Page 249 - Phase Space Optics Fundamentals and Applications

230 Chapter Seven

one on the tangent plane. To do this we begin with Eq. (7.26), namely,
−i z exp(ikr)
(x, y, z) =
r r

2 xx s + yy s
× (x s ,y s , 0) exp −i dx s dy s
r
A
Make the substitution
z z
2
2
r = = ; r = x + y + z 2
m cos
We obtain
−i 2 exp(ikz 0 /m)
P (x, y, z 0 ) = m
z 0
⎡ ⎤

2 (xx s + yy s )
× (x s ,y s , 0) exp ⎣ −i ⎦ dx s dy s
2
2
x + y + z 2
A 0
(7.41)
The symbol P is the distribution of the complex amplitude diffracted
onto the tangent plane perpendicular to the z axis at a distance
z = z 0 from the diffracting aperture. The total spectral radiant power
−1
(W Hz ) now reads
∞
∞
2
(z 0 , ) = | P (x, y, z 0 )| dx dy (7.42)
−∞ −∞
−1
2
−2
The units for | P (x, y, z 0 )| are W cm Hz . We have just estab-
lished that for radiation detection (measurement) on a plane parallel to the
aperture plane, the ensemble average of the squared modulus of the diffracted
2
field | P (x, y, z 0 )| is the spectral irradiance.
Also, it can be established that, with respect to the aperture plane,
the spectral radiant exitance is given by

2
r
r
r
r
r
M( s , 0, ) = B( s , 0, ˆ n, )md = ( s , s , 0, ) = | ( s , )|
1/2
(7.43)
The details of this calculation are not included here. It involves the
r
substitution of the definition of the spectral radiance B( s , 0, ˆ n, )from
Eq. (7.33) and evaluation of the angle integrals.
In this section, we discussed the various roles played by the en-
semble average of the squared modulus of the optical field and
the radiometric quantity it represents relative to the experimental
arrangement.