with the

experimental 10-Hz repetition rate, the decay evolution of the initial temperature

profile was followed out to 0.1 seconds, with the smallest time step set at 10−6

seconds. Values for Tmax = ∆Tmax + Troom were chosen such that throughout a

sequence of equal pulses, Tmax at any node did not exceed the temperature limit

set by the available material-parameter142 lookup table (44.9◦C maximum). This

Fig. 23. Numerical modeling of the heat diffusion: center-node temperature relaxation in timeafter first pulse.

bNumerical modeling was made by Dr. A. Schmid (University of Rochester, Laboratory for LaserEnergetics) using code of ANSYS, Inc., PA.


∆∆∆∆Tmax = 10oC ∆∆∆∆Tmax = 5oC

Short number

Cen

ter-

node

tem

pera

ture

, oC

Fig. 24. Numerical modeling of the heat diffusion: center-node temperature versus the numberof pulses.

constraint is strictly methodological and is not to imply that in an actual experiment

the temperature did not exceed, or could not have exceeded, the values chosen here.

Figure 23 shows the temperature relaxation with time after the first pulse at the

central node of the 5CB-cell (∆Tmax = 5◦C). During the first 10 milliseconds the

temperature relaxes fast, followed by slower relaxation thereafter. Even after 100

milliseconds the temperature remains ∆T ∼ 0.15◦C above room temperature.

Numerical calculations of heating as cumulative action of many, 10-Hz repetition-

rate pulses yield results presented in Fig. 24, ∆Tmax = 10◦C (solid curve) and

∆Tmax = 5◦C (dotted curve). After 60 pulses (∼ 6 s of irradiation) the tempera-

ture in the center of the LC-cell increased by ∆T = 5◦C for ∆Tmax = 10◦C, and

∆T ∼ 4.5◦C for ∆Tmax = 5◦C. The slopes of the curves increase faster for the first

15–20 pulses, and for ∆Tmax = 5◦C, ∆T is close to the temperature saturation

value at the end of the sequence.

Evaluation of the phase shift caused in a laser beam by this temperature increase

is equal to ∆Ψ = 2π∆nL/λ, where ∆n = ∆Tdn/dT . The number of rings as a

result of self-phase modulation in space on the transverse profile of a beam134 is

equal to N = ∆Ψ/2π = ∆TL(dn/dT )/λ. For 5CB in the region from 26◦C to

31◦C dn/dT ∼ 2.5 · 10−3 grad−1. For λ = 0.532 µm, L = 125 µm, and ∆T =

5◦C, N = 3. A temperature increase after single pulse irradiation (∆T ∼ 0.15◦C)

results in no diffraction rings in the beam. Only the cumulative action of many

pulses approaching a steady-state value yields a stable spatial self-phase modulation

elliptical ring picture as displayed in Figs. 16 and 17. Given the approximations

in the model, this apparent agreement between simulation and experiment cannot

serve as a rigorous proof of the cumulative heating mechanism. Instead, it provides

substantial plausibility for such a mechanism to be active in this situation.

396 S. G. Lukishova

The observation of the effect only for incident polarization parallel to the LC-

director is explained by two-photon absorption dichroism,61 and more effective LC

heating by “parallel” — polarization light, and approximately twice larger absolute

value of dn/dT for “parallel” polarization in comparison with “perpendicular”.

The scattering asymmetry and ellipticity of the ring pattern can be explained by

the anisotropy of the thermal diffusion constant, which has for 5CB a ∼ 1.6 times

larger value along the long molecular axis than perpendicular to it.142,143

The nature of the cross-structure is not yet clear. Stresses similar to those con-

sidered in Ref. 144 can create movement of LC fluid in two perpendicular directions

and destruction of the planar structure in these directions.

8. Changes of Reflectivity of Cholesteric Mirror by Light Field

Near Selective Reflection Conditions

8.1. Free-space Results

An increase in cholesteric LC-mirror transmittance was observed in free space,

i.e. outside the cavity, when laser radiation with 500-ns pulse duration and a repe-

tition rate of 4.5 kHz was focused into the cholesteric LC layer. The mirror trans-

mittance increased from 5% up to 30–80% for mirrors 1–3 and from 15% up to

25–50% for mirror 4. The intensity drop appeared 30 s–10 min after irradiation

onset (Fig. 25 for two mirrors (a) and (b)), depending on the peak power density

of the incident radiation, which was different for each mirror.

It is interesting, that the effect did not depend on the average power density. Just

as in Refs. 103–104, in the CW lasing mode no reflectance drop was observed, even

for a power density twice as high as the high-repetition rate average pulsed mode.

Moreover, even in the high-repetition-rate mode the bleaching effect was recorded

only when the reflection occurred from the side of the cholesteric LC mirror with

the strong anchoring. No bleaching was observed in the case of reflection from the

weak-anchoring side (flipping of the cell by 180◦).

It should first be noted that a high-quality beam reflection was obtained only

from the strong-surface-anchoring side of the cholesteric LC mirror, while a beam

reflected from the other, weak-anchoring side, had poor quality (e.g. amplitude

inhomogeneities were observed in beam cross-section).107

According to measurements, the temperature shift of the selective-reflection

curve of the experimental cholesteric LC mixture is approximately +1 nm/◦C, i.e. to

obtain a transmittance of ∼ 70% for CLC mirrors at λ = 1.064 µm the prepared

cholesteric LC mirrors must be heated to a temperature above 50◦C. Monitoring the

cell temperature by an attached thermocouple, no increase in the cell temperature

was recorded.

In the case of weak bleaching, it was observed that the transmittance of the

cholesteric LC mirror returned to its previous state after the laser was either

switched off or switched to the CW mode (Fig. 25(b)).


(a)

(b)

Time interval , minTr

ansm

ittan

ce, %

Time interval, min

Tran

smitt

ance

, %

Fig. 25. Dependence of two cholesteric LC mirrors ((a) and (b)) transmittance on irradiation timeunder high repetition rate, pulsed irradiation. For mirror (b) lasing action intentionally terminated(*) and restarted again (**).

8.2. Cholesteric LC Mirror in an Nd:YAG Laser Cavity

No significant difference was observed between the slope efficiency of the laser with

dielectric or cholesteric LC output mirrors when the reflection coefficients were the

same. The advantage of the cholesteric LC mirror was a lower sensitivity of the

laser operation to cavity misalignment. At low pumping rate and in the pulsed

high-repetition-rate regime, the slope efficiency and the lasing stability were the

same for cholesteric LC and dielectric mirrors. At high pumping rates, the average

output power obtained in the high-repetition-rate decayed rapidly and disappeared

after 0.5–5 min from switching, but only when the side of the cholesteric LC mirror

with strong surface anchoring was facing the gain medium. The pump threshold

(i.e. the current to the flashlamp) at which lasing was suppressed was a function of

the mirror characteristics (method of their fabrication): it was 28.5–29.5 A for the

output mirrors 1–3 with R = 95% and ∼ 24.5 A for the mirror 4 with R = 85%.

398 S. G. Lukishova

Time interval, min

Out

put P

ower

, W

t1 t2t1 t2 t2t1

Fig. 26. Time dependence of the average output power of a laser with cholesteric mirror withchanging the regime from CW to high-repetition rate (t1) and back to CW (t2). Solid and dashedcurves present results for different mirrors.

Figure 26 depicts the time dependence of the laser output power obtained for

two cholesteric LC mirrors differing in the method of their fabrication (different

degrees of surface anchoring) when a change from the CW to the pulsed regime

occurred above the pump threshold. Turning on the acousto-optical switch (at t1)

quenched the lasing action because of a drop in the reflection coefficient of the

cholesteric LC mirror. Turning off the switch (at t2) gradually restored the lasing

action. A second change to the pulsed regime suppressed lasing again and it was

restored when the laser was made to operate in the CW regime again. Adjustment

of the polarization of the radiation by rotating the λ/4 wave plate in the pulsed

regime sometimes increased the output power only slightly.

The observed reflectivity drop of the cholesteric LC mirror can be explained on

the basis of a model in which the pitch of the cholesteric helix increases in the electric

field of light pulses with high power density (intensity) ∼ (1 − 10) × 106 W/cm2.

Indeed, according to the calculations in Ref. 96, for a “thick” cholesteric LC layer

of thickness L > 2λ/(π∆n) the critical field for unwinding the cholesteric helix by

the light field is determined by the expression

|E|2cr = 4π(ω/c)2(εa/ε)K22 , (9)

where ∆n = npar − nperp, ε ∼ n2av, and the dielectric anisotropy at optical fre-

quencies is

εa = εpar − εperp = n2par − n2

perp ≈ 2∆nnav . (10)

Using the relation (10), we obtain from Eq. (9)

|E|2cr ≈ (32π3K22/λ2)(∆n/nav) . (11)


For λ = 1.064 µm, ∆n = 0.174, nav = 1.6, and K22 = 5 × 10−7 dyn we obtain

Ecr = 2.1× 104 V/cm and 2λ/(π∆n) = 3.9 µm, i.e. less than L for all four mirrors.

In the described experiments the electric field strength at the center of the beam is

∼ (2.2–7) ×104 V/cm, i.e. higher than Ecr.

The unwinding of the cholesteric helix by the field of a powerful light wave in

the region of selective reflection is substantially different from the unwinding of the

helix in a low-frequency field, since in the latter case E2cr ∼ ε−1

a (Refs. 145 and

146). In contrast to the field with a constant direction in the low-frequency case,

the direction of the electric field vector of a circularly polarized light wave changes

over the thickness of the cholesteric LC layer. The field of the light wave decays

exponentially over a distance Lc ∼ ε−1a and does not “see” the remaining thickness

of the cholesteric LC.96 Comparing results described here with the experimental

investigations of the pitch increase in a low-frequency electric field lying in the

plane of the cholesteric LC layer,145–147 it is possible to say that the electric field

reached in our experiments was much higher than the unwinding field in Refs. 145–

147 (∼ 104 V/cm).

It should be noted that in Ref. 148, the pitch of the cholesteric helix in a

cholesteric LC layer on whose surfaces the director was anchored increased in a

nonmonotonic, step-like manner. Apparently, this explains the nonmonotonic (step)

dependence of the cholesteric LC-mirror transmittance on the irradiation time in

the experiments described here (Fig. 25). The smearing of the steps is due to the ra-

dial intensity distribution in the beam and the corresponding change in electric-field

magnitude cross the beam. Deformation energy is accumulated in the cholesteric

LC layer over the interaction time of a single laser pulse (500 ns), and this energy

is partially dissipated during the “quiet” period between the pulses (200 µs), since

the relaxation time of the deformed helix is of the order of 1000 µs. Therefore,

the unwinding of a cholesteric helix in an optical field occur only over quite a long

time (accumulation of nonlinear changes149) during pulsed high-repetition rate laser

irradiation.

For a complete picture it should be noted that step-like changing the trans-

mittance was observed in Ref. 150, owing to pitch changes by heating. In difference

to our experiments, the authors of Ref. 150 used LCs with large temperature

sensitivity.

9. Conclusion and Future Developments

9.1. χ(3)(− ω,ω,ω,−ω) Measurements of Liquid Crystals

The main result of this paper is a warning of caution that, to date, results of

nanosecond and picosecond χ(3) measurements of cyanobiphenyl LCs remain in a

state of unsettlement that all but precludes the effective use of these materials in

the device engineering of power-limiting and other χ(3)-based applications. In par-

ticular, several nanosecond laser irradiation experiments involving such LCs under

400 S. G. Lukishova

two-photon absorption conditions need to be rethought. The unresolved challenges

are these:

• “Thin” (≤ 100 µm thickness) layer measurements, including the ones reported

here, tend to overestimate values of nonlinear optical parameters, in particular

when purification of “as received” materials by degassing and other means is

omitted. In the benign case, transient, micrometer-sized bubble formation will

cause wide-angle scattering that the detector will “interpret” as enhanced nonlin-

ear absorption. In the more aggravated case, microscopic carbonization will float

and remain in the medium with the potential for massive damage and device loss

later on. Both effects do “limit”, but quantification of such artifacts is tedious,

and reproducibility between devices based on such artifacts is impossible.

• Planar-nematic layers irradiated by pulses at moderate repetition-rate (2–10 Hz)

with linear polarization parallel to the director seem to experience cumulative,

laser-driven heating effects. Among the consequences of this local heating is

the important difference in LC response between first irradiation and much later

ones. LC devices, such as power limiters, that are expected to show specified

performance on the first shot without preconditioning, cannot be based on mea-

surements that do not distinguish this difference.

• Critical emphasis needs to be placed on irradiation geometry (beam-waist diam-

eter) and its precise knowledge as the nanosecond nonlinear response of “thick”

LC layers changes not only magnitude but even sign, depending on the geometry.

With much of this information absent or guessed in past measurements, reliable

χ(3) values are sparse indeed.

• It remains a vexing puzzle why the value for nonlinear absorption coefficient β in

the nanosecond case is ten- to hundred times larger than in the picosecond case.

Moreover, β depends on the incident intensity in both cases. Whether this is a

manifestation of a “clean” effect such as added triplet-state absorption or another

“dirt” effect (either bubbles or carbonization) needs to be clarified.

I would like to mark the importance of studies of materials with high two-photon

absorption, especially in optical power limiting application. It is also important to

mention that in difference with linear absorption, the two-photon absorption process

has a quadratically increase the absorption rate. This property allows activation of

photo-chemical or physical processes with high spatial resolution in 3D,151 e.g. for

data recording and storage application. Recently LCs doped with two-photon ab-

sorbing chromophores began to be used under short-pulse laser irradiation. A large

enhancement of the nonlinear absorption properties of dye molecules was observed,

dissolved in cyanobiphenyl nematic LCs.151

9.2. Cholesteric Mirrors

This paper shows the possibility to control the chiral-nematic (cholesteric) pitch

length (up to its unwinding) by a light-field alone under irradiation conditions


without either two-photon or linear absorption. Macroscopically this effect mani-

fests itself as a reflectivity drop of planar-aligned cholesteric mirrors. Previous at-

tempts to observe this drop failed because the threshold intensity (∼ MW/cm2)

must be maintained for the entire unwinding time ∼millisecond. Experiments

should also be able to distinguish between light-field-induced and thermal effects

that are present even at very low absorption coefficients of the LC medium.

Experiment described in this paper succeeded owing to specially chosen irradia-

tion conditions (∼ 4.5 kHz high-repetition rate) that enabled the cumulative action

of many pulses, each with above-threshold peak intensity. Low average power den-

sity did not permit heating of the sample.

Further development in this field should distinguish between “slow” (1) and

“fast” (2) unwinding experiments. (1) “Slow” unwinding, at near-threshold inten-

sities (I ∼ 2.2–10 MW/cm2) is presented in this paper. At such “low” intensities,

several minutes duration unwinding time offers the possibility to observe the un-

winding process in the microscope, i.e. imaging of how the chiral order changes

during the unwinding process. (2) Fast unwinding observed in Refs. 112 and 113 at

I ∼ 0.1–1 GW/cm2 opens the opportunity to use a cholesteric mirror for temporal

pulse shaping in a laser resonator with Q-switching or as the Q-switcher. It should

be added that nonlinear pitch dilation in cholesteric LC mirrors in laser resonator

gives the possibility to improve mode composition and output power stability of

CW and repetitively pulsed laser sources.

Acknowledgments

The author is very grateful to Prof. P. Palffy-Muhoray (Liquid Crystal Institute,

Kent State University) for his hospitality and help, Dr. A. Schmid (Laboratory for

Laser Energetics, the University of Rochester) for numerical simulations, providing

purified chiral mixture, and numerous discussions and help, Prof. S. Belyaev and N.

Malimonenko (Moscow Organic Intermediate and Dyes Institute (NIOPIK), Russia)

for fabrication cholesteric LC mirrors and discussions, K. Lebedev and E. Magu-

lariya (former students of Moscow Institute of Physics and Technology, Russia) for

assistance in chiral nematic experiments, Dr. T. Kosa (Liquid Crystal Institute) for

providing software and help, Dr. M. Groom (Liquid Crystal Institute) for the help

in maintaining electronics, Dr. B. Taheri (Liquid Crystal Institute) for the help in

maintaining laser system, Prof. J. Perry (University of Arizona) for providing chro-

mophore for some experiments, Prof. F. Scudiery and Dr. M. Marinelli (University

of Roma II) for providing their data in electronic form, Prof. V. Zolin, Dr. C.

Briskina, Dr. A. Lichmanov, Dr. V. Markushev, Dr. N. Ter-Gabrielyan from the

Institute of Radioengineering and Electronics of the Russian Academy of Sciences

(Moscow, Russia) for valuable advice and/or technical assistance, Dr. A. Zolot’ko

and Prof. V. Kitaeva from the P. N. Lebedev Institute of the Russian Academy of

Sciences (Moscow, Russia) for valuable references and advice.

This work was supported by NSF under ALCOM grant DMR-890147 and Cal-

ifornia Institute of Technology/AFOSR NoPC218970. In Russia this work was

402 S. G. Lukishova

partially supported by the International Science (Soros) Foundation (Grants Nos

MF2000 and MF2300), the Government of the Russian Federation (Grant No

MF2300), and the Russian Foundation for Basic Research (Grant No 96-02-18762).

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