Thin and thick diffraction gratings: Thin matrix decomposition method C. Neipp, A. Ma ´ rquez, A. Hernandez, M. L. Alvarez, A. Bele ´ ndez Departamento de Fı ´sica, Ingenierı ´a de Sistemas y Teorı ´a de la Sen ˜al, Universidad de Alicante, Ap. 99, E-03080 Alicante, Spain Abstract: A brief review of the properties of transmission diffraction gratings is presented. Two types of gratings will be analyzed: thin and volume gratings explaining how the efficiency of the different orders that propagate inside the gratings can be calculated in both cases. For thin diffraction gratings the so-called amplitude transmittance method is ap- plied in order to get the amplitude of the different orders, whereas in the case of volume gratings more complex meth- ods are needed, such as Coupled Wave or modal theories. We will comment on the thin matrix decomposition method (TMDM), firstly proposed by Alferness, which gives a very intuitive approach and connects the properties of thin grat- ings to the properties of volume ones. The thin matrix de- composition method consists in dividing the volume grating in a number of thin gratings and applying the amplitude transmittance method to each thin grating. In this way the output of a grating will be considered as the input of the next and any individual grating can be treated by the amplitude transmittance method. The novelty of this work is that a comparison is made between the analytical expressions ob- tained by Alferness using the TMDM with the numerical re- sults obtained using the coupled wave (CW) and rigorous coupled wave (RCW) theories for the efficiencies of the zero, first and second order when a plane wave incides onto a sinusoidal diffracion grating at the second on-Bragg replay angular condition. Key words: Holography – volume holograms – holographic recording materials – photographic emulsions 1. Introduction The simplest periodic structure that can be recorded on a photosensitive material is a sinusoidal diffraction grating. Therefore the basic problem in volume holo- graphy theory is to describe accurately the properties of this kind of structures [1]. A usual way to calculate the efficiencies of the different orders that propagate in the volume grating is to solve Maxwell equations for the case of an incident plane wave on a medium where the relative dielectric permittivity varies [2]. Although the idea seems clear and precise, in the literature there is a great number of models that allow to solve the problem. One of the most used models to solve the electric field in the periodic structure is the Coupled Wave (CW) theory [1, 2]. The name of this method is directly related to the way the solution of the wave equations is obtained. The most representative aspect of the CW theory is that it assumes a continuous interchange of energy between the waves that propagate inside the grating. The first study to calculate the electric field in- side a holographic dielectric grating using the coupled wave method was made by Kogelnik in 1969 [3]. Ko- gelnik assumed that only two orders propagated in the hologram, orders zero and þ1, and obtained analytical solutions for the efficiencies of the first and zero order when a plane wave impinges on a diffraction grating with a sinusoidal variation of its electro-optical proper- ties (relative dielectric permittivity and conductivity). The highly predictive character of the expressions de- rived by Kogelnik made his work one of the most cited by holographic researchers. Nonetheless Kogelnik’s theory assumed some approximations that make it in- accurate for some cases, such as dielectric gratings that are not sinusoidal or for thin gratings (outside the Bragg regime). For these cases the Rigorous Coupled Wave (RCW) theory proposed by Moharam and Gay- lord [4] in 1982 has proven to be appropriated. Moha- ram and Gaylord proposed the method to solve rigor- ously (apart for the number of orders chosen for the numerical simulations) the differential coupled wave equations that emerge from the Helmoltz equation when the coupled wave theory is used. The Rigorous Coupled Wave Theory has been ap- plied with success to volume holograms and binary gratings [5–10], photonic band structures [11], diffrac- tive lenses [12], etc. And it is also the method that should be used to test the validity of the different ap- proximations that have been done and are still doing in order to obtain analytical functions for the efficien- cies of the different orders that propagate in the holo- gram. Although exact predictions can be obtained by        Optik 115, No. 9 (2004) 385 – 392 http://www.elsevier.de/ijleo 0030-4026/04/115/09-385 $ 30.00/0 Received 5 May 2004; accepted 3 August 2004. Correspondence to: C. Neipp Fax: ++34-96-5909750 E-mail: cristian@disc.ua.es