Linear systems methods in some problems of radiative transfer between parallel planes

Douglas S. Goodman

doi:10.1364/AO.24.003240

Applied Optics
Vol. 24,
Issue 19,
pp. 3240-3248
(1985)
•https://doi.org/10.1364/AO.24.003240

Linear systems methods in some problems of radiative transfer between parallel planes

Douglas S. Goodman

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Douglas S. Goodman, "Linear systems methods in some problems of radiative transfer between parallel planes," Appl. Opt. 24, 3240-3248 (1985)

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Abstract

In practical problems of radiative transfer the methods of linear systems theory can often be used to calculate and understand better the plane-to-plane propagation of thermal radiation and scattered noncoherent radiation. In this approach the radiation on one plane due to a small surface element on the other is thought of as a spread function, its Fourier transform being a transfer function. The radiative transfer equations can then be written in either the spatial domain or the spatial frequency domain.

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A(θ)	$\tilde{S} (u)$ where u = 2πρz
1	e⁻^u
Cos¹θ	uK₁(u)
Cos²θ	(1 + u)e⁻^u
Cos³θ	½u²K₂(u) = uK₁(u) + ½u²K₀(u)
Cos⁴θ	(1 + u + ⅓u²)^e⁻^u

Distance z/p	Duty cycle
Distance z/p	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.1	96.4	95.3	93.5	90.9	87.0	81.0	71.1	54.4	28.5
0.2	79.9	77.6	73.8	68.4	61.2	51.9	40.3	26.8	12.7
0.3	55.8	53.6	49.9	44.9	38.6	31.3	23.3	15.0	7.0
0.4	35.0	33.4	30.8	27.3	23.2	18.5	13.6	8.7	4.0
0.5	20.8	19.8	18.2	16.1	13.6	10.8	7.9	5.0	2.3
0.6	12.0	11.4	10.5	9.3	7.8	6.2	4.5	2.9	1.3
0.7	6.9	6.5	6.0	5.3	4.4	3.5	2.6	1.6	0.8
0.8	3.9	3.7	3.4	3.0	2.5	2.0	1.5	0.9	0.4
0.9	2.2	2.1	1.9	1.7	1.4	1.1	0.8	0.5	0.2
1.0	1.2	1.2	1.1	0.9	0.8	0.6	0.5	0.3	0.1
1.1	0.7	0.6	0.6	0.5	0.4	0.3	0.3	0.2	0.1

Distance z/p	Duty cycle
Distance z/p	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.1	0.62	0.53	0.38	0.20	0.13	0.20	0.38	0.53	0.62
0.2	0.35	0.30	0.21	0.11	0.04	0.11	0.21	0.30	0.35
0.3	0.19	0.16	0.12	0.06	0.01	0.06	0.12	0.16	0.19
0.4	0.10	0.09	0.06	0.03	0.00	0.03	0.06	0.09	0.10
0.5	0.06	0.05	0.03	0.02	0.00	0.02	0.03	0.05	0.06
0.6	0.03	0.03	0.02	0.01	0.00	0.01	0.02	0.03	0.03
0.7	0.02	0.01	0.01	0.01	0.00	0.01	0.01	0.01	0.02
0.8	0.01	0.01	0.01	0.00	0.00	0.00	0.01	0.01	0.01
0.9	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
1.1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

A(θ)	$\tilde{S} (u)$ where u = 2πρz
1	e⁻^u
Cos¹θ	uK₁(u)
Cos²θ	(1 + u)e⁻^u
Cos³θ	½u²K₂(u) = uK₁(u) + ½u²K₀(u)
Cos⁴θ	(1 + u + ⅓u²)^e⁻^u

Distance z/p	Duty cycle
Distance z/p	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.1	96.4	95.3	93.5	90.9	87.0	81.0	71.1	54.4	28.5
0.2	79.9	77.6	73.8	68.4	61.2	51.9	40.3	26.8	12.7
0.3	55.8	53.6	49.9	44.9	38.6	31.3	23.3	15.0	7.0
0.4	35.0	33.4	30.8	27.3	23.2	18.5	13.6	8.7	4.0
0.5	20.8	19.8	18.2	16.1	13.6	10.8	7.9	5.0	2.3
0.6	12.0	11.4	10.5	9.3	7.8	6.2	4.5	2.9	1.3
0.7	6.9	6.5	6.0	5.3	4.4	3.5	2.6	1.6	0.8
0.8	3.9	3.7	3.4	3.0	2.5	2.0	1.5	0.9	0.4
0.9	2.2	2.1	1.9	1.7	1.4	1.1	0.8	0.5	0.2
1.0	1.2	1.2	1.1	0.9	0.8	0.6	0.5	0.3	0.1
1.1	0.7	0.6	0.6	0.5	0.4	0.3	0.3	0.2	0.1

Abstract

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Figures (11)

Tables (3)

Equations (53)

Applied Optics