Jones calculus

Jones calculus represents fully polarized light by a Jones vector and linear optical transformations by complex matrices.

A Jones matrix is a complex two-by-two linear transformation of a Jones vector through a deterministic nondepolarizing optical element. Successive elements multiply in reverse reading order. An ideal linear polarizer along a real transverse unit vector

a

has matrix

a a^{T}

. A common nonzero scalar phase does not change the output polarization of waves.

Circular-eigenmode Jones propagator ( $J (z) = e^{- γ^{+} z} P_{+} + e^{- γ^{-} z} P_{-}$ )

 0  0

For fixed circular eigenvectors

u_{\pm} = (1, \mp i)^{T} / 2

, let

P_{\pm} = u_{\pm} u_{\pm}^{†}

. A uniform optical medium with amplitude propagation coefficients

γ^{\pm}

has the displayed Jones matrix. Put

γ = (γ^{+} + γ^{-}) /2

and

δ = (γ^{+} - γ^{-}) /2

. Then

J (z) = e^{- γ z} (cosh (δz) i sinh (δz) - i sinh (δz) cosh (δz)) = e^{- Kz}, K = (γ - i δ i δ γ) .

(1)

The sum of the two modal exponentials supplies a hyperbolic cosine, not its reciprocal. The determinant is

e^{- 2 γ z}

, and

J (z_{1} + z_{2}) = J (z_{1}) J (z_{2})

. Real modal coefficients give circular dichroism; purely imaginary coefficients give lossless circular birefringence. Complex modal labels do not have an intrinsic numerical ordering.

Jones vector ( $J = (J_{x}, J_{y})^{T}$ )

 0  0

A Jones vector records the two transverse complex amplitudes of a completely polarized monochromatic wave. Multiplication by a common nonzero complex number preserves its polarization of waves, while changing overall amplitude and phase. With the convention

Ψ (t) = Re [J e^{iω t}]

J = (1, 2 i)^{T}

gives

(cos ω t, - 2 sin ω t)

, an ellipse with the longer axis along

y

. Handedness labels require both a viewing direction and a naming convention.

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Jones calculus is a mathematical framework used in optics to describe the polarization state of light and its transformation through optical devices. It was developed by the physicist R.W. Jones in 1941. This calculus uses a two-dimensional complex vector to represent the state of polarization of light, which can include various types of polarization such as linear, circular, and elliptical.

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Jones calculus

Jones matrix (Jout​=MJin​)

Circular-eigenmode Jones propagator (J(z)=e−γ+zP+​+e−γ−zP−​)

Jones vector (J=(Jx​,Jy​)T)

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Jones matrix ( $J_{out} = M J_{in}$ )

Circular-eigenmode Jones propagator ( $J (z) = e^{- γ^{+} z} P_{+} + e^{- γ^{-} z} P_{-}$ )

Jones vector ( $J = (J_{x}, J_{y})^{T}$ )

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