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    Eeop 6315 HW 1

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    jbenagain
    The document discusses the eikonal equation and its implications in ray optics. It derives the eikonal equation from Maxwell's equations for electromagnetic waves propagating in an isotropic medium. The eikonal equation relates the geometrical wavefronts to the resulting optical rays. It shows that in a homogeneous medium, rays travel parallel, while in an inhomogeneous medium the eikonal equation identifies surfaces of equal refractive index and the normal rays.

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    Eeop 6315 HW 1

    Uploaded by

    jbenagain
    16 views8 pages
    The document discusses the eikonal equation and its implications in ray optics. It derives the eikonal equation from Maxwell's equations for electromagnetic waves propagating in an isotropic medium. The eikonal equation relates the geometrical wavefronts to the resulting optical rays. It shows that in a homogeneous medium, rays travel parallel, while in an inhomogeneous medium the eikonal equation identifies surfaces of equal refractive index and the normal rays.

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    EEOP 6315 HW 1

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    The document discusses the eikonal equation and its implications in ray optics. It derives the eikonal equation from Maxwell's equations for electromagnetic waves propagating in an isotropic medium. The eikonal equation relates the geometrical wavefronts to the resulting optical rays. It shows that in a homogeneous medium, rays travel parallel, while in an inhomogeneous medium the eikonal equation identifies surfaces of equal refractive index and the normal rays.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    16 views8 pages

    Eeop 6315 HW 1

    Uploaded by

    jbenagain
    The document discusses the eikonal equation and its implications in ray optics. It derives the eikonal equation from Maxwell's equations for electromagnetic waves propagating in an isotropic medium. The eikonal equation relates the geometrical wavefronts to the resulting optical rays. It shows that in a homogeneous medium, rays travel parallel, while in an inhomogeneous medium the eikonal equation identifies surfaces of equal refractive index and the normal rays.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
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    Ben Jones EEOP 6315 Homework Assignment 1

    Problem 1. In words, pictures, and equations, discuss the eikonal equation and its implications in ray optics.

    Description The eikonal equation is the mathematical relation between geometrical wavefronts and the resulting optical rays each wavefront. Defining a ray as a curve that is perpendicular, or unit normal, to a given wavefront at every point r along the curvature of the wavefront. Mathematically, the ray points in the direction of . The eikonal equation is the mathematical bridge between wave optics and ray optics. In a homogeneous medium in which n(r) is equal to a constant, the rays must travel parallel to one another across the entire surface since is also

    constant. However, in an inhomogeneous medium, since n(r) is not constant, the eikonal equation identifies the surfaces with equal indices of refraction and the resultant normal rays as seen in figure 11. The surfaces defined by the equation are the geometrical wavefronts.

    Figure 1

    Derivation from Maxwells equations With the equations for a general time-harmonic field in a nonconducting isotropic medium,

    it is found that the vectors E 0 and H 0 satisfy Maxwells equations when independent of time. In regions free of charge and current ,

    where

    being the wavelength in a vacuum.

    Additionally, a homogeneous plane wave in a medium with an index of refraction of and propagating in the direction of the unit vector s can be modeled by the equations, E0 = eeik0n ( sr ),

    H 0 = heik0n ( sr ),
    where e and h are vectors with constant and mostly complex values. With this example, along with findings of a monochromatic electric dipole field in a vacuum,

    E0 = eeik0r , H 0 = heik0r ,
    where r is the distance from the dipole. In this example however, e and h are not constant vectors. But with distances very far away from the dipole relative to the wavelength andwith appropriate normalization of the dipole moment, these vectors are shown to be independent of k0 .

    These two cases lend themselves to examining regions many wavelengths away from the sources with more general types of fields such that

    E0 = e(r)eik0S ( r ), H 0 = h(r)eik0S ( r ),
    where S (r) , described as the optical path, is a real scalar position function and e(r) and h(r) are vector functions of position. Next, relating the functions S (r) , e(r) , and h(r) using various vector identities,

    andthen substituting these values into the original equations from the beginning of the derivation yields

    However, the region of interest is where k0 is very large such that the right side of the equations above go to zero.

    Making proper substitutions and eliminations yields . Substituting the identities above yields

    ( - S ) 2 = n2
    where .

    Problem 2. Using ray matrices derive the overall round trip matrix for a laser cavity comprising 2 spherical mirrors (curvatures R1 and R2) separated by a distance d. Under what conditions is this cavity stable in the Kogelnik and Li sense? Solution

    The ABCD matrix transformation is as follows: [ ] [ [ ][ ][ ][ ][ ][ ][ ] ]

    [ [ ]

    [ ] [ where [ and [ ][ ] ] [ ] [ ] ]

    In order to find the eigenvalues, the determinant of the matrix is calculated and solved in the following manner: [ ] [ ]

    [ The resulting quadratic formula is as follows: ( )

    For stability of the resonant system, the two eigenvalues must have an absolute value of 1 or less. Translating these constraints into the stability criteria that Kogelnik and Li have put forth for a stable spherical cavity which in turn leads to their findings in a simple paraxial cavity such that ( )( )

    References: 1. http://levelofdetail.wordpress.com/2007/05/ 2. Max Born, Emil Wolf, Principles of Optics 3. http://courses.engr.illinois.edu/ece455/ECE455NewNotes/02_CavityModes /02_CavityModes.pdf 4. http://www.utdallas.edu/~cantrell/ee6334/

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