Abstract

The problem of determining the parameters of a resonance signal from observations of the output of a lock-in detector is considered for the cases in which the resonance has a Lorentz or dispersion line shape. Computer techniques are applied to the more general superimposed line-shape problem accommodating modulation broadening, time-constant distortion, and the effects of multiple overlapping components. Analytical expressions are obtained for several special cases.

© 1971 Optical Society of America

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References

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  1. H. Wahlquist, J. Chem. Phys. 35, 1708 (1961).
    [CrossRef]
  2. R. C. Isler, J. Opt. Soc. Am. 59, 727 (1969).
    [CrossRef]
  3. R. Lowell Smith and T. G. Eck, Phys. Rev. A 2, 2179 (1970).
    [CrossRef]
  4. R. Lowell Smith, Ph.D. thesis, Case Western Reserve University, 1970 (University Microfilms publication number 71-19,057).
  5. P. A. Franken, Phys. Rev. 121, 508 (1961).
    [CrossRef]
  6. M. E. Rose and R. L. Carovillano, Phys. Rev. 122, 1185 (1961).
    [CrossRef]
  7. M. W. P. Strandberg, H. R. Johnson, and J. R. Eshbach, Rev. Sci. Instr. 25, 776 (1954).
    [CrossRef]
  8. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw–Hill, New York, 1969).
  9. Prototype programs incorporating both the power-series and integral methods are available from the author on request. These are presented, for simplicity, without the elaborations discussed above, but nonetheless are powerful tools for the interpretation of resonance signals under lock-in detection. Each program is roughly 100 Fortran statements in length and requires one somewhat shorter subroutine.

1970 (1)

R. Lowell Smith and T. G. Eck, Phys. Rev. A 2, 2179 (1970).
[CrossRef]

1969 (1)

1961 (3)

H. Wahlquist, J. Chem. Phys. 35, 1708 (1961).
[CrossRef]

P. A. Franken, Phys. Rev. 121, 508 (1961).
[CrossRef]

M. E. Rose and R. L. Carovillano, Phys. Rev. 122, 1185 (1961).
[CrossRef]

1954 (1)

M. W. P. Strandberg, H. R. Johnson, and J. R. Eshbach, Rev. Sci. Instr. 25, 776 (1954).
[CrossRef]

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw–Hill, New York, 1969).

Carovillano, R. L.

M. E. Rose and R. L. Carovillano, Phys. Rev. 122, 1185 (1961).
[CrossRef]

Eck, T. G.

R. Lowell Smith and T. G. Eck, Phys. Rev. A 2, 2179 (1970).
[CrossRef]

Eshbach, J. R.

M. W. P. Strandberg, H. R. Johnson, and J. R. Eshbach, Rev. Sci. Instr. 25, 776 (1954).
[CrossRef]

Franken, P. A.

P. A. Franken, Phys. Rev. 121, 508 (1961).
[CrossRef]

Isler, R. C.

Johnson, H. R.

M. W. P. Strandberg, H. R. Johnson, and J. R. Eshbach, Rev. Sci. Instr. 25, 776 (1954).
[CrossRef]

Lowell Smith, R.

R. Lowell Smith and T. G. Eck, Phys. Rev. A 2, 2179 (1970).
[CrossRef]

R. Lowell Smith, Ph.D. thesis, Case Western Reserve University, 1970 (University Microfilms publication number 71-19,057).

Rose, M. E.

M. E. Rose and R. L. Carovillano, Phys. Rev. 122, 1185 (1961).
[CrossRef]

Strandberg, M. W. P.

M. W. P. Strandberg, H. R. Johnson, and J. R. Eshbach, Rev. Sci. Instr. 25, 776 (1954).
[CrossRef]

Wahlquist, H.

H. Wahlquist, J. Chem. Phys. 35, 1708 (1961).
[CrossRef]

J. Chem. Phys. (1)

H. Wahlquist, J. Chem. Phys. 35, 1708 (1961).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. (2)

P. A. Franken, Phys. Rev. 121, 508 (1961).
[CrossRef]

M. E. Rose and R. L. Carovillano, Phys. Rev. 122, 1185 (1961).
[CrossRef]

Phys. Rev. A (1)

R. Lowell Smith and T. G. Eck, Phys. Rev. A 2, 2179 (1970).
[CrossRef]

Rev. Sci. Instr. (1)

M. W. P. Strandberg, H. R. Johnson, and J. R. Eshbach, Rev. Sci. Instr. 25, 776 (1954).
[CrossRef]

Other (3)

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw–Hill, New York, 1969).

Prototype programs incorporating both the power-series and integral methods are available from the author on request. These are presented, for simplicity, without the elaborations discussed above, but nonetheless are powerful tools for the interpretation of resonance signals under lock-in detection. Each program is roughly 100 Fortran statements in length and requires one somewhat shorter subroutine.

R. Lowell Smith, Ph.D. thesis, Case Western Reserve University, 1970 (University Microfilms publication number 71-19,057).

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Figures (2)

Fig. 1
Fig. 1

Plot of Eqs. (22) relating the true width W, observed width W0, and modulation M for isolated Lorentz or dispersion signals. The three curves correspond to an identification of W0 with WpL, Wz, or WpD, the characteristic widths indicated in Fig. 2 for typical signals.

Fig. 2
Fig. 2

Plot of the Lorentz and dispersion first Fourier amplitudes L and D as a function of for the case where β = W/M = 5/3. All of the characteristic widths of these signals, together with the peak-to-peak modulation 2M, are indicated.

Equations (74)

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S L [ Γ 2 / ( Γ 2 + Δ 2 ) ] ,             S D [ Γ Δ / ( Γ 2 + Δ 2 ) ] .
a 1 = π W [ ± W π ( 2 M ) 2 ( 2 γ - u ) 1 2 2 ( u - 2 ) 1 2 ( u - γ ) ]
a 1 = W [ 2 M ( 1 - α · ( 2 γ - u ) 1 2 + u 1 2 ( u - γ - 1 ) ( u - γ ) ( u - 2 ) 1 2 ) ] ,
γ = 1 + α 2 + β 2
u = γ + δ ,             δ = ( γ 2 - 4 α 2 ) 1 2 ,             2 < u < ,
α = [ ( B 0 + s t ) - P ] / M ,             β = W / M .
S L 1 1 + ( α / β ) 2 ,             S D α / β 1 + ( α / β ) 2 .
L = ( M / W ) a 1 ,             D = ( M / W ) a 1 ,
L = - 4 α β / [ ( u 2 - 2 u ) 1 2 δ ]
D = 2 - ( u 2 - 2 u ) 1 2 / δ .
v = γ - δ = 2 γ - u .             0 < v < 2
L = { [ 2 v 1 2 ( 2 - v ) 1 2 ] / ( v - u ) } ( α / α )
D = 2 - i { [ 2 u 1 2 ( 2 - u ) 1 2 ] / ( u - v ) } .
L = F ( u , v ) ( α / α ) ,             D = 2 - i F ( v , u ) .
u α = g ( u , v ) = h ( v , u ) v α = g ( v , u ) = h ( u , v ) ,
L α ( n ) = F α ( n ) ( u , v ) ( α / α ) ,             D α ( n ) = - i F α ( n ) ( v , u ) .
L α = D β ,             L β = - D α .
( L α , β ( n ) ) α = ( D α , β ( n ) ) β ,             ( L α , β ( n ) ) β = - ( D α , β ( n ) ) α ,
L α ( 1 ) = - 4 u 1 2 ( 2 - v ) 1 2 ( v - u ) 3 ( 2 u v - u - 3 v ) = D β ( 1 )
L α ( 2 ) = 24 v 1 2 ( 2 - v ) 1 2 ( v - u ) 5 ( 2 u 3 v + 2 u 2 v 2 - 2 u 3 - 10 u 2 v - 4 u v 2 + 5 u 2 + 10 u v + v 2 ) α α = ( D α ( 1 ) ) β
L α ( 3 ) = - 48 u 1 2 ( 2 - v ) 1 2 ( v - u ) 7 ( 8 u 4 v 2 + 24 u 3 v 3 + 8 u 2 v 4 - 12 u 4 v - 96 u 3 v 2 - 112 u 2 v 3 - 20 u v 4 + 2 u 4 + 76 u 3 v + 252 u 2 v 2 + 140 u v 3 + 10 v 4 - 5 u 3 - 105 u 2 v - 175 u v 2 - 35 v 3 ) = ( D α ( 2 ) ) β .
L = - 4 α β / δ
L α ( 1 ) = - ( 2 β u / δ 3 ) ( Q L )
L α ( 2 ) = ( 12 α β / δ 5 ) [ ( γ - 1 ) u ( Q L + γ ) - γ 2 ]
L α ( 3 ) = ( 6 β u / δ 7 ) ( f 1 Q L + f 2 + f 3 )
D = 2 - / δ
D α ( 1 ) = - ( 2 α / δ 3 u ) ( Q D )
D α ( 2 ) = ( 3 / δ 5 ) [ ( γ - 1 ) v ( Q D + γ ) - γ 2 ]
D α ( 3 ) = ( 6 α / δ 7 u ) ( f 1 Q D + f 2 - f 3 ) ,
= ( u 2 - 2 u ) 1 2 Q L = γ + v - u v ,             Q D = γ + u - u v f = 4 γ 2 - 7 γ f 1 = 2 γ f - u v [ 4 γ ( γ - 3 ) + u v + 13 / 2 ] f 2 = - 4 γ 3 ( γ - 2 ) + u v ( f - u v / 2 ) f 3 = δ ( δ 2 - u v / 2 ) .
α p 4 - 2 [ 1 + ( 5 / 3 ) β 2 ] α p 2 + ( 1 + β 2 ) 2 = 0 ,
1 / ξ = α p ,             η / ξ = β .
ξ 2 + ( η ± 1 / 3 ) 2 ± 4 3 ,
ξ = M / W 0 ,             η = W / W 0 .
ξ L 2 + ( η L - 1 / 3 ) 2 = 4 3
ξ D 2 + ( η D + 1 / 3 ) 2 = 4 3 .
ξ z 2 = - η z 2 - η z 4 3 + η z 2 3 + 1.
S m = k = 1 2 ( 1 - m ) 1 2 ( m - 1 ) S ( α + 2 k α 0 , β ) ,
S m ( α 0 ) = S m ( 0 ) + n = 1 1 n ! n S m ( 0 ) α 0 n α 0 n .
n α 0 n S ( α + 2 k α 0 , β ) α 0 = 0 = ( 2 k ) n n α n S ( α + 2 k α 0 , β ) α 0 = 0 = ( 2 k ) n S α ( n ) ,
S m = m S ( α , β ) + 1 2 ( S α ( 2 ) ) α 0 2 k = 1 2 ( 1 - m ) 1 2 ( m - 1 ) ( 2 k ) 2 ,
S α ( 1 ) + [ ( m 2 - 1 ) / 6 ] ( S α ( 3 ) ) α 0 2 = 0.
S α ( 2 ) α p ( α - α p ) + 1 2 S α ( 3 ) α p ( α - α p ) 2 + ( m 2 - 1 6 ) S α ( 3 ) α p α 0 2 = 0 ,
α - α p R p + 1 2 [ ( α - α p ) 2 + [ ( m 2 - 1 ) / 3 ] α 0 2 R p 2 ] = 0 ,
R p = S α ( 2 ) S α ( 3 ) | α p = S α ( 2 ) ( α p , β ) S α ( 3 ) ( α p , β ) .
α = α p - [ ( m 2 - 1 ) / 6 ] ( α 0 2 / R p ) .
α = α z - [ ( m 2 - 1 ) / 6 ] ( α 0 2 / R z ) ,
R p L = [ L α ( 2 ) ( α p L , β ) ] / [ L α ( 3 ) ( α p L , β ) ]
R p D = [ D α ( 2 ) ( α p D , β ) ] / [ D α ( 3 ) ( α p D , β ) ] .
R z = [ D α ( 1 ) ( α z , β ) ] / [ D α ( 2 ) ( α z , β ) ] .
R p L = - 4 α p L ( η L / 3 ) ( 1 + η L / 3 5 η L 2 + 2 3 η L - 5 )
R p D = + 4 α p D ( η D / 3 ) ( 1 - η D / 3 5 η D 2 - 2 3 η D - 5 )
R z = α z η z 2 3 ( 1 + η z 2 3 ) ( 3 - η z 2 3 ) 3 ( 5 η z 4 3 - 14 η z 2 3 + 5 ) ,
α ± = α p , z ± [ 1 - ( m 2 - 1 6 ) α 0 2 α p , z ± R p , z ± ] ,
W m = W 0 [ 1 - ( m 2 - 1 6 ) α 0 2 α p , z + R p , z + ] .
W m L = W [ 1 η L + ( m 2 - 1 3 ) ( σ W ) 2 × ( 5 η L 2 + 2 3 η L - 5 8 ( 1 + η L / 3 ) ) ] ,
W m D = W [ 1 η D + ( m 2 - 1 3 ) ( σ W ) 2 × ( 5 + 2 3 η D - 5 η D 2 8 ( 1 - η D / 3 ) ) ] ,
W m z = W [ 1 η z + ( m 2 - 1 2 ) ( σ W ) 2 × ( η z 1 3 ( - 5 η z 4 3 + 14 η z 2 3 - 5 ) ( 1 + η z 2 3 ) ( 3 - η z 2 3 ) ) ] .
W = η L W m L - ( m 2 - 1 ) σ 2 3 W m L ( 5 η L 2 + 2 3 η L - 5 8 ( 1 + η L / 3 ) )
W = η D W m D - ( m 2 - 1 ) σ 2 3 W m D ( 5 + 2 3 η D - 5 η D 2 8 ( 1 - η D / 3 ) )
W = η z W m z - ( m 2 - 1 ) σ 2 2 W m z [ η z 1 3 ( - 5 η z 4 3 + 14 η z 2 3 - 5 ) ( 1 + η z 2 3 ) ( 3 - η z 2 3 ) ] .
f ( α , β , α 0 ) [ 1 + ( α + 2 k α 0 β ) 2 ] - 2 = f ( α , β , α 0 ) ( 1 + α 2 β 2 ) - 2 ( 1 + 4 k α α 0 + 4 k 2 α 0 2 α 2 + β 2 ) - 2 .
( m - 1 ) α 0 { [ 2 α M + ( m - 1 ) α 0 ] / ( α M 2 + β 2 ) } = 1 ,
σ < W / [ ( m - 1 ) 2 ] .
S ( t ) = e - t / τ [ 0 t S ( t ) e t / τ d t τ + S ( 0 ) ] ,
S ( t ) = [ S ( 0 ) - S ( 0 ) ] e - t / τ + S ( t ) + τ S ( 0 ) t e - t / τ - τ S ( t ) t - τ 2 2 S ( 0 ) t 2 e - t / τ + τ 2 2 S ( t ) t 2 + .
S ( t ) = S ( t ) + n = 1 ( - τ ) n n S ( t ) t n
S ( t ) = S ( t ) + n = 1 ( - 1 ) n ( m = 0 n τ 1 m τ 2 n - m ) n S ( t ) t n
S = S + n = 1 ( - s M ) n ( m = 0 n τ 1 m τ 2 n - m ) S α ( n ) .
W s L = W [ 1 η L + 3 ( s W ) 2 ( τ 1 2 + τ 2 2 ) × ( 5 η L 2 + 2 3 η L - 5 8 ( 1 + η L / 3 ) ) ]
W s D = W [ 1 η D + 3 ( s W ) 2 ( τ 1 2 + τ 2 2 ) × ( 5 + 2 3 η D - 5 η D 2 8 ( 1 - η D / 3 ) ) ]
W s z = W [ 1 η z + 3 2 ( s W ) 2 ( τ 1 2 + τ 2 2 ) × ( η z 1 3 ( - 5 η z 4 3 + 14 η z 2 3 - 5 ) ( 1 + η z 2 3 ) ( 3 - η z 2 3 ) ) ] .
S ( t + Δ t ) = e - Δ t / τ S ( t ) + Δ t 2 τ [ S ( t + Δ t ) + e - Δ t / τ S ( t ) ] .
S = S - s M ( τ 1 + τ 2 ) S α ( 1 ) + ( s M ) 2 ( τ 1 2 + τ 1 τ 2 + τ 2 2 ) S α ( 2 ) ,