- Email: [email protected]
Launching of the new world of geometrical optics
Accepted Manuscript Title: Launching of the New World of Geometrical Optics Author: Pramode Ranjan Bhattacharjee PII: DOI: Reference:
S0030-4026(15)01222-X http://dx.doi.org/doi:10.1016/j.ijleo.2015.09.153 IJLEO 56358
To appear in: Received date: Accepted date:
22-10-2014 9-9-2015
Please cite this article as: P.R. Bhattacharjee, Launching of the New World of Geometrical Optics, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.09.153 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Email : [email protected]
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PRAMODE RANJAN BHATTACHARJEE Retired Principal Kabi Nazrul Mahavidyalaya Sonamura Tripura 799 131 India
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LAUNCHING OF THE NEW WORLD OF GEOMETRICAL OPTICS
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All urgent materials concerning this paper should be sent to the following Permanent residential address : Dr. P. R. Bhattacharjee 5 Mantri Bari Road P.O. Agartala, Tripura 799001 India
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Abstract: This paper reports on the discovery of the most unambiguous generalized vectorial laws of reflection and refraction along with the launching of a new world of geometrical optics with a lot of novel interesting physical insights to optical phenomena.
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INTRODUCTION
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Key Words: Geometrical optics; Reflection; Refraction; Vector algebra; Cross product; Dot product.
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Geometrical Optics is a branch of Classical Optics in which optical phenomena are studied on the basis of a few well established laws with simultaneous use of geometrical and analytical methods. It
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has been reported in [16] that the traditional definition of each of the angles of incidence, reflection and refraction [1-9] is ambiguous on account of having no rationality with the fundamental definition of
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angle in Geometry, there by leading the traditional laws of reflection and refraction [1-9] to be ambiguous as well.
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With a view to getting rid of the ambiguity present in the traditional definition of each of the angles of
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incidence, reflection and refraction, unambiguous definition of each of the aforesaid three angles has been offered in [16]. Making use of the unambiguous definition of each of the angles of incidence,
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reflection and refraction, the unambiguous generalized vectorial laws of reflection and refraction have also been developed in [16] to address the two typical problems considered in [16]. But it must be noted that, on account of the lack of specification of appropriate lower and upper bounds, the unambiguous definition of each of the aforesaid three angles offered in [16] is incomplete and hence still remains ambiguous. As a result, the generalized vectorial laws of reflection and refraction developed in [16] are also ambiguous. The same type of ambiguity also exists in each of the equivalent vector forms of reflection and refraction laws available in the traditional literature [10–15]. In view of above, there is an urgent need to refine first the definition of each of the unambiguous angles of incidence, reflection and refraction offered in [16] by incorporation of well defined realistic lower and upper bounds of each of the aforesaid three angles and then to give birth to the most unambiguous Page 2 of 16
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statement of each of the generalized vectorial laws of reflection and refraction. The discovery of the most unambiguous generalized vectorial laws of reflection and refraction has been reported in this paper along with the incorporation of the refined unambiguous definition of each of the
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angles of incidence, reflection and refraction. The incorporation of the refined unambiguous definition of each of the angles of incidence, reflection and refraction along with the discovery of the most
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unambiguous generalized vectorial laws of reflection and refraction ultimately gives birth to a new
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DEFINITIONS
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world of Geometrical Optics with a lot of novel interesting physical insights of optical phenomena.
Refined unambiguous angle of incidence (i):The angle of incidence (i) is the smaller of the angles
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between the vectors i and n subject to the condition that π/2 < i ≤ π, so long as the case considered
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is a reflection (or a refraction of light as it passes from a rarer to a denser medium). If however it is a
by the relation 0 ≤ i < π/2.
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case of refraction as light passes from a denser medium to a rarer medium, the angle i must be bounded
Refined unambiguous angle of reflection (r): The angle of reflection (r) is the smaller of the angles
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between the vectors r and n subject to the condition that 0 ≤ r < π/2. Refined unambiguous angle of refraction (R):The angle of refraction (R) is the smaller of the angles between the vectors n and R subject to the condition that, π/2 < R ≤ π when the ray of light passes from a rarer medium to a denser medium, or 0 ≤ R < π/2 when the ray of light passes from a denser medium to a rarer medium.
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AMBIGUITIES PRESENT IN THE GENERALIZED VECTORIAL LAWS OF REFLECTION AND REFRACTION
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The generalized vectorial law of reflection reported in [16] or each of its equivalent forms available in traditional literature [10-15] suffers from the lack of specification of well-defined realistic lower and
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upper bounds of the angles of incidence and reflection, and as a result of which it remains still
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ambiguous.
In a similar manner, the generalized vectorial law of refraction reported in [16] or each of its equivalent
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forms available in traditional literature [10-15] suffers from the lack of specification of well-defined realistic lower and upper bounds of the angles of incidence and refraction, and as a result of which it
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also remains still ambiguous. Furthermore, this law leads to ambiguous result regarding the refractive index of one optical medium with respect to another, computed as the ratio of the sine of the angle of
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correspond to 0º.
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incidence and the sine of the angle of refraction when in particular, both the aforesaid angles
In view of above, there is an urgent need of developing the most unambiguous statement of each of the
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generalized vectorial laws of reflection and refraction and they are being offered next along with theoretical proof of each.
THE MOST UNAMBIGUOUS GENERALIZED VECTORIAL LAWS OF REFLECTION AND REFRACTION The most unambiguous generalized vectorial law of reflection: If i and r represent unit vectors along the directions of incident ray and reflected ray respectively and if n represents unit vector along the direction of the positive unit normal to the reflector at the point of incidence then, n i = n r, where the unambiguous angle of incidence (i) is bounded by the relation, π/2 < i ≤ π, and the unambiguous angle of reflection (r) is bounded by the relation 0 ≤ r < π/2.
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Proof: The proof of the most unambiguous generalized vectorial law of reflection of light can be
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accomplished on the basis of the principle of conservation of momentum of photon as follows.
Figure 1: Diagram
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showing reflection of light by a plane mirror
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Let us consider reflection of a ray of light by a plane mirror as shown in Figure 1. It clearly follows from Figure 1 that, in real world, the unambiguous angle of incidence (i) and the unambiguous angle of
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reflection (r) must be bounded by the relations, π/2 < i ≤ π, and 0 ≤ r < π/2. Now, assuming the plane of incidence to be the XY plane and considering a right-handed coordinate system, we have from Figure 1,
n i = (0 I + 1 J + 0 K) [{sin (π ‒ i))}I Or, n i = (sin i) ( K)
{cos (π ‒ i)}J + 0 K] ...
(1),
where π/2 < i ≤ π. Again in this case, we have, n r = (0 I + 1 J + 0 K) {(cos α)I + (cos β ) J + (cos γ) K},
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where cos α, cos β, and cos γ are the direction cosines of the vector, r and 0 ≤ r < π/2. Or, n r = (cos γ) I ( cos α) K
...
(2)
have, cos γ = 0
(3)
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...
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Now, by applying the principle of conservation of momentum of photon along positive direction of Z-axis, we
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Again applying the principle of conservation of momentum of photon along the positive direction of X-axis, we get,
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90º) = (hν/C) cos α
(hν/C) cos (
Or, cos α = sin i
(4)
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...
where π/2 < i ≤ π.
...
(5),
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n r = (sin i) ( K)
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Using the relations (3) and (4) we have then from the relation (2),
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From the relations (1) and (5), we have then, n i = n r, where π/2 < i ≤ π, and 0 ≤ r < π/2. Hence proved.
The most unambiguous generalized vectorial law of refraction: If i and R represent unit vectors along the directions of the incident ray and refracted ray of a particular colour of light respectively and if n represents unit vector along the direction of the positive unit normal to the surface of separation at the point of incidence then,│(n i)│~ µ│(n R)│, where (n i) and (n R) are like parallel vectors, µ = Refractive index of the second optical medium with respect to the first optical medium for the particular colour of light under consideration, the unambiguous angle of incidence (i) being bounded by the relation, π/2 < i ≤ π, when the
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ray of light passes from a rarer medium to a denser medium, or by the relation, 0 ≤ i < π/2, when the ray of light passes from a denser medium to a rarer medium, and the unambiguous angle of refraction (R) being bounded by the relation, π/2 < R ≤ π, when the ray of light passes from a rarer medium to a denser
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medium, or by the relation, 0 ≤ R < π/2, when the ray of light passes from a denser medium to a rarer
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medium.
Proof: Let us first consider the case of refraction of light as it passes from a rarer medium to a denser
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medium as shown in Figure 2. It clearly follows from Figure 2 that, in real world, the unambiguous angle of incidence (i) and the unambiguous angle of refraction (R) must be bounded by the relations,
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π/2 < i ≤ π, and π/2 < R ≤ π.
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We shall first prove that the vectors, (n i) and (n R) are like parallel vectors. This can be done by
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applying the principle of conservation of momentum of photon as follows.
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Figure 2. Diagram showing refraction of light at a plane surface of discontinuity Assuming the plane of incidence to be the XY plane and considering a right-handed coordinate system, we have from Figure 2,
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n i = (0 I + 1 J + 0 K) [{cos (
90º)}I + { sin (
90º)}J + 0 K}]
Or, n i = (sin i) ( K)
...
(6)
(6) that, n i = (a positive quantity)( K)
(7)
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...
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Now, since in this case, π/2 < i ≤ π, the value of (sin i) is always positive. Hence, it follows from this relation
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n R = (0 I + 1 J + 0 K) {(cos α)I + (cos β ) J + (cos γ) K},
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Again in this case we have,
where cos α, cos β, and cos γ are the direction cosines of the vector R. (cos α) K
...
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Or, n R = (cos γ) I
(8)
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Now, from the principle of conservation of momentum of photon along the positive direction of Z-axis, we have,
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(hν/C1) cos 90º = (hν/C2) cos γ,
where C1 and C2 are the numerical values of the velocity of light in the first optical medium and that in the
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second optical medium respectively. Or, cos γ = 0
...
(9)
Again applying the principle of conservation of momentum of photon along the positive direction of X-axis, we have,
(hν/C1) cos (
90º) = (hν/C2) cos α
Or, cos α = (C2/C1) sin i
...
(10)
Using the relations (9) and (10), we have then from the relation (8), nR=
{(C2/C1) sin i} K
...
(11) Page 8 of 16
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Now, since in this case, π/2 < i ≤ π, and both C1 and C2 are positive quantities, it clearly follows from
the relation (11) that, n R = (a positive quantity)( K)
(12)
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...
It then readily follows from the relations (7) and (12) that the vectors, (n i), and (n R) are both directed along K). Hence, the vectors, (n i), and (n R) are like parallel vectors.
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the direction of the vector (
│(n i)│~ µ│(n R)│
(13)
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...
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Furthermore, the refined unambiguous statement of Snell's law [17] demands that,
The proof of this relation exists in [17].
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This concludes the proof of the most unambiguous statement of the generalized vectorial law of
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refraction of light as it passes from a rarer medium to a denser medium.
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Considering refraction of light as it passes from a denser medium to a rarer medium and proceeding similarly, the proof of the most unambiguous statement of the generalized vectorial law of refraction of light can be accomplished.
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RESULTS AND DISCUSSIONS
The most unambiguous generalized vectorial laws of reflection and refraction with well-defined realistic lower and upper bounds of the angles of incidence, reflection and refraction give birth to a new world of Geometrical Optics in which there is no place for the traditional ambiguous angles of incidence, reflection and refraction. Interesting physical insights of optical phenomena in this new optical world are the following. (i) The sum of the refined unambiguous angle of incidence and the refined unambiguous angle of reflection is equal to180°. This result is entirely different from the traditional result according to which
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the traditional (ambiguous) angle of incidence is equal to the traditional (ambiguous) angle of reflection. This is because in case of reflection, we have from the most unambiguous generalized vectorial law of
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reflection, n × i = n × r, where π/2 < i ≤ π, and 0 ≤ r < π/2. Thus we have, │n × i│= │n × r│. This
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implies that sin i = sin r; or, i = ( 1) p r + p π, where p = 0, ± 1, ± 2, ± 3, ... . Now, out of the various mathematical possibilities for the value of p, viz. p = 0, ± 1, ± 2, ± 3, ..., the only physically realistic
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value of p satisfying the relations, π/2 < i ≤ π, and 0 ≤ r < π/2 is p = 1. Hence in the new optical world, we must have the relation, i = π ˗ r. In other words, we must have, i + r = 180°.
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(ii) When a ray of light passes from one optical medium to another after suffering refraction, this new
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optical world demands that the ratio of the sine of the refined unambiguous angle of incidence and the sine of the refined unambiguous angle of refraction is approximately equal to the refractive index of the
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second medium with respect to the first medium [17]. This is a result not exactly identical with the
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traditional Snell's law of refraction of light according to which the ratio of the sine of the ambiguous angle of incidence and the sine of the ambiguous angle of refraction is exactly equal to the refractive index of the second optical medium with respect to the first optical medium.
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(iii) The refined unambiguous statement of Snell's law [17] in the new optical world demands further modification of the statement of the unambiguous generalized vectorial law of refraction [16] in the following sophisticated form:
“The vectors, (n × i) and (n × R) are like parallel vectors with │(n i)│~ µ│(n R)│, where π/2 < i ≤ π, π/2
(iv) In this new optical world, the refractive index of one optical medium “a” with respect to another optical medium “b” is approximately equal to the reciprocal of the refractive index of the optical
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medum “b” with respect to the optical medium “a” and vice versa [20]. These results are entirely different from those in the traditional optical literature according to which those two refractive indices are exactly equal to the reciprocal of each other.
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(v) In this new optical world, the principle of reversibility of light holds good in cases of both reflection and refraction. The proof of the principle of reversibility of light in case of reflection on the basis of the
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most unambiguous generalized vectorial law of reflection is exactly similar to that offered in [20].
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However, in case of refraction, the proof of the said principle on the basis of the most unambiguous generalized vectorial law of refraction could be accomplished as follows.
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Let a ray of light AO from the source point A in air (rarer medium) after suffering refraction at the point O of the surface of discontinuity arrive at the point B in water (denser medium with refractive index µ)
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after moving along the direction OB. Then OB is the refracted ray corresponding to the incident ray AO. Then
the most unambiguous generalized vectorial law of refraction demands that, the vectors (n × i) and (n
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× R) are like parallel vectors with │(n i)│~ µ│(n R)│, where π/2 < i ≤ π, π/2
i)} are also like parallel. Again if │(n i)│~ µ│(n R)│, then we must have, │{(n R)}│~
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(1/µ)│{(n
i)}│. This implies that if a ray of light from the point A in air (rarer medium) reaches the
point B in water (denser medium) after suffering refraction at the point O of the surface of separation, then a ray of light from the point B in water (denser medium) after suffering refraction at the point O of the surface of separation must also reach the point A in air (rarer medium). Thus the principle of reversibility of light follows from the most unambiguous generalized vectorial law of refraction when the ray of light passes from a rarer medium to a denser medium. If, however, the ray of light passes from a denser medium to a rarer medium, the proof of the said principle can be similarly accomplished with the help of the most unambiguous generalized vectorial law of refraction with relevant well defined realistic lower and upper bounds of the angles of incidence and refraction as 0 ≤ i < π/2, 0 ≤ R
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< π/2. (vi) In case of reflection of light by a plane mirror, the relation “object distance = image distance” holds good in this new optical world. The proof of this relation on the basis of the most unambiguous
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generalized vectorial law of reflection is exactly similar to that offered in [20].
(vii) This new optical world demands that, the treatment of derivation of each and every relation
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between real depth and apparent depth for normal as well as oblique observation offered in [20] should
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be made by making use of the refined unambiguous statement of Snell's law [17], Viz. the relation, │(n i)│~ µ│( n R)│, as a result of which each relation between real depth and apparent depth for normal
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as well as oblique observation should bear the “~” sign instead of the “=” sign. This implies that each such relation between real depth and apparent depth in the new optical world will only be an approximate one even in
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case of oblique observation. This is a result which is different from that found in the traditional optical literature.
(vii) It can be readily verified by applying the refined unambiguous statement of Snell's law [17], Viz.
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the relation, │(n i)│~ µ│( n R)│in the treatment offered in [20] in connection with refraction of
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light through a parallel-sided block of a refracting medium that the incident ray is not exactly parallel to the final emergent ray, a result which is not exactly identical to that of the traditional result according
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to which those two rays are exactly parallel. (viii) In addition to making approximation in respect of the small aperture of a spherical refracting surface, a further approximation must be superimposed in the treatment of derivation of the relation between object distance, image distance and focal length of a spherical refracting surface in this new optical world. This is because of the following reason. On account of the validity of the refined unambiguous statement of Snell's law, Viz. the relation, │(n i)│~ µ│( n R), each of the relations between object distance, image distance and radius of curvature of spherical refracting surfaces offered in [22] should have to be replaced by the “~” sign instead of the “=” sign. Similar change is to be made to the lens makers' formula in [22] as well. This implies
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that, in the new optical world, all such relations are only approximate ones instead of being actual equations.
(ix) In this new optical world, the relation between the critical angle (θ) and the refractive index (µ) of the denser medium with respect to the rarer medium [18] will assume the form: sin θ ~ 1/µ, instead of
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the form: sin θ = 1/µ, which exists in the long-running literature of optical physics. (x) As in [21],the most unambiguous generalized vectorial law of reflection in the new optical world
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also gives birth to the remarkable fact that the proposition “Velocity of light is unattainable” is
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incorrect. It is possible to generate velocity exceeding the velocity of light – a result not in agreement with Einstein's special theory of relativity.
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(xi) As reported in [19], on account of the incorporation of the refined unambiguous definition of each of the angles of incidence, reflection and refraction, the new optical world gives birth to novel
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expression for the deviation of a ray of light in each of the cases of reflection and refraction as well as
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through a prism.
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it gives birth to novel result for the condition of minimum deviation of the path of a ray passing
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CONCLUSION An examination of the generalized vectorial laws of reflection and refraction [16] reported by the author in 2005 has been made in this paper. It has been found that the generalized vectorial laws of
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reflection and refraction [16] as well as their equivalent forms existing in the optical physics literature
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[10-15] still remain ambiguous on account of the lack of specification of well defined realistic lower and upper bounds of each of the angles of incidence, reflection and refraction. With a view to getting
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rid of such an ambiguity, this paper reports on the discovery of the most unambiguous generalized vectorial laws of reflection and refraction by incorporating the relevant lower and upper bounds to each
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of the aforesaid three angles along with considering the refined unambiguous statement of Snell's law
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in ray optics [17].
The refined unambiguous definitions of the angles of incidence, reflection and refraction as well as the
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discovery of the most unambiguous generalized vectorial laws of reflection and refraction reported in
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this paper give birth to a new world of Geometrical Optics with a lot of interesting novel physical insights to optical phenomena. In this new optical world, there is no place for the traditional angles of incidence, reflection and refraction as well as the traditional laws of reflection and refraction [1-9].
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This new optical world is based exclusively on the refined unambiguous angles of incidence, reflection and refraction as well as the most generalized vectorial laws of reflection and refraction offered in this paper.
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REFERENCES [1] J. Morgan, 1953, Introduction to Geometrical and Physical Optics, New York: McGraw-Hill Book Company, Inc.
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[3] E. Edser, 1946, Light for Students, London: Macmillan and Co., Ltd.
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[2] F. A. Jenkins, and H. E. White, 1967, Fundamentals of Optics, 3rd ed., New York: McGraw-Hill Book Company, Inc.
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[4] R. W. Stewart, and J. Satterly, 1947, Text Book of Light, London: University Tutorial Press Ltd. [5] R. S. Longhurst, 1973, Geometrical and Physical Optics, 3rd ed., London: Orient Longman.
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[6] J. P. C. Southall, 1933, Mirrors, Prisms and Lenses, 3rd ed., New York: The Macmillan Company.
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[7] A. W. Barton, 1939, A Textbook on Light, London: Longmans Green and Co. [8] C. Curry, 1962, Geometrical Optics, London: Edward Arnold Ltd.
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[9] P. Drude, 1902, The Theory of Optics, New York: Dover Publications, Inc.
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[10] E. Hecht, 1987, Optics, 2nd ed., Addison–Wesley, Reading, Massachussetts. [11] M. Born, and E. Wolf, 1959, Principles of Optics, Oxford: Pergamon.
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[12] M. Born, and E. Wolf, 2009, Principles of Optics, 7th edition, Cambridge: Cambridge University Press. [13] M. Herzberger, 1958, Modern Geometrical Optics, New York: Interscience. [14] M. Herzberger, 1931, Strahlenoptik, Berlin: Springer. [15] O. N. Stavroudis, 2006, The Mathematics of Geometrical and Physical Optics, Weinheim: Wiley. [16] P. R. Bhattacharjee, The generalized vectorial laws of reflection and refraction, European Journal of Physics, 2005; 26(5); 901-911. [17] P. R. Bhattacharjee, Giving birth to the refined unambiguous statement of Snell's law in ray optics, Optik 2014; DOI: 10.1016/j.ijleo.2014.07.124.
[18] P. R. Bhattacharjee, Refinement of the definitions of angles of incidence, reflection, refraction, and critical angle in ray optics, Potentials IEEE (in press).
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[19] P. R. Bhattacharjee, Deviation problems in ray optics in the light of the refined unambiguous definitions of angles of incidence, reflection and refraction, Optik 2014; 125(16);4257-4261. [20] P. R. Bhattacharjee, Addressing some issues of ray optics on the basis of the newly discovered generalized vectorial laws of reflection and refraction, Optik 2013; 124(23);6250-6254.
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[21] P. R. Bhattacharjee, The generalized vectorial laws of reflection and refraction applied to the rotation poblems in ray optics, Optik 2010; 121(23);2128-2132.
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[22] P. R. Bhattacharjee, Exhaustive study of reflection and refraction at spherical surfaces on the basis of the newly discovered generalized vectorial laws of reflection and refraction, Optik 2012; 123(5); 381-386.
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S0030-4026(15)01222-X http://dx.doi.org/doi:10.1016/j.ijleo.2015.09.153 IJLEO 56358
To appear in: Received date: Accepted date:
22-10-2014 9-9-2015
Please cite this article as: P.R. Bhattacharjee, Launching of the New World of Geometrical Optics, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.09.153 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Email : [email protected]
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PRAMODE RANJAN BHATTACHARJEE Retired Principal Kabi Nazrul Mahavidyalaya Sonamura Tripura 799 131 India
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LAUNCHING OF THE NEW WORLD OF GEOMETRICAL OPTICS
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All urgent materials concerning this paper should be sent to the following Permanent residential address : Dr. P. R. Bhattacharjee 5 Mantri Bari Road P.O. Agartala, Tripura 799001 India
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Abstract: This paper reports on the discovery of the most unambiguous generalized vectorial laws of reflection and refraction along with the launching of a new world of geometrical optics with a lot of novel interesting physical insights to optical phenomena.
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INTRODUCTION
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Key Words: Geometrical optics; Reflection; Refraction; Vector algebra; Cross product; Dot product.
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Geometrical Optics is a branch of Classical Optics in which optical phenomena are studied on the basis of a few well established laws with simultaneous use of geometrical and analytical methods. It
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has been reported in [16] that the traditional definition of each of the angles of incidence, reflection and refraction [1-9] is ambiguous on account of having no rationality with the fundamental definition of
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angle in Geometry, there by leading the traditional laws of reflection and refraction [1-9] to be ambiguous as well.
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With a view to getting rid of the ambiguity present in the traditional definition of each of the angles of
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incidence, reflection and refraction, unambiguous definition of each of the aforesaid three angles has been offered in [16]. Making use of the unambiguous definition of each of the angles of incidence,
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reflection and refraction, the unambiguous generalized vectorial laws of reflection and refraction have also been developed in [16] to address the two typical problems considered in [16]. But it must be noted that, on account of the lack of specification of appropriate lower and upper bounds, the unambiguous definition of each of the aforesaid three angles offered in [16] is incomplete and hence still remains ambiguous. As a result, the generalized vectorial laws of reflection and refraction developed in [16] are also ambiguous. The same type of ambiguity also exists in each of the equivalent vector forms of reflection and refraction laws available in the traditional literature [10–15]. In view of above, there is an urgent need to refine first the definition of each of the unambiguous angles of incidence, reflection and refraction offered in [16] by incorporation of well defined realistic lower and upper bounds of each of the aforesaid three angles and then to give birth to the most unambiguous Page 2 of 16
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statement of each of the generalized vectorial laws of reflection and refraction. The discovery of the most unambiguous generalized vectorial laws of reflection and refraction has been reported in this paper along with the incorporation of the refined unambiguous definition of each of the
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angles of incidence, reflection and refraction. The incorporation of the refined unambiguous definition of each of the angles of incidence, reflection and refraction along with the discovery of the most
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unambiguous generalized vectorial laws of reflection and refraction ultimately gives birth to a new
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DEFINITIONS
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world of Geometrical Optics with a lot of novel interesting physical insights of optical phenomena.
Refined unambiguous angle of incidence (i):The angle of incidence (i) is the smaller of the angles
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between the vectors i and n subject to the condition that π/2 < i ≤ π, so long as the case considered
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is a reflection (or a refraction of light as it passes from a rarer to a denser medium). If however it is a
by the relation 0 ≤ i < π/2.
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case of refraction as light passes from a denser medium to a rarer medium, the angle i must be bounded
Refined unambiguous angle of reflection (r): The angle of reflection (r) is the smaller of the angles
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between the vectors r and n subject to the condition that 0 ≤ r < π/2. Refined unambiguous angle of refraction (R):The angle of refraction (R) is the smaller of the angles between the vectors n and R subject to the condition that, π/2 < R ≤ π when the ray of light passes from a rarer medium to a denser medium, or 0 ≤ R < π/2 when the ray of light passes from a denser medium to a rarer medium.
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AMBIGUITIES PRESENT IN THE GENERALIZED VECTORIAL LAWS OF REFLECTION AND REFRACTION
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The generalized vectorial law of reflection reported in [16] or each of its equivalent forms available in traditional literature [10-15] suffers from the lack of specification of well-defined realistic lower and
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upper bounds of the angles of incidence and reflection, and as a result of which it remains still
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ambiguous.
In a similar manner, the generalized vectorial law of refraction reported in [16] or each of its equivalent
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forms available in traditional literature [10-15] suffers from the lack of specification of well-defined realistic lower and upper bounds of the angles of incidence and refraction, and as a result of which it
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also remains still ambiguous. Furthermore, this law leads to ambiguous result regarding the refractive index of one optical medium with respect to another, computed as the ratio of the sine of the angle of
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correspond to 0º.
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incidence and the sine of the angle of refraction when in particular, both the aforesaid angles
In view of above, there is an urgent need of developing the most unambiguous statement of each of the
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generalized vectorial laws of reflection and refraction and they are being offered next along with theoretical proof of each.
THE MOST UNAMBIGUOUS GENERALIZED VECTORIAL LAWS OF REFLECTION AND REFRACTION The most unambiguous generalized vectorial law of reflection: If i and r represent unit vectors along the directions of incident ray and reflected ray respectively and if n represents unit vector along the direction of the positive unit normal to the reflector at the point of incidence then, n i = n r, where the unambiguous angle of incidence (i) is bounded by the relation, π/2 < i ≤ π, and the unambiguous angle of reflection (r) is bounded by the relation 0 ≤ r < π/2.
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Proof: The proof of the most unambiguous generalized vectorial law of reflection of light can be
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accomplished on the basis of the principle of conservation of momentum of photon as follows.
Figure 1: Diagram
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showing reflection of light by a plane mirror
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Let us consider reflection of a ray of light by a plane mirror as shown in Figure 1. It clearly follows from Figure 1 that, in real world, the unambiguous angle of incidence (i) and the unambiguous angle of
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reflection (r) must be bounded by the relations, π/2 < i ≤ π, and 0 ≤ r < π/2. Now, assuming the plane of incidence to be the XY plane and considering a right-handed coordinate system, we have from Figure 1,
n i = (0 I + 1 J + 0 K) [{sin (π ‒ i))}I Or, n i = (sin i) ( K)
{cos (π ‒ i)}J + 0 K] ...
(1),
where π/2 < i ≤ π. Again in this case, we have, n r = (0 I + 1 J + 0 K) {(cos α)I + (cos β ) J + (cos γ) K},
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where cos α, cos β, and cos γ are the direction cosines of the vector, r and 0 ≤ r < π/2. Or, n r = (cos γ) I ( cos α) K
...
(2)
have, cos γ = 0
(3)
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...
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Now, by applying the principle of conservation of momentum of photon along positive direction of Z-axis, we
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Again applying the principle of conservation of momentum of photon along the positive direction of X-axis, we get,
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90º) = (hν/C) cos α
(hν/C) cos (
Or, cos α = sin i
(4)
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...
where π/2 < i ≤ π.
...
(5),
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n r = (sin i) ( K)
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Using the relations (3) and (4) we have then from the relation (2),
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From the relations (1) and (5), we have then, n i = n r, where π/2 < i ≤ π, and 0 ≤ r < π/2. Hence proved.
The most unambiguous generalized vectorial law of refraction: If i and R represent unit vectors along the directions of the incident ray and refracted ray of a particular colour of light respectively and if n represents unit vector along the direction of the positive unit normal to the surface of separation at the point of incidence then,│(n i)│~ µ│(n R)│, where (n i) and (n R) are like parallel vectors, µ = Refractive index of the second optical medium with respect to the first optical medium for the particular colour of light under consideration, the unambiguous angle of incidence (i) being bounded by the relation, π/2 < i ≤ π, when the
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ray of light passes from a rarer medium to a denser medium, or by the relation, 0 ≤ i < π/2, when the ray of light passes from a denser medium to a rarer medium, and the unambiguous angle of refraction (R) being bounded by the relation, π/2 < R ≤ π, when the ray of light passes from a rarer medium to a denser
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medium, or by the relation, 0 ≤ R < π/2, when the ray of light passes from a denser medium to a rarer
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medium.
Proof: Let us first consider the case of refraction of light as it passes from a rarer medium to a denser
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medium as shown in Figure 2. It clearly follows from Figure 2 that, in real world, the unambiguous angle of incidence (i) and the unambiguous angle of refraction (R) must be bounded by the relations,
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π/2 < i ≤ π, and π/2 < R ≤ π.
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We shall first prove that the vectors, (n i) and (n R) are like parallel vectors. This can be done by
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applying the principle of conservation of momentum of photon as follows.
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Figure 2. Diagram showing refraction of light at a plane surface of discontinuity Assuming the plane of incidence to be the XY plane and considering a right-handed coordinate system, we have from Figure 2,
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n i = (0 I + 1 J + 0 K) [{cos (
90º)}I + { sin (
90º)}J + 0 K}]
Or, n i = (sin i) ( K)
...
(6)
(6) that, n i = (a positive quantity)( K)
(7)
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...
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Now, since in this case, π/2 < i ≤ π, the value of (sin i) is always positive. Hence, it follows from this relation
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n R = (0 I + 1 J + 0 K) {(cos α)I + (cos β ) J + (cos γ) K},
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Again in this case we have,
where cos α, cos β, and cos γ are the direction cosines of the vector R. (cos α) K
...
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Or, n R = (cos γ) I
(8)
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Now, from the principle of conservation of momentum of photon along the positive direction of Z-axis, we have,
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(hν/C1) cos 90º = (hν/C2) cos γ,
where C1 and C2 are the numerical values of the velocity of light in the first optical medium and that in the
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second optical medium respectively. Or, cos γ = 0
...
(9)
Again applying the principle of conservation of momentum of photon along the positive direction of X-axis, we have,
(hν/C1) cos (
90º) = (hν/C2) cos α
Or, cos α = (C2/C1) sin i
...
(10)
Using the relations (9) and (10), we have then from the relation (8), nR=
{(C2/C1) sin i} K
...
(11) Page 8 of 16
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Now, since in this case, π/2 < i ≤ π, and both C1 and C2 are positive quantities, it clearly follows from
the relation (11) that, n R = (a positive quantity)( K)
(12)
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...
It then readily follows from the relations (7) and (12) that the vectors, (n i), and (n R) are both directed along K). Hence, the vectors, (n i), and (n R) are like parallel vectors.
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the direction of the vector (
│(n i)│~ µ│(n R)│
(13)
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...
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Furthermore, the refined unambiguous statement of Snell's law [17] demands that,
The proof of this relation exists in [17].
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This concludes the proof of the most unambiguous statement of the generalized vectorial law of
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refraction of light as it passes from a rarer medium to a denser medium.
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Considering refraction of light as it passes from a denser medium to a rarer medium and proceeding similarly, the proof of the most unambiguous statement of the generalized vectorial law of refraction of light can be accomplished.
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RESULTS AND DISCUSSIONS
The most unambiguous generalized vectorial laws of reflection and refraction with well-defined realistic lower and upper bounds of the angles of incidence, reflection and refraction give birth to a new world of Geometrical Optics in which there is no place for the traditional ambiguous angles of incidence, reflection and refraction. Interesting physical insights of optical phenomena in this new optical world are the following. (i) The sum of the refined unambiguous angle of incidence and the refined unambiguous angle of reflection is equal to180°. This result is entirely different from the traditional result according to which
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the traditional (ambiguous) angle of incidence is equal to the traditional (ambiguous) angle of reflection. This is because in case of reflection, we have from the most unambiguous generalized vectorial law of
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reflection, n × i = n × r, where π/2 < i ≤ π, and 0 ≤ r < π/2. Thus we have, │n × i│= │n × r│. This
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implies that sin i = sin r; or, i = ( 1) p r + p π, where p = 0, ± 1, ± 2, ± 3, ... . Now, out of the various mathematical possibilities for the value of p, viz. p = 0, ± 1, ± 2, ± 3, ..., the only physically realistic
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value of p satisfying the relations, π/2 < i ≤ π, and 0 ≤ r < π/2 is p = 1. Hence in the new optical world, we must have the relation, i = π ˗ r. In other words, we must have, i + r = 180°.
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(ii) When a ray of light passes from one optical medium to another after suffering refraction, this new
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optical world demands that the ratio of the sine of the refined unambiguous angle of incidence and the sine of the refined unambiguous angle of refraction is approximately equal to the refractive index of the
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second medium with respect to the first medium [17]. This is a result not exactly identical with the
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traditional Snell's law of refraction of light according to which the ratio of the sine of the ambiguous angle of incidence and the sine of the ambiguous angle of refraction is exactly equal to the refractive index of the second optical medium with respect to the first optical medium.
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(iii) The refined unambiguous statement of Snell's law [17] in the new optical world demands further modification of the statement of the unambiguous generalized vectorial law of refraction [16] in the following sophisticated form:
“The vectors, (n × i) and (n × R) are like parallel vectors with │(n i)│~ µ│(n R)│, where π/2 < i ≤ π, π/2
(iv) In this new optical world, the refractive index of one optical medium “a” with respect to another optical medium “b” is approximately equal to the reciprocal of the refractive index of the optical
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medum “b” with respect to the optical medium “a” and vice versa [20]. These results are entirely different from those in the traditional optical literature according to which those two refractive indices are exactly equal to the reciprocal of each other.
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(v) In this new optical world, the principle of reversibility of light holds good in cases of both reflection and refraction. The proof of the principle of reversibility of light in case of reflection on the basis of the
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most unambiguous generalized vectorial law of reflection is exactly similar to that offered in [20].
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However, in case of refraction, the proof of the said principle on the basis of the most unambiguous generalized vectorial law of refraction could be accomplished as follows.
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Let a ray of light AO from the source point A in air (rarer medium) after suffering refraction at the point O of the surface of discontinuity arrive at the point B in water (denser medium with refractive index µ)
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after moving along the direction OB. Then OB is the refracted ray corresponding to the incident ray AO. Then
the most unambiguous generalized vectorial law of refraction demands that, the vectors (n × i) and (n
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× R) are like parallel vectors with │(n i)│~ µ│(n R)│, where π/2 < i ≤ π, π/2
i)} are also like parallel. Again if │(n i)│~ µ│(n R)│, then we must have, │{(n R)}│~
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(1/µ)│{(n
i)}│. This implies that if a ray of light from the point A in air (rarer medium) reaches the
point B in water (denser medium) after suffering refraction at the point O of the surface of separation, then a ray of light from the point B in water (denser medium) after suffering refraction at the point O of the surface of separation must also reach the point A in air (rarer medium). Thus the principle of reversibility of light follows from the most unambiguous generalized vectorial law of refraction when the ray of light passes from a rarer medium to a denser medium. If, however, the ray of light passes from a denser medium to a rarer medium, the proof of the said principle can be similarly accomplished with the help of the most unambiguous generalized vectorial law of refraction with relevant well defined realistic lower and upper bounds of the angles of incidence and refraction as 0 ≤ i < π/2, 0 ≤ R
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< π/2. (vi) In case of reflection of light by a plane mirror, the relation “object distance = image distance” holds good in this new optical world. The proof of this relation on the basis of the most unambiguous
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generalized vectorial law of reflection is exactly similar to that offered in [20].
(vii) This new optical world demands that, the treatment of derivation of each and every relation
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between real depth and apparent depth for normal as well as oblique observation offered in [20] should
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be made by making use of the refined unambiguous statement of Snell's law [17], Viz. the relation, │(n i)│~ µ│( n R)│, as a result of which each relation between real depth and apparent depth for normal
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as well as oblique observation should bear the “~” sign instead of the “=” sign. This implies that each such relation between real depth and apparent depth in the new optical world will only be an approximate one even in
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case of oblique observation. This is a result which is different from that found in the traditional optical literature.
(vii) It can be readily verified by applying the refined unambiguous statement of Snell's law [17], Viz.
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the relation, │(n i)│~ µ│( n R)│in the treatment offered in [20] in connection with refraction of
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light through a parallel-sided block of a refracting medium that the incident ray is not exactly parallel to the final emergent ray, a result which is not exactly identical to that of the traditional result according
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to which those two rays are exactly parallel. (viii) In addition to making approximation in respect of the small aperture of a spherical refracting surface, a further approximation must be superimposed in the treatment of derivation of the relation between object distance, image distance and focal length of a spherical refracting surface in this new optical world. This is because of the following reason. On account of the validity of the refined unambiguous statement of Snell's law, Viz. the relation, │(n i)│~ µ│( n R), each of the relations between object distance, image distance and radius of curvature of spherical refracting surfaces offered in [22] should have to be replaced by the “~” sign instead of the “=” sign. Similar change is to be made to the lens makers' formula in [22] as well. This implies
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that, in the new optical world, all such relations are only approximate ones instead of being actual equations.
(ix) In this new optical world, the relation between the critical angle (θ) and the refractive index (µ) of the denser medium with respect to the rarer medium [18] will assume the form: sin θ ~ 1/µ, instead of
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the form: sin θ = 1/µ, which exists in the long-running literature of optical physics. (x) As in [21],the most unambiguous generalized vectorial law of reflection in the new optical world
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also gives birth to the remarkable fact that the proposition “Velocity of light is unattainable” is
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incorrect. It is possible to generate velocity exceeding the velocity of light – a result not in agreement with Einstein's special theory of relativity.
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(xi) As reported in [19], on account of the incorporation of the refined unambiguous definition of each of the angles of incidence, reflection and refraction, the new optical world gives birth to novel
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expression for the deviation of a ray of light in each of the cases of reflection and refraction as well as
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through a prism.
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it gives birth to novel result for the condition of minimum deviation of the path of a ray passing
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CONCLUSION An examination of the generalized vectorial laws of reflection and refraction [16] reported by the author in 2005 has been made in this paper. It has been found that the generalized vectorial laws of
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reflection and refraction [16] as well as their equivalent forms existing in the optical physics literature
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[10-15] still remain ambiguous on account of the lack of specification of well defined realistic lower and upper bounds of each of the angles of incidence, reflection and refraction. With a view to getting
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rid of such an ambiguity, this paper reports on the discovery of the most unambiguous generalized vectorial laws of reflection and refraction by incorporating the relevant lower and upper bounds to each
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of the aforesaid three angles along with considering the refined unambiguous statement of Snell's law
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in ray optics [17].
The refined unambiguous definitions of the angles of incidence, reflection and refraction as well as the
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discovery of the most unambiguous generalized vectorial laws of reflection and refraction reported in
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this paper give birth to a new world of Geometrical Optics with a lot of interesting novel physical insights to optical phenomena. In this new optical world, there is no place for the traditional angles of incidence, reflection and refraction as well as the traditional laws of reflection and refraction [1-9].
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This new optical world is based exclusively on the refined unambiguous angles of incidence, reflection and refraction as well as the most generalized vectorial laws of reflection and refraction offered in this paper.
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REFERENCES [1] J. Morgan, 1953, Introduction to Geometrical and Physical Optics, New York: McGraw-Hill Book Company, Inc.
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[3] E. Edser, 1946, Light for Students, London: Macmillan and Co., Ltd.
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[2] F. A. Jenkins, and H. E. White, 1967, Fundamentals of Optics, 3rd ed., New York: McGraw-Hill Book Company, Inc.
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[4] R. W. Stewart, and J. Satterly, 1947, Text Book of Light, London: University Tutorial Press Ltd. [5] R. S. Longhurst, 1973, Geometrical and Physical Optics, 3rd ed., London: Orient Longman.
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[6] J. P. C. Southall, 1933, Mirrors, Prisms and Lenses, 3rd ed., New York: The Macmillan Company.
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[7] A. W. Barton, 1939, A Textbook on Light, London: Longmans Green and Co. [8] C. Curry, 1962, Geometrical Optics, London: Edward Arnold Ltd.
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[9] P. Drude, 1902, The Theory of Optics, New York: Dover Publications, Inc.
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[10] E. Hecht, 1987, Optics, 2nd ed., Addison–Wesley, Reading, Massachussetts. [11] M. Born, and E. Wolf, 1959, Principles of Optics, Oxford: Pergamon.
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[12] M. Born, and E. Wolf, 2009, Principles of Optics, 7th edition, Cambridge: Cambridge University Press. [13] M. Herzberger, 1958, Modern Geometrical Optics, New York: Interscience. [14] M. Herzberger, 1931, Strahlenoptik, Berlin: Springer. [15] O. N. Stavroudis, 2006, The Mathematics of Geometrical and Physical Optics, Weinheim: Wiley. [16] P. R. Bhattacharjee, The generalized vectorial laws of reflection and refraction, European Journal of Physics, 2005; 26(5); 901-911. [17] P. R. Bhattacharjee, Giving birth to the refined unambiguous statement of Snell's law in ray optics, Optik 2014; DOI: 10.1016/j.ijleo.2014.07.124.
[18] P. R. Bhattacharjee, Refinement of the definitions of angles of incidence, reflection, refraction, and critical angle in ray optics, Potentials IEEE (in press).
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[19] P. R. Bhattacharjee, Deviation problems in ray optics in the light of the refined unambiguous definitions of angles of incidence, reflection and refraction, Optik 2014; 125(16);4257-4261. [20] P. R. Bhattacharjee, Addressing some issues of ray optics on the basis of the newly discovered generalized vectorial laws of reflection and refraction, Optik 2013; 124(23);6250-6254.
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[21] P. R. Bhattacharjee, The generalized vectorial laws of reflection and refraction applied to the rotation poblems in ray optics, Optik 2010; 121(23);2128-2132.
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[22] P. R. Bhattacharjee, Exhaustive study of reflection and refraction at spherical surfaces on the basis of the newly discovered generalized vectorial laws of reflection and refraction, Optik 2012; 123(5); 381-386.
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