Scattering matrix

Content

The scattering matrix is a matrix whose elements ( S-parameters ) describe the physical parameters of scattering. In the microwave technology, the scattering matrix is used to describe microwave devices and by connecting a linear dependence of the complex amplitudes of the incident and reflected waves in the terminal planes of an equivalent multipole.

In quantum mechanics

In quantum mechanics, the scattering matrix or S-matrix is a matrix of quantities that describes the transition of quantum-mechanical systems from one state to another during their interaction (scattering).

The scattering matrix was first introduced by John Wheeler in 1937, '' On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure '. ^[1] In this work, Wheeler introduced the concept of a scattering matrix — a unitary matrix of coefficients that connect “the asymptotic behavior of an arbitrary particular solution of an integral equation with solutions in standard form”. ^[2] . It was later and independently introduced by Werner Heisenberg in 1943

In microwave technology

Definition

It is customary to represent each input ( port ) of a multipole in the microwave technique in the form of a cross section (“ terminal plane ”) of a transmission line with the main type of waves. The oscillation process at each i- th input can be represented as the sum of the incident (propagating towards the multipole) and reflected (propagating from the multipole) waves with amplitudes (normalized amplitudes), respectively, a _i and b _i . In a linear multipole with N ports, the amplitudes of these waves are connected by linear relationships:

{\begin{cases}b_{1}=s_{11}a_{1}+s_{12}a_{2}+\ldots +s_{1N}a_{N}\\b_{2}=s_{21}a_{1}+s_{22}a_{2}+\ldots +s_{2N}a_{N}\\\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\b_{N}=s_{N1}a_{1}+s_{N2}a_{2}+\ldots +s_{NN}a_{N}\end{cases}}

{\ displaystyle {\ begin {cases} b_ {1} = s_ {11} a_ {1} + s_ {12} a_ {2} + \ ldots + s_ {1N} a_ {N} \\ b_ {2} = s_ {21} a_ {1} + s_ {22} a_ {2} + \ ldots + s_ {2N} a_ {N} \\\ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \\ b_ {N} = s_ {N1} a_ {1} + s_ {N2} a_ {2} + \ ldots + s_ {NN} a_ {N} \ end {cases}}}

Here s _mn are the scattering coefficients that are independent of a _i and b _i . A set of equations can be written in matrix form. For this, the amplitudes of the incident and reflected waves must be represented in the form of matrix columns a and b :

a={\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{N}\end{pmatrix}};\quad b={\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{N}\end{pmatrix}}

{\ displaystyle a = {\ begin {pmatrix} a_ {1} \\ a_ {2} \\\ vdots \\ a_ {N} \ end {pmatrix}}; \ quad b = {\ begin {pmatrix} b_ { 1} \\ b_ {2} \\\ vdots \\ b_ {N} \ end {pmatrix}}}

Then the relationship between a and b has the form:

\;b=Sa

{\ displaystyle \; b = Sa}

Here S is the scattering matrix:

S={\begin{pmatrix}s_{11}&s_{12}&\cdots &s_{1N}\\s_{21}&s_{22}&\cdots &s_{2N}\\\vdots &\vdots &\ddots &\vdots \\s_{N1}&s_{N2}&\cdots &s_{NN}\end{pmatrix}}

{\ displaystyle S = {\ begin {pmatrix} s_ {11} & s_ {12} & \ cdots & s_ {1N} \\ s_ {21} & s_ {22} & \ cdots & s_ {2N} \\\ vdots & \ vdots & \ ddots & \ vdots \\ s_ {N1} & s_ {N2} & \ cdots & s_ {NN} \ end {pmatrix}}}

Physical meaning

To determine the physical meaning of the elements of the scattering matrix of the microwave multipath, it is necessary to apply an incident wave to its input (port) n , that is, excite the multipole with waves with amplitude a = (0, ..., 0, a _n , 0, ..., 0) ^T , and to all other i- th ( i ≠ n ) ports connect coordinated (non-reflecting, fully absorbing waves) loads. Then the amplitudes of the waves leaving the ports $b_{m}=s_{mn}a_{n}$ ${\ displaystyle b_ {m} = s_ {mn} a_ {n}}$ from where $\;s_{mn}=b_{m}/a_{n}$ ${\ displaystyle \; s_ {mn} = b_ {m} / a_ {n}}$ .

Thus, the elements of the scattering matrix with indices n ≠ m are the transfer coefficients to port m from port n , with the indices n = m (elements of the main diagonal of the matrix) are the reflection coefficients for the case when to all i ) absorbing loads are connected to the ports.

Scope

Unlike resistance matrices (conductivities) and transmission matrices, the scattering matrix is defined for all microwave devices. In addition, from an engineering point of view, the process of measuring S- parameters is possible for any microwave devices, since it reduces to measuring the parameters of the incident and reflected waves at the inputs of the device.

Notes

↑ John Archibald Wheeler, ' On the Mathematical Description of Light Nuclei by the Method. of Resonating Group Structure 'Phys. Rev. 52, 1107 - 1122 (1937)
↑ Jagdish Mehra , Helmut Rechenberg , The Historical Development of Quantum Theory (Pages 990 and 1031) Springer, 2001 ISBN 0-387-95086-9 , 9780387950860

Literature

Landau, L.D. , Lifshits, E.M. Quantum mechanics (nonrelativistic theory). - 4th edition. - M .: Nauka , 1989 .-- 768 p. - (“ Theoretical Physics ”, Volume III). - ISBN 5-02-014421-5 .
Sazonov D.M. Antennas and microwave devices. Textbook for radio engineering specialties of universities . - M .: Higher. shk, 1988. - P. 432. - ISBN 5-06-001149-6 .

[autogenerated1-1] John Archibald Wheeler, ' On the Mathematical Description of Light Nuclei by the Method. of Resonating Group Structure 'Phys. Rev. 52, 1107 - 1122 (1937)

[Mehra-2] Jagdish Mehra , Helmut Rechenberg , The Historical Development of Quantum Theory (Pages 990 and 1031) Springer, 2001 ISBN 0-387-95086-9 , 9780387950860