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Absorption of isotropic radiation
Volume 63, number
6
OPTICS
COMMUNICATIONS
15 September
1987
ABSORPTION OF ISOTROPIC RADIATION Wynford L. HARRIES Department of Physics, Old Dominion University, Norfolk, VA 23508, USA Received
4 May 1987
Estimates of the absorption of isotropic radiation are given for an absorbing medium parallel sided slab or a cylinder. By including the directional variation of absorption differences to the “unidirectional radiation” formulae.
The absorption of isotropic radiation such as blackbody radiation, is of interest for estimates of rates of energy deposition into a medium either as heat or molecular vibrational energy. We assume the medium is uniform and isotropic and has a characteristic absorption length A. For narrow band absorption of the radiation into molecular vibrational energy, as in laser pumping [ 11, A = (no) - ‘, where n is the density of absorbing centers and (Tthe absorption cross-section which is single valued. The fraction of radiation absorbed along a path length 1 in crossing through the medium is then F= 1 - exp( - 112), and for unidirectional radiation, 1 can be related easily to the medium geometry. However, for isotropic radiation, 1 varies with direction. Here the total fractional absorption for two medium geometries, a flat lab and a cylinder, is estimated and comparisons made with the “unidirectional formula”. The radiation energy enters the medium from the black body cavity walls according to Plan&s radiation formula and the emission and absorption coefficients of the wall are assumed unity. If the medium is contained within a transparent vessel inside the cavity, reflections from the vessel surface may occur, which are not considered. For a plane parallel infinite slab of thickness d (fig. 1) let the energy enter from a small surface dS at PQ. A spherical coordinate sytem (p, (Y,p) has origin at 0. It is convenient to work in terms of angle x=n/2-cu. For isotropic emission the amount of
whose geometrical shape is either a planepath length, the results show appreciable
cds-
Fig. 1. Infinite slab geometry with energy entering through a small area dS at PQ. Spherical coordinates (p, 01, p) have origin at 0. Isotropic radiation entering at angles less than y’ is assumed completely absorbed within an error E.
radiation in direction y is proportional to the opening, namely, dS cos (Y(Lambert’s law) = dS sin y and the fraction of solid angle between y and y + dy is cos y dy. The path length in the slab is OM =d/sin y and thus the fraction absorbed is fydy=k,
sinycosy
x[l-exp(-d/;lsiny)]dy,
(1)
where k, is a constant. To integrate numerically we assume for y < y’, all the energy is absorbed within an error e, because of the long path lengths, i.e., exp( -d/,I sin y) = t, where e is a small quantity, chosen as 10-99. Then y’=sin-’
d
Aln(l/t)
’
and the fraction of radiation absorbed for 0 < y < y ’ is f,:
0 030-4Olgf87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
361
Volume 63. number
OPTICS
6
I5 September
COMMUNICATIONS
1987
:. fi
=k, {sinycosydy
.
=tk,[d//ZIn(l/t)]’ The fraction
absorbed
(3)
for y’ < y < n/2 is .&
n/l fi=k,
J^ sinycosy
x [ 1 -exp(
-d/l
sin r)] dy .
(4)
Integration was performed using Simpson’s Rule, with 11 values of y between y’ and n/2. The total fraction absorbed is f, =fi +fiand f3is a function of the ratio A/d, so universal curves can be plotted. The normalization constant k, can be evaluated by realizing that J;“k, sin y cos y dy is the total power through dS, or 0.5 k,. The results were insensitive to t provided it was small, except where A/d-,0. For lldG0.3 it is seen f3> 1 due to numerical errors. Table 1 gives f’ versus I/d and also shown is the function F, = 1 - exp( -d/l ), for comparison. The ratiof;lF, indicates the error introduced by using the “unidirectional formula”. A more common geometry for black-body cavity experiments is a cylinder of radius r with energy entering from the sides (fig. 2). Let the cylinder and the small area dS at L be tangential to plane AFGD at AD. The amount of radiation at angles 0 and 4 to plane AFGD is proportional to dSsin 8 sin @ For section ABCD at angle 19, path length LM is 2r sin B/sin 9. As before, we require exp( - LM/A ) = t ’ [Zrsin6/I_ln(i/e)], and at Cp’, or @‘(@=sin between 19and 8 f d0 all the radiation for 0 Q @< o’ is absorbed, consisting of a fraction f4( 0):
L
A
Fig. 2. Energy enters through a small area dS at L. Plane AFGDL is tangential to cylinder. Section ABCD is at angle B to .4F, and absorption paths LM are at angle @ to LD. Isotropic radiation entering this section at @<$I’ is assumed completely absorbed. within an error t.
n/l
.A(e)=k,
j sin8 sin@
x [ 1 - exp( - 2r sin B/A sin $)I d$ ,
~sinBsin@d( 0
=k,sinB(l--cos@‘). Between @’ and 7c/2 the fraction
362
(5) absorbed
isf5( 0):
(6)
and f6( 6) =.A +.f5 is the fraction absorbed between 8 and 8 + de. For 0 d 8 < 3112 the fraction absorbed is .G. nil .f7= j.f;,odH.
J,(Q=k,
D
(7)
The value of kz is obtained by observing that the total power in through dS for 8, @between 0 and n/2 is KJC;“J,“” sin 8 sin @de d@ and Kz= 1. A double integration of f7over @and 8 was performed numerically using 11 values of @between @‘(0) and 7c/2 for each of 11 values of 8. Again .f, is a function of Air (table 1). Comparison with a “unidirectional” calculation for path length 2r, Fz = [ 1 - exp( - 2rlA )] ,
Volume 63, number
6
OPTICS
15September
COMMUNICATIONS
1987
Table 1 Estimates of the fraction of isotropic radiation absorbed versus Ud for a slab, and versus IJr for a cylinder: 1 is the absorption length, d the thickness of the slab and r the radius of the cylinder. The functionsfa, f, are the fractions alculated here, F, and F2 are the results of the “unidirectional radiation formulae”. The ratios f,/F, and f7/F2are a measure of the errors in using the latter. The values ofh for /l/d< 0.3 should be < 1. Slab geometry k!d
Cylinder h
6
F,IF,
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
1.042 I .020 1.002 0.978 0.948 0.916 0.882 0.848 0.816 0.785
1.ooo 0.993 0.964 0.918 0.865 0.811 0.760 0.713 0.671 0.532
1.04 1.03 1.04 1.07 1.10 1.13 1.16 1.19 1.22 1.24
1.10 1.20 1.30 1.40 1.50 1.60 1.70 I .80 1.90
0.756 0.728 0.702 0.678 0.655 0.633 0.613 0.594 0.576
0.597 0.565 0.537 0.510 0.487 0.465 0.445 0.426 0.409
1.27 1.29 1.31 1.33 1.35 1.36 1.38 1.39 1.41
2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
0.559 0.432 0.352 0.297 0.257 0.227 0.203 0.184 0.168
0.393 0.283 0.221 0.181 0.154 0.133 0.118 0.105 0.095
11.00 12.00 13.00 14.00
0. I54 0.143 0.133 0.124 0.117 0.110 0.104 0.099 0.094 0.090
0.087 0.080 0.074 0.069 0.064 0.061 0.057 0.054 0.05 1 0.049
15.00 16.00 17.00 18.00 19.00 20.00
Ur
f7
F2
h/F2
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.oo
1.000 0.999 0.997 0.994 0.990 0.984 0.978 0.970 0.962 0.952
1.ooo 1.000 0.999 0.993 0.982 0.964 0.943 0.918 0.892 0.865
1.00 1.00 1.oo 1.00 1.01
1.10 1.20 1.30 1.50 1.60 1.70 1.80 1.90
0.942 0.93 1 0.919 0.908 0.895 0.883 0.871 0.851 0.846
0.838 0.811 0.785 0.760 0.736 0.714 0.692 0.671 0.651
1.12 1.15 1.17 1.19 1.22 1.24 1.26 1.28 1.30
1.42 1.52 1.59 1.64 I .68 I .70 1.73 1.75 1.76
2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
0.833 0.720 0.628 0.556 0.498 0.450 0.411 0.378 0.350
0.632 0.487 0.394 0.330 0.284 0.249 0.22 1 0.199 0.181
1.60 1.69 1.76 1.81 1.86 1.90 1.93
1.78 1.79 1.80 1.81 1.81
11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00
0.326 0.305 0.287 0.270 0.256 0.243 0.231 0.220 0.211 0.202
0.166 0.154 1.143 0.133 0.125 0.118 0.111 0.105 0.010 0.095
1.96 1.99 2.01 2.03 2.05 2.07 2.08 2.09 2.11 2.12
1.82 1.83 1.83 1.84 1.84
1.40
1.02 1.04 1.06 1.08 1.10
1.32
I .45
363
Volume 63. number
6
OPTICS
COMMUNICATIONS
shows f, can be considerably greater than F2. In conclusion, estimates of the absorption of isotropic radiation should take account of the contribution of rays proceeding at angles which result in very long path lengths. The above procedure can be extended for broad band absorption if o is a known function of wavelength b,y reiterating over small wavelength ranges. The work was supported by Grant NSG- 1568 from the National Aeronautis and Space Administration
364
Langley Research Center, Hampton, ful discussions with M.D. Williams acknowledged.
15 September
I987
Virginia. Useof NASA are
References [ 11 R.J. Insuik and W.H. Christiansen, IEEE J. Quantum Electron. QE-20 (I 984) 622. [2] J.K. Roberts. Heat and thermodynamics (Blackie& Son Ltr.. 1928) p. 387.
6
OPTICS
COMMUNICATIONS
15 September
1987
ABSORPTION OF ISOTROPIC RADIATION Wynford L. HARRIES Department of Physics, Old Dominion University, Norfolk, VA 23508, USA Received
4 May 1987
Estimates of the absorption of isotropic radiation are given for an absorbing medium parallel sided slab or a cylinder. By including the directional variation of absorption differences to the “unidirectional radiation” formulae.
The absorption of isotropic radiation such as blackbody radiation, is of interest for estimates of rates of energy deposition into a medium either as heat or molecular vibrational energy. We assume the medium is uniform and isotropic and has a characteristic absorption length A. For narrow band absorption of the radiation into molecular vibrational energy, as in laser pumping [ 11, A = (no) - ‘, where n is the density of absorbing centers and (Tthe absorption cross-section which is single valued. The fraction of radiation absorbed along a path length 1 in crossing through the medium is then F= 1 - exp( - 112), and for unidirectional radiation, 1 can be related easily to the medium geometry. However, for isotropic radiation, 1 varies with direction. Here the total fractional absorption for two medium geometries, a flat lab and a cylinder, is estimated and comparisons made with the “unidirectional formula”. The radiation energy enters the medium from the black body cavity walls according to Plan&s radiation formula and the emission and absorption coefficients of the wall are assumed unity. If the medium is contained within a transparent vessel inside the cavity, reflections from the vessel surface may occur, which are not considered. For a plane parallel infinite slab of thickness d (fig. 1) let the energy enter from a small surface dS at PQ. A spherical coordinate sytem (p, (Y,p) has origin at 0. It is convenient to work in terms of angle x=n/2-cu. For isotropic emission the amount of
whose geometrical shape is either a planepath length, the results show appreciable
cds-
Fig. 1. Infinite slab geometry with energy entering through a small area dS at PQ. Spherical coordinates (p, 01, p) have origin at 0. Isotropic radiation entering at angles less than y’ is assumed completely absorbed within an error E.
radiation in direction y is proportional to the opening, namely, dS cos (Y(Lambert’s law) = dS sin y and the fraction of solid angle between y and y + dy is cos y dy. The path length in the slab is OM =d/sin y and thus the fraction absorbed is fydy=k,
sinycosy
x[l-exp(-d/;lsiny)]dy,
(1)
where k, is a constant. To integrate numerically we assume for y < y’, all the energy is absorbed within an error e, because of the long path lengths, i.e., exp( -d/,I sin y) = t, where e is a small quantity, chosen as 10-99. Then y’=sin-’
d
Aln(l/t)
’
and the fraction of radiation absorbed for 0 < y < y ’ is f,:
0 030-4Olgf87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
361
Volume 63. number
OPTICS
6
I5 September
COMMUNICATIONS
1987
:. fi
=k, {sinycosydy
.
=tk,[d//ZIn(l/t)]’ The fraction
absorbed
(3)
for y’ < y < n/2 is .&
n/l fi=k,
J^ sinycosy
x [ 1 -exp(
-d/l
sin r)] dy .
(4)
Integration was performed using Simpson’s Rule, with 11 values of y between y’ and n/2. The total fraction absorbed is f, =fi +fiand f3is a function of the ratio A/d, so universal curves can be plotted. The normalization constant k, can be evaluated by realizing that J;“k, sin y cos y dy is the total power through dS, or 0.5 k,. The results were insensitive to t provided it was small, except where A/d-,0. For lldG0.3 it is seen f3> 1 due to numerical errors. Table 1 gives f’ versus I/d and also shown is the function F, = 1 - exp( -d/l ), for comparison. The ratiof;lF, indicates the error introduced by using the “unidirectional formula”. A more common geometry for black-body cavity experiments is a cylinder of radius r with energy entering from the sides (fig. 2). Let the cylinder and the small area dS at L be tangential to plane AFGD at AD. The amount of radiation at angles 0 and 4 to plane AFGD is proportional to dSsin 8 sin @ For section ABCD at angle 19, path length LM is 2r sin B/sin 9. As before, we require exp( - LM/A ) = t ’ [Zrsin6/I_ln(i/e)], and at Cp’, or @‘(@=sin between 19and 8 f d0 all the radiation for 0 Q @< o’ is absorbed, consisting of a fraction f4( 0):
L
A
Fig. 2. Energy enters through a small area dS at L. Plane AFGDL is tangential to cylinder. Section ABCD is at angle B to .4F, and absorption paths LM are at angle @ to LD. Isotropic radiation entering this section at @<$I’ is assumed completely absorbed. within an error t.
n/l
.A(e)=k,
j sin8 sin@
x [ 1 - exp( - 2r sin B/A sin $)I d$ ,
~sinBsin@d( 0
=k,sinB(l--cos@‘). Between @’ and 7c/2 the fraction
362
(5) absorbed
isf5( 0):
(6)
and f6( 6) =.A +.f5 is the fraction absorbed between 8 and 8 + de. For 0 d 8 < 3112 the fraction absorbed is .G. nil .f7= j.f;,odH.
J,(Q=k,
D
(7)
The value of kz is obtained by observing that the total power in through dS for 8, @between 0 and n/2 is KJC;“J,“” sin 8 sin @de d@ and Kz= 1. A double integration of f7over @and 8 was performed numerically using 11 values of @between @‘(0) and 7c/2 for each of 11 values of 8. Again .f, is a function of Air (table 1). Comparison with a “unidirectional” calculation for path length 2r, Fz = [ 1 - exp( - 2rlA )] ,
Volume 63, number
6
OPTICS
15September
COMMUNICATIONS
1987
Table 1 Estimates of the fraction of isotropic radiation absorbed versus Ud for a slab, and versus IJr for a cylinder: 1 is the absorption length, d the thickness of the slab and r the radius of the cylinder. The functionsfa, f, are the fractions alculated here, F, and F2 are the results of the “unidirectional radiation formulae”. The ratios f,/F, and f7/F2are a measure of the errors in using the latter. The values ofh for /l/d< 0.3 should be < 1. Slab geometry k!d
Cylinder h
6
F,IF,
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
1.042 I .020 1.002 0.978 0.948 0.916 0.882 0.848 0.816 0.785
1.ooo 0.993 0.964 0.918 0.865 0.811 0.760 0.713 0.671 0.532
1.04 1.03 1.04 1.07 1.10 1.13 1.16 1.19 1.22 1.24
1.10 1.20 1.30 1.40 1.50 1.60 1.70 I .80 1.90
0.756 0.728 0.702 0.678 0.655 0.633 0.613 0.594 0.576
0.597 0.565 0.537 0.510 0.487 0.465 0.445 0.426 0.409
1.27 1.29 1.31 1.33 1.35 1.36 1.38 1.39 1.41
2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
0.559 0.432 0.352 0.297 0.257 0.227 0.203 0.184 0.168
0.393 0.283 0.221 0.181 0.154 0.133 0.118 0.105 0.095
11.00 12.00 13.00 14.00
0. I54 0.143 0.133 0.124 0.117 0.110 0.104 0.099 0.094 0.090
0.087 0.080 0.074 0.069 0.064 0.061 0.057 0.054 0.05 1 0.049
15.00 16.00 17.00 18.00 19.00 20.00
Ur
f7
F2
h/F2
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.oo
1.000 0.999 0.997 0.994 0.990 0.984 0.978 0.970 0.962 0.952
1.ooo 1.000 0.999 0.993 0.982 0.964 0.943 0.918 0.892 0.865
1.00 1.00 1.oo 1.00 1.01
1.10 1.20 1.30 1.50 1.60 1.70 1.80 1.90
0.942 0.93 1 0.919 0.908 0.895 0.883 0.871 0.851 0.846
0.838 0.811 0.785 0.760 0.736 0.714 0.692 0.671 0.651
1.12 1.15 1.17 1.19 1.22 1.24 1.26 1.28 1.30
1.42 1.52 1.59 1.64 I .68 I .70 1.73 1.75 1.76
2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
0.833 0.720 0.628 0.556 0.498 0.450 0.411 0.378 0.350
0.632 0.487 0.394 0.330 0.284 0.249 0.22 1 0.199 0.181
1.60 1.69 1.76 1.81 1.86 1.90 1.93
1.78 1.79 1.80 1.81 1.81
11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00
0.326 0.305 0.287 0.270 0.256 0.243 0.231 0.220 0.211 0.202
0.166 0.154 1.143 0.133 0.125 0.118 0.111 0.105 0.010 0.095
1.96 1.99 2.01 2.03 2.05 2.07 2.08 2.09 2.11 2.12
1.82 1.83 1.83 1.84 1.84
1.40
1.02 1.04 1.06 1.08 1.10
1.32
I .45
363
Volume 63. number
6
OPTICS
COMMUNICATIONS
shows f, can be considerably greater than F2. In conclusion, estimates of the absorption of isotropic radiation should take account of the contribution of rays proceeding at angles which result in very long path lengths. The above procedure can be extended for broad band absorption if o is a known function of wavelength b,y reiterating over small wavelength ranges. The work was supported by Grant NSG- 1568 from the National Aeronautis and Space Administration
364
Langley Research Center, Hampton, ful discussions with M.D. Williams acknowledged.
15 September
I987
Virginia. Useof NASA are
References [ 11 R.J. Insuik and W.H. Christiansen, IEEE J. Quantum Electron. QE-20 (I 984) 622. [2] J.K. Roberts. Heat and thermodynamics (Blackie& Son Ltr.. 1928) p. 387.