By Polyakov A.

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The two blockdiagonal matrices in the ﬁrst line do not mix creation and annihilation operators, and therefore describe linear optical interferometers. The central matrix does mix creation and annihilation operators. However, the com∗ are themselves diagonal matrices, and the ponents AD , BD , A∗D , and BD 40 The quantum theory of light a) b) U S V† Fig. 6. The Bloch-Messiah reduction: a) a general multi-port interferometer including multi-mode squeezing; b) the reduction of the interferometer in two linear optical interferometers U and V † , and a set of single-mode squeezers S.

We next introduce the paraxial wave approximation, and then ﬁnd the “minimum uncertainty” localized transverse mode. These modes can in turn be used as a generating function when we construct the Hermite-Gaussian and Laguerre-Gaussian transverse modes. When the transverse momentum is small we can treat the beam in the paraxial approximation. We start with the Helmholtz equation for the j th spatial component of the classical vector potential in the frequency domain 20 The quantum theory of light Aj (r, ω), given by the Fourier transform ∞ Aj (r, t) = dω Aj (r, ω)e−iωt .

There is a microscopic interaction of the field with the dielectric, but the details are averaged out in this Hamiltonian. 34 The quantum theory of light a) b) φ a ˆjλ c) ˆblλ ˆbj1 a ˆj1 ˆbjλ ˆbjλ a ˆjλ a ˆj2 θ ˆbj2 ζ, ϕ a ˆlλ Fig. 3. Graphical representations of optical elements: a) the phase shifter; b) the beam splitter; c) the polarization rotation. In the above example of a beam splitter we chose two modes with identical polarization λ, but diﬀerent wave vectors kj and kl . However, we can also vary the polarization (λ ∈ {1, 2}) and keep the wave vector constant: Hj (θ) = θeiϕ a ˆ†j1 a ˆj2 + θe−iϕ a ˆj1 a ˆ†j2 .