MODULE 6-11

NONLINEAR MATERIALS

General Comments of Leno Pedroitti


ã Copyright 1988 by The Center for Occupational Research and Development

All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the publisher.

The Center for Occupational Research and Development
601 C Lake Air Drive
Waco, Texas 76710

Printed in the U.S.A.

ISBN 1-55502-024-0


(1) Nonlinear optical materials are materials in which the intensity of light input, including its frequency, is not related to the intensity of light output by a simple proportionality constant. Because of this nonlinear behavior, an intense light beam propagating through a nonlinear optical material will produce new effects that can’t be seen with weak light beams. For example, an intense light beam propagating through a nonlinear material can generate, in addition, harmonics or overtones of the original light frequency. This means that the red beam from a ruby laser can create an ultraviolet beam as it passes through the nonlinear optical material, while itself still propagating as a red beam.

(2) The interaction of intense light beams with nonlinear materials has opened a large field of potential applications in optical communication systems, where many well-known radio-frequency techniques such as mixing, heterodyning, and modulation now can be transferred to the domain of optical frequencies. For these reasons, nonlinear materials play an ever-increasing role in laser applications.

(3) This module will familiarize you with the fundamental physical ideas that underlie some of the more commonly encountered nonlinear phenomena and how these phenomena depend on the properties of the nonlinear materials. The experiment at the end of this module will give you practical experience in using a nonlinear material to generate the visible second harmonic of the invisible 1.06-mm Nd:YAG laser wavelength.

(4) For a better understanding of the ideas in this niodule, you should have knowledge of algebra and some trigonometry. Before beginning the experiment at the end of this module, be sure to review laser safety.

objectiv.jpg (4761 bytes)

(5) When you complete this module, you should be able to do the following:

  • Describe the nonlinear behavior of a system by describing its linear and nonlinear behavior with appropriate equations.

  • Explain second harmonic generation of light with the aid of the induced charge polarization.

  • Discuss in general terms the need for phase matching in second harmonic light generation.

  • Describe two experimental techniques used for phase matching.

  • List the various materials and experimental parameters that determine the efficiency of second harmonic generation through the use of the correct mathematical equations.

  • Differentiate between passive and active nonlinear materials.

  • List and rate the importance of some of the more common nonlinear materials.

  • Define the field-dependent nonlinear index and its implications.

  • Produce the second harmonic (0.53 mm) of the 1.06-mm radiation from a Nd:YAG laser using a crystal of KDP or ADP.

  • Knowing the type of SHG crystal, its size and operating parameters, estimate the theoretical conversion efficiency.

 

DISCUSSION

(6) Our objective is to develop a familiarity with the behavior and application of nonlinear materials. We’ll do this by first describing the behavior of a linear system or material. Next, we’ll look at its nonlinear behavior and discuss the internal properties of the material that make it behave in a nonlinear fashion. Finally, we’ll describe in a very elementary fashion the fundamental physical idea behind one of the more common nonlinear effects and how this effect depends on the properties of the nonlinear material. We conclude this section with a brief discussion of the more common nonlinear materials.

(7) Let’s start then with a brief discussion of linear materials and linear effects. It is a fact that our everyday experiences tell us that we live in an essentially linear world. When you first encounter the mathematical expressions of elementary physical laws, you may be tempted to conclude that nature is linear. For example, you learn that gas pressure in a container will double if the temperature of the gas is doubled; or that when a given force on a spring produces a stretching of the spring of one centimeter, then doubling the force will produce a stretching of exactly two centimeters. The example of the stretching spring is shown schematically in Figure 1.

Fig. 1
Linear displacement of a spring

(8) From these and other examples you might conclude that the response to small disturbances in all physical systems follows a linear law. Doubling the cause doubles the effect, tripling the cause triples the effect, and so on.

(9) Consider once more a linear spring. The mathematical expression relating the force F on the spring to the stretching of the spring in the x-direction is

F = kx
Equation 1

where the proportionality constant k is the spring constant.

(10) In practice, as long as the stretching of the spring is small compared to the total length of the spring, the force can be calculated with Equation 1. However, when the distance through which the spring is stretched exceeds a certain limit, the magnitude of the force F no longer is related to the distance x in a simple linear manner. Then it’s more accurately expressed by the nonlinear relation.

F = kx + k'x2 + k"x3 + k"'x4 + . . .
Equation 2

where k', k", k"' are higher order constants and are much smaller than the constant k.

(11) The magnitude of the force as a function of the distance the spring is stretched is shown in Figure 2, where the linear as well as the nonlinear behavior of the spring is illustrated.

Fig. 2
Linear and nonlinear behavior at a stretched spring

(12) Now we’ve stated what we mean by "linear" and "nonlinear" behavior of a system (the spring). Let’s next discuss the nonlinear behavior of certain materials when intense light beams propagate through them. These light beams are of course electromagnetic waves that are made up of an electric field and a magnetic field. Both fields are locked in step at right angles to each other and oscillate together at the frequency of the light, as you have already learned.

(13) Consider what happens when such an oscillating field interacts with an atom, which you can think of as a positively charged nucleus surrounded by shells of electrons. The electrons in the outermost shell are more loosely bound, and their radial displacements are governed by the same force equation that describes the stretching of a spring. These outermost electrons can redistribute themselves in step with the electric field. In this way the atom becomes charge-polarized. This means that positive and negative parts separate slightly to form a small electric dipole with a so-called dipole moment pointing in the direction of the impressed field. The dipole moment is the magnitude of the displaced charge times the distance between the positive nucleus and the center of the displaced electron shell. (Note that when not displaced, the center of a spherical election shell is, of course, at the same location as the positive nucleus.)

(14) The number of electric dipoles per unit volume multiplied by the dipole moment of one atom gives the induced macroscopic charge polarization of the material. (Don’t confuse this charge polarization induced by the electric field of the light beam with the polarization of a light beam with a polarizing filter.)

(15) In any given material, the magnitude of the induced charge polarization depends on the magnitude the applied electric field E. Specifically, we can express the charge polarization P in a series of powers of E and write

P = aE + dE2 + d'E3 + . . .
Equation 3

where: a = Polarizability coefficient of the material
dd', etc. = Higher-order nonlinear optical coefficients that are much less than a

The similarity between Equation 3 and Equation 2 should be clear.

(16) The electrons in the outermost shell not only are subject to the pulling of the applied external field, but also are subjected to the pull of the internal field due to the positive nucleus. The magnitude of this internal electric field is enormously large, somewhere in the neighborhood of 109 volts/cm. For comparison, the electric field of sunlight at the surface of the earth is only about 10 volts/cm. It turns out that, when the applied electric field is many orders of magnitude less than the atom’s internal electric field—which is always the case for ordinary light sources—the linear approximation P = aE of Equation 3 is very accurate.

(17) With the development of the laser in 1960 we were able to build light sources capable of producing optical electric fields of 106 to 107 volts/cm. When some optical materials are irradiated by such intense fields, the contribution to the charge polarization by the higher-order terms in Equation 3 no longer can be neglected.

(18) Let’s see what happens when we use only the first two terms in Equation 3. For simplicity, let’s also assume that the electric field points in the x-direction. Then the total induced charge polarization in the x-direction is

Px = aEx + dE2x
Equation 4

(19) For plane polarized light of frequency f and amplitude Exo, the electric field at some given point in an optical material is described by

Ex = Exo cos(2p ft) = Exo cos (w t)
Equation 5

where w º 2pf º the "angular frequency" in rad/sec

and the induced charge polarization is

Px = aExo cos (w t) + dE2xo cos2 (w t)
Equation 6

Using the trigonometric identity

Equation 7

Equation 6 can be written as:

or

Px = Px (w) + Px(2w) + constant term (DC)

Equation 8

(20) An inspection of the above equation reveals that the first term Px(w) is a charge polarization of angular frequency w. The second term Px(2w) is a charge polarization of angular frequency 2w, twice the fundamental frequency w. Finally, the third term has no frequency dependence whatsoever. It provides a "DC" charge polarization. The three charge polarization terms are shown in Figure 3. Equation 8 then tells us that a rapidly oscillating electric field in the neighborhood of 1014 to 1015 Hz (optical frequencies) will induce two types of oscillating charge polarization; one of frequency w and another of frequency 2w. The material consists, then, of a collection of oscillating dipoles that absorb and reradiate light waves. The total energy in the beam of light is not significantly altered.

(21) One effect of the induced charge polarization and subsequent reradiation is a decrease in the speed of the light in the medium. The decrease in speed is reflected in the increased value of the index of refraction n, which is given by the ratio of the speed of light in vacuum c divided by the speed of light in the medium v, that is, Equation 9,

Equation 9

Fig. 3
Frequency associated with charge-polarization terms

(22) Another effect, predicted by Equation 8, is the reradiation of energy at the frequency 2w, twice the frequency of the incident radiation. This doubling of frequency is known as "second harmonic generation" (SHG) or "frequency doubling."

(23) Frequency doubling was observed for the first time in 1961 by Professor Peter Franken and some graduate students at the University of Michigan. They irradiated a quartz crystal with the beam from a ruby laser that operated at 694.3 nm. A very small amount of the light striking the crystal was converted to light with a wavelength of 347.2 nm. This wavelength lies in the ultraviolet region of the spectrum and is of course exactly half the wavelength and twice the frequency of the incident laser light.

(24) The details of this famous experiment are shown in Figure 4. As you might expect, this experiment initiated a search for materials in which this effect occurs strongly. As a result, the experiment has been duplicated many times with a host of different nonlinear materials.

Fig. 4
Experimental arrangement for second harmonic generation

(25) What are the properties of a crystal that are responsible for efficient SHG? To answer this question let’s first note that, whatever these properties may be, they manifest themselves in the expansion coefficient d of the second-order term in Equation 3.

(26) To find out how d is governed by the internal properties of the crystal, we need to examine the oscillatory motion of the dipoles. This is best done by adding the three curves in Figure 3. The resultant charge polarization is shown in Figure 5, along with the applied electric field.

(27) The curve for the total polarization shows mainly the contributions of the fundamental polarization and steady DC polarization terms. The contribution of the second harmonic term would cause a slight change in the slopes of the rise and fall of the total polarization, but not in the location of the maximum, zero, and minimum points. So its effect has not been shown in the lower curve in Figure 5.

Fig. 5
Impressed electric field and resulting total polarization

(28) It’s clear from Figure 5 that the individual dipoles, whose oscillatory motions lead to the charge polarization, do not faithfully reproduce the sinusoidal electric field. Note especially that the total polarization is asymmetric in the upward direction. This is because the outer electrons of the atoms in a crystal are more easily pushed in one direction than in the opposite direction.

(29) Figure 5 shows that an intense electric field in the up-direction is more effective in polarizing the medium than the same field in the down-direction. Such a situation can occur only in a crystal with no internal symmetry. Only about 10% of the crystals found in nature fall in this category.

(30) The distorted charge polarization wave produced by the intense radiation field travels at the same speed as the incident light wave. This light, which is reradiated by the oscillating dipoles, contains, as we saw above, two discrete frequencies w and 2w. Since the refractive index for the light of frequency 2w is usually larger than it is for light of frequency w, the speed of the component with frequency 2w will be just a bit slower than the component with frequency w, in accordance with Equation 9.

(31) The difference in propagation speed between the fundamental frequency and the second harmonic frequency means that the second harmonic light radiated from one part of the crystal, say P(1), will not be in phase with the second harmonic light radiated from another part of the crystal P(2). As a result, the two waves from P(1) and P(2) will interfere destructively at some point P(3). The distance that it takes for a second harmonic charge polarization wave and its radiated light to get 180° out of phase is called the "coherence length." This distance is usually on the order of 10 µm. If the crystal length happens to be equal to an odd multiple of the coherence length, virtually no second harmonic radiation will emerge from the crystal.

(32) This problem of destructive interference can be overcome by making use of the double-refraction birefringence exhibited by certain crystals. It turns out that in such crystals the velocity of light depends on the direction of polarization of the electric field, the propagation direction, and the wavelength.

(33) For example, the speed of light in the nonlinear material potassium dihydrogen phosphate (KDP) for the horizontally and perpendicularly polarized electric field components is shown in Figure 6. As you can see, the ordinary fundamental that we have taken to be the light from a ruby laser at 694.3 nm, polarized perpendicular to the plane of the page, travels at exactly the same speed as the extraordinary second harmonic light at 347.2 nm, when both light waves propagate in a direction that’s inclined 50° to the optic axis of the crystal. The optic axis of a doubly refracting crystal is that direction along which the ordinary and extraordinary waves of the same frequency propagate at the same speed.

Fig. 6
Velocities of ordinary and extraordinary light in KDP

(34) When the direction of propagation of the fundamental is chosen so that the index of refraction of the fundamental and second harmonic are the same, both waves will stay in phase. The crystal then is said to be phase-matched or "index-matched." The second harmonic wave produced at any given point in the beam now will interfere constructively with the second harmonic wave produced further "downstream" in the beam. As we will see later, when the light travels along the index-matching direction, the efficiency for SHG can be improved very significantly.

(35) Phase-matching also can be achieved in some crystals by varying their temperature. The reason is that the extraordinary index is in general more temperature-dependent than the ordinary index. So by varying the temperature you can adjust the birefringence of the crystal until phase matching is obtained. The exact temperature for phase matching depends on the exact chemical composition of the crystal. This effect was discovered by accident when it was noted that different crystals of lithium niobate required different phase-matching temperatures under otherwise similar experimental conditions.

(36) For the niobates in which the phase-matching is achieved by controlling the temperature, the apparatus shown in Figure 7 is used to determine the phase-matching temperature for any given crystal. A laser such as Nd:YAG operating at 1.06 mm is used to illuminate the crystal. The intensity of the second harmonic is monitored and displayed with a pen recorder as the temperature of the crystal is tuned through the phase-matching temperature.

Fig. 7
Apparatus for determining phase-matching temperature in
birefrigent crystals

(37) It turns out that a perfect crystal should give a plot of second harmonic power versus temperature that has a [(sin x) / x]2 dependence. However, because of slightly varying chemical compositions at different sites of the crystal, different portions of the crystal will phase-match at slightly different temperatures. The result is a broadening of the central peak and a reduction in height. Furthermore, the side lobes are obscured. A typical curve of second harmonic power versus temperature for an ideal crystal and a nonideal crystal is shown in Figure 8.

Fig. 8
Second harmonic power output versus temperature

(38) The conversion efficiency, h SHG defined as the ratio of the emerging second harmonic power P(2w) to the incident power P(w) is one of the most useful measures of the performance of a nonlinear crystal. It is given by

Equation 10

where: Dk = k2w – 2kw
k = 2p/l
eo and mo = Permittivity and the magnetic permeability of free space, respectively
n = Index of refraction
w = Angular frequency of the incident light
d = Second-order coefficient
= Length of the crystal
A = Beam area

(39) The factor Dk represents the amount of phase mismatch between the second harmonic wavefronts generated at different points in the crystal. If Dk is zero (proper phase-matching) then the "interference" term,

reaches a maximum value that in turn maximizes the efficiency of the SHG process. Note that the "interference" term is squared, making the power converted into the second harmonic very sensitive to phase mismatches.

(40) The efficiency of second harmonic generation is directly proportional to the intensity of the incident fundamental radiation. This is why second harmonic generation often is done in an intracavity configuration since the higher power of the fundamental produces much higher intensities than are achievable in external convertors.

 

Example A: Compute the Theoretical SHG Conversion Efficiency with a Doubling Crystal External to the Laser Cavity

Given:

Q-switched, Nd:YAG laser, output power = 1.2 MW, with a 2.00-mm-diameter beam inside KDP SHG crystal. Length of optical path inside crystal is 2.35 cm. Assume proper phase-matching and n » 1.49.

Find: Estimated SHG efficiency.

Solution:

Equation 10,

Note: if phase-matched

Work in mks (SI) units:

(See Table 1)

 

(41) Occasionally, in the case of intracavity generation, an alternative definition of efficiency is used to specify performance. This definition relates the output of the second harmonic (for a cavity designed for maximum second harmonic output) to the output of the fundamental (with the cavity designed for maximum fundamental output). Since mirrors can be fabricated with essentially 100-percent transmittance at one frequency and 100-percent reflectance at some other frequency, it’s entirely feasible to design a cavity that retains the fundamental while coupling out all of the second harmonic. In this type of system, according to the above definition of efficiency, it’s possible to achieve 100-percent conversion. Bear in mind, however, that the conversion refers to the available intensity (obtainable as output) and not to the actual intensity at the crystal itself.

(42) Before we conclude our discussion of second harmonic generation and describe other nonlinear effects, we need to discuss one important complexity that we’ve ignored. We stated earlier that all nonlinear effects arise directly from higher-order polarization induced in the nonlinear material by the incident radiation. For simplicity we assumed that the induced polarization would have the same direction as the electric field of the incident radiation. This was the assumption made when we presented Equations 3 and 4.

(43) In reality, the second-order polarization P (2w) is a vector quantity, each component of which may depend on all three components of the incident electric field. For example, you can see that the x-component of the second-order polarization, Px(2w) given by Equation 11 is dependent on all components of the electric field (Ex, Ey, Ez) in a rather complicated way.

Equation 11

In addition; before there was only one second-order coefficient (d) in the relation between P and E (Equation 3 or Equation 4). Now there are six coefficients (dxxx---dxxy) for Px alone. As you might guess, there are also six coefficients in the equation for Py(2w) and six for Pz(2w). Some values for nonlinear optical coefficients of common crystals are given on Table 1. You can find values for other coefficients in selected references given at the end of this module. Estimates for the damage threshold of some common SHG crystals are given in Table 2.


Table 1. The Nonlinear Optical Coefficients of a Number of Crystals*

Crystal dijk (2w) in units of 1/9 × 10–22 (mks)
LilO3 d31 = 0.46 ± 0.1
NH4H2PO4 d36 = d3l2 = 0.45
(ADP) d14 = d123 = 0.45 ± 0.02
KH2PO4 d36 = d312 = 0.45 ± 0.03
(KDP) d14 = d123 = 0.45 ± 0.03
KD2PO4 d36 = d312 = 0.42 ± 0.02
(KD*P) d14 = d123 = 0.42 ± 0.02
KH2ASO4 d36 = d312 = 0.48 ± 0.03
d14 = d123 = 0.51 ± 0.03
Quartz d11 = d111 = 0.37 ± 0.02
AIPO4 d11 = d111 = 0.38 ± 0.03
ZnO d33 = d333 = 6.5 ± 0.2
d31 = d311 = 1.95 ± 0.2
d15 = d113 = 2.1 ± 0.2
CdS d33 = d333 = 28.6 ± 2
d31 = d311 = 14.5 ± 1
d15 = d113 = 16 ± 3
GaP d14 = d123 = 80 ± 14
GaAs d14 = d123 = 107 ± 30
BaTiO3 d33 = d333 = 6.4 ± 0.5
d31 = d311 = 18 ± 2
d15 = d113 = 17 ± 2
LiNbO3 d31 = d311 = 4.76 ± 0.5
d22 = d222 = 2.3 ± 1.0
Te d11 = d111 = 730 ± 230
Se d11 = d111 = 130 ± 30
Ba2NaNb5O15 d33 = d333 = 10.4 ± 0.7
d32 = d322 = 7.4 ± 0.7
Ag3AsS3 d22 = d222 = 225
(Proustite) d36 = d312 = 135
* These coefficients are defined with 1 = x, 2 = y, 3 = z. From Yariv.

 

Table 2. Approximate Damage Thresholds for
Common SHG Materials at lo = 1.06 mm
for Q-switched Operation

Material

Approximate Damage Threshold
in (GW/cm2)

ADP

0.50

Ba2NaNb5O15

0.001

CDA

0.50

CD* A

0.36

KDP

0.20

KD* P

0.50

LiNbO3

0.02

LiIO3

0.05

Precise values will depend on many factors including laser pulse width. Data courtesy of Quantum Technology, Inc., Sanford, Florida.

 

(44) Common substances that exhibit SHG are potassium dihydrogen phosphate (KDP), barium titanate (BaTiO3) and lithium niobate (LiNbO3). In barium titanate, for example, only five of the second-order coefficients are non-zero, leading to the following simplified forms for Px(2w ), Py(2w ) and Pz(2w ):

Px (2w ) = 2dxxz Ex Ez

Py(2w ) = 2dyyz Ey Ez

Pz (2w ) = dzxx Ex2 + dzyy Ey2 + dzzz Ez2

If, in addition, the incident electric field is polarized in the x-direction, (Ey = 0, Ez = 0), the equations above reduce to only one:

Pz(2w) = dzxx Ex2

Equation 12

(45) Equation 12 tells us that an electric field polarized in the x-direction, incident on barium titanate, will produce a second harmonic charge-polarization in the z-direction. The orientation of this crystal in terms of its optic axis and the electric field polarization of the incident beam is shown in Figure 9.

 

Fig. 9
Typical electric-field-optic-axis orientations in a second-harmonic crystal

(46) Now let’s turn our attention from SHG, which was the first nonlinear effect observed with a laser beam, to some other important nonlinear effects. Depending on the particular nonlinear material, these effects can be catalogued into two groups—passive and active.

(47) Materials producing passive optical effects are those that act essentially as catalysts without imposing their characteristic internal resonance frequencies onto the particular nonlinear effect. Passive nonlinear effects include harmonic generations, frequency mixing, optical rectification, and self-focusing of light.

(48) The latter results from a field-dependent change in refractive index as shown in Equation 13.

n = no + n2E2

Equation 13


where: n = Effective refractive index
no = Ordinary refractive index
n2 = Ordinary refractive index
E = Electric field magnitude

(49) The quantity n2 comes from other considerations of polarizabilities of materials as we discussed for second harmonic generation. This nonlinear behavior can lead to intensity-dependent lensing (focusing or defocusing) of propagating beams. If focusing occurs, the decreased beam diameter can cause power densities (W/cm2) exceeding the damage threshold of the material and result in breakdown failure or permanent damage to the material. If defocusing occurs, beam spread results.

(50) Materials producing active nonlinear effects do impose their characteristic resonance frequencies onto an incident beam of light. The optical nonlinear effects produced by these active materials include two-photon absorption, stimulated Raman, Rayleigh, and Brillouin scattering.

(51) We won’t discuss the numerous active nonlinear materials, nor will we describe the related effects in this module. If you’re interested, you will find it worth your time to read the paper by J. A. Giordmaine in the April 1964 issue of the Scientific American, where some of these effects are described in an appropriately elementary fashion.

(52) We will briefly describe some of the passive nonlinear effects.

 

Sum and Difference Frequency Generation

(53) Two light beams of different frequencies w1 and w2 are mixed in a nonlinear crystal. The light leaving the crystal then will have two additional frequencies w1 + w2 and w1w2. Generation of these additional frequencies requires intense unidirectional sources of light. Although lasers commonly are used in this application, their monochromaticity and coherence play no essential role. In fact, sum and difference frequencies have been obtained with very intense incoherent light sources.

(54) You can understand this particular phenomenon best in terms of the quantum theory of radiation, according to which light propagates in discrete packages of energy called photons. For example, two photons can be annihilated to produce a new photon that carries the energy of the two that disappeared. Since the frequency is proportional to its energy, the new photon will exhibit the sum frequency of the two annihilated photons.

(55) In the interaction of a photon of frequency w1 with a photon of frequency w2 to produce a photon of frequency w3 = w1w2, the w1 photon is annihilated. In its place appears another w2 photon. The w1 photon gives up sufficient energy to create one w3 photon. The total energy is conserved. This particular interaction will produce as many new w2 photons as w3 photons.

 

Optical Rectification

(56) In Equation 8 we saw that a DC signal should be produced across a nonlinear crystal when an intense polarized laser beam passes through the crystal. This optical rectification has indeed been observed.

(57) You can detect the DC signal by connecting conductors to opposite sides of a piezoelectric crystal that is transmitting the light output from a laser. The DC signal appears as a voltage pulse that’s proportional to the intensity of the laser beam. This effect is, of course, analogous to that of a semiconductor diode used to rectify an AC electrical signal.

 

Beam-trapping

(58) As we said previously, the index of refraction of a dielectric material is actually a function of the light intensity. This has been observed in a variety of liquids and solids.

(59) One manifestation of this is the well known Kerr effect, where a strong applied electric field aligns the molecules of a liquid and thereby produces birefringence. A second effect is electrostriction which is due to a strain experienced by dielectrics when they are subjected to an electric field. The strain depends on the square of the electric field. Because of this field-induced strain, electrostriction involves a change in the density of the material. But the index of refraction depends on the density, which in turn implies that the index of refraction is intensity-dependent.

(60) In liquids, both the Kerr effect and electrostriction are of equal importance in changing the refractive index. In solids, however, molecules can’t be aligned easily so that electrostriction is the dominant cause of an intensity-dependent index of refraction, except at very short pulse lengths.

(61) If a laser beam is focused so that the focal volume lies within a piece of glass, the beam, if sufficiently intense, can increase the index of refraction so much that the slender, intense beam is trapped because of total internal reflection. The trapped beam has been observed to travel through transparent materials along a thin filament and leave a trail of ionization damage in the form of microscopic bubbles and cracks.

(62) Because scientists want to understand and apply the many different nonlinear effects, they started a search for materials in which these effects occur strongly. They have learned how to find these nonlinear materials and also how to calculate and predict their properties.

(63) Before we discuss some of the more common nonlinear materials, let’s briefly list some properties common to all useful nonlinear optical crystals.

(64) Foremost on the list is the fact that nonlinear materials belong to a group of non-centrosymmetric crystals. This means that the internal motion of electrons in an oscillating electric field is not symmetric. The material should furthermore exhibit a large index of refraction since there is, as a rule, a strong dependence of nonlinear optical properties on the refractive index. It also should have a large birefringence for easy index-matching.

(65) From a practical point of view the material should have a low susceptibility for optical quality, can be optically polished, is not hygroscopic, and has a wide range of spectral transmission. This is a big order! Yet, certain materials such as lithium niobate and barium sodium niobate tend to exhibit several of these properties. So the challenging problems encountered when growing these materials now are receiving a lot of attention. The nonlinear properties of literally hundreds of nonlinear materials are catalogued already. Let’s turn our attention then to a few of the more common ones.

 

ADP and KDP

(66) Ammonium dihydrogen phosphate (ADP) and potassium dihydrogen phosphate (KDP) were among the first nonlinear crystals used for the generation of phase-matched second harmonic light. Because of their similarity we’ll discuss them together.

(67) Both materials first were used in piezoelectric applications such as ultrasonic transducers. Since these materials are easy to grow in aqueous solutions that produce high-optical-quality single crystals, you can purchase specimens as large as 10 cm on one side. The optical transmission band extends from the ultraviolet to the near infrared, typically from 200 nm to 1500 nm.

(68) ADP will deteriorate when heated to temperatures of over 100°C and has a tendency toward cracking upon cooling. But KDP is relatively stable and can be heated and cooled. Both materials are sufficiently resistant to laser damage for most applications.

(69) Isomorphs, which are substances of different chemical composition that crystallize in the same form as ADP or KDP, also have been used to produce nonlinear optical effects. The best-known of these is deuterated KDP, which is designated KD*P and called: "K-D-star-P." The shortcomings of KDP and ADP are a relatively small nonlinear coefficient and poor optical transmission in the infrared region of the spectrum.

 

Lithium niobate (LiNbO3)

(70) This is one of the more important of the nonlinear materials that have appeared in recent years. It’s transparent from 400 nm to 5 mm, and its nonlinear coefficient is about ten-times larger than that of KDP. This means that lithium niobate is two orders of magnitude more efficient than KDP, and SHG efficiencies close to 100 percent are possible. It’s one of the few nonlinear materials available in large sizes in commercial quantities.

(71) Since it has a very large birefringence in the visible and near-infrared region, it allows phase-matching of the fundamental and harmonic waves. Phase-matching usually is accomplished by heating the crystal to the phase-matching temperature for the particular laser being used.

(72) Crystals are available in all sizes up to a few centimeters in diameter and more than 10 centimeters in length. The crystals are clear as water and insoluble in water. This material is used in second harmonic generation; parametric oscillation, which is an application of sum and difference frequency generation; and a host of other applications.

(73) One problem with lithium niobate is that it has a very low damage threshold. In some samples, this damage disappears by itself soon after the laser beam is turned off. But in other samples the characteristic "tracks" can remain for days. Fortunately, the damage can be reversed by heating the crystal to about 200°C. This is one reason for using temperature-controlled phase-matching.

 

Barium sodium niobate (BaNaNb5O15)

(74) This nonlinear material—which also is known as "banana"—is similar to lithium niobate. But it doesn’t suffer as much from optical damage when its temperature is maintained above room temperature. Barium sodium niobate has a nonlinear coefficient that’s about three times larger than lithium niobate.

(75) BaNaNb5O15 is optically transparent from 370 nm to about 5 mm. Some samples don’t have a good optical quality due to a marked brown discoloration. Phase-matching for the second harmonic generation of 1.06 mm radiation from an Nd:YAG laser occurs around 100°C. The exact temperature depends on the stoichiometric composition.

(76) This material has been used for very efficient generation of the second harmonic of 1.06 mm radiation and in the building of parametric oscillators.

 

Proustite (Ag3AsS3)

(77) Proustite is a naturally occurring crystal found in mineral deposits. It also can be produced synthetically. The one characteristic that distinguishes this material from all other nonlinear materials is its transmission band, which extends from about 600 nm to beyond 13 mm. This and its large birefringence make it a prime candidate for the study of phase-matched interactions between the infrared region and the visible range. Furthermore, the appropriate coefficients that are a measure of its efficiency for producing nonlinear effects are relatively large. In fact, they are about 300 times larger than the analogous coefficient of KDP.

(78) The material has been used in experiments that involve the mixing of the 10.6 mm radiation from a CO2 laser with a visible laser. Proustite crystals of good optical quality and dimensions of several centimeters are grown in an aqueous solution. Some disadvantages of this material are its inability at times to accept dielectric coatings and its resistance to optical polishing.

(79) Aside from the few materials listed above, a host of other materials are available to the experimenter, and new nonlinear materials constantly are added to the already large list. An excellent source that lists most of these materials is the CRC Handbook of Lasers.

(80) This concludes the discussion of nonlinear materials and nonlinear effects. The field of nonlinear optics is in its infancy, with a promise for a bright future. As electro-optic technology expands and the applications of coherent light waves become more numerous, nonlinear materials and effects will be of great importance in exploiting the intense laser beam.

exercise.jpg (4815 bytes)

  1. Describe linear and nonlinear behavior of a crystal through the use of appropriate equations.

  2. Describe second harmonic generation using the concept of induced charge polarization and the need for phase-matching. Indicate two experimental methods for achieving phase-matching conditions.

  3. List the various material and experimental parameters that determine the efficiency of second harmonic generation through the use of the correct mathematical equation.

  4. Differentiate between passive and active nonlinear materials.

  5. List and rate the importance of some of the more common nonlinear materials.

  6. Describe mathematically the field-dependent nonlinear index. Explain how this quantity is made manifest phenomenologically in damaging optical components subjected to high-peak-power laser radiation.

  7. Compute the approximate length of a KD*P SHG crystal to be used with a 2.0-mw Q-switched Nd:YAG laser that has a beam diameter within the crystal of 5.00 mm. Assume that we want 20% conversion efficiency when properly phase-matched.

  8. According to Equation 10, if the incident fundamental power is doubled, what increase should we expect in the power of the second harmonic? What increase in conversion efficiency should we expect? Discuss.

  9. A 200-mJ Q-switched Nd:YAG laser with a pulse width of 20 nsec is doubled with a 2.00-cm-long KDP crystal. The beam diameter is 4.00 mm. What will be the approximate irradiance produced by the second harmonic assuming proper phase-matching?

  10. Use Equation 3 including up to the third term in the expression for the induced charge polarization,

    Px = aEx + dEx2 + d 'Ex3

    Assume an oscillating electric field Ex like in Equation 5. Using appropriate trigonometric identities derive an expression for Px in terms of the angular frequency w of the incident light. Group the terms according to the frequency dependence: DC term, fundamental term, second harmonic term, third harmonic term. Compare this new expression to Equation 8 and discuss the physical significance of each term.

material.jpg (4606 bytes)

Q-switched Nd:YAG laser

SHG crystal with mount

Crystal mounting, alignment device with angular readout

Screen

Energy meter

Photodetector and oscilloscope (to measure pulse width)

Lens of about 10-cm focal length

530.0-nm filter

1.06-mm filter

procedur.jpg (4925 bytes)

(82) Before you start the experiment, you should review safety procedures that apply to the Nd:YAG laser. You will assemble the apparatus and observe the second harmonic of the 1.06 mm radiation from the Nd:YAG laser on a screen.

(83) The main object of this experiment is to familiarize you with one nonlinear material and one nonlinear effect. Specifically, in the experiment you will observe the second harmonic radiation at 530 nm that’s produced with a crystal of KDP and ADP when the beam from a 1.06 mm Nd:YAG laser is focused into the crystal. Use care not to damage the crystal.

(84) The experimental setup is quite simple. The output from the Q-switched laser is focused into the crystal, which is mounted on an angular orientation mount. If you know the phase-matching direction for the crystal, mount it accordingly. If you don’t know the phase-matching direction, you can find it by trial and error.

(85) To do this, start with an arbitrary orientation of the crystal and observe the screen as the laser is fired. If a green flash of light splashes on the screen, the crystal is mounted in a position that isn’t too far removed from the desired phase-matching orientation. If you don’t see a green light, change the orientation of the crystal slightly and fire the laser again. Repeat this procedure until a bright green flash is visible on the screen every time the laser is Q-switched. Use care when viewing the green light. Never try to observe the second harmonic by direct viewing of the light as it exits the crystal.

(86) Adjust either the angle and/or the temperature of the doubling crystal to maximize the SHG effficiency. Using the energy meter and photodetector, measure both the power of the fundamental and the power of the second harmonic. Compute the conversion efficiency.

(87) Compare your measured efficiency to the estimated theoretical value given by Equation 10.

 

OPTIONAL:

(88) If you have time, investigate the dependence of light intensity in the second harmonic as a function of crystal orientation. To do this, carefully rotate the crystal about a vertical axis, degree by degree, on either side of the orientation for which you got optimum phase-matching. Measure the intensity at each angle with a sensitive energy meter. Plot the intensity versus angle. Compare this plot with Figure 8 and explain why the curves are similar.

(89) Next, when the crystal has been oriented for maximum output, vary the input intensity from the YAG laser. Measure the resultant change in output. You can vary the input intensity with appropriate filters. Plot results as IYAG input, ISHG output at appropriate filter wavelengths.

referenc.jpg (4806 bytes)

Adhav, R. S., and A. D. Vlassopoules. "Guide to Efficient Doubling," Laser Focus, May 1974.

Baldwin,G.C. An Introduction to Nonlinear Optics. New York: Plenum Press, 1969.

Bass, M.; P. A. Franken; A. E. Hill; C. W. Peters; and G. Weinreich. "Optical Mixing," Physical Review Letters, V. 8, 1961.

CRC Handbook on Lasers, Chemical Rubber Co., Cleveland, OH.

Giordmain, J. A. "Nonlinear Optics," Physics Today, January 1969.

________. "Nonlinear Optical Properties of Liquids," Physical Review Letters, Volume 138,1965.

________. "The Interaction of Light with Light," Scientific American, April 1964.

Hulme, K. F.; O. Jones; P. H. Davies; and M. V. Hobden. "Synthetic Proustite: A New Crystal For Optical Mixing," Applied Phvsics Letters, V. 10,1967.

Kaminow, I. P. "Pararmetric Principles in Optics," IEEE Spectrum, 1965.

Weinberg, D. L. "Tunable Optical Parametric Amplifiers and Generators," Laser Focus, April 1969.

Yariv, A. Introduction to Optical Electronics. San Francisco: Holt, Rinehart, and Winston, Inc., 1971.

--------------------------------------------------------------

---Course Contents---