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OPTICS

DOI: 10.1615/AtoZ.o.optics

Optics is a branch of science that deals with light propagation in vacuum and in various media. By "light," it is usually understood a radiation visible to the human eye (spectra! range 400–700 nm), although the laws of optics are valid for a much larger wavelength range. From the point of view of classical physics, light is transverse electromagnetic waves; the propagation velocity of such waves (light velocity) in the vacuum, c, is constant in any coordinate system and represents one of the fundamental constants. The velocity of light in a medium, v, is always less than in a vacuum. The ratio c/v = n is called the refractive index. From the point of view of quantum physics, light is a flow of "light quantums," or photons. Radiation with a single wavelength (λ) which is described by an infinite sinusoid, is called monochromatic. Near-monochromatic radiation is generated by some types of lasers. Nonmonochromatic radiation can be represented as a superposition of monochromatic waves . If the phase of monochromatic vibration is the same at each point of a certain surface, then this surface is called a wave surface, or wave front. Usually, the form of wave front varies when the wave transmits through a medium and/or optical system.

An important concept in optics is radiation coherence, which means a conservation of constant phase difference of electric (or magnetic) field at two different points in the radiation field during a time period Δt. Spatial and temporal coherence should be distinguished. If a pair of points is chosen on a wave front surface (which at time instant t = 0 is, by definition, a locus of equiphase points) they are said to be spatially coherent during time interval Δtf. If a pair of points is chosen along the line of radiation propagation, they are said to be temporarily coherent during time interval Δtt . A value cΔtt is named the coherence length.

Polarization of monochromatic radiation is characterized by the spatial-temporal behavior of the electric field vector, .

If the Cartesian components of vector have a form:

then a vector = rotates in a transversal plane with a rate of angular motion, ω, relative to a center of coordinates and its end describes an ellipse (elliptic polarization, Figure 1). At φ = π/2 and Ex = Ey, an ellipse degenerates to a circumference (circular polarization); at φ = 0 it degenerates to a straight line (linear polarization). Natural ("white") light is nonpolarized. It represents a collection of light waves with all possible vector directions, simultaneously existing and replacing each other at random. Partially polarized light is characterized by a predominant vibration direction, but not exclusively. After transmittance of natural light through some crystals, it polarizes partially or completely. Polarization also takes place after light reflection or refraction at a boundary of two dielectrics under the condition of oblique incidence of a light ray on the boundary.

Typically, light propagation is described by the wave equation, which follows from electromagnetic field theory (Maxwell's equations). This rigorous approach is the basis of wave (physical) optics. However, many very important results from the practical point of view can be obtained more simply with the use of a geometrical optics approach.

Elliptic polarization.

Figure 1. Elliptic polarization.

Geometrical optics is a name of system of optics in which the limiting case of λ = 0 is considered. In this case, the optics laws can be formulated in terms of geometrical concepts. Geometrical optics deals with infinitely fine light beams, namely light rays, along which light propagates. A boundary between light and shadow is considered as absolutely sharp. Light intensity changes occur due only to change of the light beam cross-section. The main laws of geometrical optics are as follows:

  1. Law of rectilinear light propagation: in a uniform medium the light rays are straight lines.

  2. Law of independent action of light beams: action produced by one beam doesn't depend on the presence of another beam.

  3. Law of light reflection: the incident ray, reflected ray and the normal to the reflecting surface all lie on the same plane, and the angles between these rays and the normal are equal, i.e., an angle of incidence is equal to an angle of reflection.

  4. Law of light refraction: if ray transits from one medium to another, then it changes its direction (refracts). The incident ray, refracted ray and a normal to a separating surface of the two media at the point of ray incidence all lie on the same plane. The angle of incidence, γ, and angle of refraction, θ, (relative to the normal) are connected to each other by equation sin γ/sin θ = nθγ, here nθγ is an index of refraction of the second medium with respect to the first. If it is assumed that nθγ = −1, then it leads to the law of reflection γ = −γ. Thus, any equation for calculation of a refracting system can be applied to a reflecting system.

Commonly, an index of refraction depends on the wavelength. This dependence is called dispersion; it explains light decomposition to a color spectrum after light transmittance through a prism.

An optical system for image formation or for object illumination consists of a set of lenses and mirrors. A lens is called thin if its maximal thickness is significantly less than the curvature radii of its surfaces. A straight line which passes through the lens center and is normal to both lens surfaces is called the main optical lens axis. A point where all rays parallel to the main optical axes converge is called the focus and the length between focus and lens center is called focal length, f. An equation of a thin lens is −1/a1 + 1/a2 = 1/f; here a1 , a2 are the lengths from lens center to an object and its image, respectively. The value of a is assumed positive if the direction with respect to the lens center coincides with the light propagation direction, and negative in the opposite case (Figure 2). For off-axes point A (object AB correspondingly) image formation of any two rays amongst three special rays are used. Ray 1 is parallel to the main optical axis, and after refraction in a lens, it passes across a rear focus F2 . Ray 2 passes across a center 0 of a thin lens without refraction. Ray 3 passes across a front focus F1 and after refraction it passes parallel to the main optical axis. The point where these rays intersect, A′ , is an image of point A. If the rays do not intersect at A′ (this case corresponds to object location between lens and its front focus), then their extensions intersect to the left of a lens and the A image is a virtual one.

An optical system is called ideal if an image of a point also represents the point. Such an image is called stigmatic. Real optic systems have errors, or aberrations, attributed to violation of stigmatic image conditions. The most important are spherical aberration (attributed to nonideal shapes of lenses and mirror surfaces leading to deviation from a stigmatic image), coma and astigmatism (both attributed to violation of the stigmatic image of a point not lying on a main optical axis), distortion (violation of proportions at the object image) and chromatic aberration in nonmonochromatic illumination (colored halos attributed to dispersion of the index of refraction).

During light propagation in a medium, there is light intensity decrease along a propagation direction, attributed to two phenomena, namely light absorption and light scattering. Decrease of monochromatic radiation intensity, during light propagation through a uniform layer with thickness d is described by Bouguer’s law: I = IO exp[−(ka + ks)δ]; here IO is intensity of incident radiation, ka, ks are the absorption and scattering indexes, respectively, which usually depend on the wavelength. The value ke = ka + ks is called the extinction index. In contrast to light scattering, absorption is connected with the transfer of a part of the light energy to the medium. The absorption and refraction indexes depend only on the physical-chemical properties of the material and are called its optical constants. Light scattering is attributed to two phenomena:

  1. deflection from rectilinear light propagation in a medium with nonuniform distribution of index of refraction (e.g., in a gas medium with nonuniform density distribution);

  2. light reflection and diffraction in a heterogeneous medium (e.g., in emulsions and suspensions).

A medium with large quantity of fine inhomogeneities is called turbid. The product K = kaδ is called optical thickness of a layer. If K << 1, then a layer is optically thin, i.e., light propagates through it practically without absorption. For large K a layer is termed optically thin. Light scattering by fine particles attributed to the diffraction phenomenon is described by Mie theory and is called Mie Scattering.

Wave Optics

In wave optics, the law of independent action of light beams is usually invalid. If coherent beams superimpose, then an interference occurs, which simply appears as alternating light and dark bands or rings on a screen. A phase difference of two coherent waves with the same polarization, arriving to a point of observation, A, depends on initial phases of the waves and on a length difference, Δl, between point A and wave sources S1 and S2 (Figure 3). If the initial wave phases are the same and Δl is a multiple of wavelength, then the waves arrive to point A with the same phase, field strengths E1 and E2 are added and light intensity, which is proportional to (E1 + E2)2, correspondingly increases. If l is a multiple of odd number of half-waves, λ/2, then the waves arrive to A with the opposite phases and the field strengths are subtracted. At E1 = E2 light intensity at point A is equal to 0. At all points B equidistant from S1 and S2, an interference leads to light intensity increase.

Optical system for image formation or for object illumination.

Figure 2. Optical system for image formation or for object illumination.

Phase difference of coherent waves with the same polarization.

Figure 3. Phase difference of coherent waves with the same polarization.

If a parallel (collimated) beam of monochromatic light is incident with inclination on a wedge-shaped transparent plate (Figure 4), then due to interference of the beams reflected from front and rear plate faces, an interference picture appears in the form of alternating light and dark bands parallel to wedge edge. If the difference between BA and BCDA is multiple to an odd number of half-waves, then point A lies on a light band; otherwise, it lies on a dark one. These bands are called the uniform thickness bands, because they correspond to a collection of the points where the thicknesses of wedge are the same. Distortion of a straight shape of the bands reveals defects of plate faces. If a plate with parallel faces is illuminated by a monochromatic diverging light beam, then alternating light and dark concentric rings (or its parts) occur when the reflected light is projected onto a screen. They also appear as a result of interference of the beams, reflecting from two plate faces and are called uniform inclination bands.

Wedge-shaped transparent plate.

Figure 4.  Wedge-shaped transparent plate.

Based on interference phenomenon, optical methods and instruments (Michelson, Mach-Zender, Twaiman-Green and other interferometers) are widely used for determining the quality of optical elements, of optical and other materials media homogeneity, for precise measuring of distances and for many other purposes. The most convenient are the laser interferometers in which a laser is used as a source of light.

Diffraction

In wave optics, the law of rectilinear light propagation in a homogeneous medium is invalid. If a plane monochromatic wave falls on an aperture of an arbitrary shape in an opaque screen, then for a small distance behind the screen the cross-section shape of the light beam replicates the aperture shape, i.e., there is a sharp boundary between light and shadow. As the distance increases, and the light spot enlarges, its shape deforms and the boundary between light and shadow becomes less and less sharp. This is due to the light diffraction phenomenon. Light intensity distribution in beam cross-section behind a screen can be approximately calculated by the application of the Fresnel-Huygens principle. According to this principle, each point of wave front is a coherent source of a semispherical elementary wave. An envelope of these waves taking into account their interference represents a new wave front at the next moment. If the condition that the diffraction zone is remote from the beam source is valid (Fraunhofer diffraction), L >> D2/λ, where L is the distance from aperture to the screen, D is a characteristic aperture size, then the distribution curve of light intensity in a diffraction pattern does not depend on the value of L. Diffraction in a near zone is called Fresnel diffraction. In the wave diffraction of a plane monochromatic wave by a round aperture, the remote zone diffraction pattern forms a central light circle, surrounded by alternating dark and light rings.

Figure 5 shows relative local intensity, Ī = I/IO, versus relative radius, = πDr/λL and also the total radiation power, , passing through a circle with radius . The value (φ of corresponding to the central circle is determined by the equation φ = 2.44 λ/D and 84 percent of total radiation energy passes through this circle. In diffraction of a parallel monochromatic light beam by a narrow long slit, a set of equidistant light and dark bands are formed on a screen, and a distance between the bands is proportional to the wavelength. An assembly of a large number of narrow parallel equidistant slits is called diffraction grating. When nonmonochromatic radiation is incident on a diffraction grating, spectral dispersion of the radiation occurs; thus such a grating is used as a dispersing element in many spectral instruments.

Light intensity distribution on a screen irradiated by monochromatic light which has passed through a small aperture at some distance from the screen.

Figure 5. Light intensity distribution on a screen irradiated by monochromatic light which has passed through a small aperture at some distance from the screen.

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