17 Birefringence

Dr. Ajit K. Mahapatro

Learning Objectives

From this module students may get to know about the following

i. Introduction to Second harmonic generation (SHG)

ii. Details of Phase Matching

iii. Birefringence

1. Introduction

Second harmonic generation (also called frequency doubling and abbreviated as SHG) is a nonlinear optical process, in which photons with same frequencies interact with a nonlinear material, “combined” to generate new photons with twice the energy, and thence twice the frequency and half the wavelength of the initial photons. Here, the dipoles oscillating with the applied electric field of frequency ω, radiates electric field with frequency 2ω as well as ω.

In Second Harmonic Generation, the applied and driving field at frequency ω produces nonlinear polarization at frequency 2ω. The total SHG field generated at point r is the resultant sum of interfering waves generated between edge of the medium r = 0 and r is given as

Many phase-sensitive nonlinear processes, especially, the parametric processes including frequency doubling, sum and difference frequency generation, and parametric amplification and oscillation, require phase matching to be efficient. For optimum nonlinear frequency conversion, it is required to have proper phase relationship between the interacting waves, maintained along the propagation direction. This condition is fulfilled when the amplitudes from different locations are all in phase with the resultant total wave at the end of the nonlinear crystal. An effective nonlinear interaction is possible with phase mismatch close to zero. The phase matching of frequency doubling is given by ∆k=2k₁-k₂ =0, where k₁ and k₂ are the wavenumbers of the fundamental and second-harmonic beam, respectively. Without chromatic dispersion, k₂ = 2 k₁ would hold, so that the phase mismatch vanishes.

Figure 1. Schematics with illustrations for (a) high efficiency in phase matching and (b) low efficiency in phase mismatching.

In Figure 1, the arrows indicate the phase corresponding to the amplitude contributions from different parts of the nonlinear crystal to the harmonic wave. These contributions added constructively only when the phase matching is achieved, and results high power conversion efficiency. Else, the direction of energy transfer changes periodically with the change in phase between the interacting waves, during passaging through the crystal.

The energy then oscillates between the waves without being transferred in a constant direction. The effect on the second-harmonic power conversion in a crystal along the propagation direction is demonstrated in Figure 2. Here, the solid curve is observed during phase-matching that indicates growth of power proportional to the square of the propagation distance. The dashed curve represents the power during non phase-matching with the second-harmonic power oscillating between zero and a small value.

Figure 2. Second-harmonic power in a crystal along the propagation direction.

3. Effect of Temperature in the SHG Power Efficiency

The crystal temperature around the optimum point develops phase mismatch and variation in the conversion efficiency. Figure 3 plots the second-harmonic power versus temperature deviation from the optimum point with homogeneous temperature distribution in a crystal. The high conversion efficiency is inversely proportional to the crystal length and also relies on the temperature dependence of the refractive indices involved. Similar relations apply to other nonlinear frequency conversion processes. Similar curves are obtained e.g. for critical phase matching when the angular orientation of the crystal is varied.

Figure 3. The second-harmonic power versus temperature deviation from the optimum point with homogeneous temperature distribution in a crystal.

In practice, the phase-matching curve is not as symmetric as shown in Figure 3. For example, it becomes asymmetric if the crystal temperature is not homogeneously distributed throughout the length or lower in temperature values at the end faces compared to the middle of the crystal. The temperature homogeneity in a crystal could be quantified from the phase-matching curve and the conversion efficiency could be optimized by fabricating crystals of desirable dimension.

4. Coherence Length

The oscillation caused by interference between SHG waves generated at different points along the path, severely limits the SHG intensity unless the phase matching conditions, ∆?= 2?1 − ?2 = 0 and the index matching n1(?1) = n2(?2), are fulfilled, in equivalent conditions.

As most of the optical media are dispersive, n1≠n2

∆?= 2?1 − ?2 ≠ 0 (6)

Hence SHG amplitude oscillates as ??(2?1−?2)^r with periodicity lc = ?/ (2?1−?2 ) along the path. The distance ‘lc’ is called coherence length and is typically several wavelengths in condensed media.

In a SHG medium, the coherence length is infinite when it satisfies the phase or index matching, and it leads to complete conversion of fundamental into SH.

Efficient SHG requires phase matching (momentum conservation) satisfying the index matching n1(ω1)= n2(2ω1). The difficulty in materials is that they are usually dispersive and the refractive index increases with frequency with n2 > n1, in normal dispersion.

5. Phase-matching Techniques

The chromatic dispersion of an optical medium is the phenomenon where the phase velocity and group velocity of light propagating in a medium depend on the optical frequency. Absence of chromatic dispersion could lead to k₂ = 2k1 and the phase mismatch vanishes. Due to chromatic dispersion, the wavenumber of the second harmonic is larger than twice of the fundamental wave. Dispersion generally causes a non-zero phase mismatch as shown in Figure 4. Special measures could be taken to avoid this by choosing a different polarization in a birefringent crystal.

Figure 4. Phase mismatch for second-harmonic generation and birefringent phase matching.

6. Birefringence

The most common technique for achieving phase matching is the natural birefringence (double refraction) of uniaxial anisotropic crystals. Birefringence is the property of non-isotropic material with refractive index dependance on the polarization direction (i.e., direction of the electric field), favoring the material capable of exhibiting double refraction by shining an unpolarized light beam. It is referred as the double refraction of light in a transparent, molecularly ordered material, manifested by the existence of orientation-dependent differences in refractive index. These optically anisotropic materials with directionally dependent property are termed as birefringent. The birefringence could be quantified as the maximum difference between refractive indices of the material. Crystals with asymmetric crystal structures are often birefringent.

6.1 Isotropic and Anisotropic Materials

Optically isotropic materials have the properties of identical refractive index in all directions throughout the crystalline lattice. Examples of isotropic solids are glass, table salt (sodium chloride), polymers, and a wide variety of both organic and inorganic compounds. The isotropic properties remain symmetrical, regardless of the direction of measurement, with each type of probe providing identical results.

The term anisotropy refers to a non-uniform spatial distribution of properties, resulting for sample materials probed from various directions within the same material. Generally, the observed properties are dependent on the particular probe being used and often vary depending on the phenomena for optical, acoustical, thermal, magnetic, or electrical events.

Generally, the anisotropic crystals (viz. quartz, calcite, and tourmaline) have crystallographically distinct axes and interact with light by a mechanism that depends on the orientation of the crystalline lattice with respect to the incident light angle.

6.2 Double refraction

When an unpolarized light beam is incident on a calcite crystal, it usually splits up into two linearly polarized beams. The beam which travels undeviated is known as the ordinary ray (o-ray) and obeys Snell’s Laws of refraction. The second one, does not obey Snell’s Law is known as the extra-ordinary ray (e-ray). The Figure 5 represents the schematic diagram for the polarization of light by double refraction. The appearance of two beams is due to double refraction and a crystal like calcite is referred as a double refracting crystal.

Figure 5. The process of double refraction.

The velocity of the o-ray is same in all directions, the velocity of e -ray is different in different direction. A substance (like quartz, calcite) that exhibits different properties in different directions is called anisotropic substance. The axis along which the two velocities are equal is known as the optic axis of crystal. In a crystal like calcite, two rays have the same speed only along one direction, such crystals are known as uniaxial crystals.

6.3 Types of Birefringent Crystals

6.3.1 Uniaxial crystals

The simplest type of birefringence could be described as uniaxial, where the optical anisotropy is exhibited in a single and all the directions perpendicular to it or, at a given angle to it, are optically equivalent. When the material is rotated around this axis, the optical behavior remains same. This special direction is termed as the “optic axis” of the material. However, for most of the other polarization direction will be partly in the direction of the optic axis, and referred as extraordinary ray. The ordinary ray will always experience a refractive index of no, whereas the refractive index of the extraordinary ray could be between no and n_e, depending on the ray direction as described by the index ellipsoid. The magnitude of the difference Δn = n_e – n₀ is quantified by the birefringence.

6.3.2 Biaxial Materials

Biaxial crystals are somewhat complex and are characterized by three refractive indices corresponding to three principal axes of the crystal. For most ray directions, both the polarizations would be classified as extraordinary rays but with different effective refractive indices. Being extraordinary waves, the direction of power flow is not identical to the direction of the wave vector in either case.

The two refractive indices can be determined using the index ellipsoids for given directions of the polarization. For biaxial crystals, the index ellipsoid will not be an ellipsoid of revolution (“spheroid”) but is described by three unequal principle refractive indices with values nα, nβ, and nγ. Thus there is no axis around which a rotation leaves the optical properties invariant (as observed in uniaxial crystals whose index ellipsoid is a spheroid)

6.4 Categories of Birefringence

Birefringence is an inherent property of many anisotropic crystals including calcite and quartz, or, it can also arise from other factors, such as structural ordering, physical stress, deformation, and strain. The naturally occurring materials with asymmetry in direction dependent refractive are referred to have “intrinsic birefringence”. These materials include many anisotropic natural and synthetic crystals, minerals, and chemicals.

Structural birefringence is used to identify a wide spectrum of anisotropic formations, including biological macromolecular assemblies such as chromosomes, muscle fibers, microtubules, liquid crystalline DNA, and fibrous protein structures such as hair. Unlike many other forms, the structural birefringence is sensitive to the refractive index fluctuations or gradients present in the surrounding medium. The synthetic materials including fibers, long-chain polymers, and composites, exhibit structural birefringence.

Stress and strain birefringence arises when the external forces or deformations act on materials that are not intrinsic birefringent. For example, stretched films and fibers, deformed glass and plastic lenses, and stressed polymer castings. In a naturally isotropic medium, birefringence can be induced by inhomogeneous mechanical stress. This can be observed by placing a piece of acrylic between two crossed polarizers and applying stress to the acrylic. Colored patterns could be observed resulting from the wavelength-dependent effect of stress-induced birefringence.

Generally, the optical fibers are not birefringent, but some birefringence can result from bending (that causes bend losses) and from random perturbations as observed in bent optical fibers, and also due to thermal effects in laser crystals, which can lead to depolarization loss.

Flow birefringence is the one that occur due to induced alignment of materials such as asymmetric polymers that become ordered in presence of fluid flow. Rod-shaped and plate-like molecules and macromolecular assemblies, such as high molecular weight DNA and detergents, are often utilized as candidates in flow birefringence studies.

6.5 Examples of Birefringence

The phenomenon of birefringence occurs in non-isotropic crystals. Below are few examples:
Laser crystals (e.g. vanadate or tungstate crystals) belong to intrinsic birefringent. This is helpful in observing a linearly polarized output without depolarization loss.
The nonlinear crystals for nonlinear frequency conversion are birefringent.
Birefringent crystals are also used for making polarizers.
Straight optical fibers exhibit a small degree of random birefringence, which can disturb the polarization state of guided light over propagation distance. There are polarization-maintaining fibers with strong artificial birefringence and are used for suppressing such effects.

6.6 Quantification of Birefringence

The magnitude of birefringence can be estimated in various ways:

• In bulk crystals, the difference of refractive indices for the two polarization directions is considered to provide birefringence.

• In optical fibers and other waveguides, the difference of effective refractive indices provides more appropriate to consider. This can be directly estimated from the difference in imaginary values of the propagation constants.

• In a waveguide, if the waves with different polarization directions propagate together, their phase relation could be restored after integer multiples of the propagation beat length, lp = 2π/(difference of the propagation constants).

Summary

The efficiency of nonlinear optical processes could be determined from the nonlinear susceptibility of the medium and the phase mismatch parameter. To achieve optimal energy conversion efficiency, the phase-matching condition ∆?= 2?1 − ?2 = 0, has to be fulfilled.
These relations can be translated into relations between the refractive indices at the respective frequencies. For example, in case of SHG, phase matching requires n1(?1) = n2(?2 = 2?1).
Except for very specific cases, material dispersion usually does not allow to fulfill such kind of relations. The most commonly used technique of phase matching relies on utilizing birefringence.
Birefringence is a phenomenon manifested by an asymmetry of properties that may be optical, electrical, mechanical, acoustical, or magnetic in nature.
A wide spectrum of materials display varying degrees of birefringence, but the ones of specific interest to the optical microscopist are those specimens that are transparent and readily observed in polarized lights.

you can view video on Birefringence

References

Geoffrey New, Introduction to Non linear optics (Cambridge, UK University Press, 2011)
G.C. Baldwin, An Introduction to NonLinear Optics(Plenum Press, New York 1969