- Email: [email protected]
(i – δ) curve of a prism in ray optics in the light of the refined unambiguous angles of incidence and refraction
Optik 126 (2015) 3193–3196
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
(i – ␦) curve of a prism in ray optics in the light of the refined unambiguous angles of incidence and refraction Pramode Ranjan Bhattacharjee ∗ Retired Principal, Kabi Nazrul Mahavidyalaya, Sonamura, Tripura 799 131, India
a r t i c l e
i n f o
Article history: Received 16 June 2014 Accepted 18 July 2015 Keywords: Geometrical optics Refraction Prism Deviation
a b s t r a c t This paper makes use of the unambiguous definitions of angles of incidence and refraction offered by the author in 2005 to make an extensive study of the theoretical behavior of the (i – ␦) curve of a prism in ray optics. Novel relations have been achieved for the explicit dependence of the final angle of emergence (i2 ) as well as the net deviation (ı) on the unambiguous primary angle of incidence (i1 ). Considering a glass prism with angle, A = 60◦ and refractive index, = 1.5, the theoretical relations developed have been subsequently employed to predict the graphical behavior of each of the (i1 versus i2 ) curve, (i1 versus (i2 − i1 )) curve, and (i1 versus ı) curve. Though the quantitative interpretation of the peculiar nature of the (i1 versus ı) curve of a prism follows from the relevant relation derived in the paper, qualitative interpretation regarding the peculiar trend of variation of the net deviation (ı) with the unambiguous primary angle of incidence (i1 ) has also been provided on the basis of the (i1 versus i2 ) as well as (i1 versus (i2 − i1 )) curves. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction The discovery of ambiguity in the traditional definitions of angles of incidence, reflection, and refraction in ray optics (which make the traditional laws of reflection and refraction also ambiguous), along with the introduction of the unambiguous definitions of those three angles as well as the generalized vectorial laws of reflection and refraction reported in [8] ultimately resulted in a new optical world in which there is no place for the traditional ambiguous angles of incidence, reflection, and refraction. This new optical world is based exclusively on the unambiguous angles of incidence, reflection, and refraction [8,9]. At the same time, in this new optical world there is also no place for the traditional ambiguous laws of reflection and refraction. Only the unambiguous generalized vectorial laws of reflection and refraction are of concern to this new optical world. Now that the long-running definitions of angles of incidence, reflection, and refraction as well as the traditional laws of reflection and refraction [1–7] stand ambiguous [8], immediate replacement/refinement of each theoretical treatment/discussion in ray optics in traditional literature (which is based on those ambiguous
∗ Correspondence to: 5 Mantri Bari Road, P.O. Agartala, Tripura 799 001, India. Tel.: +91 03812312288. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.07.085 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
angles of incidence, reflection, and refraction, and which makes use of the traditional ambiguous laws of reflection and refraction) on the basis of the refined unambiguous definitions of angles of incidence, reflection, and refraction making use of the unambiguous generalized vectorial laws of reflection and refraction is essential. As a first attempt, the deviation problems in ray optics have been considered in [10]. Refined unambiguous definitions of the aforesaid three angles have been employed in [10] for the calculation of deviation in various cases of reflection and refraction. The condition of minimum deviation suffered by a ray of light in passing through a prism has also been achieved in [10] in the light of the newly introduced unambiguous angles of incidence and refraction [8,9]. This paper considers the theoretical study of the peculiar nature of the (i – ␦) curve of a prism in ray optics in the light of the new optical world making use of the refined unambiguous angles of incidence and refraction, and the generalized vectorial law of refraction. Attempt has been made to find the explicit dependence of the final angle of refraction (angle of emergence) as well as the net deviation suffered by a ray of light in passing through a prism on the unambiguous primary angle of incidence. The theoretical relations derived have been employed to find the nature of a few curves by considering a glass prism with angle, A = 60◦ and refractive index, = 1.5. Qualitative as well as quantitative interpretations of the peculiar nature of the (i – ␦) curve of a prism has been subsequently offered.
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Considering modulus on both sides of this relation, we have sin i1 = sin R1 , where 90◦ < i1 ≤ 180◦ , and 90◦ < R1 ≤ 180◦ or,
R1 = sin−1 sin i1 /
(3)
Similarly for refraction at the point Q of the second refracting face of the prism shown in Fig. 1, we have from the generalized vectorial law of refraction, n2 × R1 =
1 (n2 × i2 ) ,
where 0◦ ≤ i2 < 90◦ ,
and 0◦ ≤ R2 < 90◦
Considering modulus on both sides of this relation we have,
Fig. 1. Diagram showing the deviation of a ray of light in passing through a prism.
sin i2 = sin R2 , where 0◦ ≤ i2 < 90◦ , and 0◦ ≤ R2 < 90◦
1.1. Derivation of the expression for the net deviation suffered by a ray of light in passing through a prism in terms of unambiguous primary angle of incidence and final angle of refraction (i.e., angle of emergence)
or,
R2 = sin−1 sin i2 /
(4)
Using the relations (3) and (4), the relation (2) may be written From Fig. 1, we have, ı1 = R1 − i1 and ı2 = i2 − R2 . As shown in Fig. 1, let us now consider a right handed system of coordinates with O as origin and OX, OY, and OZ as the axes of coordinates. Then we have,
sin−1 sin i1 / − sin−1 sin i2 / = − A
i1 = cos ı1 I + sin ı1 J = cos (R1 − i1 ) I + sin (R1 − i1 ) J
sin−1 sin i2 / = − − sin−1 sin i1 / − A
and
or,
i2 = cos ı2 I − sin ı2 J = cos (i2 − R2 ) I − sin (i2 − R2 ) J
sin i2 / = − sin − sin−1 sin i1 / + A
Now, i1 · i2 =
· cos (i2 − R2 ) I
cos ı = cos (R1 − i1 ) + (i2 − R2 )
sin i2 / = − sin sin−1 sin i1 / + A
− sin (i2 − R2 ) J or,
or,
or,
cos (R1 − i1 ) I − sin (R1 − i1 ) J
as,
or,
sin i2 = − sin sin−1 sin i1 / + A
or,
or,
i2 = − sin−1 sin sin−1 sin i1 / + A
ı = (R1 − R2 ) − (i1 − i2 )
(5)
Using the relation (5), the expression for the net deviation (ı) suffered by a ray of light in passing through a prism can then be obtained from the relation (1) as,
or,
ı = ( − A) − (i1 − i2 ) , since from triangle OQM in Fig. 1 it follows that, R1 − R2 = − A. Thus the net deviation (ı) suffered by a ray of light in passing through a prism is given by, ı = ( − A) − (i1 − i2 ), where, A is the angle of the prism. 1.2. Theoretical treatment for the derivation of the explicit dependence of the net deviation suffered by a ray of light in passing through a prism on the unambiguous primary angle of incidence
ı = ( − A) − i1 − sin−1 sin sin−1 sin i1 / + A
(6)
The relation (5) represents the explicit dependence of the angle of emergence (i.e., the final unambiguous angle of refraction) on the primary unambiguous angle of incidence and is a novel one. The relation (6) represents the explicit dependence of the net deviation suffered by a ray of light while passing through a prism in terms of the unambiguous primary angle of incidence and is also a novel relation. 2. Condition of no emergence from a prism
We have seen that the net deviation suffered by a ray of light in passing through a prism is given by, ı = ( − A) − (i1 − i2 )
(1)
where, R1 − R2 = − A
(2)
With reference to Fig. 1, we have in case of refraction of a ray of light passing through a prism, the relation, R1 − R2 = − A
(7)
Now, on putting R2 = c , where c is the critical angle for the pair of media separated by the refracting face of the prism, we have,
Now considering refraction at the point O of the first refracting face of the prism shown in Fig. 1, we have, from the generalized vectorial law of refraction,
R1 − c = − A
n1 × i1 = (n1 × R1 )
R1 = c + − A
or,
P.R. Bhattacharjee / Optik 126 (2015) 3193–3196
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Table 1 Display of values of different angles. Admissible values of i1 (in degrees)
Values of i2 (in degrees)
Values of (i2 − i1 ) (in degrees)
Values of deviation (ı) (in degrees)
95 100 105 110 115 120 125 130 135 140 145 150
28.234 29.174 30.723 32.866 35.588 38.877 42.738 47.210 52.380 58.466 65.999 77.097
−66.766 −70.826 −74.277 −77.134 −79.412 −81.123 −82.262 −82.79 −82.62 −81.534 −79.001 −72.903
53.234 49.174 45.723 42.866 40.588 38.877 37.738 37.210 37.380 38.466 40.999 47.097
Fig. 2. Diagram showing the theoretical behavior of the unambiguous primary angle of incidence (i1 ) versus final angle of emergence (i2 ) curve for a ray of light passing through a prism.
Now for refraction at the first refracting face of the prism, we have, sin i1 = sin R1 or,
i1 = sin−1 sin c + − A
(8)
For a given prism with known refractive index () and known refracting angle (A), the value of i1 can then be calculated from the relation (8) by substituting the value of the critical angle ( c ) for the pair of media separated by the refracting face of the prism. It then readily follows from the relations (7) and (8) that, when i1 > sin−1 {sin( c + – A)}, the rays after having been incident at the second refracting face of the prism by suffering refraction at the first refracting face of the prism, will all undergo total internal reflection at the second refracting face of the prism and there will be no emergent ray from the prism for each of those rays incident at the second refracting face. Thus the condition of no emergence is given by, i1 > sin−1 {sin( c + − A)}. For example, let us consider a glass prism kept in air with the following specifications: A = 60◦ , = 1.5. Also, for such a glass prism kept in air, we may take the value of the critical angle ( c ), as c = 41.8◦ . Then from the relation (8), we have,
i1 = sin−1 1.5 sin (41.8 + 180 − 60)
with the help of a scientific calculator for different admissible values of i1 , not exceeding the value 152.063, such that 90◦ < i1 ≤ 180◦ . The values so computed are displayed in Table 1. Using these computed values of i2 , (i2 − i1 ), and ı, one can then proceed to obtain the graphical behavior of (i1 versus i2 ), (i1 versus (i2 − i1 )), and (i1 versus ı) curves, which will be as shown in Fig. 2, Fig. 3, and Fig. 4 respectively. Making use of these tabulated values of i1 , i2 , (i2 − i1 ), and ı, three different curves, viz. (i1 versus i2 ) curve, i1 versus (i2 − i1 ) curve, and (i1 versus ı) curve are drawn and they are represented in Fig. 2, Fig. 3, and Fig. 4 respectively. A careful look at the (i1 versus i2 ) curve as shown in Fig. 2 reveals that i2 increases more slowly as compared to the increase of i1 from the initial stage. Thereafter, the increase of i2 becomes faster as compared to that of i1 . Thus during the increase of i1 starting from
or, i1 = sin−1 (1.5 sin 161.8) or, i1 = sin−1 0.46850
Fig. 3. Diagram showing theoretical behavior of unambiguous angle of incidence (i1 ) versus (i2 − i1 ) curve of a prism, where i2 represents the final angle of emergence.
or, i1 = 152.063 Thus it readily follows from above that for a glass prism with the aforesaid specifications, there will exist no emergent ray from the prism when rays are incident on such a prism kept in air with unambiguous angle of incidence at the first refracting face exceeding the value 152.063◦ . 3. Theoretical behavior of (unambiguous angle of incidence versus net deviation) curve It appears that the relations (5) and (6) are very much complicated and quite difficult to handle manually. However, an idea of the various curves can be obtained by considering typical values of the parameters A and for a glass prism kept in air. Thus considering A = 60◦ , = 1.5, the values of i2 , (i2 − i1 ), and ı can be computed
Fig. 4. Diagram showing the theoretical behavior of the unambiguous primary angle of incidence (i1 ) versus net deviation (ı) curve for a ray of light passing through a prism.
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the initial stage, the quantity (i2 − i1 ) will become more and more negative. Thereafter, the quantity (i2 − i1 ) will become more and more positive with subsequent increase of the value of i1 . This very trend of the (i1 versus i2 ) curve reflects qualitatively the fact that the net deviation ı, where ı = ( − A) − (i1 − i2 ) = ( − A) + (i2 − i1 ), will first decrease and then increase with increase of the value of the unambiguous angle of incidence i1 . Similarly, a qualitative idea about the nature of the (i1 versus ı curve) can be obtained from the nature of the i1 versus (i2 − i1 ) curve shown in Fig. 3. It appears from Fig. 3 that the value of (i2 − i1 ) goes on decreasing with the increase of the value of i1 from the initial stage until it reaches a minimum value for a particular value of i1 . On further increase of the value of i1 , the value of (i2 − i1 ) goes on increasing. It is then clear from this very trend of the i1 versus (i2 − i1 ) curve and the relation, ı = ( − A) + (i2 − i1 ), that with the increase of the value of i1 , the net deviation (ı) suffered by a ray of light in passing through a prism, will first go on decreasing so as to attain a minimum value for a particular value i1 , beyond which it will go on increasing with further increase of the value of i1 . Fig. 4 represents the quantitative behavior of the (i1 versus ı) curve of a prism kept in air with the specifications: A = 60◦ , = 1.5, and is obtained by making use of the relation (6). 4. Conclusion This paper makes use of the refined unambiguous definitions of angles of incidence and refraction [8,9] to give birth to novel theoretical treatment for the interpretation of the peculiar nature of the (i – ␦) curve of a prism. Considering a glass prism with typical values of the angle (A) as 60◦ and the refractive index () as 1.5, the novel relations developed have been employed to predict the theoretical nature of three different curves, viz. the (i1 versus i2 ) curve, (i1 versus (i2 − i1 )) curve, and (i1 versus ı) curve, where i1 , i2 ,
and ı correspond to the unambiguous primary angle of incidence, unambiguous final angle of refraction (angle of emergence), and net deviation suffered by the ray of light in passing through the prism respectively. While qualitative interpretation of the peculiar nature of the (i1 versus ı) curve follows from the nature of each of the (i1 versus i2 ) curve and the (i1 versus (i2 − i1 )) curve, quantitative nature of the (i1 versus ı) curve follows from the novel theoretical relation involving ı and i1 . Incorporation of the refined unambiguous definitions of angles of incidence, and refraction [9], which are much clearer leaving no room for confusion, into the theoretical treatment for the prediction of the behavior of (i1 versus ı) curve of a prism is novel, original and at the same time it will enhance and sophisticate the study of optical physics unlike the tradition literature. References [1] L. Laurance, General and Practical Optics, School of Optics, Macmillan Company, London, York, 1918 (3rd and revised edition). [2] F.A. Jenkins, H.E. White, Fundamentals of Optics, 3rd ed., McGraw Hill Book Company, New York, Toronto, London, 1957. [3] J.P.C. Southall, Mirrors, Prisms, and Lenses—A Text Book of Geometrical Optics, The Macmillan Company, New York, 1918. [4] L.S. Pedrotti, Basic Geometrical Optics, Module 1.2, SPIE Press, spie.org, USA. [5] D. Halliday, R. Resnick, in: J. Walker (Ed.), Principles of Physics, International Student Version, 9th ed., John Wiley & Sons Inc., Wiley India Pvt. Ltd, Cleveland State University, New Delhi, India, 2013 (Authorized reprint by). [6] E. Hecht, A.R. Genesan, Optics, 4th ed., Pearson, Delhi, India, 2008. [7] A. Ghatak, Optics, 4th ed., Tata McGraw-Hill Education Private Limited, New Delhi, India, 2011 (7th ed.). [8] P.R. Bhattacharjee, The generalized vectorial laws of reflection and refraction, Eur. J. Phys. 26 (2005) 901–911. [9] P.R. Bhattacharjee, Refinement of the definitions of angles of incidence, reflection, refraction, and critical angle in ray optics, to appear in Potentials IEEE. [10] P.R. Bhattacharjee, Deviation problems in ray optics in the light of the refined unambiguous definitions of angles of incidence, reflection and refraction, to appear in Optik.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
(i – ␦) curve of a prism in ray optics in the light of the refined unambiguous angles of incidence and refraction Pramode Ranjan Bhattacharjee ∗ Retired Principal, Kabi Nazrul Mahavidyalaya, Sonamura, Tripura 799 131, India
a r t i c l e
i n f o
Article history: Received 16 June 2014 Accepted 18 July 2015 Keywords: Geometrical optics Refraction Prism Deviation
a b s t r a c t This paper makes use of the unambiguous definitions of angles of incidence and refraction offered by the author in 2005 to make an extensive study of the theoretical behavior of the (i – ␦) curve of a prism in ray optics. Novel relations have been achieved for the explicit dependence of the final angle of emergence (i2 ) as well as the net deviation (ı) on the unambiguous primary angle of incidence (i1 ). Considering a glass prism with angle, A = 60◦ and refractive index, = 1.5, the theoretical relations developed have been subsequently employed to predict the graphical behavior of each of the (i1 versus i2 ) curve, (i1 versus (i2 − i1 )) curve, and (i1 versus ı) curve. Though the quantitative interpretation of the peculiar nature of the (i1 versus ı) curve of a prism follows from the relevant relation derived in the paper, qualitative interpretation regarding the peculiar trend of variation of the net deviation (ı) with the unambiguous primary angle of incidence (i1 ) has also been provided on the basis of the (i1 versus i2 ) as well as (i1 versus (i2 − i1 )) curves. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction The discovery of ambiguity in the traditional definitions of angles of incidence, reflection, and refraction in ray optics (which make the traditional laws of reflection and refraction also ambiguous), along with the introduction of the unambiguous definitions of those three angles as well as the generalized vectorial laws of reflection and refraction reported in [8] ultimately resulted in a new optical world in which there is no place for the traditional ambiguous angles of incidence, reflection, and refraction. This new optical world is based exclusively on the unambiguous angles of incidence, reflection, and refraction [8,9]. At the same time, in this new optical world there is also no place for the traditional ambiguous laws of reflection and refraction. Only the unambiguous generalized vectorial laws of reflection and refraction are of concern to this new optical world. Now that the long-running definitions of angles of incidence, reflection, and refraction as well as the traditional laws of reflection and refraction [1–7] stand ambiguous [8], immediate replacement/refinement of each theoretical treatment/discussion in ray optics in traditional literature (which is based on those ambiguous
∗ Correspondence to: 5 Mantri Bari Road, P.O. Agartala, Tripura 799 001, India. Tel.: +91 03812312288. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.07.085 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
angles of incidence, reflection, and refraction, and which makes use of the traditional ambiguous laws of reflection and refraction) on the basis of the refined unambiguous definitions of angles of incidence, reflection, and refraction making use of the unambiguous generalized vectorial laws of reflection and refraction is essential. As a first attempt, the deviation problems in ray optics have been considered in [10]. Refined unambiguous definitions of the aforesaid three angles have been employed in [10] for the calculation of deviation in various cases of reflection and refraction. The condition of minimum deviation suffered by a ray of light in passing through a prism has also been achieved in [10] in the light of the newly introduced unambiguous angles of incidence and refraction [8,9]. This paper considers the theoretical study of the peculiar nature of the (i – ␦) curve of a prism in ray optics in the light of the new optical world making use of the refined unambiguous angles of incidence and refraction, and the generalized vectorial law of refraction. Attempt has been made to find the explicit dependence of the final angle of refraction (angle of emergence) as well as the net deviation suffered by a ray of light in passing through a prism on the unambiguous primary angle of incidence. The theoretical relations derived have been employed to find the nature of a few curves by considering a glass prism with angle, A = 60◦ and refractive index, = 1.5. Qualitative as well as quantitative interpretations of the peculiar nature of the (i – ␦) curve of a prism has been subsequently offered.
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P.R. Bhattacharjee / Optik 126 (2015) 3193–3196
Considering modulus on both sides of this relation, we have sin i1 = sin R1 , where 90◦ < i1 ≤ 180◦ , and 90◦ < R1 ≤ 180◦ or,
R1 = sin−1 sin i1 /
(3)
Similarly for refraction at the point Q of the second refracting face of the prism shown in Fig. 1, we have from the generalized vectorial law of refraction, n2 × R1 =
1 (n2 × i2 ) ,
where 0◦ ≤ i2 < 90◦ ,
and 0◦ ≤ R2 < 90◦
Considering modulus on both sides of this relation we have,
Fig. 1. Diagram showing the deviation of a ray of light in passing through a prism.
sin i2 = sin R2 , where 0◦ ≤ i2 < 90◦ , and 0◦ ≤ R2 < 90◦
1.1. Derivation of the expression for the net deviation suffered by a ray of light in passing through a prism in terms of unambiguous primary angle of incidence and final angle of refraction (i.e., angle of emergence)
or,
R2 = sin−1 sin i2 /
(4)
Using the relations (3) and (4), the relation (2) may be written From Fig. 1, we have, ı1 = R1 − i1 and ı2 = i2 − R2 . As shown in Fig. 1, let us now consider a right handed system of coordinates with O as origin and OX, OY, and OZ as the axes of coordinates. Then we have,
sin−1 sin i1 / − sin−1 sin i2 / = − A
i1 = cos ı1 I + sin ı1 J = cos (R1 − i1 ) I + sin (R1 − i1 ) J
sin−1 sin i2 / = − − sin−1 sin i1 / − A
and
or,
i2 = cos ı2 I − sin ı2 J = cos (i2 − R2 ) I − sin (i2 − R2 ) J
sin i2 / = − sin − sin−1 sin i1 / + A
Now, i1 · i2 =
· cos (i2 − R2 ) I
cos ı = cos (R1 − i1 ) + (i2 − R2 )
sin i2 / = − sin sin−1 sin i1 / + A
− sin (i2 − R2 ) J or,
or,
or,
cos (R1 − i1 ) I − sin (R1 − i1 ) J
as,
or,
sin i2 = − sin sin−1 sin i1 / + A
or,
or,
i2 = − sin−1 sin sin−1 sin i1 / + A
ı = (R1 − R2 ) − (i1 − i2 )
(5)
Using the relation (5), the expression for the net deviation (ı) suffered by a ray of light in passing through a prism can then be obtained from the relation (1) as,
or,
ı = ( − A) − (i1 − i2 ) , since from triangle OQM in Fig. 1 it follows that, R1 − R2 = − A. Thus the net deviation (ı) suffered by a ray of light in passing through a prism is given by, ı = ( − A) − (i1 − i2 ), where, A is the angle of the prism. 1.2. Theoretical treatment for the derivation of the explicit dependence of the net deviation suffered by a ray of light in passing through a prism on the unambiguous primary angle of incidence
ı = ( − A) − i1 − sin−1 sin sin−1 sin i1 / + A
(6)
The relation (5) represents the explicit dependence of the angle of emergence (i.e., the final unambiguous angle of refraction) on the primary unambiguous angle of incidence and is a novel one. The relation (6) represents the explicit dependence of the net deviation suffered by a ray of light while passing through a prism in terms of the unambiguous primary angle of incidence and is also a novel relation. 2. Condition of no emergence from a prism
We have seen that the net deviation suffered by a ray of light in passing through a prism is given by, ı = ( − A) − (i1 − i2 )
(1)
where, R1 − R2 = − A
(2)
With reference to Fig. 1, we have in case of refraction of a ray of light passing through a prism, the relation, R1 − R2 = − A
(7)
Now, on putting R2 = c , where c is the critical angle for the pair of media separated by the refracting face of the prism, we have,
Now considering refraction at the point O of the first refracting face of the prism shown in Fig. 1, we have, from the generalized vectorial law of refraction,
R1 − c = − A
n1 × i1 = (n1 × R1 )
R1 = c + − A
or,
P.R. Bhattacharjee / Optik 126 (2015) 3193–3196
3195
Table 1 Display of values of different angles. Admissible values of i1 (in degrees)
Values of i2 (in degrees)
Values of (i2 − i1 ) (in degrees)
Values of deviation (ı) (in degrees)
95 100 105 110 115 120 125 130 135 140 145 150
28.234 29.174 30.723 32.866 35.588 38.877 42.738 47.210 52.380 58.466 65.999 77.097
−66.766 −70.826 −74.277 −77.134 −79.412 −81.123 −82.262 −82.79 −82.62 −81.534 −79.001 −72.903
53.234 49.174 45.723 42.866 40.588 38.877 37.738 37.210 37.380 38.466 40.999 47.097
Fig. 2. Diagram showing the theoretical behavior of the unambiguous primary angle of incidence (i1 ) versus final angle of emergence (i2 ) curve for a ray of light passing through a prism.
Now for refraction at the first refracting face of the prism, we have, sin i1 = sin R1 or,
i1 = sin−1 sin c + − A
(8)
For a given prism with known refractive index () and known refracting angle (A), the value of i1 can then be calculated from the relation (8) by substituting the value of the critical angle ( c ) for the pair of media separated by the refracting face of the prism. It then readily follows from the relations (7) and (8) that, when i1 > sin−1 {sin( c + – A)}, the rays after having been incident at the second refracting face of the prism by suffering refraction at the first refracting face of the prism, will all undergo total internal reflection at the second refracting face of the prism and there will be no emergent ray from the prism for each of those rays incident at the second refracting face. Thus the condition of no emergence is given by, i1 > sin−1 {sin( c + − A)}. For example, let us consider a glass prism kept in air with the following specifications: A = 60◦ , = 1.5. Also, for such a glass prism kept in air, we may take the value of the critical angle ( c ), as c = 41.8◦ . Then from the relation (8), we have,
i1 = sin−1 1.5 sin (41.8 + 180 − 60)
with the help of a scientific calculator for different admissible values of i1 , not exceeding the value 152.063, such that 90◦ < i1 ≤ 180◦ . The values so computed are displayed in Table 1. Using these computed values of i2 , (i2 − i1 ), and ı, one can then proceed to obtain the graphical behavior of (i1 versus i2 ), (i1 versus (i2 − i1 )), and (i1 versus ı) curves, which will be as shown in Fig. 2, Fig. 3, and Fig. 4 respectively. Making use of these tabulated values of i1 , i2 , (i2 − i1 ), and ı, three different curves, viz. (i1 versus i2 ) curve, i1 versus (i2 − i1 ) curve, and (i1 versus ı) curve are drawn and they are represented in Fig. 2, Fig. 3, and Fig. 4 respectively. A careful look at the (i1 versus i2 ) curve as shown in Fig. 2 reveals that i2 increases more slowly as compared to the increase of i1 from the initial stage. Thereafter, the increase of i2 becomes faster as compared to that of i1 . Thus during the increase of i1 starting from
or, i1 = sin−1 (1.5 sin 161.8) or, i1 = sin−1 0.46850
Fig. 3. Diagram showing theoretical behavior of unambiguous angle of incidence (i1 ) versus (i2 − i1 ) curve of a prism, where i2 represents the final angle of emergence.
or, i1 = 152.063 Thus it readily follows from above that for a glass prism with the aforesaid specifications, there will exist no emergent ray from the prism when rays are incident on such a prism kept in air with unambiguous angle of incidence at the first refracting face exceeding the value 152.063◦ . 3. Theoretical behavior of (unambiguous angle of incidence versus net deviation) curve It appears that the relations (5) and (6) are very much complicated and quite difficult to handle manually. However, an idea of the various curves can be obtained by considering typical values of the parameters A and for a glass prism kept in air. Thus considering A = 60◦ , = 1.5, the values of i2 , (i2 − i1 ), and ı can be computed
Fig. 4. Diagram showing the theoretical behavior of the unambiguous primary angle of incidence (i1 ) versus net deviation (ı) curve for a ray of light passing through a prism.
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the initial stage, the quantity (i2 − i1 ) will become more and more negative. Thereafter, the quantity (i2 − i1 ) will become more and more positive with subsequent increase of the value of i1 . This very trend of the (i1 versus i2 ) curve reflects qualitatively the fact that the net deviation ı, where ı = ( − A) − (i1 − i2 ) = ( − A) + (i2 − i1 ), will first decrease and then increase with increase of the value of the unambiguous angle of incidence i1 . Similarly, a qualitative idea about the nature of the (i1 versus ı curve) can be obtained from the nature of the i1 versus (i2 − i1 ) curve shown in Fig. 3. It appears from Fig. 3 that the value of (i2 − i1 ) goes on decreasing with the increase of the value of i1 from the initial stage until it reaches a minimum value for a particular value of i1 . On further increase of the value of i1 , the value of (i2 − i1 ) goes on increasing. It is then clear from this very trend of the i1 versus (i2 − i1 ) curve and the relation, ı = ( − A) + (i2 − i1 ), that with the increase of the value of i1 , the net deviation (ı) suffered by a ray of light in passing through a prism, will first go on decreasing so as to attain a minimum value for a particular value i1 , beyond which it will go on increasing with further increase of the value of i1 . Fig. 4 represents the quantitative behavior of the (i1 versus ı) curve of a prism kept in air with the specifications: A = 60◦ , = 1.5, and is obtained by making use of the relation (6). 4. Conclusion This paper makes use of the refined unambiguous definitions of angles of incidence and refraction [8,9] to give birth to novel theoretical treatment for the interpretation of the peculiar nature of the (i – ␦) curve of a prism. Considering a glass prism with typical values of the angle (A) as 60◦ and the refractive index () as 1.5, the novel relations developed have been employed to predict the theoretical nature of three different curves, viz. the (i1 versus i2 ) curve, (i1 versus (i2 − i1 )) curve, and (i1 versus ı) curve, where i1 , i2 ,
and ı correspond to the unambiguous primary angle of incidence, unambiguous final angle of refraction (angle of emergence), and net deviation suffered by the ray of light in passing through the prism respectively. While qualitative interpretation of the peculiar nature of the (i1 versus ı) curve follows from the nature of each of the (i1 versus i2 ) curve and the (i1 versus (i2 − i1 )) curve, quantitative nature of the (i1 versus ı) curve follows from the novel theoretical relation involving ı and i1 . Incorporation of the refined unambiguous definitions of angles of incidence, and refraction [9], which are much clearer leaving no room for confusion, into the theoretical treatment for the prediction of the behavior of (i1 versus ı) curve of a prism is novel, original and at the same time it will enhance and sophisticate the study of optical physics unlike the tradition literature. References [1] L. Laurance, General and Practical Optics, School of Optics, Macmillan Company, London, York, 1918 (3rd and revised edition). [2] F.A. Jenkins, H.E. White, Fundamentals of Optics, 3rd ed., McGraw Hill Book Company, New York, Toronto, London, 1957. [3] J.P.C. Southall, Mirrors, Prisms, and Lenses—A Text Book of Geometrical Optics, The Macmillan Company, New York, 1918. [4] L.S. Pedrotti, Basic Geometrical Optics, Module 1.2, SPIE Press, spie.org, USA. [5] D. Halliday, R. Resnick, in: J. Walker (Ed.), Principles of Physics, International Student Version, 9th ed., John Wiley & Sons Inc., Wiley India Pvt. Ltd, Cleveland State University, New Delhi, India, 2013 (Authorized reprint by). [6] E. Hecht, A.R. Genesan, Optics, 4th ed., Pearson, Delhi, India, 2008. [7] A. Ghatak, Optics, 4th ed., Tata McGraw-Hill Education Private Limited, New Delhi, India, 2011 (7th ed.). [8] P.R. Bhattacharjee, The generalized vectorial laws of reflection and refraction, Eur. J. Phys. 26 (2005) 901–911. [9] P.R. Bhattacharjee, Refinement of the definitions of angles of incidence, reflection, refraction, and critical angle in ray optics, to appear in Potentials IEEE. [10] P.R. Bhattacharjee, Deviation problems in ray optics in the light of the refined unambiguous definitions of angles of incidence, reflection and refraction, to appear in Optik.