BRIEF COMMUNICATIONS AND LETTERS TO THE EDITOR NATURE OF THE INTENSITY DISTRIBUTION IN A REFRACTED BEAM FORMED BY RAYS INITIALLY MOVING ALONG THE INTERFACE OF TWO OPTICALLY HOMOGENEOUS MEDIA. I Yu. I. Terent'ev
UDC 535
The present paper is a further step in the investigation of the refraction of grazing light rays occurring in the dispersion zone of an optically less dense medium according to [l], which is located above the interface of two optically homogeneous media. The present report is devoted to clarifying factors governing the ray intensity disbribution inthe refracted beam and its divergence. Let us direct a parallel beam of monochromatic light along the narrow face of a plate HC6 in a cuvette with dimethylphthalate (Fig. i [l]), and we place a screen in the path of refracted rays at a large range from the cuvette, then an interference pattern is seen within the limits of the refracted beam projection. This pattern consists of the brightest central strip and less wide and intense strips on either side. Rays refracted at the limit angle correspond to the center of the pattern. According to the preceding report, rays deflected into the dispersion zone of the plate go over into the denser medium over the whole surface of its narrow face. Moreover, as test shows, the angular beam width of the refracted rays is practically inversely proportional to t h e w i d t h of the face. Because of these facts, small sections of the plate face can, in all probability, he considered sources of refracted rays leaving at different angles relative to the ray refracted at the limitangle, and therefore, able to be incident from any surface element of the narrow face onto any point of the refracted beam projection. Starting from the above, the Fresnel zone method should be applicable to the determination of the positions of the maxima and minima of the interference pattern. On the basis of this assumption, let us derive a formula characterizing the location of the interference fringes. To do this, we use the diagram in Fig. i, where i) is the HC6 plate of thickness t, is the spacing between the plate and its emergence from the cuvette, L) is the spacing between the cuvette and the screen, h) is the spacing between the centers of the strips to the projection of the narrow face of the plate on the screen, n~) is the refractive index of the dimethylphthalate, n) is the relative refractive index of the interface, and n=) is the plate index of refraction. As is seen from Fig. l, Ao/y = sin BI, A3/y = sin el, hence A3 = niAo. Therefore, for any values of h the rays i and 2 from the section pm to the point 0 pass over an identical optical path, and the path difference A between them can only occur on their paths from the fact to pm~ g Evidently the greatest illumination will be at the point 0 when there is no path difference between rays i and 2. The equality tn2 = An~ corresponds to this, or equivalently, An = t. For rays leaving at the limit angle after refraction, sin 8 = i/n. The angles 83 and B2 differ from BI by the angle y/2, which is quite small compared to them for t << Lz. Consequently, the rays i and 2 are practically parallel to the middle ray of the beam within the cuvette. In that case sin 8 = A/t, i/n = A/t, t = An, i.e., the illumination at the center of the first maximum is caused by that light beam whose middle ray is at the limit refraction angle, From the point b (l + Aa)n~, while ray them is Aon~ • Agn~. lations, we find Ao =
to the exit from the cuvette the ray i traverses the optical path tn2 + 2 traverses the path (t + ~ + A 2 ) n ~ hence the path difference between Writing these expressions as an equality and performing simple maniput(n --l)/n + (A2 " At) ~ Ag.
Institute of Optics of the Atmosphere, Siberian Branch, Academ F of Sciences of the USSR, Translated from Izvestiya Vysshikh Uchebnykh Zavedenil, Fizika, No, 7, pp. i12-i15, July, 1979. Original article submitted April 12, 1978.
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0038-5697/79/2207- 0788507.50
9 1980 Plenum Publishing Corporation
I'~I
,t
;
h
~I
Fig. i. Diagram to observe the interference of rays refracted from a grazing beam, Since Z --I+A
t+ t
tt--t
thenAx-- A
1 --t+l+A~' ~,2 - - At
A
y _ a
Ao
At . (t _ . ~- .-~ ) '
f --
t(t + h ,
(nt
A~--
--
(l)
(t + t),
(z)
A) 9 Ag;
n.A
yA
t a
a=---~-; Ao=V'y'J'a2==Yvt2--A'-'. '
Because of
A
(2)
t
and (2),
~gt
t~ (nt - - A)
(3)
Y--nAVt.~_~ T - V ~ . According to Fig, i, r/hi = (11 +l), (r+y)/h~ = ( l + t ) / ( L z + l + t ) . Let us transform these formulas relative to h~ and let us equate the expressions obtained r(Ll4-l)_ t
Since l = and I t
<<
1L,, y
(r+y)(Lt+l+t) t+t
rLtt
'
,where Y ~ l L t + l ~ + l t "
rt/l
:
r+y
y(2t+t)
2 --
2"--'---~- -
2hA V t2 - A~
(4)
_ ~
t ( , , t - a ) t 2 t + t) ~
v' ~-----~
'
'
L ,,A
q;
Aa _
~
(5) (6)
,
L
Substituting their values from (i), (5) and (4), instead of the terms indicated in (6), we obtain h=
n~L V't~ -- A~ ]/t2_tz~(t=.M)
In the general case A = t/n • A g .
t (nt - - A) (2l + t) +
2nA]/t-7-~A2
Ag ( 2 / + t) =F 2 ] : ~ - - - Z ~ ' 9
(7)
Taking account of this the formula for h becomes:
789
~g (2l + l) n n
~
~:
n
_:
2t V n ~
--
1 T- 2~g
n_
t
(8)
The second member of the formula is converted to the form
~gn
I :F V'n-v'~- l (2t 4- t)
( n : - I) t
(1+"~)
V'
":
(n2 -- I) t
Here the last factor is practically equal to one, while the third component is much less than the others, and hence can be neglected. Consequently
n*LVn:--I:~2Aa't
V'~--
I (2/+
t)
) + Since the wavelength of light in the medium equals I/n,, the path difference occurring between rays i and 2 in dimethylphthalate is Ag = Kl/2n,, where K is the number of half-waves included in Ag.
At the same time, K will equal the number of Fresnel zones in the width of
the plate face if the path difference A n between rays i and 2 and the middle ray is considerably less than the half-wave length because of some lack of parallelism between the rays of the beam arriving at the point 0. Substituting the value of Ag in (9), we obtain after manipulation
r
~.I
(n2 - - 1) ~
For K = 0, 3, 5, ..., h corresponds lumination.
(lO)
I/.--r~-I ( ~ + 0
/12
It = n , t
--n~ 4-
"'""1.#I
to maximums and for K = 2, 4, 6 to minimums of il-
In case the interference fringes are measured relative to the center of the first maximum , the formula becomes
H
=
n2
ntL
n:
--"~--
--
.
----,~ I
_ _
(n'
For A g . = 0 formula
,~,
(n)
n:--
1) :F nit
(9) characterizes only the value of h max
I/'n-7~-~ I (2l q- t)
n, V n : - - l . L hmaxl
=
I/#1 = -- n~ (.n:~--
1,)
-j-
2
*
(12) '
where ni V n ~ - - 1. L
I / n " - - n t (.'-' - - l)
= L tg =, = ao.
(13)
Radiation of a laser LG-56 with I = 06328 um, was used in the tests formulated; here n, = 1.5115; n = 1.0088; t = 2.15 mm; ~ = 5.3 mm; L = 2760 mm. Then tg ~, = 0.2033 accord-
790
TABLE 1 Heap, Hcomp, mm ID/n max! min, max~ rain2 max3 min3 max4 min4 maxs
to (13), or a, = ii~
0
0 2 3 4 5 6
7 8 9
0 6,6 6,~2 9,85 9,9 13,15 13,14 16,4 16,51 19,7 19,84 23 23,10 26,15 .26,5 29, 4 29,83
Hp--He , rnI-n
0 --0,08 --0,05 --0,01 O, l l 0,14 0,1 0,35 0,43
', which agrees with the experimental value ~
Since i and t << L, A
= i!,5 a.
= 6~/8S, ~ = t/n /~-L----l./n = n~(n2--1) = 0.28 mm; s = n L2;/2J=2820 mm, A n = 0.0035 Bm. Therefore, the condition A n << (X/2) is satisfied, Experimental and computed values of H are contained in Table i. good agreement between the computation results and test,
(ho = +
As is seen there is
Therefore, under the conditions considered, the intensity distribution in the refracted beam is determined by the interference between the rays refracted from the whole interfacial surface of the connected media. The formula (3) obtained in [2] does not take account of the dependence of the location of the interference fringes on the extent of the interface, which is one reason for the disagreement with test. LITERATURE CITED l,
2.
Yu. I. Terent'ev, Izv. Vyssh. 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).
791