crystal optics [¦krist·əl ′äp·tiks]

The study of the propagation of light, and associated phenomena, in crystalline solids.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

Warning! The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Crystal Optics

an area between optics and crystal physics, encompassing the study of the propagation of light in crystals. Phenomena characteristic of crystals that are studied in crystal optics include double refraction, polarization of light, rotation of the plane of polarization, and pleochroism.

The phenomenon of double refraction was first observed in crystals of Iceland spar by the Danish scientist E. Bartholin in 1669. This date is considered the beginning of crystal optics. Problems of the absorption and emission of light by crystals are studied in crystal spectroscopy. The effect of electric and magnetic fields on the optical properties of crystals is studied in electro-optics and magneto-optics, which are based on the fundamental laws of crystal optics.

Since the lattice constant (of the order of 10 angstroms [Å]) is much smaller than the wavelength of visible light (4,000–7,000 Å), a crystal may be regarded as a homogeneous but anisotropic medium. The optical anisotropy of crystals is caused by the anisotropy of the force field of particle interaction. The nature of the field is related to crystal symmetry. All crystals, except crystals of the cubic system, are optically anisotropic.

The optical anisotropy of transparent nonmagnetic crystals is due to the anisotropy of the dielectric constant c. In isotropic mediums the electric induction vector D is related to the electric field vector E by the equation D = ∊E, where € is a scalar quantity that depends on the frequency in the case of alternating fields. Thus, the vectors D and E have the same direction in isotropic mediums. In crystals the directions of the vectors D and E do not coincide, and the relationship between D and E is more complex, since the dielectric constant e, a tensor quantity, depends on direction in the crystal. A result of this is the ob-served anisotropy of the optical properties of crystals, particularly the dependence on direction of the rate of propagation v of waves and of the index of refraction n. The dependence of the components of the dielectric constant tensor on wave frequency explains the dispersion of the optical properties of crystals.

The dependence on direction of the dielectric constant —and, consequently, of the index of refraction n—can be represented graphically. If radius vectors r with magnitudes r = n = √є, where ∊ is the dielectric constant in direction r, are drawn in all directions from an arbitrary point O of the crystal, then the ends of the vectors r lie on the surface of an ellipsoid, called the optical indicatrix or index ellipsoid (Figure 1). The axes of symmetry of the ellipsoid define the three mutually perpendicular principal directions in the crystal. In a rectangular Cartesian coordinate system, whose axes coincide with the principal directions, the equation of the optical indicatrix is

where nx, ny, and nz are the values of n along the principal directions (principal values of the tensor of the dielectric constant and of n). The optical axis of the crystal is a straight line that passes through a given point O of the crystal at right angles to the plane of a circular section of the optical indicatrix.

Figure 1. Optical indicatrix of a biaxial crystal, a triaxial ellipsoid: (ox), (oy), and (oz) axes of symmetry of the ellipsoid, called the principal axes of the indicatrix; (nx), (ny), and (nz) indexes of refraction along the principal axes; (1) and (2) two circular sections of the ellipsoid; (O1,O1) and (O2O2) optical axes of the crystal

In the case of cubic crystals, which are optically isotropic, ∊ is independent of direction, and the optical indicatrix becomes a sphere with radius r = n = √є. In crystals of intermediate systems (trigonal, tetragonal, and hexagonal), one of the principal directions coincides with the main axis of symmetry of the crystal. The optical indicatrix in such crystals is an ellipsoid of revolution, and the crystals have only one optical axis, which coincides with the axis of rotation of the ellipsoid. Such crystals are called uniaxial. A uniaxial crystal is called optically positive (+) if its optical axis coincides with the long axis of the optical indicatrix (the ellipsoid is elongated along the axis of rotation) and optically negative (−) if the ellipsoid is compressed along the axis of rotation. Crystals of the lower systems (rhombic, monoclinic, and triclinic) are called biaxial. Their optical indicatrix is a triaxial ellipsoid, which has two circular sections and two optical axes (Figure 1).

Because of the noncoincidence of the direction of the vectors D and E, a polarized, plane monochromatic wave in a crystal is characterized by two triads of mutually perpendicular vectors DHv and EH, v′ (Figure 2). The velocity v′ coincides in direction with the Poynting vector S and is equal to the rate of energy transfer by the wave. It is called the ray velocity of the wave. The velocity v is called the normal velocity of the wave. It is equal to the rate of propagation of the phase and the wave front in the direction of the normal vector N to the front. The quantities v and v′ are connected by the relation v′ =v/cos α, where α is the angle between the vectors D and E.

Figure 2

The normal and ray velocities of the wave are determined from the Fresnel equation, which is the basic equation of crystal optics:

Here Nx, Ny, and Nz are projections of the vector normal N onto the principal directions of the crystal; vx = c/nyvy = c/nz are the principal phase velocities of the wave; c is the velocity of light in a vacuum; and nx, ny, nz are the principal indexes of refraction of the crystal.

Since the Fresnel equation is quadratic with respect to v, for any direction N there are two values of the normal velocity, v1 and v2, which coincide only in the direction of the optical axes of the crystal. If vectors of the corresponding normal velocities v are plotted from a point O in all directions N, then the ends of the vectors lie on a surface called the normal surface. It is a double-sheeted surface, each sheet of which corresponds to one of the values of vfor a given direction N. In the case of a uniaxial crystal, one surface is a sphere and the other is an ovaloid, which touches the sphere at its two points of intersection with the optical axis. In the case of a biaxial crystal, the surfaces intersect at four points located on the two optical axes (binormals).

Similarly, the geometric locus of points at a distance v′ from a point O is called the ray surface, or wave surface. It is the wave surface for waves propagating within the crystal from a point source located at point O. This is also a double-sheeted surface. In uniaxial crystals, one surface is a sphere and the other is an ellipsoid of revolution about the optical axis Oz. The sphere and the ellipsoid touch in the points of their intersection with the optical axis. In positive crystals the ellipsoid is inscribed within the sphere (Figure 3,a), whereas in negative crystals the sphere is inscribed within the ellipsoid (Figure 3,b). In biaxial crystals, the surfaces intersect at four points located in pairs on two straight lines that intersect at point O (biradials).

Figure 3. Ray surfaces of uniaxial crystals: (a) positive, (b) negative, (OZ) optical axis of the crystal, (v0) and (vc) phase velocities of two waves propagating in the crystals

Thus, two plane waves that are polarized in two mutually perpendicular planes may propagate in an arbitrary direction N in crystals. The directions of the vectors D1 and D2 of the waves coincide with the axes of the ellipse resulting from intersection of the optical indicatrix with the plane perpendicular to Nand passing through the point O. The normal velocities of the waves are v1 = c/n1 and v2 = c/n2. The vectors E1 and E2 of the waves are also located on two perpendicular planes, and there are two corresponding ray vectors, Si and S2, and two values of the ray velocities, v1 = v1′ cos α and v2 = v2′ cos α. Similarly, for a given direction of the ray S there are two possible directions for the vector E (E1 ⊥ E2), corresponding to the two values of the ray velocity,v1′ and v2′.

The dependence of the ray velocity of a plane wave propagating in a crystal on the direction of propagation and the nature of polarization of the wave leads to the splitting of light rays in crystals. In a uniaxial crystal, one of the refracted rays obeys the usual laws of refraction and is therefore called the ordinary ray, whereas the other ray does not (it does not lie in the plane of incidence) and is called the extraordinary ray. In a biaxial crystal, both rays are extraordinary.

During propagation in a crystal, the two waves arising upon refraction acquire a path difference because of differences in the indexes of refraction and the geometric path, although they re-main coherent. A polarization device may be used to bring the directions of oscillation of the waves emerging from the crystal into the same plane and to observe their interference. Interference of linearly polarized white light is manifested as coloration of the crystal, which depends on the path difference acquired by the beam. Characteristic interference patterns (”conoscopic patterns”) whose shape depends on the orientation of the crystal may sometimes be observed.

In crystals of some symmetry classes, rotation of the plane of polarization is possible in addition to double refraction. Two elliptically polarized waves (with opposite senses of rotation), each with its own index of refraction, may propagate in each direction in such crystals. The polarization is circular only in the direction of the optical axis, which leads to rotation of the plane of polarization of linearly polarized light incident on the crystal.

In the case of strongly absorbing crystals, a linearly polarized wave propagating in the crystal is split into two elliptically polarized waves with the same sense of rotation. Such crystals exhibit different absorption of waves with different polarization.

Each crystal has its own characteristic set of optical proper-ties, which can be used for identification. The most important properties for uniaxial crystals are the indexes of refraction n0 of the ordinary wave and ne of the extraordinary wave and the difference Δn between them (magnitude of the double refraction), as well as the dependence of the characteristics on the wave-length (various types of dispersion). Biaxial crystals are characterized by a more complex set of properties. In applied crystal optics, which is concerned with analysis of minerals and rocks, various methods for measuring these quantities were developed for various preparations of minerals in the form of powders and thin plates (sections). The principal methods are the immersion method, for determining the index of refraction using special liquids or melts with known indexes of refraction, and Fedorov’s method, for the determination of the orientation of the indicatrix, using a microscope stage that rotates the crystal about various axes. Most crystal-optics measurements are performed with a polarization microscope. There are reference books containing collected information on the optical properties of most known minerals.

The methods of crystal optics are of great importance in physics research (for example, in the generation of polarized light and analysis of elliptically polarized light; in various instruments for controlling beams of light) and in chemical engineering (analysis of materials; optical activity).

REFERENCES

Born, M., and E. Wolf. Osnovy optiki. Moscow, 1970. (Translated from English.)
Landsberg, G. S. Optika, 4th ed. Moscow, 1957. (Obshchii kurs fiziki, vol. 3.)
Fedorov, F. I. Optika anizotropnykh sred. Minsk, 1958.
Shubnikov, A. V. Osnovy opticheskoi kristallografii. Moscow, 1958.
Beliankin, D. S., and V. P. Petrov. Kristallooptika, 4th ed. Moscow, 1951.
Tatarskii, V. B. Kristallooptika i immersionnyi metod issledovaniia mineralov. Moscow, 1965.
Ditchburn, R. Fizicheskaia optika. Moscow, 1965. (Translated from English.)

V. B. TATARSKII and B. N. GRECHUSHNIKOV

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.

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Crystal optics

The study of the propagation of light, and associated phenomena, in crystalline solids. For a simple cubic crystal the atomic arrangement is such that in each direction through the crystal the crystal presents the same optical appearance. The atoms in anisotropic crystals are closer together in some planes through the material than in others. In anisotropic crystals the optical characteristics are different in different directions. In classical physics the progress of an electromagnetic wave through a material involves the periodic displacement of electrons. In anisotropic substances the forces resisting these displacements depend on the displacement direction. Thus the velocity of a light wave is different in different directions and for different states of polarization. The absorption of the wave may also be different in different directions. See DichroismTrichroism

In an isotropic medium the light from a point source spreads out in a spherical shell. The light from a point source embedded in an anisotropic crystal spreads out in two wave surfaces, one of which travels at a faster rate than the other. The polarization of the light varies from point to point over each wave surface, and in any particular direction from the source the polarization of the two surfaces is opposite. The characteristics of these surfaces can be determined experimentally by making measurements on a given crystal.

In the most general case of a transparent anisotropic medium, the dielectric constant is different along each of three orthogonal axes. This means that when the light vector is oriented along each direction, the velocity of light is different. One method for calculating the behavior of a transparent anisotropic material is through the use of the index ellipsoid, also called the reciprocal ellipsoid, optical indicatrix, or ellipsoid of wave normals. This is the surface obtained by plotting the value of the refractive index in each principal direction for a linearly polarized light vector lying in that direction (see illustration). The different indices of refraction, or wave velocities associated with a given propagation direction, are then given by sections through the origin of the coordinates in which the index ellipsoid is drawn. These sections are ellipses, and the major and minor axes of the ellipse represent the fast and slow axes for light proceeding along the normal to the plane of the ellipse. The length of the axes represents the refractive indices for the fast and slow wave, respectively. The most asymmetric type of ellipsoid has three unequal axes. It is a general rule in crystallography that no property of a crystal will have less symmetry than the class in which the crystal belongs.

Index ellipsoid, showing construction of directions of vibrations of D vectors belonging to a wave normal s enlarge picture

Index ellipsoid, showing construction of directions of vibrations of D vectors belonging to a wave normals

Accordingly, there are many crystals which, for example, have four- or sixfold rotation symmetry about an axis, and for these the index ellipsoid cannot have three unequal axes but is an ellipsoid of revolution. In such a crystal, light will be propagated along this axis as though the crystal were isotropic, and the velocity of propagation will be independent of the state of polarization. The section of the index ellipsoid at right angles to this direction is a circle. Such crystals are called uniaxial and the mathematics of their optical behavior is relatively straightforward.

In crystals of low symmetry the index ellipsoid has three unequal axes. These crystals are termed biaxial and have two directions along which the wave velocity is independent of the polarization direction. These correspond to the two sections of the ellipsoid which are circular. See Crystallography

The normal to a plane wavefront moves with the phase velocity. The Huygens wavelet, which is the light moving out from a point disturbance, will propagate with a ray velocity. Just as the index ellipsoid can be used to compute the phase or wave velocity, so can a ray ellipsoid be used to calculate the ray velocity. The length of the axes of this ellipsoid is given by the velocity of the linearly polarized ray whose electric vector lies in the axis direction.See Phase velocity

The refraction of a light ray on passing through the surface of an anisotropic uniaxial crystal can be calculated with Huygens wavelets in the same manner as in an isotropic material. For the ellipsoidal wavelet this results in an optical behavior which is completely different from that normally associated with refraction. The ray associated with this behavior is termed the extraordinary ray. At a crystal surface where the optic axis is inclined at an angle, a ray of unpolarized light incident normally on the surface is split into two beams: the ordinary ray, which proceeds through the surface without deviation; and the extraordinary ray, which is deviated by an angle determined by a line drawn from the center of one of the Huygens ellipsoidal wavelets to the point at which the ellipsoid is tangent to a line parallel to the surface. The two beams are oppositely linearly polarized.

McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.


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1. Paraxial Geometrical Optics and the System Matrix

Refraction and Reflection

The law of refraction (Snell’s Law): When a ray of light is refracted at an interface between two uniform media, the transmitted ray remains in the plane of incidence and the sine of the angle of refraction is directly proportional to the sine of the angle of incidence.

The law of reflection: When a ray of light is reflected at an interface dividing two uniform media, the reflected ray remains in the plane of incidence, and the angle of reflection equals the angle of incidence. The plane of incidence is defined by the incident ray and the surface normal vector at the point of incidence.

Snell’s Law is usually written:
n1 sin θ1 = n2 sin θ2,
with n1, n1 the refractive indices of the two media, where the speed of light vin a medium of refractive index n is v = c/n if c is the speed of light in a vacuum. Slower lightspeed in second medium, more refraction. Here’s a picture of reflection and refraction.

snell

Using these two laws we can analyze systems of lenses and mirrors like the all-lens system below: lenses

After all this, we’d like to use Snell’s Law in our geometrical optics approximation to reality. Linearize problem with paraxial rays, those that never make large angles with the optical axis. I.e. assume
sinθ = tan θ = θ
The paraxial Snell’s Law is
n1 θ1 = n2 θ2 .

This is a “small angle approximation”. Suc linear approximations describe a given function (in some locality) as a linear function. Infinite series are in your future! Here,
sin(x) = x – x3/3! + x5/5! – …, near 0, and for small x we’re only considering the first term.

Thin Lenses

Different rays travel different distances through lenses, which are fatter and thinner. So just what happens depends on just where and at what angle the ray enters the lens. Inconvenient, nonlinear. The thin lens assumption is: “consider an infinitely thin lens” with all the refractive power and none of the annoying size of an actual object.

thinlens

Geometrical optics uses pretty simple algebra (but pretty complicated diagrams) to derive elegant formulae (in terms of things like radii of curvature, indices of refraction, and distances) that describe spherical mirrors, refraction at spherical surfaces, thin lenses, thick lenses etc.

Focal Length

Focal length f is the (signed) distance to the image it forms of an object at infinity. Convex lenses have positive focal lengths, Concave negative).

The lensmaker’s equation predicts the focal length of a lens in terms of its refractive index, that of the medium it is in, and its radii of curvature. In air (of refractive index n = 1), one version of the thin-lens equation states that an object at distance s is imaged at point s’ by a lens of focal length f if
1/s + 1/s’ = 1/f.

The power P of a lens measures how strongly it bends light, and is defined as 1/f. In an x-y plane,

We measure angles that rays deviate from the optical system (x) axis (rays parallel to the x-axis are at 0 radians). Rays from the left rising in the y direction are positive. A lens bends an incoming ray of angle θ, according to our approximations, to give it a new angle:
θ’ = θ – yP
if the ray impinges on the lens at height y. Thus
θ’ = θ -(1/f)y

The Method

  • Describe linear optical system as the product of matrices for its component parts acting in series.
  • Describe a ray moving left to right by its instantaneous height and angle of flight: [y,θ]T.
  • Describe paraxial optical elements by 2×2 matrix transformations (ray tranfer functions) representing what happens to a ray at an optical element.
  • The product of the transfer matrices describes the optical system as a system transfer function.

Ray Diagrams

Diagram the system as thin lenses, a 2-D plot of Y versus X, with the X axis being the optical axis. An object being imaged is considered to be in an input plane at the left of the diagram (below, it’s at x = 0), and the output plane is wherever we want to compute the final height of the ray. Can compute (inter alia):

  • front and back focal length. The latter is just the effective focal length of the entire system, and the former is similarly the focus point of angle-0 rays coming in from the right, headed left.
  • Axial image point of an object point nearer than infinity on the optic axis. All the rays from that point are focussed there, and all points from that input plane are the image plane contining the axial image point.
  • Linear magnification of the system is yf/y0, the relative height of a ray at the input and output planes.

Ray Transfer Matrices

Consider a ray [y0, θ0 ]T moving (translating) through a homogeneous medium. If it moves axially by L at an angle θ0, Its new description is
[y0 + L tan θ0, θ0]T. The paraxial assumption is that tan(x) = sin(x) = x, so we can describe the resulting linear tranformation of the ray’s description by

|y

1

| = |1 L|  |y

0

|
|θ

1

| = |0 1|  |θ

0

|

This simple matrix is how we describe the change in the ray as if moves through some uniform medium for an axial distance.

Remember
θ’ = θ -(1/f)y.
So…

|y

1

| = |1    0|  |y

0

|
|θ

1

| = |-1/f 1|  |θ

0

|

The transfer matrices for spherical or refraction interfaces, spherical mirrors, and thick lenses are similarly simple.

System Transfer Matrix

Express a paraxial system with elements described by 2×2 matrices, say M1, M2, M3, M4 in order from left to right. Then for ray r, M1*r is the ray after the first element, M2*M1*r, M3*M2*M1*r, and M4*M3*M2*M1 is the complete transformation and the system transfer matrix.
Note the system components appear in reverse order.

Use ray-transfer technique find: focal points, nodal points and first and second principal planes.

The four elements of the transfer matrix have discernable semantics, which can be illuminated by considering the physical meaning of setting each one to zero. For now, though, we’re done.

Example: Focal Point

Consider three lenses of focal length 200, -50, and 50 mm. The first and second are separated by 100 mm. the second and third by 50 mm. The input plane is 200 mm. in front of the first lens. What is the (back, effective) focal length of the system? focal

Send a ray parallel to the axis into the 6-element system (translation, lens, translation, lens, translation, lens). The last ray we get is the one emerging from the last lens at X=350. It turns out to be (3, -.04)T. We’re after the focal length g= y/tan(θ), (the base of the final triangle in the figure from X= 350 to X= 425). With the small angle assumption, the tangent of the angle is the angle, so this ray intercepts the axis at g= y/θ, or 75mm. We can then add a final, seventh, transfer of this distance, which should bring the ray down to the axis at the focal point, and that final system produced the figure above.

Example: Image Plane, Axial Image Point

The axial image point of a system is where the image of a ray starting out at the origin crosses the optic axis. It’s like the focal point, and calculated the same way, only using a different initial ray. It turns out that all the system’s image points from an object at the origin will fall in the plane at that distance, so that is where an in-focus image will be formed.

Use slightly different 3-lens system: lenses are spaced out by 300, 100, and 50 of focal lengths 200, -50, and 50. Shooting out a ray from the origin at angle .01 and calculating the axial image distance (it’s 95 mm. out from the third lens) we can compose a final translation after the 3rd lens of 95 mm to get this plot:

axial

Several Rays: imageplane

Use the y height at the image plane to get the ratio object-height/image-height, or linear magnification, as .2 (the image height is upside-down).

2. Ray-Casting and Systems of Linear Equations

The Problem and the Method

Ray-casting: given a point of view and direction of gaze (ray), compute the color and intensity of the light in that direction. In geometrical optics we follow light-paths (rays) through optical systems with geometry, some algebra, and a few physical laws.

Basic case: given a 3-D point of origin for the ray and its direction, where does it intersect a given plane in space? No different from the high-school algebra “line-intersect plane” problem, and in optics sometimes called “intersecting a ray with a plane mirror”.

As usual, represent points p, x, r etc.in 3-D by (x,y,z)T vectors. Let’s represent directions α, β etc. in 3-D by (x,y,z)T unit vectors. They form a family that lives on a sphere of unit radius centered on the origin. You can see that every direction corresponds to a vector (point) whose head is on this Gaussian Sphere.

Lines and Planes

The standard way to intersect a vector with something is to stretch it out in its direction until it hits; the resulting length is all we need. So: Any point on a ray can be written
Ray: r = r0 + d α,
with d the length, r0 the ray’s origin, α its direction. This vector equation represents three linear equations in x,y,z.

The plane equation is linear, and a 3-D version of the familiar line equation:
Plane: Ax + By + Cz + D = 0
Written like this, [A, B, C]T is a vector specifying a direction, in fact the direction normal to the plane. Scaling the whole equation so that this direction is a unit vector gives the new, scaled value of D a meaning: it’s the perpendicular distance from the plane to the origin.

Force our infinite mirror, or plane, to pass through the origin, so we can describe it with a linear equation: one strictly in (x,y,z) (D = 0: no pesky constant).
Plane through Origin: Ax + By + Cz = 0, or
(A/C)x + (B/C)y + z = 0.

For a more general raycasting project, still pretty easy, see Pinholes and Beachballs (or spherical chickens!).

Intersection

The equations Ray and Plane above are four linear equations. A solution to them gives thed at which the ray intersects the plane, and we’re done. Preview: we write down the equations, do a quick massage, write them in matrix form, and the answer is obvious.

Ray origin: (x0,y0,z0).
Ray direction: (α123).
3 Ray Equations, 1 Plane Equation:
x = x0 + α1d
y = y0 + α2d
z = z0 + α3d
(A/C)x + (B/C)y + z = 0.

We need four equations for unknowns x,y,z,d, but clearly we only really need three numbers, x,y,d say, from which we can easily solve for z using the Plane through Origin equation. Premptively using that equation first, we get an expression for z to substitute into the third ray equation and we can rewrite the system as:
x0 = x – α1d
y0 = y – α2d
z0 = -(A/C)x – (B/C)y – α3d.

Matrix Formulation

Express above system as matrix equation:
[x0,y0,z0]T = M [x,y,d]T , where M is

| 1             0   -α1 |
| 0             1   -α2 |
| -(A/C) -(B/C) α3 | .

If u = M v, then v = M-1 u, So we can solve (x,y,d) in terms of what we know or can easily compute ( x0,y0,z0, M-1).

Then ignore d, put x,y into the Plane through Origin equation, get z, and we’ve got our intersection-point.

References

  1. Tom Brown; Course Notes, Optics 211 2008, Part I.
  2. F.L. Pedrotti and L.S. Pedrotti; Introduction to Optics, Prentice-Hall 1987.
  3. Wikipedia.
  4. R. W. Ditchburn; Light, Wiley Interscience, N.Y. 1963.


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