# Fredholm Integral Equation for Finite Fresnel Transform

### Tomohiro Aoyagi, Kouichi Ohtsubo, Nobuo Aoyagi

#### Abstract

The fundamental formula in an optical system is Rayleigh diffraction integral. In practice, we deal with Fresnel diffraction integral as approximate diffraction formula. We seek the function that its total power is maximized in finite Fresnel transform plane, on condition that an input signal is zero outside the bounded region. This problem is a variational one with an accessory condition. This leads to the eigenvalue problems of Fredholm integral equation of the first kind. The kernel of the integral equation is Hermitian conjugate and positive definite. Therefore, eigenvalues are nonnegative and real number. By discretizing the kernel, the problem depends on the eigenvalue problem of Hermitian conjugate matrix in finite dimensional vector space. By using the Jacobi method, we compute the eigenvalues and eigenvectors of the matrix. We applied it to the problem of approximating a function and evaluated the error.

Download#### Paper Citation

#### in Harvard Style

Aoyagi T., Ohtsubo K. and Aoyagi N. (2018). **Fredholm Integral Equation for Finite Fresnel Transform**.In *Proceedings of the 6th International Conference on Photonics, Optics and Laser Technology - Volume 1: PHOTOPTICS,* ISBN 978-989-758-286-8, pages 286-291. DOI: 10.5220/0006709202860291

#### in Bibtex Style

@conference{photoptics18,

author={Tomohiro Aoyagi and Kouichi Ohtsubo and Nobuo Aoyagi},

title={Fredholm Integral Equation for Finite Fresnel Transform},

booktitle={Proceedings of the 6th International Conference on Photonics, Optics and Laser Technology - Volume 1: PHOTOPTICS,},

year={2018},

pages={286-291},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006709202860291},

isbn={978-989-758-286-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 6th International Conference on Photonics, Optics and Laser Technology - Volume 1: PHOTOPTICS,

TI - Fredholm Integral Equation for Finite Fresnel Transform

SN - 978-989-758-286-8

AU - Aoyagi T.

AU - Ohtsubo K.

AU - Aoyagi N.

PY - 2018

SP - 286

EP - 291

DO - 10.5220/0006709202860291