other beams. This gives rise to a reflected beam g at an angle of reflection B and a refracted beam h at an angle of refraction C, where, of course, from Snell's Law,

R

where N is the index of refraction of the second medium (glass) relative to the first (air). If we send the light on to the interface backwards along the direction of

g, we

produce a beam backwards along f. refracted beam j at an angle C cident light is sent backwards along the beam direction h, the beam j is excited by reflection and f by refraction, while incident light along j excites beams h and 8. Thus we have a situation analogous to a hybrid circuit in electrical circuit theory, the best known special case of which is the microwave "Magic Tee" junction.

The beam h is not produced, but we produce a different measured in the other direction. Similarly if the in

In general for calculating the transmissivity T and the reflectivity R, we must distinguish between the case where the electric vector of the incident radiation is perpindicular to the plane of incidence (denoted by the subscript e) and when it is parallel (denoted by the subscript a). As can be found in many textbooks, [4]

and

e

=

T (sin 2B) (sin 2C)/(sin2 (B+C) cos2 (B-C),

a

(7)

In principle, by the use of Snell's Law, eq (3), these coefficients can be given as functions of B and N rather than of B and C, but the form of these equations is not

convenient.

When B and C are small, the sines and tangents can be approximated by the appro

However, at normal incidence itself, B = 0, C = 0, and eqs (4-7) become indeterminant.

Also, for all cases, in accordance with conservation of energy,

C.

It is to be noted, that in eqs (4-7), R and T are functions only of BC and the magnitude (not the sign) of B Also in eqs (8) and (9), R and T are invariant with the replacement of N by 1/N. Therefore, for a beam reflected in medium 1 at an angle B, R has the same value as for a beam reflected in medium 2 with an angle

C, where B and C Similarly, for a beam incident at an angle B in medium 1 and transmitted into medium 2, T

are related by eq (3).

has the same value for a wave incident at an angle C in

medium 2 and transmitted into medium 1.

Therefore, in principle, all of the R and T coefficients involved in a high order F can be determined experimentally. For example, if we want to and T at the point f in figure 1, we send in a calibrated beam backwards

beam attenuation factor

know R

along the direction that the third order beam had emerged. This generates a reflected beam

at

f (which is not shown in the diagram). If the power in this beam is measured by a

suitable calibrated detector, and if this power is divided by that in the incident beam, the

At the same time the transmission coefficient at f for the emerging third order beam is

where

Τ1

is the transmission coefficient at the point of entrance of the incident beam, is the transmission coefficient at the point of exit of the m'th order beam, and R is

T

m

181

k'th order beam. These co

the reflection coefficient at the point of emergence of the efficients, of course, are assumed to be evaluated at the appropriate respective angles of incidence.

Now let us suppose that the incident beam had been backwards along the direction of the original emergent beam of order m. From the discussion of the properties of a single interface, it can be inferred that a beam would emerge in the direction of the original incident beam. Since the order of the factors in eq (11) is unimportant, the attenuation factor F is the same as for the original configuration. In other words, in the language of circuit

theory, the beam splitter is a reciprocal device.

2

m

Next let us multiply numerator and denominator of eq (11) by T where m' is internediate between 0 and m, and group the factors as follows:

In principle, each quantity in brackets can be determined experimentally. The first PRA quantity is just the attenuation factor of the

m'

order beam. The last factor is the

attenuation factor which applies when the incident light is sent backwards along the direcion of the m order beam, and when the observation is made on a beam emerging from m ort (but leaving the surface with an opposite angle to the normal). The middle quantitity

can be found by sending a beam backwards along the original m beam direction and observing its -1 reflection.

This gives R directly, and Tm'

can be found from it by eq (10).

m

8

limited range of the instruments, it can be broken down into the product of quantities which

can be brought within range, and yet it is not necessary to measure individually every coefficient, as had been suggested earlier.

In applying the formulas that have been given for calculating the individual R and coefficients and thus for calculating the attenuation factor, it remains for us to develop formulas for the angles of incidence. For this purpose we refer to figure 4, which duplicates the first few beams of figure 1 with the same notation but, for clarity, with an expanded scale and with the angles made much larger.

T

By applying the theorem that the sum of the angles of a triangle is 180° to the triangl abc, we find that

By applying this theorem to the triangle bcd, we find that

(13)

G1 = 2E1 -C
G1

= C + 2A.

Thus, G1 exceeds C by 2A.

We can see by inspection that if we were to repeat this argument to the triangles cde

It is important to point out that there is an implied sign convention with regard to

the angles B, C, E and G. These angles are all positive as shown in figures 1 and 4.

the incident beam associated with any of these beams should be on the opposite side of the respective normal to the interface, it should be considered as a negative angle.

If

For negative values of the angle of incidence of the entering beam B, some of these angles may become negative, and in extreme cases the light may be propagated internally towards the apex of the prism. However, ultimately, when L becomes large enough, there are

enough increments of 2A to make E and G1 positive, and the light is reversed in direc

tion. Then the higher order beams emerge directed towards the base. In the usual situation prevalent under practical conditions, all even order beams emerge directed towards the plane of the base, and, the odd order ones emerge in the sequence shown in figure 1.

Since

and GL are smaller for negative values of B than for positive ones, the

minimum polarization effects occur at small negative values of B, as to be shown later by Yo the graphs which have been based upon calculations from the theory.

YOF

17

ARI

Therefore, especially,

in working with higher order beams, one should avoid using a positive value of B.

It is convenient to record the relationship between the beam order m and the index L. For odd orders,

ΛΙ

уна

pressions for explicit formulas for these in terms of A, B, and N, but if the exact form

10