Page 29 - Phase Space Optics Fundamentals and Applications

10 Chapter One

light can be written in the form

t

r 1 + r 2 G 1 −iH r 1 + r 2
(r 1 , r 2 ) = 2 det G 1 exp − t
2 r 1 − r 2 −iH G 2 r 1 − r 2
(1.18)
where we have chosen a representation that enables us to determine
the Wigner distribution of such light in an easy way. The exponent
shows a quadratic form in which a four-dimensional column vector
t
t t
[(r 1 +r 2 ) , (r 1 −r 2 ) ] arises together with a symmetric 4×4 matrix. This
t
matrix consists of four real 2×2 submatrices G 1 , G 2 , H, and H , where,
moreover, matrices G 1 and G 2 are positive definite symmetric. The
special form of the matrix is a direct consequence of the fact that the
cross-spectral density is a nonnegative definite Hermitian function.
The Wigner distribution of such Gaussian light takes the form 20,21

t −1 t −1
det G 1 r G 1 + HG H −HG 2 r
2
W(r, q) = 4 exp −2 −1 t −1
det G 2 q −G H G 2 q
2
(1.19)
In a more common way, the cross-spectral density of general
Gaussian light (with 10 degrees of freedom) can be expressed in the
form

t
1
(r 1 , r 2 ) = 2 det G 1 exp − (r 1 − r 2 ) G 0 (r 1 − r 2 )
2
t 1 t
× exp − r 1 G 1 − i (H + H ) r 1
2
t 1 t
× exp − r 2 G 1 + i (H + H ) r 2
2

t t
× exp − r i(H − H ) r 2 (1.20)
1
where we have introduced the real, positive definite symmetric 2 × 2
matrix G 0 = G 2 − G 1 . Note that the asymmetry of matrix H is a mea-
sure for the twist 22–26 of Gaussian light, and that general Gaussian
light reduces to zero-twist Gaussian Schell-model light 27,28 if the
t
matrix H is symmetric, H−H = 0. In that case, the light can be consid-
eredasspatiallystationarylightwithaGaussiancross-spectraldensity
√
1
t
2 det G 1 exp[− (r 1 − r 2 ) G 0 (r 1 − r 2 )], modulated by a Gaussian
2
t
modulator with modulation function exp[− r (G 1 − iH) r]. We re-
mark that such Gaussian Schell-model light (with nine degrees of
freedom) forms a large subclass of Gaussian light; it applies, for in-
stance, in
t
• the completely coherent case (H = H , G 0 = 0, G 1 = G 2 )
• the (partially coherent) one-dimensional case (g 0 = g 2 − g 1 ≥ 0)