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Experimental results on the longitudinal displacement of light beams near total reflection
Volume 45A, number 3
PHYSICS LETTERS
24 September 1973
EXPERIMENTAL RESULTS ON THE LONGITUDINAL DISPLACEMENT OF LIGHT BEAMS NEAR TOTAL REFLECTION M. GREEN, P. KIRKBY and R.S. TIMSIT Department of Physics, University of Toronto, Ontario, Canada Received 17 July 1973 Accurate measurements of the longitudinal displacement of a light beam on total internal reflection have been made by determining the spatial shift of the intensity profile of the totally reflected beam. The results can be explained solely on classical grounds.
Recently de Brogue and Vigier [1] have proposed a new quantum interpretation for the displacement of light beams on total reflection. Troup et al [21have refuted their argument and maintained that the effect may be explained on purely classical grounds. The resuIts reported on this letter demonstrate that the measured shifts are indeed consistent with those predicted classically, The measurements were carried out by reflecting a light beam, linearly polarized in the plane of incidence, from an optical flat using the experimental arrangement shown in fig. 1. The incident beam was generated by passing the light emitted by a He—Ne laser (6328 A) through optical accessories and finally through a slit 1.03 mm in width and 20 mm in height. The intensity distribution at the slit was approximately square. The beam was sufficiently long in the vertical direction to cover the height of the optical flat (n = 1.54). The faces of the flat were parallel to within LASER AND
two seconds of arc, polished to betterthan X/lO and were half coated with silver strips. The portion of the incident beam reflected off the silver-glass interfaces was used to generate a reference beam. The longitudinal shift of the internally reflected beam was determined by recording with a photocell the intensity distributions of both beams after 27 reflections in the optical flat and measuring the distance D between the centroids of the two profiles. The experimental results are shown in fig. 2. The average displacement for one reflection, taken as D/27, increases slowly as the critical angle is approached and reaches a relatively steady value of approximately 35 X. For angles of incidence lying approximately 0.7 mr. below the critical angle the internally reflected beam had vanished. These results are the most accurate reported to date. Earlier workers had relied upon the use of photographic techniques to measure beam displacements [3,4]. We have found that this approach is unreliable and leads at best to estimates of the beam shifts. The m~tKoddescribed in
COLLIMATING OPTICS
/fi#
REFERENCE BEAM_\’\~7
,)~-\
INTERNALLY
PLAN
:L~.
PHOTOCELL
~ ~
SIDE
SCANNING SLIT-AND-
VIEW
I
STRIP ON
I
BOTH FACES OF FLAT
Fig. 1. Schematic diagram of the experimental apparatus. The flat is mounted on a rotatable platform so that both the mcident and reflected beams lie in the horizontal plane.
-
__2_
-
50
-o.i
-----
40
o
/
—i
r.~iiiriiii
—0.01
0.01
1pI~new~ve approximation gaussian profIle
square profile I II 111111
I
0.1
I
10
100
e e~(milliradians) —
Fig. 2. Average longitudinal displacement versus deviation of the angle of incidence 0 from the critical angle o~.
259
Volume 45A, number 3
PHYSICS LETTERS
this letter is superior in that it utilizes well-resolved beam intensity profiles and thus offers direct comparison of experiment with theory. For the calculation of the longitudinal displacement it is assumed that the intensity variation in the verticaldirection in fig. 1 (y-direction) may be neglected. The electric field distribution of the linearly polarized incident light beam may be expressed, in all space, as
24 September 1973
The spectral amplitude ~(k~) may be determined from the field distribution at the source (z0) by
mental situation. For comparison, the result obtained from the classical limit of an infinite plane wave [61 is also illustrated. For angles of incidence exceeding the critical angle by more than 1 mr. the three curves yield almost identical results in good agreement with the experimental data. For smaller angles, the displacement becomes increasingly dependent upon the profile of the incident beam. In this region the curve computed for an incident beam of square profile follows the pattern produced by the experimental points. This result is consistent with our knowledge of the characteristics of the incident beam. The only discrepancy between theory and experiment occurs in the region of angles smaller than the critical angle. We surmise that this deviation is an indication of the degree of validity of the
ø(kx)=fEx(x,O)exp(_ikxx)dx.
two-dimensional approximation in the calculation. We conclude that the longitudinal displacement of
E~(x,z)= ~f~(k~)
exp {i(k~x+k~z)}dk~, ~
where k~= ~
k = 2ir/ X.
(2)
The effect of the reflecting medium on E~(x,z)is taken into account by inserting into eq. (1) the appropriate reflection coefficient R(k~)and by expressing the spatial variables with reference to a new coordinate system x’, y’ and z’ defined with respect to the outgoing beam. After n reflections we can thus write the reflected field at the observation slit as E~(x’,z’)=~IR~1(kx)ø(kx)exp{i(kxx~+kzz~)}dkx where (x’,z’) are the coordinates of the detector. For the reference beam R(k~)=—1. For the internally reflected beam R(k~)is identical with the Fresnel coefficient [5]. The displacement between the centroids of the intensity profiles of the two reflected beams was thus calculated and the average displacement for one reflection deduced. The results of the calculations are shown by the curves of fig. 2. The calculations were carried out for the cases of two incident beams having respectively a gaussian profile with a standard deviation of 0.51 mm and a square profile with a width of 1.03 mm at the source. These widths were consistent with the experi-
260
a linearly polarized light beam on total internal reflection is adequately explained on classical grounds. Since the experimental method described in this note does allow direct comparison of experiment with theory, it should be extended to the measurement of the lateral displacement [4]. We are grateful to the National Research Council of Canada and to the University of Toronto for financial support. References [1] L. de Brogue and J.P. Vigier, Phys. Rev. Lett. 28 (1972) 1001. [2] G.J. Troup, J.L.A. Francey, R.G. Turner and A. Tirkel, Phys. Rev. Lett. 28 (1972) 1540. [3] H.K.V. Lotsch, Optik 32 (1970—71) 116, 189, 299, 553. This is a review paper that gives an exhaustive description of the effect. [4] C. Imbert, Phys. Rev. D5 (1972) 787. [51 M. Born and E. Wolf, Principles ofoptics (Pergamon Press, 1959) page 46. [6] 586. B.R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61(1971)
PHYSICS LETTERS
24 September 1973
EXPERIMENTAL RESULTS ON THE LONGITUDINAL DISPLACEMENT OF LIGHT BEAMS NEAR TOTAL REFLECTION M. GREEN, P. KIRKBY and R.S. TIMSIT Department of Physics, University of Toronto, Ontario, Canada Received 17 July 1973 Accurate measurements of the longitudinal displacement of a light beam on total internal reflection have been made by determining the spatial shift of the intensity profile of the totally reflected beam. The results can be explained solely on classical grounds.
Recently de Brogue and Vigier [1] have proposed a new quantum interpretation for the displacement of light beams on total reflection. Troup et al [21have refuted their argument and maintained that the effect may be explained on purely classical grounds. The resuIts reported on this letter demonstrate that the measured shifts are indeed consistent with those predicted classically, The measurements were carried out by reflecting a light beam, linearly polarized in the plane of incidence, from an optical flat using the experimental arrangement shown in fig. 1. The incident beam was generated by passing the light emitted by a He—Ne laser (6328 A) through optical accessories and finally through a slit 1.03 mm in width and 20 mm in height. The intensity distribution at the slit was approximately square. The beam was sufficiently long in the vertical direction to cover the height of the optical flat (n = 1.54). The faces of the flat were parallel to within LASER AND
two seconds of arc, polished to betterthan X/lO and were half coated with silver strips. The portion of the incident beam reflected off the silver-glass interfaces was used to generate a reference beam. The longitudinal shift of the internally reflected beam was determined by recording with a photocell the intensity distributions of both beams after 27 reflections in the optical flat and measuring the distance D between the centroids of the two profiles. The experimental results are shown in fig. 2. The average displacement for one reflection, taken as D/27, increases slowly as the critical angle is approached and reaches a relatively steady value of approximately 35 X. For angles of incidence lying approximately 0.7 mr. below the critical angle the internally reflected beam had vanished. These results are the most accurate reported to date. Earlier workers had relied upon the use of photographic techniques to measure beam displacements [3,4]. We have found that this approach is unreliable and leads at best to estimates of the beam shifts. The m~tKoddescribed in
COLLIMATING OPTICS
/fi#
REFERENCE BEAM_\’\~7
,)~-\
INTERNALLY
PLAN
:L~.
PHOTOCELL
~ ~
SIDE
SCANNING SLIT-AND-
VIEW
I
STRIP ON
I
BOTH FACES OF FLAT
Fig. 1. Schematic diagram of the experimental apparatus. The flat is mounted on a rotatable platform so that both the mcident and reflected beams lie in the horizontal plane.
-
__2_
-
50
-o.i
-----
40
o
/
—i
r.~iiiriiii
—0.01
0.01
1pI~new~ve approximation gaussian profIle
square profile I II 111111
I
0.1
I
10
100
e e~(milliradians) —
Fig. 2. Average longitudinal displacement versus deviation of the angle of incidence 0 from the critical angle o~.
259
Volume 45A, number 3
PHYSICS LETTERS
this letter is superior in that it utilizes well-resolved beam intensity profiles and thus offers direct comparison of experiment with theory. For the calculation of the longitudinal displacement it is assumed that the intensity variation in the verticaldirection in fig. 1 (y-direction) may be neglected. The electric field distribution of the linearly polarized incident light beam may be expressed, in all space, as
24 September 1973
The spectral amplitude ~(k~) may be determined from the field distribution at the source (z0) by
mental situation. For comparison, the result obtained from the classical limit of an infinite plane wave [61 is also illustrated. For angles of incidence exceeding the critical angle by more than 1 mr. the three curves yield almost identical results in good agreement with the experimental data. For smaller angles, the displacement becomes increasingly dependent upon the profile of the incident beam. In this region the curve computed for an incident beam of square profile follows the pattern produced by the experimental points. This result is consistent with our knowledge of the characteristics of the incident beam. The only discrepancy between theory and experiment occurs in the region of angles smaller than the critical angle. We surmise that this deviation is an indication of the degree of validity of the
ø(kx)=fEx(x,O)exp(_ikxx)dx.
two-dimensional approximation in the calculation. We conclude that the longitudinal displacement of
E~(x,z)= ~f~(k~)
exp {i(k~x+k~z)}dk~, ~
where k~= ~
k = 2ir/ X.
(2)
The effect of the reflecting medium on E~(x,z)is taken into account by inserting into eq. (1) the appropriate reflection coefficient R(k~)and by expressing the spatial variables with reference to a new coordinate system x’, y’ and z’ defined with respect to the outgoing beam. After n reflections we can thus write the reflected field at the observation slit as E~(x’,z’)=~IR~1(kx)ø(kx)exp{i(kxx~+kzz~)}dkx where (x’,z’) are the coordinates of the detector. For the reference beam R(k~)=—1. For the internally reflected beam R(k~)is identical with the Fresnel coefficient [5]. The displacement between the centroids of the intensity profiles of the two reflected beams was thus calculated and the average displacement for one reflection deduced. The results of the calculations are shown by the curves of fig. 2. The calculations were carried out for the cases of two incident beams having respectively a gaussian profile with a standard deviation of 0.51 mm and a square profile with a width of 1.03 mm at the source. These widths were consistent with the experi-
260
a linearly polarized light beam on total internal reflection is adequately explained on classical grounds. Since the experimental method described in this note does allow direct comparison of experiment with theory, it should be extended to the measurement of the lateral displacement [4]. We are grateful to the National Research Council of Canada and to the University of Toronto for financial support. References [1] L. de Brogue and J.P. Vigier, Phys. Rev. Lett. 28 (1972) 1001. [2] G.J. Troup, J.L.A. Francey, R.G. Turner and A. Tirkel, Phys. Rev. Lett. 28 (1972) 1540. [3] H.K.V. Lotsch, Optik 32 (1970—71) 116, 189, 299, 553. This is a review paper that gives an exhaustive description of the effect. [4] C. Imbert, Phys. Rev. D5 (1972) 787. [51 M. Born and E. Wolf, Principles ofoptics (Pergamon Press, 1959) page 46. [6] 586. B.R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61(1971)