# Diffraction Grating Monochromatic Light

Experiment to determine the wavelength of monochromatic light using a plane transmission grating. The wavelength of a spectral line can be very accurately determined with the help of a diffraction grating and spectrometer. Initially all the preliminary adjustments of the spectrometer are made. Calculate the wavelength of the monochromatic light where the second order image is diffracted through an angle of 25 o using a diffraction grating with 300 lines per millimetre. Grating spacing (e) = 10 -3 /300 m = 3.3x10 -6 m Wavelength (l) = esin25/2 = 3.3x10 -6 x 0.42/2 = 6.97x10 -7 m = 697 nm 2. Diffraction gratings: holographic 1000 linee/mm – 600 linee/mm Collimating lens slit 100micron / slits 0.1–0.5-1mm built with razor blades / micrometric adjustable slit Portable Spectrometer Theory of Diffraction Grating A monochromatic light beam that is incident on a grating gives rise to a transmitted beam and various.

If the incident beam of light that is incident in the diffraction grating is monochromatic, then the diffraction pattern will produce an energy density distribution occurring at angles that correspond to the integer values such as -3, -2, -1, 0, 1, 2, 3, etc. The diagram below shows a section of a diffraction grating. Monochromatic light of wavelength λ is incident normally on its surface. Light waves diffracted through angle θ form the second order image after passing through a converging lens (not shown). A, B and C are adjacent slits on the grating.

﻿UDC 519.6, 535.4

The Propagation of Polarized Monochromatic Light in

Periodic Media

D. V. Divakov, A. A. Tyutyunnik

Telecommunication Systems Department Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, 117198, Russia

The work is dedicated to analysis of photonic band gaps of one-dimensional photonic crystal and simulating diffraction on one-dimensional binary diffraction grating using RCWA method. The results of simulating diffraction using RCWA are compared with spectropho-tometrical data.

Key words and phrases: RCWA, periodic media, diffraction, photonic crystal, Bloch-waves.

1. Introduction

Periodic structures as photonic crystals are widely used in modern laser devices, communication technologies and for creating various beam splitters and filters. Diffraction gratings are applied for creating 3D television sets, DVD and Blu-ray drives and reflective structures (Berkley mirror). It is important to simulate diffraction on such structures to design optical systems with predetermined properties based on photonic crystals and diffraction gratings. Methods of simulating diffraction on periodic structures uses theory of Floquet-Bloch and rigorous coupled-wave analysis (RCWA) [1,2]. Current work is dedicated to analysis of photonic band gaps and simulating diffraction on one-dimensional binary diffraction grating using RCWA. The Maxwell's equations for isotropic media and constitutive relations based on the cgs system were used as a model.

2. One-Dimensional Photonic Crystal

The photonic crystal is a dielectric structure with periodical refractive index along one or more directions. A distinctive feature of these structures is the presence of so-called photonic band gaps, preventing the propagation of waves of a certain frequency. Due to this property, with the help of photonic crystals it's possible to create devices that can reflect or transmit light with fixed wavelength.

We consider the formation of photonic band gaps by example of monochromatic waves propagating along z-axis, which is perpendicular to the direction of periodicity of one-dimensional photonic crystal. Its dielectric permittivity is a periodic function of z:

( £, (n — 1)d < z < nd — a,

£ (*)=i A A (!)

[ e2, na — a < z < na, where n is a number of the cell and e (z) = e (z + d), where d is a period and a is the width of layer with dielectric permittivity s1.

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Since the mediums are isotropic, the TE and TM polarizations propagate independently and they can be considered separately. Consider the case of waves of TE polarization. We can reduce Maxwell's equations to the following wave equation:

d2Ey/dz1 + kll3Ey = 0, (p = £ — kl/kl) . (2)

From this equation, we find a general solution for the field components in a uniform layer, and then obtain the system considering the condition that the tangential field

This work was supported by research topic PFUR №020617-1-174.

components at the interface between two media with z = (n — 1)d and z = nd — a, from which we find equations relating the undetermined coefficients from (n — l)-th cell with n-th cell in a layer with a dielectric constant e2:

= (m 11 mi 2fA(n2) ) (

U-J Vm2i m22) U?7 '

where m; j depends of geometrical and optical properties of layers of each crystal cell.

Next, we find the general periodic solution for TE polarized waves using the theory of Floquet-Bloch [l], according to which a decision of wave equation in a layered periodic medium can be found in the form E (z) = E (z) e^Kz~k), where E (z) is a periodic function with the period of d. Constant K is called the Bloch wave number.

From (3) and the condition of periodicity, we obtain the following system:

(mu mi2(A(i) = . Ké

U21 ™22) [b^) 6 ' (4)

## Diffraction Grating Monochromatic Lightning

In this system, the factor of elKd is an eigenvalue of the coefficient matrix. Then solving the characteristic equation of this system with respect to factor éKd, given that the coefficient matrix is unimodular, we obtain:

2 cos Kd = m11 + m22. (5)

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Substituting the values of the coefficient matrix to the (5), we obtain the dispersion equation in implicit form, which establishes the relationship between the Bloch wave number K, frequency w, and ^-component kx of wave vector. The dispersion equation for TE waves:

1 Pi +P2

2 VM2

cos Kd = cos cos (ko^foa^ — 2 sin sin (ko^foa^ . (6)

If (toii + TO22) /2 > 1, K takes complex values, the Bloch wave is evanescent, and the propagation of electromagnetic waves is impossible in this region, it is called band gap. If (mn + m22) /2 < 1, K is real, this is the case of permitted zones, in this area Bloch waves will propagate.

The dependence of the wave vector kx and frequency u is shown below in fig. 1. Dark areas correspond to areas of transmission. Wave frequencies with the value of this area will be distributed in the environment. Bright areas correspond to the band gaps. These frequencies are forbidden and light cannot propagate in the medium. Such waves will be reflected from the structure.

3. One-Dimensional Binary Diffraction Grating

A linearly polarized electromagnetic field incident at A-periodic binary grating at an angle © (Fig.1.). The wave-vector of the incident field is obtained from the

2 tY

geometry: ki = k (sin 6, 0, cos 0) = nik0 (sin 6, 0, cos 0) with k0 = -r— and ni —

Ao

refractive index in region I. The wave-vectors of reflected and transmitted diffracted orders can be determined from the Floquet condition:

kx j = ko ^ni sin 6 - j j^j , (7)

Figure 2. Geometry for the Diffraction Problem

кL,zj —

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J(konL)2 - k2xj, Re ((k0nL)2 - k2x^ > 0 - iJk2xj - (konL)2, Re {{k0nL)2 - k2x^ < 0,

L — I, II.

(8)

The grating is bound by two media: input media with refractive index m = ^Jei, as a rule vacuum, and output media with refractive index nu = -JeJi, which corresponds to substrate. Relative permittivity in the grating region is a A-periodic function e (x) = £ (x + toA), to = 0, ±1, ±2, ..., which is expandable in a Fourier series. Electromagnetic fields in regions I and II are satisfied to Rayleigh expansion on the form:

— exp (-ik0ni (sin 6x + cos 6z)) + ^ Rj exp (—г (kXjx — ki,Zjz))

(9)

3 = -*x — Ti exp (— (kxjX — ku,Zj(z — d)))

(10)

J = -<X>

U

where u corresponds to Ey in TE case and Hy in TM case, Rj and Tj are amplitudes of the reflected and transmitted diffracted orders, which corresponds to Ey in TE case and Hy in TM case.

The problem of planar diffraction is decomposed into two independent problems: TE- and TM-polarization. The general problem is to find Rj and Tj.

The case of TE-polarization is fully described in article . We will consider the case of TM-polarization. Main steps of algorithm are presented below.

1. Fourier series for tangential electric and magnetic fields in the grating region:

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Hy = Y,Uyj (z)exp(-ikXjX), Ex = i^Sxj (z)exp(-ikxjx). (11)

2. Substituting Fourier series for tangential electric and magnetic fields into corresponding equations:

J2ei-p dyzP -koSxj, ^ dz ZSj-P ^Uyp) Uyj. (12)

p p /

3. Previous coupled-wave equations can be reduced to:

92Uy/9(/)2] - [FB] [Uy] , (13)

where z' - ko z, B - KxF-1Kx-I, Kx - diag {kx-n/k0,...,kx a/k0,...kxN /ko}, I — identity matrix, F-1 — is a toeplitz matrix of components of the Fourier series of function 1/e (x), F — inverse to F-1.

4. Solving previous equation we obtain:

n

Uyj exp(-ko qmz ) + c' exp [ko qm (z -d)]} , (14)

ra=1 n

S-xj (z) vj™{-exp (-koqmz) + c' exp [koqm (z - d)}} , (15)

where Wjm and qm are the elements of the eigenvector matrix W and the positive square root of the eigenvalues of matrix FB; Vjm is the element of the matrix V - F-1WQ, Q -diag {qu q2,..., qn}.

5. Using boundary conditions at the input boundary we obtain:

5jo + Rj wj™ + cm exp (-koqmd)] , (16)

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m=1

_ ki,zj R

ni jo kon2 j

- vjm [cm - cm exp (-koqmd)] , (17)

ra=1

and at the output boundary:

n

wjm [crn exp (-koqmd) + c'] - Tj, (18)

ra=1

^ vjm [c+ exp ( koqm<i) - cm] - i ( k^T ) Tj. (19)

m=1 ^ 11 '

6. Eliminating Rj and Tj from previous equations we obtain:

n =

/ m=1 I ki>zj , cos в' .k0nj

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I T^ +

• kl, zj I

1 и „I wjm + Vjm fto '7

+

+ cm exP (-koqmd)

m=1

.ki,z

## Diffraction Grating Orders

konj

(20)

Y^ ¿m exp(-koqmd)

rn=1

. kii,zj — Wjm

+ V

jm

+

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+

/ - CmWim

m=l

—г

.kii,z

k0n2II

= 0. (21)

7. Solving the system of linear equations we obtain c+ and c^. Substituting c+ and c' into corresponding equations we determine Rj and Tj. The efficiencies of the diffracted orders are given by:

,j = lRj1 Re

T = IT,

I ko ni cos 0 I '

B a Re (^ j / (^).

(22)

The results of comparison between simulation and spectrophotometrical data for copper binary grating with period 200000nm, spike width 110000nm and spike height 187nm in the case of TM-polarization with incidence angle 8° are presented below.

Figure 3. Comparison Between Simulation and Spectrophotometrical Data

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References

1. Neviere M., Popov E. Light Propagation in Periodic Media. Differential Theory and Design. — New York: MARCEL DEKKER, 2003. — 410 p.

2. Formulation for Stable and Efficient Implementation of the Rigorous Coupled-Wave Analysis of Binary Gratings / M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord // J. Opt. Soc. Am. — 1995. — Vol. 12, No 5. — Pp. 1077-1086.

УДК 519.6, 535.4

Распространение поляризованного монохроматического света в периодических структурах Д. В. Диваков, А. А. Тютюнник

Кафедра систем телекоммуникаций Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198

Работа посвящена анализу запрещённых зон одномерного фотонного кристалла и моделированию дифракции света на одномерной бинарной дифракционной решётке методом ИСШЛ. Метод БСШЛ описан для одномерных фотонных кристаллов и одномерных бинарных дифракционных решёток. Приведено сравнение результатов моделирования со спектрофотометрическими данными.

## Diffraction Grating Explained

Ключевые слова: ИСШЛ, периодические среды, дифракция, фотонный кристалл, блоховские волны.