Research article · 2019 · Full text
International Journal of Statistics and Applied MathematicsAnalytical treatment of Fresnel’s theory
Volume 4(6) · Pages 152–153
Research areasGeophysics and Sensor TechnologyOptical and Acousto-Optic Technologies
Abstract
Author-writtenIn this paper, problem of wave theory of light solved analytically by applying mathematical analysis to Fresnel’s theories. This paper based on principal that plane polarizes light on entering crystal along optical axis decomposed into two circular polarized vibration s rotating in clockwise and anticlockwise direction with same frequency and having zero initial phase difference and find angle of rotation analytically by applying mathematical computation.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Applying mathematical analysis to Fresnel's theories is used to resolve various problems within the wave theory of light. View evidence ↗
The question
- The objective of the paper is to analytically solve problems in the wave theory of light by applying mathematical analysis to Fresnel's theories. View evidence ↗
What they did
- The analysis is based on the principle that plane-polarized light entering a crystal along its optical axis decomposes into two circular polarized vibrations rotating in opposite directions with the same frequency and zero initial phase difference. View evidence ↗
- The incident vibration is modeled as x=2b cos wt, where 2b represents the amplitude and w represents the angular velocity. View evidence ↗
- The incident vibration is mathematically separated into two equal and opposite circular motions, defined by the equations y1=b sinwt, z1=b coswt and y2=-b sinwt, z2=b coswt. View evidence ↗
What they found
- When vibrations are transmitted through a crystal plate, they travel at different velocities and emerge with a phase difference denoted as Φ. View evidence ↗
- The angle of rotation δ is found to be equal to half of the phase difference (Φ/2), as derived from the tangent of the ratio of y to z displacements. View evidence ↗
- The vibration plane of incident plane-polarized light is rotated in an anticlockwise direction by an angle equal to half the phase difference between the two emergent circular polarized vibrations. View evidence ↗
- The path of retardation is calculated as t(µS- µT), where Vb is the velocity of light in air. View evidence ↗
- The phase difference Φ is expressed as (2π/λ)d(µS- µT), where λ is the wavelength of light in air and d is the thickness of the crystal. View evidence ↗
- For a crystal where µs > µT, the angle of rotation δ is determined to be (π d (µS- µT))/λ. View evidence ↗
What it means
- The authors conclude that linear vibration results from two opposite circular motions of the same frequency, and the rotation of the vibration plane depends on which circular vibration is faster, equaling half the phase difference upon emergence. View evidence ↗
Visual summary
1 visualsIllustrations explain the paper; they do not show study data.
Decomposition of Plane Polarized Light
AI-generated conceptual illustration (Plane polarized light beam entering a birefringent crystal). It explains the idea and does not show study data.
View source evidence ↗Introduction
A brief review is given of the relevant point about the role of mathematics in physics so applying mathematical analysis to Fresnel theories we removed many of problem of wave theory of light. Shappard and Hrynevch (1992) [6] studied the generalization of Fresnel diffraction theory. BB Baker et al. (2003) give the mathematical theory of Huygenis principal in optics and its application to theory of diffraction. BA Kamp (2011) [3] studied the implications for electromagnetic wave theory light in matter.
V.Lakshmiharyanan (2012) studied the mathematical optics. In the present paper we show the problem of wave theory of light solved analytically by applying mathematical analysis to Fresnel’s theories. 2. Formulation of Problem In an optically inactive crystal, we obtained the result by considering the equation of vibrations. Let the incident vibration Z’OZ be represented by x=2b cos wt where 2b is amplitude and w is angular velocity.
These incident vibration is separated into two equal and opposite circular motion which is given by y1=b sinwt z1=b coswt (1.1) and y2=-b sinwt z2=b coswt (1.2) here equation (1.1) represents the right handed circular motion and equation (1.2) represents the Left handed circular motion .These equation represent individually in circular motion because pair gives y12+z12=b2 y22+z22=b2 (1.3) On superposition they give rise Resultant displacement along y-axis http://www.mathsjournal.com ΔT=TS-TT =(t/vS)-(t/vT) (2.1) Vb is the velocity of light in air ,then path of retardation = Vb ΔT=t((t/vS)-(t/vT)) =t(µS- µT) (2.2) Where µS and µT are the refractive index of substance for right handed and left handed vibration respectively. Φ=(2π/λ)d(µS- µT) (2.3) Where λ is wavelength of light in air. Therefore for the crystal µs> µT, the angle of rotation of plane of vibration is δ=(Φ/2)= (π d (µS- µT))/λ (2.4)
Conclusions
This paper is based on the principal that a linear vibration may be taken as resultant of two opposite circular motion of same frequency .The rotation of vibration plane will be right or left according as right handed or left handed circular vibration is faster and equal to half of the phase difference on emergence between two circular motion.
Acknowledgments
Author is thankful to university grants commission, New Delhi, for financial support.
References
6 in the original paper- Anton H, Busby R. Contemporary linear algebra, 2003.
- John Wiley, Sons NJ, Aoyagi N. Theoretical study of optical Fresnel transformation, Dr. Thesis., Tokyo Institute of Technology, Tokyo, 1973.
- Kemp BA. Resolution of Abraham–minkowski debate: Implication for the electromagnetic wave theory in matter. Journal of Applied Physics. 2011, 109, 111101.
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- Reed M, Simon B. Method of modern mathematical physics, Aceadmic Press, New York, 1972, 1.
- Shappard, Hrynevych. Diffraction by a Circular Aperature A Generalization of Fresnel Diffraction Theory. J Opt. Soc. Am. 1992, 9. y=y1+y2 =b sinwt - b sinwt =0 (1.4) Resultant displacement along z-axis z=z1+z2 =b coswt + b coswt =2b coswt (1.5) These (1.4) and (1.5) components give the resultant displacement z=2b coswt when the vibration transmitted through crystal plate they travel with different velocities, and when they emerge from crystal plate they emerge with phase difference Φ between them. If the anti-clockwise components advance in front of other. The emergent circular components given by y1=b sinwt z1=b coswt (1.6) and y2=-b sin(wt+Φ) z2=b cos(wt+Φ) (1.7) equation (1.6) represent the right hand system and equation(1.7) represent the left hand system the resultant displacement along x axis and y axis are y=y1+y2 =b (sinwt - sin(wt+Φ)) =2 b sin(Φ/2) cos(wt+(Φ/2)) (1.8) And take z= z1+z2 =b (coswt + cos(wt+Φ)) =2 b cos(Φ/2) cos(wt+(Φ/2)) (1.9) These two equation (1.8) and (1.9) represent the mutually perpendicular vibration having same phase, when compounded together give resultant linear vibration make an angle δ with y-axis, Such that Tanδ=(y/z)=tan(Φ/2) (2.0) Here δ=(Φ/2) represent the angle of rotation. Equation(2.0) represents the vibration plane of incident plane polarized light has been rotated in the anticlockwise direction by an angle which is equal to half the phase difference between the two emergent circular polarized vibration. Similarly same relation is proved for right handed system. 2.1 Angle of rotation The magnitude of angle of rotation is calculated in term of refractive indices of right and left handed vibration represent µS and µT respectively. If d is thickness of the crystal along the optic al axis then the time of rotation between two circular components traveling left-handed crystal is.