For description of optical radiations in a geometrical optics the notions of optical ray and optical beam are used. Optical ray is a line, along which the energy of optical radiations spreads. An optical beam is the aggregate of optical rays. The theory of geometrical optics is based on the laws of rectilinear propagation, independence of propagation and reversibility of motion of rays, reflection and refraction of radiations on the boundary of two environments.

In obedience to the law of rectilinear propagation of radiations, in a homogeneous environment an optical ray spreads rectilinearly with permanent velocity. If an environment is heterogeneous, in different regions the velocity of propagation of the ray is different, and the straightforwardness of propagation of the ray is then violated. The law of rectilinear propagation of radiations is not executed also, when on his way there is the obstacle as an opaque screen or diaphragm with the very small hole. In this case the diffraction occurs.

The law of independent propagation of different rays determines: the separate rays going out from different centers of radiations do not influence one on other. If a few cones of coherent rays, going by various ways, meet in one point, their energies are summed, appearing in interference, strengthening or diminishing of radiations.

Principle of reversibility of motion of rays determines the independence of direction of propagation of the ray on the same way. Naturally, it is necessary to take into account the losses of radiations as a result of absorption, dispersion and reflection.

The reflection and refraction of radiations arise up as a result of passing of the ray through heterogeneous environments. Flat boundary of division of two endless homogeneous environments, in which the radiations propagates with velocities of v₁ and v₂, is the simplest heterogeneity_.

The ray 1(fig. 2,a), falling from the first environment under the angle of ε to the perpendicular, on the boundary of section is divided into the reflected ray 2 and refracted ray 3. The reflected ray 2 propagates in the first environment under an angle (- ε) with the same velocity of v₁, as the ray 1 does. A refracted ray 3 spreads in the second environment under the angle of ε' to the perpendicular. The geometrical arrangement of falling, reflected and refracted rays is determined by three Snellius laws:

1) the angle of incidence is equal to the angle of reflection: e = - e;

2) the falling, reflected and refracted rays lie in one plane with the normal, set in the point of falling on the boundary of environments;

3) the ratio of sine of angle of incidence to the sine of angle of reflection is a permanent value: sinε/sinε' = n_2,1 = const, where: n_2,1 – relative index of refraction of two environments, equal to the ratio of absolute indexes of refraction of these environments to the vacuum, in which velocity of propagation of optical beam is equal c. It gives: n_2,1 = n₂/n₁ = (c/v_opt2)/(c/v_opt1) =  v_opt1/v_opt2 = sinε/sinε'.

Determination of indexes of refraction of matter in relation to a vacuum is a difficult task; therefore they are determined in relation to air. The index of refraction of air relies on a temperature and atmospheric pressure: at the calculations of the optical systems it is usually adopted equal to 1.

For transparent and not absorbing environments the optical beam, falling on the boundary of division fissions onto reflected and refracted beams. The coefficient of reflection can be calculated according to the formulas of electromagnetic theory; it relies on the indexes of refraction of both environments n₁ and n₂, angle of incidence ε, and also depends on the degree of polarization of the ray. If n₂ = n₁, the coefficient of reflection is equals to zero, and a boundary of division of two environments becomes invisible.

In transition of the ray from an environment with the greater index of refraction n₁ into an environment with the smaller index of refraction n₂ in obedience to fig.1, sinε'/sinε = n₁/n₂ > 1 and a refracted ray is removed from the perpendicular (fig. 1,b).

At gradual growth of angle of incidence a moment comes, when sinε'=1, ε'=90º, and a refracted ray travels along the boundary of division. In this case ε = ε_TIR, we have sinε_TIR = n₂/n₁.   

At the subsequent increase of angle of incidence (ε > ε_TIR) only the reflected ray remains. This phenomenon is called the total internal reflection (TIR), and the angle ε = ε_TIR is called the angle of the total internal reflection. When a ray passes from an environment with the index of refraction of n into air, the following correlation is just to: sinε_TIR = 1/n.   

The special interest represents an environment with the gradient change of index of refraction in space. For the analysis of physical process of propagation of the ray in such environment, imagine an environment, consisting of row of vertical layers of small thickness Δz, the indexes of refraction of which differ on a small size Δn. At every boundary of next layers (fig.1,b) there will be the partial weak reflection and partial refraction, and a ray will gradually deviate from primary direction. 

Written by Vasil Sidorov on August 04, 2010 in queltanews.com

Technopark QUELTA,

Nizhyn Laboratories of Scanning Devices

sidorovvasil@gmail.com

References

1.       Rebrin Y.K., Sidorov V.I. // Optical Deflectors. Kiev: Tékhnika, 1988. 136 pp.

2.       Rebrin Y.K., Sidorov V.I. Optical mechanical and holographic deflectors // Results in science and technology. Radio engineering. Vol. 45. - Moscow: VINITI, 1992. - 252 pp.

3.     Rebrin Y.K., Sidorov V.I. Holographic devices of control of an optical ray. – Kiev:  KHMAES, 1986. - 124 pp.