Abstract

In this paper, we describe experimental and theoretical investigations of two variations of frequency modulation (FM) spectroscopy that use two electrooptic modulators. In the first variation, both modulators are frequency modulators (FM–FM), and, in the second, one is a frequency modulator and one is an amplitude modulator (FM–AM). The essential advantage of FM–FM and FM–AM spectroscopy is that sensitive lowbandwidth detectors, such as photomultiplier tubes, can be used to detect signals generated by the absorption of sidebands displaced from the carrier by frequencies far above the detector cutoff frequency. These two variations are complementary in the sense that, in situations where optical power is at a premium, the FM–FM scheme is most appropriate, and in situations where modulator drive power is at a premium, the FM–AM scheme is most appropriate. Using either of these variations, we have detected the absorption of 700-MHz sidebands with photomultiplier tubes whose cutoff frequencies lie below 100 MHz.

© 1985 Optical Society of America

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References

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  1. G. C. Bjorklund, “Frequency-Modulation Spectroscopy: A New Method for Measuring Weak Absorptions and Dispersions,” Opt. Lett. 5, 15 (1980).
    [CrossRef] [PubMed]
  2. D. E. Cooper, T. F. Gallagher, “Frequency-Modulation Spectroscopy with a Multimode Laser,” Opt. Lett. 9, 451 (1984).
    [CrossRef] [PubMed]
  3. N. H. Tran, R. Kachru, P. Pillet, H. B. van Linden van den Heuvell, T. F. Gallagher, J. P. Watjen, “Frequency-Modulation Spectroscopy with a Pulsed Dye Laser: Experimental Investigations of Sensitivity and Useful Features,” Appl. Opt. 23, 1353 (1984).
    [CrossRef]
  4. S. Y. Wang, D. M. Bloom, D. M. Collins, “20-GHz Bandwidth GaAs Photodiode,” Appl. Phys. Lett. 42, 190 (1983).
    [CrossRef]
  5. G. M. Carter, “Tunable High Efficiency Microwave Frequency-Shifting of Infrared Lasers,” Appl. Phys. Lett. 32, 810 (1978).
    [CrossRef]
  6. D. L. Spears, “Theory and Status of High-Performance Heterodyne Detectors,” Proc. Soc. Photo-Opt. Instrum. Eng. 300, 174 (1981).
  7. H. Lotem, “Extension of the Spectral Coverage Range of Frequency Modulation Spectroscopy by Double Frequency Modulation,” J. Appl. Phys. 54, 10 (1983).
    [CrossRef]
  8. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (NBS Applied Math. Series 55, Dec.1972).
  9. A. Yariv, Quantum Electronics (John Wiley & Sons, New York, 1975).

1984 (2)

1983 (2)

S. Y. Wang, D. M. Bloom, D. M. Collins, “20-GHz Bandwidth GaAs Photodiode,” Appl. Phys. Lett. 42, 190 (1983).
[CrossRef]

H. Lotem, “Extension of the Spectral Coverage Range of Frequency Modulation Spectroscopy by Double Frequency Modulation,” J. Appl. Phys. 54, 10 (1983).
[CrossRef]

1981 (1)

D. L. Spears, “Theory and Status of High-Performance Heterodyne Detectors,” Proc. Soc. Photo-Opt. Instrum. Eng. 300, 174 (1981).

1980 (1)

1978 (1)

G. M. Carter, “Tunable High Efficiency Microwave Frequency-Shifting of Infrared Lasers,” Appl. Phys. Lett. 32, 810 (1978).
[CrossRef]

1972 (1)

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (NBS Applied Math. Series 55, Dec.1972).

Bjorklund, G. C.

Bloom, D. M.

S. Y. Wang, D. M. Bloom, D. M. Collins, “20-GHz Bandwidth GaAs Photodiode,” Appl. Phys. Lett. 42, 190 (1983).
[CrossRef]

Carter, G. M.

G. M. Carter, “Tunable High Efficiency Microwave Frequency-Shifting of Infrared Lasers,” Appl. Phys. Lett. 32, 810 (1978).
[CrossRef]

Collins, D. M.

S. Y. Wang, D. M. Bloom, D. M. Collins, “20-GHz Bandwidth GaAs Photodiode,” Appl. Phys. Lett. 42, 190 (1983).
[CrossRef]

Cooper, D. E.

Gallagher, T. F.

Kachru, R.

Lotem, H.

H. Lotem, “Extension of the Spectral Coverage Range of Frequency Modulation Spectroscopy by Double Frequency Modulation,” J. Appl. Phys. 54, 10 (1983).
[CrossRef]

Pillet, P.

Spears, D. L.

D. L. Spears, “Theory and Status of High-Performance Heterodyne Detectors,” Proc. Soc. Photo-Opt. Instrum. Eng. 300, 174 (1981).

Tran, N. H.

van Linden van den Heuvell, H. B.

Wang, S. Y.

S. Y. Wang, D. M. Bloom, D. M. Collins, “20-GHz Bandwidth GaAs Photodiode,” Appl. Phys. Lett. 42, 190 (1983).
[CrossRef]

Watjen, J. P.

Yariv, A.

A. Yariv, Quantum Electronics (John Wiley & Sons, New York, 1975).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

S. Y. Wang, D. M. Bloom, D. M. Collins, “20-GHz Bandwidth GaAs Photodiode,” Appl. Phys. Lett. 42, 190 (1983).
[CrossRef]

G. M. Carter, “Tunable High Efficiency Microwave Frequency-Shifting of Infrared Lasers,” Appl. Phys. Lett. 32, 810 (1978).
[CrossRef]

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1)

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (NBS Applied Math. Series 55, Dec.1972).

J. Appl. Phys. (1)

H. Lotem, “Extension of the Spectral Coverage Range of Frequency Modulation Spectroscopy by Double Frequency Modulation,” J. Appl. Phys. 54, 10 (1983).
[CrossRef]

Opt. Lett. (2)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. L. Spears, “Theory and Status of High-Performance Heterodyne Detectors,” Proc. Soc. Photo-Opt. Instrum. Eng. 300, 174 (1981).

Other (1)

A. Yariv, Quantum Electronics (John Wiley & Sons, New York, 1975).

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Figures (8)

Fig. 1
Fig. 1

Optical power spectrum of (a) pure FM light at frequency 2ω + σ and modulation index M1, and (b) the light produced by directing this FM light through a second frequency modulator operating at frequency ω and modulation index M2.

Fig. 2
Fig. 2

Experimental configuration used for FM–AM spectroscopy and for FM–FM spectroscopy when no dc bias voltage is applied to the second modulator crystal and the polarizers are removed.

Fig. 3
Fig. 3

Optical power spectrum of pure FM light with a 1460-MHz modulation frequency.

Fig. 4
Fig. 4

FM–AM signal at 60 MHz resulting from absorption of sidebands at 700 MHz obtained with 65 nW of optical power incident on an RCA 931 PMT biased at 900 V.

Fig. 5
Fig. 5

FM–AM signal at 60 MHz resulting from absorption of sidebands at 500 MHz obtained with 0.5 mW of optical power incident on a HP 4220 photodiode.

Fig. 6
Fig. 6

Optical power spectra of (a) unmodulated laser light and FM light from the second modulator (ω = 700 MHz) with rf drive power adjusted to give modulation indices of (b) 1.7, (c) 2.2, and (d) 3.2, respectively.

Fig. 7
Fig. 7

FM–FM signal at 10 MHz resulting from absorption of sidebands at 700 MHz obtained with 16 nW of optical power incident on a EMI 9558 PMT biased at 1000 V and gated on during the rf pulse.

Fig. 8
Fig. 8

FM–FM signal at 10 MHz resulting from absorption of sidebands at 700 MHz obtained with 0.5 mW of optical power incident on an EG&G FND 100 photodiode.

Equations (10)

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E FM FM ( t ) = ½ E 0 p , q = J p ( M 1 ) J q ( M 2 ) exp i ( ω L + p Ω 1 + q Ω 2 ) t + c . c . ,
T ( t ) = E 0 p , q = J p ( M 1 ) J q ( M 2 ) T ( ω L + p Ω 1 + q Ω 2 ) × exp i ( ω L + p Ω 1 + q Ω 2 ) t .
P s ( t ) = c 8 π T ( t ) T * ( t ) = c E 0 2 8 π p , q p , q J p ( M 1 ) J q ( M 2 ) J p ( M 1 ) J q ( M 2 ) × T p q T p q * exp i [ ( p p ) Ω 1 + ( q q ) Ω 2 ] t .
P s ( k σ t ) = c E 0 2 8 π p , q J p ( M 1 ) J q ( M 2 ) × [ J p k ( M 1 ) J q + l ( M 2 ) T p q T p k , q + l * exp i ( k Ω 1 l Ω 2 ) t + J p + k ( M 1 ) J q l ( M 2 ) T p q T p + k , q l * × exp i ( k Ω 1 l Ω 2 ) t ] .
P s ( σ t ) = c E 0 2 8 π p , q J p ( M 1 ) J q ( M 2 ) × [ J p 1 ( M 1 ) J q + 2 ( M 2 ) T p q T p 1 , q + 2 * exp ( i σ t ) + J p + 1 ( M 1 ) J q 2 ( M 2 ) T p q T p + 1 , q 2 * exp ( i σ t ) ]
P s ( σ t ) = c E 0 2 8 π J 0 ( M 1 ) J 1 ( M 1 ) ( J 1 ( M 2 ) J 1 ( M 2 ) [ ( δ ω + σ + 2 δ ω + δ ω σ ) ( δ ω σ + 2 δ ω + δ ω + σ ) ] cos σ t + { J 1 ( M 2 ) J 1 ( M 2 ) [ ( Φ ω + Φ ω ) ( Φ ω + σ + Φ ω σ ) ] + J 1 ( M 2 ) J 3 ( M 2 ) [ ( Φ ω σ + Φ ω + σ ) ( Φ ω + Φ ω ) ] } sin σ t ) .
δ ω + σ = δ ω = δ ω σ δ + , Φ ω + σ = Φ ω = Φ ω σ = Φ + , δ ω σ = δ ω = δ ω + σ δ ̅ , Φ ω σ = Φ ω = Φ ω + σ = Φ ̅ ,
P s ( σ t ) = c E 0 2 8 π 2 M 1 J 1 2 ( M 2 ) Δ δ cos σ t ,
E FM AM ( t ) = ½ E 0 [ p , q = J p ( M 1 ) J q ( M 2 ) exp i ( ω L + p Ω 1 + q Ω 2 ) t p = J p ( M 1 ) exp i ( ω L + p Ω 1 ) t ] + c . c .
T ( t ) = E 0 p = J p ( M 1 ) T ( ω L + p Ω 1 ) exp i ( ω L + p Ω 1 ) t .

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