Abstract

Analytical expressions are presented for the frequency dependence of loss factor, conductivity, loss tangent, amplitude attenuation per radian (absorption index), imaginary refractive index, amplitude and power attenuation per unit length (absorption coefficient), and reflectivity for a damped harmonic oscillator system. Equations are given for the displacement of maxima of these parameters as a function of line strength and width (intensity and damping). Some limiting value relations are given, and the functional dependences of the parameters illustrated graphically. Applications to physical problems are discussed briefly.

© 1959 Optical Society of America

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References

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  1. W. Heitler, The Quantum Theory of Radiation (Clarendon Press, Oxford, 1954).
  2. E. P. Gross, Phys. Rev. 97, 395 (1955).
    [Crossref]
  3. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954).
  4. A. von Hippel, Dielectrics and Waves (John Wiley & Sons, Inc., New York, 1954).
  5. T. H. Havelock, Proc. Roy. Soc. (London) A86, 1 (1911);Proc. Roy. Soc. (London) A105, 488 (1924).
    [Crossref]
  6. I. Simon, J. Opt. Soc. Am. 41, 336 (1951).
    [Crossref]
  7. The rationalized mks system is employed in this presentation.
  8. The velocity of light υ=∂x/∂t at constant phase. From the equation for Ey for constant phase, x=ωt/β+const, then υ=ω/β=1/real part of((∊*μ*)12=2/{[(∊′μ′−∊″μ″)2+(∊′μ″−∊″μ′)2]12+(∊′μ′−∊″μ″)})12.
  9. The appropriate field E in this expression is discussed later.
  10. For real media with additional susceptibility terms, we may writePtotal=∑qχq*∊0Eq=(κ*−1)∊0E,κ*=1+∑qχq*rq=1+1V0+∑q≠q0χq*rq=a(1+1/V).where Eq is the local electric field associated with χq*,rq=|Eq/E0|,V=aV0=(1−f2+ilf)/X,a=1+∑q≠q0χq*rq, and X=X0/a.
  11. n* has been multiplied by the scale factor a=1.5 for proper reflectivity.
  12. The theory of impedance-circle diagrams is important in electric transmission-line engineering, and is discussed in several texts, e.g., J. C. Slater, Microwave Transmission (McGraw-Hill Book Company, Inc.New York, 1942).
  13. In Havelock’s notation q1′, g, k, and y are equivalent to a, X, l, and f2− 1 = −s, respectively.
  14. In his notation the coefficient of the term q1′k2y should be 32 instead of 16.
  15. gj is 1/∊0times an anisotropy factor h, depending on the detailed structure of the oscillator system. For an isotropic array of dipoles with large interoscillator distance, h=−23 and gj= −2/(3∊0).

1955 (1)

E. P. Gross, Phys. Rev. 97, 395 (1955).
[Crossref]

1951 (1)

1911 (1)

T. H. Havelock, Proc. Roy. Soc. (London) A86, 1 (1911);Proc. Roy. Soc. (London) A105, 488 (1924).
[Crossref]

Born, M.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954).

Gross, E. P.

E. P. Gross, Phys. Rev. 97, 395 (1955).
[Crossref]

Havelock, T. H.

T. H. Havelock, Proc. Roy. Soc. (London) A86, 1 (1911);Proc. Roy. Soc. (London) A105, 488 (1924).
[Crossref]

Heitler, W.

W. Heitler, The Quantum Theory of Radiation (Clarendon Press, Oxford, 1954).

Huang, K.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954).

Simon, I.

Slater, J. C.

The theory of impedance-circle diagrams is important in electric transmission-line engineering, and is discussed in several texts, e.g., J. C. Slater, Microwave Transmission (McGraw-Hill Book Company, Inc.New York, 1942).

von Hippel, A.

A. von Hippel, Dielectrics and Waves (John Wiley & Sons, Inc., New York, 1954).

J. Opt. Soc. Am. (1)

Phys. Rev. (1)

E. P. Gross, Phys. Rev. 97, 395 (1955).
[Crossref]

Proc. Roy. Soc. (London) (1)

T. H. Havelock, Proc. Roy. Soc. (London) A86, 1 (1911);Proc. Roy. Soc. (London) A105, 488 (1924).
[Crossref]

Other (12)

W. Heitler, The Quantum Theory of Radiation (Clarendon Press, Oxford, 1954).

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954).

A. von Hippel, Dielectrics and Waves (John Wiley & Sons, Inc., New York, 1954).

The rationalized mks system is employed in this presentation.

The velocity of light υ=∂x/∂t at constant phase. From the equation for Ey for constant phase, x=ωt/β+const, then υ=ω/β=1/real part of((∊*μ*)12=2/{[(∊′μ′−∊″μ″)2+(∊′μ″−∊″μ′)2]12+(∊′μ′−∊″μ″)})12.

The appropriate field E in this expression is discussed later.

For real media with additional susceptibility terms, we may writePtotal=∑qχq*∊0Eq=(κ*−1)∊0E,κ*=1+∑qχq*rq=1+1V0+∑q≠q0χq*rq=a(1+1/V).where Eq is the local electric field associated with χq*,rq=|Eq/E0|,V=aV0=(1−f2+ilf)/X,a=1+∑q≠q0χq*rq, and X=X0/a.

n* has been multiplied by the scale factor a=1.5 for proper reflectivity.

The theory of impedance-circle diagrams is important in electric transmission-line engineering, and is discussed in several texts, e.g., J. C. Slater, Microwave Transmission (McGraw-Hill Book Company, Inc.New York, 1942).

In Havelock’s notation q1′, g, k, and y are equivalent to a, X, l, and f2− 1 = −s, respectively.

In his notation the coefficient of the term q1′k2y should be 32 instead of 16.

gj is 1/∊0times an anisotropy factor h, depending on the detailed structure of the oscillator system. For an isotropic array of dipoles with large interoscillator distance, h=−23 and gj= −2/(3∊0).

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Figures (9)

F. 1
F. 1

Locus of dielectric constant κ*= κ′iκ″= a[1 + X/(1 − f2 +ilf)] on complex plane for a damped-harmonic oscillator (X = 1.62, l = 0.1).

F. 2
F. 2

Locus of refractive index n * = n i k = n ( 1 i k ) = ( k * ) 1 2 on complex plane for a damped-harmonic oscillator (X = 1.62, l = 0.1).

F. 3
F. 3

Frequency dependence of the maximum of the loss factor (∊″) on band width (l2) for a damped-harmonic oscillator.

F. 4
F. 4

Frequency dependence of the maximum of the loss tangent (tanδ) on intensity X and band width l2 for a damped-harmonic oscillator. For points below the dashed line, δmax<π/2, the loss tangent remaining finite.

F. 5
F. 5

Frequency dependence of the maximum of the absorption index (k) on intensity X and band width l2 for a damped-harmonic oscillator.

F. 6
F. 6

Frequency dependence of the maximum of the imaginary refractive index k = nk on the intensity X and band width l2 for a damped-harmonic oscillator.

F. 7
F. 7

Frequency dependence of the maximum of the absorption coefficient A = 2α on the intensity X and band width l2 for a damped-harmonic oscillator.

F. 8
F. 8

Frequency dependence of the maximum of reflectivity R on the intensity X and band width l2 for a damped-harmonic oscillator with b = 0. b = 1 − 1/κ, where κ = limf→∞κ.

F. 9
F. 9

Dependence on b of contour lines of constant frequency shift for reflectivity. b = 1 − 1/κ, where κ = limf→∞κ.

Equations (46)

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I C ( ω ) = a ω 2 ( ω 0 2 ω 2 ) 2 + b ω 2 a ( ω ω 0 ) 2 ( 1 + ω 0 / ω ) 2 + b ,
× E = B * / t · D = ρ × H = D * / t · B = 0 { where D = * E and B = μ * H .
2 E = * μ * 2 E / t 2 , 2 H = * μ * 2 H / t 2 ,
E y = E 0 exp ( i ω t γ x ) = E 0 exp [ i ( ω t β x ) α x ) , H z = H 0 exp ( i ω t γ x ) ,
k = [ 1 2 κ tan ( δ / 2 ) ] 1 2 = 1 2 [ κ + ( κ 2 + κ 2 ) 1 2 ] 1 2 .
α = k ω / c = ( ω 0 f / c 2 ) [ κ + ( κ 2 + κ 2 ) 1 2 ] 1 2 .
A = 2 α = ( ω 0 k f 2 / c ) [ κ + ( κ 2 + κ 2 ) 1 2 ] 1 2
R = r r ¯ = 1 ( κ * ) 1 2 1 + ( κ * ) 1 2 1 ( κ ¯ * ) 1 2 1 + ( κ ¯ * ) 1 2 = 1 2 [ ( κ * ) 1 2 + ( κ ¯ * ) 1 2 ] ( κ * κ ¯ * ) 1 2 + ( κ * ) 1 2 + ( κ ¯ * ) 1 2 + 1 = 1 2 2 [ κ + ( κ 2 + κ 2 ) 1 2 ] 1 2 ( κ 2 + κ 2 ) 1 2 + 2 [ κ + κ 2 + κ 2 ) 1 2 ] 1 2 + 1 .
m d 2 y d t 2 + p d y d t K y = e E y = e E 0 exp ( i ω t ) , 9
P = D * 0 E = ( * 0 ) E = ( κ * 1 ) 0 E = χ * 0 E = N e y 10
χ * = κ * 1 = N e y / 0 E = N e 2 / 0 m ω 0 2 1 f 2 + i l f = X 0 1 f 2 + i l f = 1 / V 0 ,
tan δ = κ / κ = V / ( V + V 2 + V 2 ) = X l f / [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 ] ,
k = cot δ + ( cot 2 δ + 1 ) 1 2 = X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 X l f + { [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 ] 2 ( X l f ) 2 + 1 } 1 2 , k = ( κ k 2 ) 1 2 = { [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 ] + [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 2 + ( X l f ) 2 ] 1 2 2 [ ( 1 f 2 ) 2 + l 2 f 2 ] } 1 2 , α = ω 0 f k c = ω 0 f c { [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 ] + [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 2 + ( X l f ) 2 ] 1 2 2 [ ( 1 f 2 ) 2 + l 2 f 2 ] } 1 2 A = 2 α = 2 ω 0 f c { [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 ] + [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 2 + ( X l f ) 2 ] 1 2 2 [ ( 1 f 2 ) 2 + l 2 f 2 ] } 1 2 R = 1 2 2 [ ( 1 f 2 ) 2 + l 2 f 2 ] 1 2 { X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 + [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 2 + ( X l f ) 2 ] 1 2 } 1 2 1 2 numerator + { [ X ( 1 f 2 ) + ( 1 f 2 ) 2 + l 2 f 2 ] 2 + ( X l f ) 2 } 1 2 + ( 1 f 2 ) 2 + l 2 f 2 .
d d f = 0 X l { ( 1 f 2 ) 2 + l 2 f 2 f [ 2 ( 1 f 2 ) ( 2 f ) + 2 l 2 f ] } [ ( 1 f 2 ) 2 + l 2 f 2 ] 2 = 0 ,
f = { 2 l 2 6 + [ ( 2 l 2 6 ) 2 + 1 3 ] 1 2 } 1 2 .
d σ d f = 0 ω 0 X l { [ ( 1 f 2 ) 2 + l 2 f 2 ] 2 f f 2 [ 2 ( 1 f 2 ) ( 2 f ) + 2 l 2 f ] } [ ( 1 f 2 ) 2 + l 2 f 2 ] 2 = 0 ,
κ = 0 or X ( 1 f 2 ) + ( 1 + f 2 ) 2 + l 2 f 2 = 0 ,
f tan δ = { 1 + X 2 l 2 2 ± [ ( 1 + X 2 l 2 2 ) 2 1 X ] 1 2 } 1 2 .
l X , 1 , f tan δ { [ 1 + X l 2 ( 1 + 1 / X ) ] 1 2 1 + l 2 / 2 X .
d n * / d f n * = d κ * d f 2 κ * = d V / d f 2 V ( V + 1 ) = ( d V / d f ) [ V ¯ ( V ¯ + 1 ) ] 2 V V ¯ ( V + 1 ) ( V ¯ + 1 ) ,
( d n * / d f ) / n * = Real number × ( 2 f + i l ) [ ( 1 f 2 ) i l f ] [ ( 1 f 2 ) i l f + X ] .
f k = { 2 l 2 + X 6 + [ ( 2 l 2 + X 6 ) 2 + 1 + X 3 ] 1 2 } 1 2 ,
d n * d f = d κ * / d f 2 κ * 1 2 = d V / d f 2 ( V 4 + V 3 ) 1 2 = ( d V / d f ) ( V ¯ 4 + V ¯ 3 ) 1 2 2 ( V 4 + V 3 ) 1 2 ( V ¯ 4 + V ¯ 3 ) 1 2 .
( d n * / d f ) 2 = Real number × ( 2 f + i l ) 2 ( 1 f 2 i l f ) 3 ( 1 f 2 i l f + X ) .
12 s 4 + ( 16 + 8 X 4 l 2 ) s 3 + ( 12 X 5 X l 2 8 l 2 ) s 2 + l 2 ( 4 + l 2 ) ( 4 + X ) s + l 2 ( 4 X 4 l 2 X l 2 ) = 0 .
α 2 = ω 0 2 f 2 2 c 2 [ κ + ( κ 2 + κ 2 ) 1 2 ] d α 2 d f = ω 0 2 f 2 2 c 2 { 2 κ ( κ 2 + κ 2 ) 1 2 + f [ d κ d f + κ ( d κ / d f ) + κ ( d κ / d f ) ( κ 2 + κ 2 ) 1 2 ] } ,
4 ( κ 2 + κ 2 ) ( κ + f d κ d f ) + f [ κ ( d κ d f ) 2 + 2 κ d κ d f d κ d f κ ( d κ d f ) 2 ] = 0 , or ( V V ¯ ) 5 { 4 V V ¯ [ ( V V ¯ + V ) 2 + V 2 ] [ V V ¯ V + f V V ¯ V f V ( V V ¯ ) f ] + f 2 ( V [ ( V V ¯ ) 2 V f 2 2 V V ¯ ( V V ¯ ) f V V f + ( V V ¯ ) f 2 V 2 ] + 2 ( V V ¯ + V ) [ ( V V ¯ ) 2 V f V f + V V ¯ ( V V ¯ ) f ( V V f + V V f ) + ( V V ¯ ) f 2 V V ] ( V V ¯ ) 2 V f 2 + 2 V V ¯ ( V V ¯ ) f V V f ( V V ¯ ) f 2 V 2 ) } = 0 .
8 s 6 + ( 16 + 16 l 2 4 X ) s 5 + [ 48 l 2 8 l 4 + X ( 4 + 12 l ) 2 ] s 4 + [ 32 l 2 + 32 l 4 + X ( 16 52 l 2 ) + X 2 ( 4 + 3 l 2 ) ] s 3 + [ 40 l 4 + X ( 56 l 2 + 4 l 4 ) + X 2 ( 12 15 l 2 + l 4 ) ] s 2 + [ 16 l 4 + X ( 16 l 2 8 l 4 ) + X 2 ( 16 l 2 2 l 4 ) s + 4 X l 4 + X 2 ( 4 l 2 + l 4 ) ] = 0 .
r = 1 κ * 1 2 1 + κ * 1 2 , d r d f = d κ * / d f κ * 1 2 ( 1 + κ * 1 2 ) 2 , d r / d f r = d κ * / d f ( κ * 1 ) κ * 1 2 ,
( d r / d f ) r = d U / d f a 1 2 ( U + 1 ) ( U 2 + b U ) 1 2 = ( ā ) 1 2 ( d U / d f ) ( Ū + 1 ) ( Ū 2 + b ¯ Ū ) 1 2 ( a ā ) 1 2 ( U + 1 ) ( Ū + 1 ) ( U 2 + b U ) 1 2 ( Ū 2 + b ¯ Ū ) 1 2 ,
[ ( d r / d f ) / r ] 2 = Real number × ā ( d U / d f ) 2 ( Ū + 1 ) 2 ( Ū 2 + b ¯ Ū ) = ā ( 2 f + i l ) 2 ( s i l f + Y ) 2 [ ( s i l f ) 2 + b Y ( 4 i l f ) ] ,
12 s 4 + [ 16 4 l 2 + 8 ( 2 + b ) Y ] s 3 + [ 8 l 2 + Y ( 2 + b ) ( 12 5 l 2 ) + 4 Y 2 ( 1 + 2 b ) ] S 2 + [ 16 l 2 + 4 l 4 + Y ( 2 + b ) ( 4 l 2 + l 4 ) + Y 2 ( 1 + 2 b ) × ( 8 2 l 2 ) ] s + [ 4 l 4 + Y ( 2 + b ) ( 4 l 2 l 4 ) + 4 Y 2 ( 1 + 2 b ) l 2 + Y 3 b ( 4 + l 2 ) ] = 0 .
χ j * = N α j 0 ( 1 N α j g j ) or α j = 0 N χ j * 1 + h χ j ,
χ j * = 1 / V 0 j = X 0 j 1 f 2 + i l f , α j = 0 N ( V 0 l + h ) = 0 X 0 j / N ( 1 + h X 0 j ) 1 f 2 / ( 1 + h X 0 j ) + i l f / ( 1 + h X 0 j )
ξ = 0 X 0 j / N ( 1 + h X 0 j ) , f = f / ( 1 + h X 0 j ) 1 2 , d = l / ( 1 + h X 0 j ) 1 2 .
P j = [ g R C ( δ R C / N R α R ) ] ( 1 δ C j ) δ C j g R C ( δ R C / N R α R ) E 0 ,
δ a b = { 0 for a b 1 for a = b .
P j 0 E 0 = X j * = [ g R C ( δ R C / N R α R ) ] ( 1 δ C j ) δ C j 0 g R C ( δ R C / N R α R ) ,
P j = χ j * 0 E j = X j * 0 E 0 , E j = E 0 X j * χ j * = E 0 [ g R C ( δ R C / N R α R ) ] ( 1 δ C j ) δ C j ( 1 N j α j g j j ) g R C ( δ R C / N R α R ) N j α j .
l 2 = P ± ( P 2 + Q ) 1 2 ,
P = 4 s 3 + [ 8 5 Y ( 2 + b ) ] s 2 + [ 16 + 4 Y ( 2 + b ) 2 Y 2 ( 1 + 2 b ) ] s + 4 Y ( 2 + b ) + 4 Y 2 ( 1 + 2 b ) + Y 3 b 2 [ 4 + Y ( 2 + b ) ] ( 1 s ) , Q = 12 s 4 + [ 16 + 8 Y ( 2 + b ) ] s 2 + [ 12 Y ( 2 + b ) 4 Y 2 ( 1 + 2 b ) ] s 2 8 Y 2 ( 1 + 2 b ) s 4 Y 3 b [ 4 + Y ( 2 + b ) ] ( 1 s ) .
X = 12 s 4 + ( 16 + 4 l 2 ) s 3 + 8 l 2 s 2 4 l 2 ( 4 + l 2 ) s + 4 l 4 8 s 3 + ( 12 5 l 2 ) s 2 + l 2 ( 4 + l 2 ) s + 4 l 2 l 4 ;
S = 2 s 5 + ( 2 + 6 l 2 ) s 4 + ( 8 26 l 2 ) s 3 + ( 28 l 2 + 2 l 4 ) s 2 + ( 8 l 2 4 l 4 ) s + 2 l 4 ( 4 3 l 2 ) s 3 + ( 12 + 15 l 2 l 4 ) s 2 + ( 16 l 2 + 2 l 4 ) s + 4 l 2 l 4 , T = 8 s 6 + ( 16 + 16 l 2 ) s 5 + ( 48 l 2 8 l 4 ) s 4 + ( 32 l 2 + 32 l 4 ) s 3 40 l 4 s 2 + 16 l 4 s ( 4 3 l 2 ) s 3 + ( 12 + 15 l 2 l 4 ) s 2 + ( 16 l 2 + 2 l 4 ) s + 4 l 2 l 4 .
((*μ*)12=2/{[(μμ)2+(μμ)2]12+(μμ)})12.
Ptotal=qχq*0Eq=(κ*1)0E,κ*=1+qχq*rq=1+1V0+qq0χq*rq=a(1+1/V).
rq=|Eq/E0|,V=aV0=(1f2+ilf)/X,a=1+qq0χq*rq,