Abstract

This paper presents a theoretical analysis of the line shapes and signal-to-noise ratios obtained in two-tone optical heterodyne spectroscopy with tunable lead-salt diode lasers. The theory is described in terms of the frequency-modulation (FM) index β, the amplitude-modulation (AM) index M, their relative phase shift ψ, and the ratio of modulation frequency to the absorption-line half-width ν¯m. Synthetic spectra are presented for both Gaussian and Lorentzian line shapes and show considerable structural variation with the theoretical parameters. Experimental two-tone optical heterodyne spectra were obtained by modulating a specially modified lead-salt diode laser in the radio-frequency region. The experimental spectra obtained from NH3 absorption lines confirm the theoretical results.

© 1987 Optical Society of America

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  1. G. R. Janik, C. B. Carlisle, T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 3, 1070–1074 (1986).
    [CrossRef]
  2. W. Lenth, “Optical heterodyne spectroscopy with frequency-and amplitude-modulated semiconductor lasers,” Opt. Lett. 8, 575–577 (1983).
    [CrossRef] [PubMed]
  3. M. Gehrtz, W. Lenth, A. T. Young, H. S. Johnston, “High frequency-modulation spectroscopy with lead-salt diode lasers,” Opt. Lett. 11, 132–134 (1986).
    [CrossRef]
  4. D. E. Cooper, J. P. Watjen, “Two-tone optical heterodyne spectroscopy with a tunable lead-salt diode laser,” Opt. Lett. 11, 606–608 (1986).
    [CrossRef] [PubMed]
  5. J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
    [CrossRef]
  6. G. C. Bjorklund, M. D. Levinson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
    [CrossRef]
  7. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).
  8. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, NBS Applied Mathematics Series #55 (National Bureau of Standards, Gaithersburg, Md., 1972).
  9. R. S. Eng, J. F. Butler, K. J. Linden, “Tunable diode laser spectroscopy: an invited review,” Opt. Eng. 19, 945–960 (1980).
    [CrossRef]
  10. D. E. Jennings, “Absolute line strengths in ν412CH4: a dual-beam diode laser spectrometer with sweep integration,” Appl. Opt. 19, 2695–2700 (1980).
    [CrossRef] [PubMed]
  11. D. T. Cassidy, J. Reid, “High-sensitivity detection of trace gases using sweep integration and tunable diode lasers,” Appl. Opt. 21, 2527–2530 (1982).
    [CrossRef] [PubMed]
  12. D. E. Cooper, T. F. Gallagher, “Frequency-modulation spectroscopy with a CO2laser: results and implications for ultrasensitive point monitoring of the atmosphere,” Appl. Opt. 24, 710–716 (1985).
    [CrossRef]
  13. A. Erdelyi, W. Magnus, F. Oberhettnger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953).
  14. J. M. Osterwalder, B. J. Rickett, “Frequency modulation of GaAlAs injection lasers at microwave frequency rates,” IEEE J. Quantum Electron. QE-16, 250–252 (1980).
    [CrossRef]
  15. C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor a of semiconductor lasers,” Appl. Phys. Lett. 42, 328–330 (1983).
    [CrossRef]
  16. W. T. Tsang, N. A. Olsson, “Enhanced frequency modulation in cleave-coupled-cavity semiconductors lasers with reduced spurious intensity modulation,” Appl. Phys. Lett. 43, 527–529 (1983).
    [CrossRef]
  17. K. J. Linden, R. E. Reeder, “Operation of cleave-coupled-cavity Pb-salt diode lasers in the 4–5 μ m spectral region,” Appl. Phys. Lett. 44, 377–379 (1984).
    [CrossRef]
  18. W. Lo, “Cleave-coupled-cavity lead-salt diode lasers,” Appl. Phys. Lett. 44, 1118–1119 (1984).
    [CrossRef]

1986 (3)

1985 (1)

1984 (2)

K. J. Linden, R. E. Reeder, “Operation of cleave-coupled-cavity Pb-salt diode lasers in the 4–5 μ m spectral region,” Appl. Phys. Lett. 44, 377–379 (1984).
[CrossRef]

W. Lo, “Cleave-coupled-cavity lead-salt diode lasers,” Appl. Phys. Lett. 44, 1118–1119 (1984).
[CrossRef]

1983 (4)

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor a of semiconductor lasers,” Appl. Phys. Lett. 42, 328–330 (1983).
[CrossRef]

W. T. Tsang, N. A. Olsson, “Enhanced frequency modulation in cleave-coupled-cavity semiconductors lasers with reduced spurious intensity modulation,” Appl. Phys. Lett. 43, 527–529 (1983).
[CrossRef]

G. C. Bjorklund, M. D. Levinson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

W. Lenth, “Optical heterodyne spectroscopy with frequency-and amplitude-modulated semiconductor lasers,” Opt. Lett. 8, 575–577 (1983).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

1980 (3)

R. S. Eng, J. F. Butler, K. J. Linden, “Tunable diode laser spectroscopy: an invited review,” Opt. Eng. 19, 945–960 (1980).
[CrossRef]

D. E. Jennings, “Absolute line strengths in ν412CH4: a dual-beam diode laser spectrometer with sweep integration,” Appl. Opt. 19, 2695–2700 (1980).
[CrossRef] [PubMed]

J. M. Osterwalder, B. J. Rickett, “Frequency modulation of GaAlAs injection lasers at microwave frequency rates,” IEEE J. Quantum Electron. QE-16, 250–252 (1980).
[CrossRef]

Bjorklund, G. C.

G. C. Bjorklund, M. D. Levinson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Butler, J. F.

R. S. Eng, J. F. Butler, K. J. Linden, “Tunable diode laser spectroscopy: an invited review,” Opt. Eng. 19, 945–960 (1980).
[CrossRef]

Carlisle, C. B.

G. R. Janik, C. B. Carlisle, T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 3, 1070–1074 (1986).
[CrossRef]

Cassidy, D. T.

Cooper, D. E.

Eng, R. S.

R. S. Eng, J. F. Butler, K. J. Linden, “Tunable diode laser spectroscopy: an invited review,” Opt. Eng. 19, 945–960 (1980).
[CrossRef]

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettnger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953).

Gallagher, T. F.

Gehrtz, M.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Harder, C.

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor a of semiconductor lasers,” Appl. Phys. Lett. 42, 328–330 (1983).
[CrossRef]

Janik, G. R.

G. R. Janik, C. B. Carlisle, T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 3, 1070–1074 (1986).
[CrossRef]

Jennings, D. E.

Johnston, H. S.

Labrie, D.

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

Lenth, W.

Levinson, M. D.

G. C. Bjorklund, M. D. Levinson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Linden, K. J.

K. J. Linden, R. E. Reeder, “Operation of cleave-coupled-cavity Pb-salt diode lasers in the 4–5 μ m spectral region,” Appl. Phys. Lett. 44, 377–379 (1984).
[CrossRef]

R. S. Eng, J. F. Butler, K. J. Linden, “Tunable diode laser spectroscopy: an invited review,” Opt. Eng. 19, 945–960 (1980).
[CrossRef]

Lo, W.

W. Lo, “Cleave-coupled-cavity lead-salt diode lasers,” Appl. Phys. Lett. 44, 1118–1119 (1984).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettnger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953).

Oberhettnger, F.

A. Erdelyi, W. Magnus, F. Oberhettnger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953).

Olsson, N. A.

W. T. Tsang, N. A. Olsson, “Enhanced frequency modulation in cleave-coupled-cavity semiconductors lasers with reduced spurious intensity modulation,” Appl. Phys. Lett. 43, 527–529 (1983).
[CrossRef]

Ortiz, C.

G. C. Bjorklund, M. D. Levinson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Osterwalder, J. M.

J. M. Osterwalder, B. J. Rickett, “Frequency modulation of GaAlAs injection lasers at microwave frequency rates,” IEEE J. Quantum Electron. QE-16, 250–252 (1980).
[CrossRef]

Reeder, R. E.

K. J. Linden, R. E. Reeder, “Operation of cleave-coupled-cavity Pb-salt diode lasers in the 4–5 μ m spectral region,” Appl. Phys. Lett. 44, 377–379 (1984).
[CrossRef]

Reid, J.

D. T. Cassidy, J. Reid, “High-sensitivity detection of trace gases using sweep integration and tunable diode lasers,” Appl. Opt. 21, 2527–2530 (1982).
[CrossRef] [PubMed]

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

Rickett, B. J.

J. M. Osterwalder, B. J. Rickett, “Frequency modulation of GaAlAs injection lasers at microwave frequency rates,” IEEE J. Quantum Electron. QE-16, 250–252 (1980).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettnger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953).

Tsang, W. T.

W. T. Tsang, N. A. Olsson, “Enhanced frequency modulation in cleave-coupled-cavity semiconductors lasers with reduced spurious intensity modulation,” Appl. Phys. Lett. 43, 527–529 (1983).
[CrossRef]

Vahala, K.

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor a of semiconductor lasers,” Appl. Phys. Lett. 42, 328–330 (1983).
[CrossRef]

Watjen, J. P.

Yariv, A.

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor a of semiconductor lasers,” Appl. Phys. Lett. 42, 328–330 (1983).
[CrossRef]

Young, A. T.

Appl. Opt. (3)

Appl. Phys. B (2)

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

G. C. Bjorklund, M. D. Levinson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Appl. Phys. Lett. (4)

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor a of semiconductor lasers,” Appl. Phys. Lett. 42, 328–330 (1983).
[CrossRef]

W. T. Tsang, N. A. Olsson, “Enhanced frequency modulation in cleave-coupled-cavity semiconductors lasers with reduced spurious intensity modulation,” Appl. Phys. Lett. 43, 527–529 (1983).
[CrossRef]

K. J. Linden, R. E. Reeder, “Operation of cleave-coupled-cavity Pb-salt diode lasers in the 4–5 μ m spectral region,” Appl. Phys. Lett. 44, 377–379 (1984).
[CrossRef]

W. Lo, “Cleave-coupled-cavity lead-salt diode lasers,” Appl. Phys. Lett. 44, 1118–1119 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. M. Osterwalder, B. J. Rickett, “Frequency modulation of GaAlAs injection lasers at microwave frequency rates,” IEEE J. Quantum Electron. QE-16, 250–252 (1980).
[CrossRef]

J. Opt. Soc. Am. B. (1)

G. R. Janik, C. B. Carlisle, T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 3, 1070–1074 (1986).
[CrossRef]

Opt. Eng. (1)

R. S. Eng, J. F. Butler, K. J. Linden, “Tunable diode laser spectroscopy: an invited review,” Opt. Eng. 19, 945–960 (1980).
[CrossRef]

Opt. Lett. (3)

Other (3)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions, NBS Applied Mathematics Series #55 (National Bureau of Standards, Gaithersburg, Md., 1972).

A. Erdelyi, W. Magnus, F. Oberhettnger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953).

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Figures (14)

Fig. 1
Fig. 1

Experimental arrangement for two-tone optical heterodyne spectroscopy with tunable lead-salt diode lasers. LO, local oscillator; RF, radio frequency; IF, intermediate frequency.

Fig. 2
Fig. 2

Theoretical two-tone spectra of a Lorentzian line shape for various values of normalized modulation frequency ν ¯ mcalculated with β = 0.9, M = 0.064, α0 = 0.01, and ψ = π/2.

Fig. 3
Fig. 3

Theoretical two-tone spectra of a Gaussian line shape for various values of normalized modulation frequency ν ¯ m calculated with β = 0.9, M = 0.064, α0 = 0.01, and ψ = π/2.

Fig. 4
Fig. 4

Two-tone spectra of a Lorentzian line shape versus FM index (β) for νm = 10, α0 = 0.01, M = 0.01, ν ¯ m= α0 = 0.01, M = 0.1, and ψ = π/2.

Fig. 5
Fig. 5

Ratio of first five sideband peaks to central peak versus β for ν ¯ m = 10, M = 0, and α0 = 0.01, using a Gaussian line shape.

Fig. 6
Fig. 6

Dependence of two-tone spectra on AM index (M) for ν ¯ m = 10, β = 1, α0 = 0.01, and ψ = π/2.

Fig. 7
Fig. 7

Variation of minimum detectable absorption with FM index (β) for AM indices (M) of 0, 0.01, 0.02, and 0.03. (Parameters listed in Table 1 were used in the calculation.)

Fig. 8
Fig. 8

Minimum detectable absorption plotted as a function of AM index for β = 1.15 and various laser powers using the parameters given in Table 1.

Fig. 9
Fig. 9

Lead-salt diode-laser mount modified for high-frequency operation and installed in a cold head.

Fig. 10
Fig. 10

(a) Gas-absorption cell transmission as a single laser mode is swept through an isolated NH3 line in the 930-cm−1 region. (b) A repeat trace with −0.5 dBm of 2.23-GHz RF power supplied to the diode laser.

Fig. 11
Fig. 11

Experimental arrangement used for two-tone optical heterodyne measurements.

Fig. 12
Fig. 12

Experimental two-tone spectra obtained from a pair of Doppler-broadened NH3 lines in the 930-cm−1 region using modulation frequencies from 110 MHz to 1.1 GHz.

Fig. 13
Fig. 13

Two-tone spectrum of an isolated NH3 line near 930 cm−1 obtained under (a) Doppler-broadened and (c) pressure-broadened conditions, with corresponding theoretical fits (b) and (d) using Eq. (13).

Fig. 14
Fig. 14

Two-tone spectra from a Doppler-broadened NH3 line obtained using various RF drive levels to the laser diode.

Tables (1)

Tables Icon

Table 1 Parameters Used in Detection-Sensitivity Analysis

Equations (58)

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E 1 ( t ) = E 0 ( t ) [ 1 + M 1 sin ( 2 π ν 1 t + ψ 1 ) ] × [ 1 + M 2 sin ( 2 π ν 2 t + ψ 2 ) ] × exp [ i β 1 sin ( 2 π ν 1 t ) + i β 2 sin ( 2 π ν 2 t ) ] ,
exp [ i β sin 2 π ν i t ] = n = - J n ( β ) exp [ i 2 π n ν i t ] ,             i = 1 , 2.
[ 1 + M sin ( 2 π ν i t + ψ ) ] exp [ i β sin 2 π ν i t ] = n = - r n exp [ i 2 π n ν i t ] ,             i = 1 , 2 ,
r n ( β , M , ψ ) J n ( β ) + M 2 i [ exp ( i ψ ) J n - 1 ( β ) - exp ( - i ψ ) J n + 1 ( β ) ]
= k = - 1 1 a k J n - k ( β ) ,
E 1 ( t ) = E 0 ( t ) n , m r n r m exp [ i 2 π ( n ν 1 + m ν 2 ) t ] .
E ˜ 1 ( ν ) - d t E 1 ( t ) exp ( - i 2 π ν t ) = n , m r n r m - d t E 0 ( t ) × exp [ - i 2 π ( ν - n ν 1 - m ν 2 ) t ] = n , m r n r m E ˜ 0 ( ν - n ν 1 - m ν 1 )
E ˜ 2 ( ν ) = E ˜ 1 ( ν ) exp [ - α ( ν ) - i ϕ ( ν ) ] ,
E 2 ( t ) = n , m r n r m exp [ i 2 π ( n ν 1 + m ν 2 ) t ] - d ν E ˜ 0 ( ν ) exp [ i 2 π ν t - α ( ν - n ν 1 + m ν 2 ) - i ϕ ( ν + n ν 1 + m ν 2 ) ] .
E 2 ( t ) E 0 ( t ) n , m r n r m exp [ i 2 π ( n ν 1 + m ν 2 ) t - α ( ν 0 + n ν 1 + m ν 2 ) - i ϕ ( ν 0 + n ν 1 + m ν 2 ) ] .
E 2 ( t ) E 0 ( t ) n , m r n r m exp { i 2 π ( n ν 1 + m ν 2 ) t - α [ ν 0 + ( n + m ) ν m ] - i ϕ [ ν 0 + ( n + m ) ν m ] } .
I 2 ( t ) = c 8 π E 2 ( t ) 2 = c 8 π E 0 ( t ) 2 n , m n , m r n r m r * n r * m × exp { - 2 π [ ( n - n ) ν 1 + ( m - m ) ν 2 ] t } × exp { - α [ ν 0 + ( n + m ) ν m ] - α [ ν 0 + ( n + m ) ν m ] - i ϕ [ ν 0 + ( n + m ) ν m ] + i ϕ [ ν 0 + ( n + m ) ν m ] } .
I Ω ( t ) = I 2 ( t ) Ω = c 8 π E 0 ( t ) 2 2 cos 2 π Ω t n , m r n r m r * n - 1 r * m + 1 × exp { - 2 α [ ν 0 + ( n + m ) ν m ] } .
n r n r * n - 1 = k , l = - 1 1 a k a * l n J n - k ( β ) J n - 1 - l ( β ) = k , l = - 1 1 a k a * l J k - l - 1 ( 0 ) = k = - 1 1 a k a * k - 1 = M i exp ( i ψ ) ,
m r m r * m + 1 = - M i exp ( - i ψ )
I Ω ( t ) = c 8 π E 0 ( t ) 2 2 M 2 cos ( 2 π Ω t )             ( α = 0 ) .
I Ω ( t ) = c 8 π E 0 ( t ) 2 2 cos ( 2 π Ω t ) n , m J n J m J n - 1 J m + 1 × exp { - 2 α [ ν 0 + ( n + m ) ν m ] }             ( M = 0 ) .
r n = k = - 1 1 a k J n - k ( 0 ) = a n - 1 n 1 = 0 , n > 1.
n , m r n r m r * n - 1 r * m + 1 exp { - 2 α [ ν 0 + ( n + m ) ν m ] } = a - 1 2 × exp [ - 2 α ( ν 0 - ν m ) ] + ( a 1 a - 1 + a * 1 a * - 1 ) × exp [ - 2 α ( ν 0 ) ] + a 1 2 exp [ - 2 α ( ν 0 + ν m ) ]
I Ω ( t ) = c 8 π E 0 ( t ) 2 M 2 2 cos ( 2 π Ω t ) { 2 exp [ - 2 α ( ν 0 ) ] + exp [ - 2 α ( ν 0 - ν m ) ] + exp [ - 2 α ( ν 0 + ν m ) ] } ( β = 0 ) .
r 0 J 0 1 , r ± 1 J ± 1 ± M 2 i exp [ ± i ψ ] J 0 ½ ( β + M i ) exp ( ± i ψ ) .
I Ω ( t ) c 8 π E 0 ( t ) 2 ½ cos ( 2 π Ω t ) { 2 ( M 2 - β 2 ) exp [ - 2 α ( ν 0 ) ] + ( β 2 - 2 M β sin ψ + M 2 ) exp [ - 2 α ( ν 0 - ν m ) ] + ( β 2 + 2 M β sin ψ + M 2 ) exp [ - 2 α ( ν 0 + ν m ) ] } ( β , M 1 ) .
I Ω ( t ) c 8 π E 0 ( t ) 2 cos ( 2 π Ω t ) [ 2 M 2 + 2 α 0 ( β 2 - M 2 ) + α + ( β 2 + 2 M β sin ψ + M 2 ) - α - ( β 2 - 2 M β sin ψ + M 2 ) ] .
α G ( ν ¯ ) = α 0 exp [ - ν ¯ 2 ln 2 ]             ( Gaussian line shape ) ,
α L ( ν ¯ ) = α 0 1 + ν ¯ 2             ( Lorentzian line shape ) ,
I peak = c 8 π E 0 2 2 cos 2 π Ω t n , m r n r m r * n - 1 r * m + 1 × exp { - 2 α [ ( n + m ) ν ¯ m ] } ,
i ( t ) = e η h ν 0 A d I peak + i n ,
i s ( t ) = e η h ν 0 2 P 0 Q ( α ) cos ( 2 π Ω t ) ,
i b ( t ) = e η h ν 0 2 P 0 M 2 cos ( 2 π Ω t ) ,
P 0 A d c 8 π E 0 2
Q ( α ) n , m r n r m r * n - 1 r * m + 1 exp { - 2 α [ ( n + m ) ν ¯ m ] } - M 2 .
SNR = i s ( t ) 2 ¯ Var ( i b ) + Var ( i n )
SNR = CNR 1 + CNR / SBR ,
CNR = i s ( t ) 2 ¯ / Var ( i n )
SBR i s ( t ) 2 ¯ / Var ( i b ) .
i s ( t ) 2 ¯ = ( e η h ν 0 ) 2 2 P 0 2 Q ( α ) 2 ,
Var ( i b ) i b 2 ( t ) - i b ( t ) 2 = ( e η h ν ) 2 2 M 4 σ P 0 2 ,
i sn 2 = 2 e Δ f ( e η h ν 0 ) P | n | r n 2 2 ,
n r n 2 = k = - 1 1 a k 2 = 1 + M 2 / 2.
I p = 2 k T eff e R L ,
CNR = CNR 0 Q ( α ) 2 ,
CNR 0 ( e η h ν 0 ) 2 2 P 0 2 2 e Δ f [ e η h ν 0 P 0 ( 1 + M 2 2 ) 2 + I p ] .
SBR = SBR 0 Q ( α ) 2 ,
SBR 0 P 0 2 σ P 0 2 M 2 .
SNR = CNR 0 Q ( α ) 2 1 + CNR 0 / SBR 0 .
SNR CNR 0 Q ( α ) 2 .
SNR SBR 0 Q ( α ) 2 .
Q ( α ) - 2 n , m r n r m r * n - 1 r * m + 1 exp [ α ( n + m ) ν ¯ m ] - 2 α ( 0 ) n r n 2 r * n - 1 2 - 2 α ( 0 ) n J n 2 ( β ) J n - 1 2 ( β )
α min 1 2 n J n 2 J n - 1 2 [ 1 CNR 0 + M 4 σ P 0 2 P 0 2 ] 1 / 2 .
α min 2.1 [ 1 CNR 0 M 4 σ P 0 2 P 0 2 ] 1 / 2 .
I k ( β ) n = - J n 2 ( β ) J n - k 2 ( β ) .
n = - cos ( n ϕ ) J n 2 ( β ) = J 0 ( 2 β sin ϕ 2 ) .
n , m cos ( n ϕ ) cos ( m ϕ ) J n 2 ( β ) J m 2 ( β ) = J 0 2 ( 2 β sin ϕ 2 ) .
cos ( n ϕ ) cos ( m ϕ ) = ½ { cos [ ( n - m ) ϕ ] + cos ( n + m ) ϕ } ,
1 2 k = - cos ( k ϕ ) n = - ( J n 2 J n - k 2 + J n 2 J k - n 2 ) = k cos ( k ϕ ) n J n 2 J n - k 2 = J 0 2 ( 2 β sin ϕ 2 ) .
I k = 1 2 π - π π d ϕ J 0 2 ( 2 β sin ϕ / 2 ) cos ( k ϕ ) .
J 0 2 ( z ) = m = 0 ( 2 m ) ! ( m ! ) 4 ( - z 2 / 4 ) m ,
I k ( β ) = ( - 1 ) k m = k [ ( 2 m ) ! ] 2 ( m ! ) 4 ( m + k ) ! ( m - k ) ! ( - β 2 4 ) m .

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