Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory: errata James E. Harvey, Cynthia L. Vernold, Andrey Krywonos, and Patrick L. Thompson
OCIS codes: 070.2580, 050.1960, 030.5630.
1. Introduction
We are chagrined at having made a rather blatantly inaccurate statement in the derivation of Eq. 共21兲 on page 6472 of our recent paper.1 Clearly, the integrands of two definite integrals are not, in general, equal just because the integrals are equal. Although our result was correct, we wish to provide an accurate derivation. Rather than integrating radiant intensity over the entire hemisphere as we did in Eq. 共16兲, and then applying Parseval’s theorem and erroneously equating the resulting integrands to obtain Eq. 共18兲, we can apply the well-known change-of-variables theorem2 as follows, starting immediately after Eq. 共14兲. For the coordinate transformation defined by Eq. 共14兲, the Jacobian determinant in the change-ofvariables theorem is given by sin cos , and the differential solid angle can be expressed as dc ⫽ sin dd ⫽ d␣d兾␥. The radiant power obtained by integration of Eq. 共12兲 over some arbitrary solid angle, B, can be written as P⫽
兰 兰兰 兰
I共, 兲dc
that the aperture diameter is small compared with the distance r 共it is not required that the angle be small, i.e., no paraxial limitation兲. We can therefore bring the cos outside the integral over As and apply the change-of-variables theorem to obtain P⫽
兰兰 兰 兰兰 兰 B
⫽
B
B⬘
(16)
Here B⬘ is the region in direction cosine space corresponding to the arbitrary solid angle B. Similarly, integrating Eq. 共13兲 with respect to dAc, noting that dAc ⫽ r2dc, and applying the change-of-variables theorem, we obtain P⫽
兰兰 兰兰 兰兰
Ec共, 兲dAc
Ac
⫽
r2
Ec共, 兲 sin cos dd cos
r2
Ec共␣, 兲 d␣d. ␥
B
L共, , x, y兲cos sdAs sin dd.
(15)
As
However, from Fig. 2 it is clear that, for a diffracting aperture in a plane, s ⫽ does not vary significantly from surface element to surface element provided The authors are with the Center for Research and Education in Optics and Lasers 共CREOL兲, P.O. Box 162700, 4000 Central Florida Boulevard, The University of Central Florida, Orlando, Florida 32816. J. E. Harvey’s e-mail address is
[email protected]. Received 28 August 2000. 0003-6935兾00兾346374-02$15.00兾0 © 2000 Optical Society of America 6374
L共␣, , x, y兲dAsd␣d.
As
B
⫽
L共, , x, y兲dAs sin cos dd
As
APPLIED OPTICS 兾 Vol. 39, No. 34 兾 1 December 2000
⫽
(17)
B⬘
However, the irradiance on the observation hemisphere is given by 兩U共␣, ; rˆ兲兩2兾␥; hence from Eq. 共6兲 we can write P⫽
兰兰 兰兰
r2兩U共␣, ; rˆ兲兩2 d␣d ␥2
B⬘
⫽
B⬘
2兩Ᏺ兵U0共xˆ, yˆ兲其兩2d␣d.
(18)
The right-hand sides of Eqs. 共16兲 and 共18兲 are equal for identical arbitrary limits 共including each and every infinitesimal region兲; hence the integrands themselves must be equal. We thus obtain the following general relationship:
兰
L共␣, , x, y兲dAs ⫽ 2兩Ᏺ兵U0共xˆ, yˆ兲其兩2.
(19)
As
The derivation of Eqs. 共21兲 and 共22兲 now proceeds as in Ref. 1; however, the final sentence of paragraph 1 on page 6473 needs to be changed to read as follows:
where PT is the total radiant power emanating from the diffracting aperture. Note that the entire hemisphere is represented by a unit circle in direction cosine space, and recall that only those plane-wave components that lie inside the unit circle are real and propagate. Those that lie outside the unit circle are imaginary and are referred to as evanescent waves 共and thus do not propagate兲. Equation 共27兲 in Ref. 1 now follows directly with the help of Eq. 共21兲.
From Rayleigh’s 共Parseval’s兲 theorem, and with the help of Eq. 共18兲, we can now write PT ⫽ 2 ⫽
兰兰 兰 兰 ⬁
⬁
⫺⬁
⫺⬁
1
␣⫽⫺1
References
兩U0共xˆ, yˆ兲兩2dxˆdyˆ 共1⫺␣2兲1兾2
⫽⫺共1⫺␣2兲1兾2
兩Ᏺ兵U0共xˆ, yˆ兲其兩2d␣d,
(25)
1. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469 – 6481 共1999兲. 2. H. F. Davis, Vector Analysis 共Allyn and Bacon, Boston, 1961兲.
1 December 2000 兾 Vol. 39, No. 34 兾 APPLIED OPTICS
6375
