Krawtchouk matrices

The transfer-matrix method is a method used in optics and acoustics to analyze the propagation of electromagnetic or acoustic waves through a stratified (layered) medium.^[1] This is for example relevant for the design of anti-reflective coatings and dielectric mirrors.

The reflection of light from a single interface between two media is described by the Fresnel equations. However, when there are multiple interfaces, such as in the figure, the reflections themselves are also partially transmitted and then partially reflected. Depending on the exact path length, these reflections can interfere destructively or constructively. The overall reflection of a layer structure is the sum of an infinite number of reflections, which is cumbersome to calculate.

The transfer-matrix method is based on the fact that, according to Maxwell's equations, there are simple continuity conditions for the electric field across boundaries from one medium to the next. If the field is known at the beginning of a layer, the field at the end of the layer can be derived from a simple matrix operation. A stack of layers can then be represented as a system matrix, which is the product of the individual layer matrices. The final step of the method involves converting the system matrix back into reflection and transmission coefficients.

Formalism for electromagnetic waves

Below is described how the transfer matrix is applied to electromagnetic waves (for example light) of a given frequency propagating through a stack of layers at normal incidence. It can be generalized to deal with incidence at an angle, absorbing media, and media with magnetic properties. We assume that the stack layers are normal to the $z$ axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with wave number $k$ ,

E (z) = E_{r} e^{i k z} + E_{l} e^{- i k z}

.

Because it follows from Maxwell's equation that $E$ and $F = d E / d z$ must be continuous across a boundary, it is convenient to represent the field as the vector $(E (z), F (z))$ , where

F (z) = i k E_{r} e^{i k z} - i k E_{l} e^{- i k z}

.

Since there are two equations relating $E$ and $F$ to $E_{r}$ and $E_{l}$ , these two representations are equivalent. In the new representation, propagation over a distance $L$ into the positive $z$ direction is described by the matrix

M = (\begin{matrix} \cos k L & \frac{1}{k} \sin k L \\ - k \sin k L & \cos k L \end{matrix}),

and

(\begin{matrix} E (z + L) \\ F (z + L) \end{matrix}) = M \cdot (\begin{matrix} E (z) \\ F (z) \end{matrix})

Such a matrix can represent propagation through a layer if $k$ is the wave number in the medium and $L$ the thickness of the layer: For a system with $N$ layers, each layer $j$ has a transfer matrix $M_{j}$ , where $j$ increases towards higher $z$ values. The system transfer matrix is then

M_{s} = M_{N} \cdot \dots \cdot M_{2} \cdot M_{1} .

Typically, one would like to know the reflectance and transmittance of the layer structure. If the layer stack starts at $z = 0$ , then for negative $z$ , the field is described as

E_{L} (z) = E_{0} e^{i k_{L} z} + r E_{0} e^{- i k_{L} z}

,

where $E_{0}$ is the amplitude of the incoming wave, $k_{L}$ the wave number in the left medium, and $r$ is the amplitude (not intensity!) reflectance coefficient of the layer structure. On the other side of the layer structure, the field consists of a right-propagating transmitted field

E_{R} (z) = t E_{0} e^{i k_{R} z}

,

where $t$ is the amplitude transmittance and $k_{R}$ is the wave number in the rightmost medium. If $F_{L} = d E_{L} / d z$ and $F_{R} = d E_{R} / d z$ , then we can solve

(\begin{matrix} E (z_{R}) \\ F (z_{R}) \end{matrix}) = M \cdot (\begin{matrix} E (0) \\ F (0) \end{matrix})

in terms of the matrix elements $M_{m n}$ of the system matrix $M_{s}$ and obtain

t = 2 i k_{L} e^{- i k_{R} L} [\frac{M_{11} M_{22} - M_{12} M_{21}}{- M_{21} + k_{L} k_{R} M_{12} + i (k_{R} M_{11} + k_{L} M_{22})}]

and

r = [\frac{(M_{21} + k_{L} k_{R} M_{12}) + i (k_{L} M_{22} - k_{R} M_{11})}{(- M_{21} + k_{L} k_{R} M_{12}) + i (k_{L} M_{22} + k_{R} M_{11})}]

.

The transmittance and reflectance (i.e., the fractions of the incident intensity ${| E_{0} |}^{2}$ transmitted and reflected by the layer) are often of more practical use and are given by $T = | t |^{2}$ and $R = | r |^{2}$ , respectively (at normal incidence).

Example

As an illustration, consider a single layer of glass with a refractive index n and thickness d suspended in air at a wave number k (in air). In glass, the wave number is $k^{'} = n k$ . The transfer matrix is

M = (\begin{matrix} \cos k^{'} d & \sin (k^{'} d) / k^{'} \\ - k^{'} \sin k^{'} d & \cos k^{'} d \end{matrix})

.

The amplitude reflection coefficient can be simplified to

r = \frac{(1 / n - n) \sin k^{'} d}{(n + 1 / n) \sin k^{'} d + 2 i \cos (k^{'} d)}

.

This configuration effectively describes a Fabry–Pérot interferometer or etalon: for $k^{'} d = 0, π, 2 π, \dots$ , the reflection vanishes.

Acoustic waves

It is possible to apply the transfer-matrix method to sound waves. Instead of the electric field E and its derivative F, the displacement u and the stress $σ = C d u / d z$ , where $C$ is the p-wave modulus, should be used.

Abeles matrix formalism

The Abeles matrix method is a computationally fast and easy way to calculate the specular reflectivity from a stratified interface, as a function of the perpendicular momentum transfer, Q_z.

Application

Q_{z} = \frac{4 π}{λ} \sin θ = 2 k_{z}

Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE

To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.

Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.

The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.

As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com

Where θ is the angle of incidence/reflection of the incident radiation and λ is the wavelength of the radiation. The measured reflectivity depends on the variation in the scattering length density (SLD) profile, (ρ(z)) perpendicular to the interface. Although the scattering length density profile is normally a continuously varying function, the interfacial structure can often be well approximated by a slab model in which layers of thickness (d_n), scattering length density (ρ_n) and roughness (σ_n,n+1) are sandwiched between the super- and sub-phases. One then uses a refinement procedure to minimise the differences between the theoretical and measured reflectivity curves, by changing the parameters that describe each layer.

In this description the interface is split into n layers. Since the incident neutron beam is refracted by each of the layers the wavevector, k, in layer n, is given by:

k_{n} = \sqrt{{k_{z}}^{2} - 4 π (ρ_{n} - ρ_{0})}

The Fresnel reflection coefficient between layer n and n+1 is then given by:

r_{n, n + 1} = \frac{k_{n} - k_{n + 1}}{k_{n} + k_{n + 1}}

Since the interface between each layer is unlikely to be perfectly smooth the roughness/diffuseness of each interface modifies the Fresnel coefficient and is accounted for by an error function, as described by Nevot and Croce (1980).

r_{n, n + 1} = \frac{k_{n} - k_{n + 1}}{k_{n} + k_{n + 1}} \exp (- 2 k_{n} k_{n + 1} {σ_{n, n + 1}}^{2})

A phase factor, β is introduced, which accounts for the thickness of each layer.

β_{0} = 0

β_{n} = i k_{n} d_{n}

where $i^{2} = - 1$ . A characteristic matrix, c_n is then calculated for each layer.

c_{n} = [\begin{matrix} \exp (β_{n}) & r_{n, n + 1} \exp (β_{n}) \\ r_{n, n + 1} \exp (- β_{n}) & \exp (- β_{n}) \end{matrix}]

The resultant matrix is defined as the product of these characteristic matrices

M = \prod_{n} c_{n}

from which the reflectivity is calculated

R = {| \frac{M_{10}}{M_{00}} |}^{2}

References

↑ Born, M.; Wolf, E., Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Oxford, Pergamon Press, 1964.

Abeles matrix formalism

O. S. Heavens. Optical Properties of Thin Films. Butterworth, London (1955).
L. Nevot, P. Croce, Revue de physique appliquée, 15, 761 (1980).
F. Abeles, Le Journal de Physique et le Radium, "La théorie générale des couches minces", 11, 307–310 (1950).

External links

Derivations and explanations

Lecture notes by Bo Sernelius, see Lecture 13. This derivation is more general than the one above, including light that is incident at a non-normal angle, and with specific light polarizations.
Manual for a transfer-matrix calculating program lists and explains the equations used, including formulas to calculate the absorption and Poynting vector at any given point.
A self-contained first-principles derivation of the transmission and reflection probabilities from a multilayer with complex indices of refraction.

Computer programs

There are a number of computer programs that implement this calculation:

FreeSnell is a stand-alone computer program that implements the transfer-matrix method, including more advanced aspects such as granular films.
Thinfilm is a web interface that implements the transfer-matrix method, outputting reflection and transmission coefficients, and also ellipsometric parameters Psi and Delta.
Luxpop.com is another web interface that implements the transfer-matrix method.
Transfer-matrix calculating programs in Python and in Mathematica.
EMPy ("Electromagnetic Python") software.
motofit is a program for analysing neutron and X-ray reflectometry data.

[1] Born, M.; Wolf, E., Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Oxford, Pergamon Press, 1964.

[1]

Krawtchouk matrices

Contents

Formalism for electromagnetic waves

Example

Acoustic waves

Abeles matrix formalism

Application

See also

References

Abeles matrix formalism

External links

Derivations and explanations

Computer programs

Navigation menu