FORMATION OF A REFRACTED BEAM DURING LIGHT RAY MOTION ALONG THE INTERFACE OF OPTICALLY HOMOGENEOUS MEDIA UNDER CONDITIONS OF HIGH VALUES OF THE RELATIVE REFRACTIVE INDEX. II Yu. I. Terent'ev

UDC 535.0

As investigations show, the light ray intensity distribution in a refracted beam formed by the rays moving initially along the interface of two optically homogeneous media in the optically less dense medium, is described correctly by (ii) in [i] for not all values of t and n . The reason is that the passage of the rays being deflected into the dispersion zone into an optically denser medium along the whole interfaclal surface holds only for n ~ n , where cr ncr is the critical value of the relative index of refraction on the boundary of the connected media, which decreases as n grows. In case n > ncr the source of the refracted rays is just the part of the interface of length tu. This circumstance can be explained on the basis of the formula sin S = I/n [2], by the increase in the efficiency of ray deflection from the initial direction as n grows, consequently, the rays incident in the dispersion zone reach the interface on a path tu (Fig. i) less than the total length of the zone. The appropriate diagram is shown in Fig. 2; here t is the length of the face along which the parallel beam of monochromatic rays is propagated, t is the length of its active part. G As is seen from the figure, for the possibility of the refracted rays emerging into the air, the plate is replaced by a prism with the hypotenuse face perpendicular to the rays refracted at the limit angle. Because of the absence of differences, in principle, between the present scheme and the scheme in [i], the rays being deflected in the dispersion zone of air should go into the glass at different relative angles even on small sections of the face. Hence, we use the Fresnel zone method here also in the computation of the total effect of rays refracted from different sections of the active part of the face at points of the screen. As is easy to conceive, the refracted beam has maximum intensity at those points of its section at which there is no path difference between the rays. For this it is necessary that An = tu. Let us substitute the value of tu in the formula sin S = A/tu, whereupon we arrive at the formula sin ~ = i/n, which shows that the refracted beam is propagated in conformity w l t h t h e law of refraction. Therefore, under these conditions the direction of refracted ray propagation is determined by the same causes as for ordinary refraction. According to the figure m/t = sin ~ = l/n, m = t/n; ~ = t -- ta/n; a=ta

h = r l -l- r =

]/.~;

r (m + nR) nR

s (1 q- n R ) H~r2q-s-=

r.!=sin~,,

r hn~-R r~ t ~, n 2 R , l

, r=

HnR .

S - - - -

l q- n R

sin= l

H=~a--h,

'

s--

rra

rra

s

sinl3~' R =stna'z' (a

--

(t a ~ - -

h) R n --

l -4- R n

sl r2 - - n R "

(1)

I --nh) Rn

(t - - t~) -t" Rn~

Substituting their values instead of I and s in (I), we obtain r2='

(t - - t.) (t.. I/n--~-- 1 [(t - - t a ) W R n2] n

,h)

Institute of Optics of the Atmosphere, Siberian Branch, Academy of Sciences of the USSR, Translated from Izvestiya Vysshlkh Uchebnykh Zavedenll, Fizlka, No. 7, pp. I15-117, July, 1979. Original article submitted April 12, 1978,

792

0038-5697/79/2207- 0792507.50

9 1980 Plenum Publishing Corporation

i

! Fig. i

Fig, 2

Fig. i. Diagram of ray deflection into the dispersion zone for a large relative index of refraction of the interface, Fig. 2. Diagram of the observation of interference between the refracted rays as a parallel light beam moves in air along the prism face.

Let us find the expressions for Ax, A2, Aa, and A,: s2 AI -- 2R -r~

(ta ] / n-F'-~-~ 1 - - nhF R n ~ 2 [(t - - ta) + Rn=] 2 ' A2 th~n

r~

~3 = 2"-'m= '2 (t q- n~-R) ~ ' A , -

2l

h~Rn~ 2 (t + Rn~) '

(t - - ta) (ta l / ' ~ - - 1 - - nh)2 -

2 [(t

- - ta) + Rn=] ~ n

In the general case the optical path difference between rays i and 2 is determined by the expression (m + ~3) n = (R + ~ ) - - t~ - - (l + ~,) n - - (R + ~ ) = ~

~',

from which

<21

A3n q- A2 - - A~n - - A1 = tc ~ . 2

A3n q- A2 - -

n~h2

<3)

2 (t + n=R) '

t a~ ( n = - - 1) - - 2t a l/'n-i-~--- lnh q- n=h2

~4n q- A1 =

<4)

2 [(t - - ta) -{- Rn2]

Let us substitute (3) and (4) into (2), and after manipulation we have h~

2 ] I n ' 7 - - 1 (t + .qn "~)h + t a (n 2 - - 1~ ( / + n=R) .4:.~ (t q- n~R) [(t - - ta) q- n =R] =0. n n~ n=/a

from which

I,= v.=-i

n

+.--m [i- ]/'i

tr t + n~R

[(t-

ta) q- n2R]

l

( n ~ . - 1) (t -t- n~R) ta ij ,

(5)

793

TABLE i h (3)comp, .1211i1 maxl mini max2 min~ max3

0,303 0,429 0,525 0,606 0,68

h exptl. mm

h (6), mlTl

0,49 0,81 t,015 t,17 1,~

0,455 0,804 0,981 1,165 1,331

where k is the number of half-waves included in the optical path difference between rays i and 2, and k will also equal the number of Fresnel zones on the surface Ap if the path difference between the outer rays of the beam and the middle ray refracted at the limit angle is considerably less than l/2. The first member of the quadratic equation is much less than the second, hence, it can be neglected; then

h-- t.V.-:~-~- ~ + ~ [ ( t - t=~ + .~.,~] 2n

2nta "I/n=----I

(6) '

w h e r e K = O, 3 , 5 , . . . corresponds to maxima, and ~ = 2, 4, 6, . .. to minima of the In this case the first term turns out to equal h max~, while the second determines tion of the succeeding maxima relative to max,.

intensity. the posi-

In order to verify the formula, the fringe locations in the interference pattern were computed for A = 0.63 pm; R = 500 mm; t = 20 mm; n = 1.713 for ta = 1.12 n~n. The computation results are contained in Table i . Placed there also are the data of an experiment formulated on the basis of the graph in [3], and the results of a computation made on the basis of (3) therein. Analysis of the tabulated data shows the good agreement between the values of h computed by (6) and experiment. Therefore, the formula obtained correctly characterizes the intensity distribution in the refracted beam. At the same time, the sharp discrepancy between the results of the computation using the formula from [3] and experiment is seen. This should have been expected since the former does not reflect the dependence of the fringes on t and even more so on ta . As follows from a comparison between t and t, the active part of the face is a small part of the surface for this 9 of n. u The author is grateful to S. D. Tvorogov for attention to the research and discussion of its results. LITERATURE CITED

l. 2. 3.

794

Yu. I. Terent'ev, Izv. Vyssh~ Uchebn. Zaved., Fizika, No. 7, 112 (1979). Yu. I. Terent'ev, Izv. Vy~sh. Uchebn. Zaved., Fizika, No. 8, 48 (1977). M. P. Kolomiev, P. N. Svirkunov, and S. S, Khmelevtsov, Pisma Zh. Eksp. Teor. Fiz., 26, No. 3, 153 (1977).