MODULE 6-11

NONLINEAR MATERIALS

(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.

• 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

(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

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.

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

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

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

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

and the induced charge polarization is

Using the trigonometric identity

Equation 7

Equation 6 can be written as:

(20) An inspection of the above equation reveals that the first term P_x(w) is a charge polarization of angular frequency w. The second term P_x(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

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

(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, P_x(2w) given by Equation 11 is dependent on all components of the electric field (E_x, E_y, E_z) 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 (d_xxx---d_xxy) for P_x alone. As you might guess, there are also six coefficients in the equation for P_y(2w) and six for P_z(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*

(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.

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,

    P_x = aE_x + dE_x² + d 'E_x³

    Assume an oscillating electric field E_x like in Equation 5. Using appropriate trigonometric identities derive an expression for P_x 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.

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

(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 I_YAG input, I_SHG output at appropriate filter wavelengths.