Abstract

A new type of laser-induced thermal lens effect measurement is demonstrated in which the thermooptic impulse response of a weakly absorbing sample is recovered in the time delay domain through excitation with a fast linear frequency sweep. The autocorrelation function of the excitation sweep approximates to a Dirac delta function to a time resolution limited by the modulation bandwidth of the sweep, thereby permitting the fast recovery of high quality frequency and impulse response information. Impulse response data recovered in the time delay domain showed good agreement with the results predicted from Fresnel diffraction theory, indicating an equivalence to the response recovered in a typical pulsed measurement. The FM time delay technique, however provides enhanced measurement dynamic range and an overall reduced peak excitation power when compared with the pulsed measurement.

© 1990 Optical Society of America

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References

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  1. J. P. Gordon, R. C. C. Leite, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
    [Crossref]
  2. N. J. Dovichi, J. M. Harris, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
    [Crossref]
  3. D. S. Burgi, N. J. Dovichi, “Submicron Resolution Images of Absorbance and Thermal Diffusivity with the Photothermal Microscope,” Appl. Optics 26, 4665–4669 (1987).
    [Crossref]
  4. A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra using Thermal Blooming: Parts I and II,” Chem. Phys. 20, 253–264 (1977).
    [Crossref]
  5. K. Fuke, M. Ueda, M. Itoh, “Thermal Lensing Study of Singlet Oxygen Reactions,” J. Am. Chem. Soc. 105, 1091–1096 (1983).
    [Crossref]
  6. K. Daree, “Photochemical Blooming of Laser Beams,” Opt. Commun. 4, 238–242 (1971).
    [Crossref]
  7. L. Chen, T. C. Wang, T. L. Ricca, A. G. Marshall, “Phase-Modulated Stored Waveform Inverse Fourier Transform Excitation for Trapped Ion Mass Spectrometry,” Anal. Chem. 59, 449–454 (1987).
    [Crossref] [PubMed]
  8. A. Mandelis, L. M-L. Borm, J. Tiessinga, “Frequency Modulated (FM) Time Delay Photoacoustic and Photothermal Wave Spectroscopies: Parts I–III,” Rev. Sci. Instrum. 57, 617–635 (1986).
    [Crossref]
  9. J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection (Parts I–III),” Rev. Sci. Instrum. 58, 2018–2043 (1987).
    [Crossref]
  10. A. Mandelis, J. F. Power, “Frequency-Modulated Impulse Response Photothermal Detection through Optical Reflectance (Part 1–2),” Appl. Opt. 27, 3397–3417 (1988).
    [Crossref] [PubMed]
  11. J. F. Power, “Pulsed Mode Thermal Lens Effect Detection in the Near Field via Thermally Induced Probe Beam Spatial Phase Modulation,” (in press).
  12. Copyright: Adaptable Laboratory Software (McMillan Software Company, NY).
  13. J. S. Bendat, A. G. Piersol, in Engineering Applications of Correlation and Spectral Analysis (Wiley, New York, 1980).
  14. S. Stearns, Digital Signal Analysis (Hayden, Rochelle Park, N.J., 1975), Chap. 4.
  15. Y. Sugitani, A. Uejima, K. Kato, “Correlation Photoacoustic Spectroscopy,” J. Photoacoust. 1, 217–236 (1982).
  16. G. F. Kirkbright, R. M. Miller, A. Rzadkiewicz, “Applications of Cross-Correlation Signal Recovery in Photoacoustics,” J. Phys. (Paris) 44, C6–249–252 (1983).
    [Crossref]
  17. A. Mandelis, “Time-Delay Domain and Pseudorandom Noise Photoacoustic and Photothermal Wave Processes: A Review of the State of the Art,” IEEE Trans. UFFC-33, 590–617 (1986).
  18. H. Biering, O. Z. Pedersen, “System Analysis and Time Delay Spectrometry,” Bruel and Kjaer Tech. Rev. 2, 5–50 (1983).
  19. A. Yariv, Introduction to Optical Electronics (Holt, Rhinehart & Winston, New York, 1971), Chap. 11.

1988 (1)

1987 (3)

J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection (Parts I–III),” Rev. Sci. Instrum. 58, 2018–2043 (1987).
[Crossref]

D. S. Burgi, N. J. Dovichi, “Submicron Resolution Images of Absorbance and Thermal Diffusivity with the Photothermal Microscope,” Appl. Optics 26, 4665–4669 (1987).
[Crossref]

L. Chen, T. C. Wang, T. L. Ricca, A. G. Marshall, “Phase-Modulated Stored Waveform Inverse Fourier Transform Excitation for Trapped Ion Mass Spectrometry,” Anal. Chem. 59, 449–454 (1987).
[Crossref] [PubMed]

1986 (2)

A. Mandelis, L. M-L. Borm, J. Tiessinga, “Frequency Modulated (FM) Time Delay Photoacoustic and Photothermal Wave Spectroscopies: Parts I–III,” Rev. Sci. Instrum. 57, 617–635 (1986).
[Crossref]

A. Mandelis, “Time-Delay Domain and Pseudorandom Noise Photoacoustic and Photothermal Wave Processes: A Review of the State of the Art,” IEEE Trans. UFFC-33, 590–617 (1986).

1983 (3)

H. Biering, O. Z. Pedersen, “System Analysis and Time Delay Spectrometry,” Bruel and Kjaer Tech. Rev. 2, 5–50 (1983).

G. F. Kirkbright, R. M. Miller, A. Rzadkiewicz, “Applications of Cross-Correlation Signal Recovery in Photoacoustics,” J. Phys. (Paris) 44, C6–249–252 (1983).
[Crossref]

K. Fuke, M. Ueda, M. Itoh, “Thermal Lensing Study of Singlet Oxygen Reactions,” J. Am. Chem. Soc. 105, 1091–1096 (1983).
[Crossref]

1982 (1)

Y. Sugitani, A. Uejima, K. Kato, “Correlation Photoacoustic Spectroscopy,” J. Photoacoust. 1, 217–236 (1982).

1980 (1)

N. J. Dovichi, J. M. Harris, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
[Crossref]

1977 (1)

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra using Thermal Blooming: Parts I and II,” Chem. Phys. 20, 253–264 (1977).
[Crossref]

1971 (1)

K. Daree, “Photochemical Blooming of Laser Beams,” Opt. Commun. 4, 238–242 (1971).
[Crossref]

1965 (1)

J. P. Gordon, R. C. C. Leite, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Bendat, J. S.

J. S. Bendat, A. G. Piersol, in Engineering Applications of Correlation and Spectral Analysis (Wiley, New York, 1980).

Biering, H.

H. Biering, O. Z. Pedersen, “System Analysis and Time Delay Spectrometry,” Bruel and Kjaer Tech. Rev. 2, 5–50 (1983).

Borm, L. M-L.

A. Mandelis, L. M-L. Borm, J. Tiessinga, “Frequency Modulated (FM) Time Delay Photoacoustic and Photothermal Wave Spectroscopies: Parts I–III,” Rev. Sci. Instrum. 57, 617–635 (1986).
[Crossref]

Burgi, D. S.

D. S. Burgi, N. J. Dovichi, “Submicron Resolution Images of Absorbance and Thermal Diffusivity with the Photothermal Microscope,” Appl. Optics 26, 4665–4669 (1987).
[Crossref]

Chen, L.

L. Chen, T. C. Wang, T. L. Ricca, A. G. Marshall, “Phase-Modulated Stored Waveform Inverse Fourier Transform Excitation for Trapped Ion Mass Spectrometry,” Anal. Chem. 59, 449–454 (1987).
[Crossref] [PubMed]

Daree, K.

K. Daree, “Photochemical Blooming of Laser Beams,” Opt. Commun. 4, 238–242 (1971).
[Crossref]

Dovichi, N. J.

D. S. Burgi, N. J. Dovichi, “Submicron Resolution Images of Absorbance and Thermal Diffusivity with the Photothermal Microscope,” Appl. Optics 26, 4665–4669 (1987).
[Crossref]

N. J. Dovichi, J. M. Harris, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
[Crossref]

Fuke, K.

K. Fuke, M. Ueda, M. Itoh, “Thermal Lensing Study of Singlet Oxygen Reactions,” J. Am. Chem. Soc. 105, 1091–1096 (1983).
[Crossref]

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Harris, J. M.

N. J. Dovichi, J. M. Harris, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
[Crossref]

Itoh, M.

K. Fuke, M. Ueda, M. Itoh, “Thermal Lensing Study of Singlet Oxygen Reactions,” J. Am. Chem. Soc. 105, 1091–1096 (1983).
[Crossref]

Kato, K.

Y. Sugitani, A. Uejima, K. Kato, “Correlation Photoacoustic Spectroscopy,” J. Photoacoust. 1, 217–236 (1982).

Kirkbright, G. F.

G. F. Kirkbright, R. M. Miller, A. Rzadkiewicz, “Applications of Cross-Correlation Signal Recovery in Photoacoustics,” J. Phys. (Paris) 44, C6–249–252 (1983).
[Crossref]

Kliger, D. S.

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra using Thermal Blooming: Parts I and II,” Chem. Phys. 20, 253–264 (1977).
[Crossref]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Mandelis, A.

A. Mandelis, J. F. Power, “Frequency-Modulated Impulse Response Photothermal Detection through Optical Reflectance (Part 1–2),” Appl. Opt. 27, 3397–3417 (1988).
[Crossref] [PubMed]

J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection (Parts I–III),” Rev. Sci. Instrum. 58, 2018–2043 (1987).
[Crossref]

A. Mandelis, “Time-Delay Domain and Pseudorandom Noise Photoacoustic and Photothermal Wave Processes: A Review of the State of the Art,” IEEE Trans. UFFC-33, 590–617 (1986).

A. Mandelis, L. M-L. Borm, J. Tiessinga, “Frequency Modulated (FM) Time Delay Photoacoustic and Photothermal Wave Spectroscopies: Parts I–III,” Rev. Sci. Instrum. 57, 617–635 (1986).
[Crossref]

Marshall, A. G.

L. Chen, T. C. Wang, T. L. Ricca, A. G. Marshall, “Phase-Modulated Stored Waveform Inverse Fourier Transform Excitation for Trapped Ion Mass Spectrometry,” Anal. Chem. 59, 449–454 (1987).
[Crossref] [PubMed]

Miller, R. M.

G. F. Kirkbright, R. M. Miller, A. Rzadkiewicz, “Applications of Cross-Correlation Signal Recovery in Photoacoustics,” J. Phys. (Paris) 44, C6–249–252 (1983).
[Crossref]

Pedersen, O. Z.

H. Biering, O. Z. Pedersen, “System Analysis and Time Delay Spectrometry,” Bruel and Kjaer Tech. Rev. 2, 5–50 (1983).

Piersol, A. G.

J. S. Bendat, A. G. Piersol, in Engineering Applications of Correlation and Spectral Analysis (Wiley, New York, 1980).

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Power, J. F.

A. Mandelis, J. F. Power, “Frequency-Modulated Impulse Response Photothermal Detection through Optical Reflectance (Part 1–2),” Appl. Opt. 27, 3397–3417 (1988).
[Crossref] [PubMed]

J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection (Parts I–III),” Rev. Sci. Instrum. 58, 2018–2043 (1987).
[Crossref]

J. F. Power, “Pulsed Mode Thermal Lens Effect Detection in the Near Field via Thermally Induced Probe Beam Spatial Phase Modulation,” (in press).

Ricca, T. L.

L. Chen, T. C. Wang, T. L. Ricca, A. G. Marshall, “Phase-Modulated Stored Waveform Inverse Fourier Transform Excitation for Trapped Ion Mass Spectrometry,” Anal. Chem. 59, 449–454 (1987).
[Crossref] [PubMed]

Rzadkiewicz, A.

G. F. Kirkbright, R. M. Miller, A. Rzadkiewicz, “Applications of Cross-Correlation Signal Recovery in Photoacoustics,” J. Phys. (Paris) 44, C6–249–252 (1983).
[Crossref]

Stearns, S.

S. Stearns, Digital Signal Analysis (Hayden, Rochelle Park, N.J., 1975), Chap. 4.

Sugitani, Y.

Y. Sugitani, A. Uejima, K. Kato, “Correlation Photoacoustic Spectroscopy,” J. Photoacoust. 1, 217–236 (1982).

Tiessinga, J.

A. Mandelis, L. M-L. Borm, J. Tiessinga, “Frequency Modulated (FM) Time Delay Photoacoustic and Photothermal Wave Spectroscopies: Parts I–III,” Rev. Sci. Instrum. 57, 617–635 (1986).
[Crossref]

Twarowski, A. J.

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra using Thermal Blooming: Parts I and II,” Chem. Phys. 20, 253–264 (1977).
[Crossref]

Ueda, M.

K. Fuke, M. Ueda, M. Itoh, “Thermal Lensing Study of Singlet Oxygen Reactions,” J. Am. Chem. Soc. 105, 1091–1096 (1983).
[Crossref]

Uejima, A.

Y. Sugitani, A. Uejima, K. Kato, “Correlation Photoacoustic Spectroscopy,” J. Photoacoust. 1, 217–236 (1982).

Wang, T. C.

L. Chen, T. C. Wang, T. L. Ricca, A. G. Marshall, “Phase-Modulated Stored Waveform Inverse Fourier Transform Excitation for Trapped Ion Mass Spectrometry,” Anal. Chem. 59, 449–454 (1987).
[Crossref] [PubMed]

Whinnery, J. R.

J. P. Gordon, R. C. C. Leite, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rhinehart & Winston, New York, 1971), Chap. 11.

Anal. Chem. (2)

N. J. Dovichi, J. M. Harris, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A–706A (1980).
[Crossref]

L. Chen, T. C. Wang, T. L. Ricca, A. G. Marshall, “Phase-Modulated Stored Waveform Inverse Fourier Transform Excitation for Trapped Ion Mass Spectrometry,” Anal. Chem. 59, 449–454 (1987).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Optics (1)

D. S. Burgi, N. J. Dovichi, “Submicron Resolution Images of Absorbance and Thermal Diffusivity with the Photothermal Microscope,” Appl. Optics 26, 4665–4669 (1987).
[Crossref]

Bruel and Kjaer Tech. Rev. (1)

H. Biering, O. Z. Pedersen, “System Analysis and Time Delay Spectrometry,” Bruel and Kjaer Tech. Rev. 2, 5–50 (1983).

Chem. Phys. (1)

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra using Thermal Blooming: Parts I and II,” Chem. Phys. 20, 253–264 (1977).
[Crossref]

IEEE Trans. (1)

A. Mandelis, “Time-Delay Domain and Pseudorandom Noise Photoacoustic and Photothermal Wave Processes: A Review of the State of the Art,” IEEE Trans. UFFC-33, 590–617 (1986).

J. Am. Chem. Soc. (1)

K. Fuke, M. Ueda, M. Itoh, “Thermal Lensing Study of Singlet Oxygen Reactions,” J. Am. Chem. Soc. 105, 1091–1096 (1983).
[Crossref]

J. Appl. Phys. (1)

J. P. Gordon, R. C. C. Leite, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

J. Photoacoust. (1)

Y. Sugitani, A. Uejima, K. Kato, “Correlation Photoacoustic Spectroscopy,” J. Photoacoust. 1, 217–236 (1982).

J. Phys. (Paris) (1)

G. F. Kirkbright, R. M. Miller, A. Rzadkiewicz, “Applications of Cross-Correlation Signal Recovery in Photoacoustics,” J. Phys. (Paris) 44, C6–249–252 (1983).
[Crossref]

Opt. Commun. (1)

K. Daree, “Photochemical Blooming of Laser Beams,” Opt. Commun. 4, 238–242 (1971).
[Crossref]

Rev. Sci. Instrum. (2)

A. Mandelis, L. M-L. Borm, J. Tiessinga, “Frequency Modulated (FM) Time Delay Photoacoustic and Photothermal Wave Spectroscopies: Parts I–III,” Rev. Sci. Instrum. 57, 617–635 (1986).
[Crossref]

J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection (Parts I–III),” Rev. Sci. Instrum. 58, 2018–2043 (1987).
[Crossref]

Other (5)

A. Yariv, Introduction to Optical Electronics (Holt, Rhinehart & Winston, New York, 1971), Chap. 11.

J. F. Power, “Pulsed Mode Thermal Lens Effect Detection in the Near Field via Thermally Induced Probe Beam Spatial Phase Modulation,” (in press).

Copyright: Adaptable Laboratory Software (McMillan Software Company, NY).

J. S. Bendat, A. G. Piersol, in Engineering Applications of Correlation and Spectral Analysis (Wiley, New York, 1980).

S. Stearns, Digital Signal Analysis (Hayden, Rochelle Park, N.J., 1975), Chap. 4.

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Figures (14)

Fig. 1
Fig. 1

Schematic representation of the optical model solved in Ref. 11 for dual beam aberrant thermal lens effect.

Fig. 2
Fig. 2

Block diagram of the experimental FM-time delay thermal lens effect spectrometer used in this work: (a) optical layout; (b) signal processing system. In (a) AOM is the acoustooptic modulator; KE, knife edge; DS, dichroic beam splitters; BPS, narrow linewidth filter; PH, pinhole; and PD, phototransistor/photodetector.

Fig. 3
Fig. 3

Examples of typical (a) excitation (x(t)) and (b) response (y(t)) signals recorded in an FM time delay thermal lens effect measurement. Sweep bandwidth was 0–1525 Hz at a sweep rate of 4.9 KHz/sec. Each recording is the result of 100 averages. Note that each time record was multiplied with a quarter-sinusoid data window with Tw = 0.0804 s.

Fig. 4
Fig. 4

Quarter sinusoid data windows (see text for details) recorded with different values of the parameter Tw: curve (i) Tw = 0.0804 s; curve (ii) Tw = 0.201 s; curve (iii) Tw = 0.402 s.

Fig. 5
Fig. 5

Autospectral density functions for wideband measurements of thermal defocusing effect; (a) Gxx(f) and (b) Gyy(f) (corresponding time records are shown in Fig. 1).

Fig. 6
Fig. 6

Autocorrelation functions for (a) input Rxx(τ) and (b) output Ryy(τ) demonstrating impulse characteristics available from experimental wideband measurement (corresponding autospectral data are given in Fig. 5).

Fig. 7
Fig. 7

Comparison between (a) frequency response, (b) cross spectrum obtained for wideband thermal defocusing effect (data of Fig. 3), (c) phase response data corresponding to (b), and (d) frequency response magnitude from Gyy(f)1/2. Note that the phase response is obtained from the cross-spectral measurement only (Eq. 4).

Fig. 8
Fig. 8

Impulse response (a) and (b) cross-correlation function observed for thermal defocusing effect. Corresponding frequency domain functions are shown in Fig. 7.

Fig. 9
Fig. 9

Frequency response data (a) magnitude and (b) phase observed for thermal defocusing effect by FM time delay measurement (—) and narrow band harmonic measurement (*) using lock-in amplification. Sweep conditions were the same as in Fig. 3 for FM time delay results with N = 100 averages.

Fig. 10
Fig. 10

Frequency response/transfer function measurement of preamplifier input prefilter recorded by FM time delay technique (excitation conditions were same as in Fig. 3), (a) magnitude and (b) phase.

Fig. 11
Fig. 11

Corrected frequency and impulse response data for thermal defocussing effect: (a) frequency response data deconvoluted from input prefilter response; (b) phase response (c) impulse response.

Fig. 12
Fig. 12

Theoretical time dependence of near field intensity profile measured at various delays past excitation. Time delays are as follows from (i) to (vii): 5, 10, 15, 20, 30, 40, 64 μm, past excitation; w1 = 1.020 × 10−3 m; wo = 1 × 10−4 m; z1 = 1.0 m; z2 = 1.25m; b = 0.1 cm−1. Other values were: k = 1.5w/m-degK; dn/dt = −1 × 10−5/k; α = 1 × 10−6 m2/5.

Fig. 13
Fig. 13

(a) Theoretical impulse response functions I(r, z ¯,t) predicted from Eq. (1) for experimental conditions. (see text for details) and (b) experimental impulse response data, h(t), observed for different radial detector offsets: curve (i), r = 0.0 m; curve (ii), r = 2 × 10−4 m; curve (iii), r = 4 × 10−4 m; curve (iv), r = 7 × 10−4 m.

Fig. 14
Fig. 14

Comparison of normalized experimental and theoretical impulse response profiles for the thermal defocusing effect. (All geometric and thermooptical parameters are described in the text.); (a) beam center measurements, r = 0 and (b) off center measurements, r = 7 × 10−4 m.

Equations (33)

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U 2 ( r , z , t ) = 2 π A 1 n = 0 Λ n n ! · ( 1 j k { 1 q pn + 1 z ¯ } ) exp [ - j k r 2 2 ( q pn + z ¯ ) ] .
1 q pn = 1 q 1 + 2 n j k ( ω o 2 + 4 α t )
Λ = - j k d n d T α A o π k ( ω o 2 + 4 α t ) .
S x x ( f ) = lim T 1 T X * ( f ) X ( f ) ;
S y y ( f ) = lim T 1 T Y * ( f ) Y ( f ) ;
S x y ( f ) = lim T 1 T X * ( f ) Y ( f ) .
R x x ( τ ) = - exp ( 2 π j f τ ) S x x ( f ) d f ;
R y y ( τ ) = - exp ( 2 π j f τ ) S y y ( f ) d f ;
R x y ( τ ) = - exp ( 2 π j f τ ) S x y ( f ) d f .
H ( f ) = S x y ( f ) / S x x ( f ) = H ( f ) exp [ j ϕ ( f ) ] ;
H ( f ) = { S y y ( f ) S x x ( f ) } 1 / 2 .
R x y ( τ ) = h ( τ ) * R x x ( τ ) .
R x x ( τ ) δ ( t ) ,
G x y ( f ) H ( f ) ,
{ G y y ( f ) } 1 / 2 H ( f ) ,
R x y ( τ ) h ( τ ) .
x ( t ) = { A o cos ( π S t 2 + 2 π f a t + θ a ) , 0 t T , 0 t > T
x ( t ) = x + ( t ) + x - ( t ) = ½ { exp [ j ( π S t 2 + θ a + ω a ) ] + exp [ - j ( π S t 2 + θ a + ω a ) ] } ;
R x x ( τ ) = R x + x + ( τ ) + R x - x - ( τ ) + R x - x + τ R x + x - ( τ ) .
R x x ( τ ) R x + x + ( τ ) + R x - x ( τ )
R x x ( τ ) = sin ( 2 π f τ ) ( 4 π Δ f τ ) .
x ( t ) = { cos ( π S t 2 + θ a + 2 π f a t ) 0 τ T , 0 t > T
w ( t ) = p 1 ( t ) sin [ 2 π ( t 4 T w ) ] + p 2 ( t ) + p 3 ( t ) sin [ - π 2 T w ( t - ( T - T w ) ) ] ,
p 1 ( t ) = { 1 0 t T w / 4 0 t > T w / 4 p 2 ( t ) = { 1 T w / 4 t T - T w / 4 0 t > T w / 4 or t > T - T w / 4 p 3 ( t ) = { 1 t - T w / 4 t T 0 t < T - T w / 4
1 q 1 = 1 R 1 - j λ π ω 1 2 .
A o = 4.606 A . E p .
L ( a 1 f 1 + a 2 f 2 ) = a 1 L ( f 1 ) + a 2 L ( f 2 ) ,
L [ f ( t ) ] = 0 t f ( τ ) h ( t - τ ) d τ L [ f 1 ( t ) + f 2 ( t ) ] = 0 t f 1 ( τ ) h ( t - τ ) d τ + 0 t f 2 ( τ ) h ( t - τ ) d τ = L [ f 1 ( t ) ] + L [ f 2 ( t ) ] .
U 2 ( r , z , t ) 2 π A 1 { exp ( - j ϕ 1 ) exp [ - j ( k / 2 ) ] ( ξ 1 - j χ 1 ) r 2 R M 1 - ( θ M 1 p ) exp ( - j ϕ 1 p ) exp [ - j ( k / 2 ) ] ( ξ 1 p - j χ 1 p ) r 2 }
1 ω 1 p 2 = 1 ω 1 2 + 1 ( ω o 2 + 4 α t ) θ = d n d T × α A o π K ( ω o 2 + 4 α t ) M 1 ( 1 p ) 2 = ( 1 R 1 + 1 z ¯ ) 2 + ( 2 k ω 1 ( 1 p ) 2 ) 2 ϕ 1 ( 1 p ) = tan - 1 { 2 k ω 1 ( 1 p ) 2 ( 1 R 1 + 1 z ¯ ) } ξ 1 ( 1 p ) = [ ( k ω 1 ( 1 p ) 2 ) 2 + ( 2 R 1 ) 2 ] [ ( k ω 1 ( 1 p ) 2 ) 2 R 1 + z ¯ ] [ ( k ω 1 ( 1 p ) 2 ) 2 + ( 2 R 1 ) 2 ] [ ( k ω 1 ( 1 p ) 2 ) 2 R 1 + [ ( k ω 1 ( 1 p ) 2 ) 2 + ( 2 R 1 ) 2 z ¯ ] 2 + [ 2 R 1 2 ( k ω 1 ( 1 p ) 2 ) ] 2 χ 1 ( 1 p ) = [ ( k ω 1 ( 1 p ) 2 ) 2 + ( 2 R 1 ) 2 ] 2 R 1 2 ( k ω 1 ( 1 p ) 2 ) [ ( k ω 1 ( 1 p ) 2 ) 2 R 1 + [ ( k ω 1 ( 1 p ) 2 ) 2 + ( 2 R 1 ) 2 ] z ¯ ] 2 + [ 2 R 1 2 ( k ω 1 ( 1 p ) 2 ) ] 2
I 2 ( r , z , t ) = 2 π A 1 2 [ exp ( - k χ 1 r 2 ) k 2 M 1 2 + ( θ M 1 p ) 2 exp ( - k χ 1 p r 2 ) { cos 2 ( Ψ 1 p ) - sin 2 ( Ψ 1 p ) } - 2 θ k M 1 M 1 p exp [ - ( k / 2 ) ( χ 1 + χ 1 p ) r 2 ] sin ( Ψ 1 - Ψ 1 p ) ] ,
Ψ 1 = ϕ 1 + k 2 ξ 1 r 2             Ψ 1 p = ϕ 1 p + k 2 ξ 1 p r 2 .
Δ I 2 ( r , z , t ) - 2 π A 1 2 { 2 θ k M 1 M 1 p exp [ - ( k / 2 ) ( χ 1 + χ 1 p ) r 2 sin ( Ψ 1 p - Ψ 1 ) }

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