Abstract

Several easily implemented devices for doing ultrasensitive optical measurements with noisy lasers are presented. They are all-electronic noise cancellation circuits that largely eliminate excess laser intensity noise as a source of measurement error and are widely applicable. Shot-noise-limited optical measurements can now easily be made at baseband with noisy lasers. These circuits are especially useful in situations where strong intermodulation effects exist, such as current-tuned diode laser spectroscopy. These inexpensive devices (parts cost ≈$10) can be optimized for particular applications such as wideband or differential measurements. Although they cannot eliminate phase noise effects, they can reduce amplitude noise by 55–70 dB or more, even in unattended operation, and usually achieve the shot-noise limit. With 1-Hz signal-to-noise ratios of 150–160 dB, they allow performance equal or superior to a complex heterodyne system in many cases, while using much simpler dual-beam or homodyne approaches. Although these devices are related to earlier differential and ratiometric techniques, their noise cancellation performance is much better. They work well at modulation frequencies from dc to several megahertz and should be extensible to ≈100 MHz. The circuits work by subtracting photocurrents directly, with feedback applied outside the signal path to continuously adjust the subtraction for perfect balance; thus the excess noise and spurious modulation ideally cancel at all frequencies, leaving only the shot noise. The noise cancellation bandwidth is independent of the feedback bandwidth; it depends only on the speeds of the photodiodes and of the bipolar junction transistors used. Two noise-canceled outputs are available; one is a high-pass filtered voltage proportional to the signal photocurrent and the other is a low-pass filtered voltage related to the log ratio of the signal and comparison photocurrents. For reasonable current densities, the noise floors of the outputs depend only on the shot noise of the signal beam. Four variations on the basic circuit are presented: low noise floor, high cancellation, differential high power, and ratio-only. Emphasis is placed on the detailed operation and design considerations, especially performance extension by compensation of the nonideal character of system components. Experience has shown that some applications advice is required by most users, so that is provided as well.

© 1997 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. S. Okamura, M. Maruyama, “Improvement on the sensitivity of electro-optical system for electric field strength measurements,” Trans. Inst. Electron. Commun. Eng. Jpn. E 65, 702 (1982).
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1995 (1)

1993 (1)

1992 (1)

B. Wandernoth, “20 photon/bit 565 Mbit/s PSK homodyne receiver using synchronization bits,” Electron. Lett. 28 (4), 387–388 (1992).
[CrossRef]

1991 (4)

P. C. D. Hobbs, “Reaching the shot noise limit for $10,” Opt. Photon. News 2 (4) , 17–23 (1991).
[CrossRef]

M. A. Newkirk, K. J. Vahala, “Amplitude-phase decorrelation: a method for reducing intensity noise in semiconductor lasers,” IEEE J. Quantum Electron. 27, 13–22 (1991).
[CrossRef]

D. L. Heinz, J. S. Sweeney, P. Miller, “A laser heating system that stabilizes and controls the temperature: diamond anvil cell applications,” Rev. Sci. Instrum. 62, 1568–1575 (1991).
[CrossRef]

R. P. Moeller, W. K. Burns, “1.06 µm all-fiber gyroscope with noise subtraction,” Opt. Lett. 16, 1902–1904 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

1986 (2)

R. Stierlin, R. Bättig, P.-D. Henchoz, H. P. Weber, “Excess noise suppression in a fibre-optic balanced heterodyne detection system,” Opt. Quantum Electron. 18, 445–454 (1986).
[CrossRef]

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

1982 (1)

S. Okamura, M. Maruyama, “Improvement on the sensitivity of electro-optical system for electric field strength measurements,” Trans. Inst. Electron. Commun. Eng. Jpn. E 65, 702 (1982).

1980 (1)

1978 (1)

R. L. Forward, “Wideband laser-interferometer gravitational radiation experiment,” Phys. Rev. D 17 (2) , 379–390 (1978).
[CrossRef]

1975 (1)

1957 (1)

Bättig, R.

R. Stierlin, R. Bättig, P.-D. Henchoz, H. P. Weber, “Excess noise suppression in a fibre-optic balanced heterodyne detection system,” Opt. Quantum Electron. 18, 445–454 (1986).
[CrossRef]

Bjorklund, G. C.

Bonin, K. D.

Burns, W. K.

Carlisle, C. B.

Collins, C. L. A.

Dubnitshev, Yu. N.

Forward, R. L.

R. L. Forward, “Wideband laser-interferometer gravitational radiation experiment,” Phys. Rev. D 17 (2) , 379–390 (1978).
[CrossRef]

Gallagher, T. F.

Haller, K. L.

K. L. Haller, P. C. D. Hobbs, “Double beam laser absorption spectroscopy: shot-noise limited performance at baseband with a novel electronic noise canceller,” in Optical Methods for Ultrasensitive Detection and Analysis: Techniques and Applications, B. L. Fearey, ed., Proc. SPIE1435, 298–309 (1991).
[CrossRef]

Harvey, G. T.

Heinz, D. L.

D. L. Heinz, J. S. Sweeney, P. Miller, “A laser heating system that stabilizes and controls the temperature: diamond anvil cell applications,” Rev. Sci. Instrum. 62, 1568–1575 (1991).
[CrossRef]

Henchoz, P.-D.

R. Stierlin, R. Bättig, P.-D. Henchoz, H. P. Weber, “Excess noise suppression in a fibre-optic balanced heterodyne detection system,” Opt. Quantum Electron. 18, 445–454 (1986).
[CrossRef]

Heutmaker, M. S.

Hobbs, P. C. D.

P. C. D. Hobbs, “ISICL: in situ coherent lidar for particle detection in semiconductor processing equipment,” Appl. Opt. 34, 1579–1590 (1995).
[CrossRef] [PubMed]

P. C. D. Hobbs, “Reaching the shot noise limit for $10,” Opt. Photon. News 2 (4) , 17–23 (1991).
[CrossRef]

P. C. D. Hobbs, “Shot noise limited optical measurements at baseband with noisy lasers,” in Laser Noise, R. Roy, ed., Proc. SPIE1376, 216–221 (1991).
[CrossRef]

P. C. D. Hobbs, “Noise cancelling circuitry for optical systems, with signal dividing and combining means,” U.S. patent5,134,276 (28July1992).

K. L. Haller, P. C. D. Hobbs, “Double beam laser absorption spectroscopy: shot-noise limited performance at baseband with a novel electronic noise canceller,” in Optical Methods for Ultrasensitive Detection and Analysis: Techniques and Applications, B. L. Fearey, ed., Proc. SPIE1435, 298–309 (1991).
[CrossRef]

Ingelstam, E.

Janik, G. R.

Kadar Kallen, M. A.

Keller, U. H.

Koronkevich, V. P.

Levenson, M. D.

Maruyama, M.

S. Okamura, M. Maruyama, “Improvement on the sensitivity of electro-optical system for electric field strength measurements,” Trans. Inst. Electron. Commun. Eng. Jpn. E 65, 702 (1982).

Miller, P.

D. L. Heinz, J. S. Sweeney, P. Miller, “A laser heating system that stabilizes and controls the temperature: diamond anvil cell applications,” Rev. Sci. Instrum. 62, 1568–1575 (1991).
[CrossRef]

Miller, P. J.

P. J. Miller, “Methods and applications for intensity stabilization of pulsed and cw lasers from 257 nm to 10.6 microns,” in Laser Noise, R. Roy, ed., Proc. SPIE1376, 180–191 (1991).
[CrossRef]

Moeller, R. P.

Newkirk, M. A.

M. A. Newkirk, K. J. Vahala, “Amplitude-phase decorrelation: a method for reducing intensity noise in semiconductor lasers,” IEEE J. Quantum Electron. 27, 13–22 (1991).
[CrossRef]

Nuss, M. C.

Okamura, S.

S. Okamura, M. Maruyama, “Improvement on the sensitivity of electro-optical system for electric field strength measurements,” Trans. Inst. Electron. Commun. Eng. Jpn. E 65, 702 (1982).

Perlmutter, S. H.

Peters, C. J.

C. J. Peters, “Laser communications system employing narrow band noise cancellation,” U.S. patent3,465,156 (2Sept.1969).

Reuter, B.

B. Reuter, N. Talukder, “A new differential laser microanemometer,” in 1980 European Conference on Optical Systems and Applications, D. J. Kroon, ed., Proc. SPIE236, 226–230 (1980).
[CrossRef]

Richardson, W. H.

Smith, P. R.

Sobolev, V. S.

Stierlin, R.

R. Stierlin, R. Bättig, P.-D. Henchoz, H. P. Weber, “Excess noise suppression in a fibre-optic balanced heterodyne detection system,” Opt. Quantum Electron. 18, 445–454 (1986).
[CrossRef]

Stolpovski, A. A.

Sweeney, J. S.

D. L. Heinz, J. S. Sweeney, P. Miller, “A laser heating system that stabilizes and controls the temperature: diamond anvil cell applications,” Rev. Sci. Instrum. 62, 1568–1575 (1991).
[CrossRef]

Talukder, N.

B. Reuter, N. Talukder, “A new differential laser microanemometer,” in 1980 European Conference on Optical Systems and Applications, D. J. Kroon, ed., Proc. SPIE236, 226–230 (1980).
[CrossRef]

Utkin, E. N.

Vahala, K. J.

M. A. Newkirk, K. J. Vahala, “Amplitude-phase decorrelation: a method for reducing intensity noise in semiconductor lasers,” IEEE J. Quantum Electron. 27, 13–22 (1991).
[CrossRef]

Vasilenko, Yu. G.

Wandernoth, B.

B. Wandernoth, “20 photon/bit 565 Mbit/s PSK homodyne receiver using synchronization bits,” Electron. Lett. 28 (4), 387–388 (1992).
[CrossRef]

Weber, H. P.

R. Stierlin, R. Bättig, P.-D. Henchoz, H. P. Weber, “Excess noise suppression in a fibre-optic balanced heterodyne detection system,” Opt. Quantum Electron. 18, 445–454 (1986).
[CrossRef]

Wilmshurst, T. H.

T. H. Wilmshurst, Signal Recovery from Noise in Electronic Instrumentation (Hilger, Boston, 1985).
[CrossRef]

Appl. Opt. (2)

Electron. Lett. (1)

B. Wandernoth, “20 photon/bit 565 Mbit/s PSK homodyne receiver using synchronization bits,” Electron. Lett. 28 (4), 387–388 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. A. Newkirk, K. J. Vahala, “Amplitude-phase decorrelation: a method for reducing intensity noise in semiconductor lasers,” IEEE J. Quantum Electron. 27, 13–22 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (5)

Opt. Photon. News (1)

P. C. D. Hobbs, “Reaching the shot noise limit for $10,” Opt. Photon. News 2 (4) , 17–23 (1991).
[CrossRef]

Opt. Quantum Electron. (1)

R. Stierlin, R. Bättig, P.-D. Henchoz, H. P. Weber, “Excess noise suppression in a fibre-optic balanced heterodyne detection system,” Opt. Quantum Electron. 18, 445–454 (1986).
[CrossRef]

Phys. Rev. D (1)

R. L. Forward, “Wideband laser-interferometer gravitational radiation experiment,” Phys. Rev. D 17 (2) , 379–390 (1978).
[CrossRef]

Rev. Sci. Instrum. (1)

D. L. Heinz, J. S. Sweeney, P. Miller, “A laser heating system that stabilizes and controls the temperature: diamond anvil cell applications,” Rev. Sci. Instrum. 62, 1568–1575 (1991).
[CrossRef]

Trans. Inst. Electron. Commun. Eng. Jpn. E (1)

S. Okamura, M. Maruyama, “Improvement on the sensitivity of electro-optical system for electric field strength measurements,” Trans. Inst. Electron. Commun. Eng. Jpn. E 65, 702 (1982).

Other (9)

National Semiconductor, Inc., “LM13600 data sheet,” in Operational Amplifiers Data Book (National Semiconductor, Santa Clara, Calif., 1990).

Precision Monolithics, Inc., PM-1008 data sheet, in Analog IC Data Book (Precision Monolithics, Inc., Santa Clara, Calif., 1990), Vol. 10, pp. 5-556 to 5-566.

B. Reuter, N. Talukder, “A new differential laser microanemometer,” in 1980 European Conference on Optical Systems and Applications, D. J. Kroon, ed., Proc. SPIE236, 226–230 (1980).
[CrossRef]

C. J. Peters, “Laser communications system employing narrow band noise cancellation,” U.S. patent3,465,156 (2Sept.1969).

P. C. D. Hobbs, “Shot noise limited optical measurements at baseband with noisy lasers,” in Laser Noise, R. Roy, ed., Proc. SPIE1376, 216–221 (1991).
[CrossRef]

P. C. D. Hobbs, “Noise cancelling circuitry for optical systems, with signal dividing and combining means,” U.S. patent5,134,276 (28July1992).

K. L. Haller, P. C. D. Hobbs, “Double beam laser absorption spectroscopy: shot-noise limited performance at baseband with a novel electronic noise canceller,” in Optical Methods for Ultrasensitive Detection and Analysis: Techniques and Applications, B. L. Fearey, ed., Proc. SPIE1435, 298–309 (1991).
[CrossRef]

T. H. Wilmshurst, Signal Recovery from Noise in Electronic Instrumentation (Hilger, Boston, 1985).
[CrossRef]

P. J. Miller, “Methods and applications for intensity stabilization of pulsed and cw lasers from 257 nm to 10.6 microns,” in Laser Noise, R. Roy, ed., Proc. SPIE1376, 180–191 (1991).
[CrossRef]

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Figures (22)

Fig. 1
Fig. 1

Block diagram of a generic all-electronic laser noise suppression scheme. The beam itself is unmodified; the noise improvement comes from combining signal and comparison photocurrents.

Fig. 2
Fig. 2

Simplified schematic diagram showing the use of a BJT differential pair as a variable current splitter. The splitting ratio depends on ΔV BE but not on i comp, so the fluctuations split just the same way as the dc.

Fig. 3
Fig. 3

Schematic diagram of the basic noise canceller. The BJT current divider Q 1/Q 2 is controlled by servo amplifier A 2 so that the dc output of transresistance amplifier A 1 is zero. This ensures that i signal and i C2 are equal. Because the fluctuations are proportional to the dc, the excess noise cancels identically at all frequencies of interest, independently of the feedback bandwidth.

Fig. 4
Fig. 4

Performance of the basic noise canceler of Fig. 3 with a 40-mW, 532-nm diode-pumped Nd:YAG laser exhibiting significant low-frequency noise and several peaks in the 1–2-MHz region. Q 1 and Q 2 were matched Motorola MRF904’s, and D 1 and D 2 were a Hamamatsu type S1722-01. The signal and comparison beam powers were 5.6 and 7.2 mW, respectively, and i signal = 1.77 mA. The 5–100-kHz noise voltage spectral density averages 88.7 nV/√Hz, which (after accounting for 29-nV/√Hz instrument noise and a factor of 0.5 that is due to a 50-Ω load) is within 0.15 dB of the predicted shot-noise level and shows a 1-Hz dynamic range of 154 dB. (a) dc to 100 kHz; the upper curve was taken in I-V (transresistance) mode (comparison beam blocked), the lower curve with the canceler operating normally. (b) 0 to 2 MHz; the top trace was taken in I-V mode, the middle curve with the canceler operating normally, and the bottom curve with both beams blocked (instrument noise).

Fig. 5
Fig. 5

Frequency response of the ultimate cancellation performance of the same canceller used in Fig. 4 with small (0.3%) intensity modulation. The signal beam power was 1.86 mW and i signal = 0.583 mA. In (a)–(c), the top curve was taken in I-V mode and the bottom curve with the canceller operating normally. This canceler version is optimized for good cancellation at the highest collector currents and low frequencies. (a) V log = +1.88 V (i comp ≈ 1.4 i signal). The cancellation is 55–60 dB throughout the baseband range shown. (b) V log = 0.38 V (i comp ≈ 1.83i signal). The cancellation is not as good at this setting. (c) V log = -0.145 V (i comp ≈ 2.08i signal). The cancellation is seriously degraded at this value of the comparison current. (d) 50-kHz, 2-MHz cancellation response showing the effect of the choice of comparison current on the ac response. Bottom right trace: V log = 2.70 V (i comp ≈ 1.25 i signal); bottom left trace: V log = 0.394 V (i comp ≈ 1.82i signal). Note the effect on the cancellation bandwidth of starving Q 1 of collector current.

Fig. 6
Fig. 6

50-kHz to 10-MHz frequency response of the ultimate cancellation performance of the canceller used in Fig. 5 with the exception that Q 3 here is a MM4049 rf device selected for good beta linearity and that the circuit is operated at lower current. In each plot, the top trace is the beam modulation, made in I-V mode. (a) i signal = 0.235 mA. Middle trace: V log = 2.87 V (i comp ≈ 1.24i signal); bottom trace: V log = 0.388 V (i comp ≈ 1.82i signal). The deterioration of the cancellation with frequency is less rapid than in Fig. 5, and the low-frequency cancellation behavior is better as well—almost 70 dB. (b) i signal = 0.153 mA. Middle trace: V log = +0.296 V (i comp ≈ 1.86i signal); bottom trace: V log = 0.00 V (i comp ≈ 2.00 i signal). Poorer low-frequency behavior is offset by excellent high-frequency response, although A 1’s bandwidth is only 1.8 MHz and the feedback bandwidth <100 Hz, the cancellation is >50 dB to 2 MHz, and >40 dB to at least 8 MHz, where the cancelled signal drops below the shot-noise floor (1 kHz BW).

Fig. 7
Fig. 7

Calculated limit to cancellation performance that is due to equal 0.5-Ω emitter bulk resistances r E in Q 1/Q 2 as a function of ΔV BE and i signal.

Fig. 8
Fig. 8

Measured 1-kHz cancellation performance of the circuit of Fig. 3 as a function of the log ratio output voltage that reflects i comp/i signal. The green Nd:YAG laser beam was sinusoidally modulated at 1 kHz by using an acousto-optical modulator, a variable attenuator, a Glan–Taylor prism to control the polarization, a Wollaston prism to split the beams, and a rotatable Glan–Thompson prism in the comparison beam to adjust the relative beam intensities. All were slightly misaligned to control étalon fringes. The interaction of r E degeneration and beta nonlinearity make the details of the cancellation performance difficult to predict a priori, but there is a clear trend toward better cancellation at lower currents.

Fig. 9
Fig. 9

Cancellation performance at 1 kHz of the circuit of Fig. 3 with a temperature-stabilized MAT-04 matched array used for Q 1 and Q 2. Excellent beta linearity of this device and elimination of temperature errors result in much more predictable cancellation performance. The deep minima near 0 V are due to symmetric cancellation of r E degeneration. The sharp deterioration with negative V log is due to the rapidly increasing current in Q 1, which increases its r E nonlinearity.

Fig. 10
Fig. 10

One implementation of low-noise current splitting, to reduce the 3-dB noise penalty resulting from the uncorrelated shot noise of i C2 and i signal. With i comp = 5i signal, the noise degradation resulting from i comp is now a factor of 1.33 (1.25 dB) instead of 2 (3 dB) and can be reduced further by one increasing the relative size of the comparison beam.

Fig. 11
Fig. 11

Noise floor of the low-noise canceller of Fig. 10 compared with that of Fig. 4. Q 1 and Q 2 were part of a MAT-04 array, with five diode-connected (base and collector shorted) MAT-04 devices in each emitter. The value of i signal was 0.189 mA and i comp/i signal was 5.0. The noise floor is 1.7 dB lower, in excellent agreement with the expected value of 1.75 dB calculated from Eq. (13).

Fig. 12
Fig. 12

Application of regenerative feedback to the bases of Q 1 and Q 2 for r E compensation.

Fig. 13
Fig. 13

Cancellation performance of the high spurious suppression canceller of Fig. 12 (solid curve), under the same conditions as in Fig. 9, as a function of the strength of i C 2. The data of Fig. 9 (dashed curve) are provided for comparison. The compensation circuit makes the behavior of the canceller stable over a wide range of signal and comparison currents. (a) i signal = 0.142 mA. The curves are nearly identical because r E is not a significant limitation yet. (b) i signal = 0.665 mA. The compensated device is several decibels better nearly everywhere and as much as 15 dB near null. (c) i signal = 1.26 mA. The benefits of r E compensation become evident, with as much as 28-dB improvement. Cancellation is >60 dB almost everywhere.

Fig. 14
Fig. 14

Ratio-only version of the noise canceler. The photocurrent now goes directly to the summing junction of A 2. The time constant of A 2 has been reduced. Feedback goes to the base of Q 1 now, since the sign of the loop gain was changed by the elimination of inverting amplifier A 1. Schottky diode D 3 protects Q 1 from excessive base currents when A 1 is saturated.

Fig. 15
Fig. 15

Noise floor and frequency response of the ratio-only noise canceller of Fig. 14. (a) Flatband (3–10 kHz) noise voltage spectral density compared with the prediction of Eq. (6) (with the sign of ΔV BE inverted). (b) Noise floor with V A 2 = 0.00 V, with i signal = 931 µA (corresponding to the rightmost point of the lower curve in (a). The flatband extends down to the low audio frequencies. (c) Differential signal frequency response of the ratio-only noise canceller with an incandescent light supplying the comparison current so that all the modulation appears as signal (see text). V A 2 was 0.00 V. (d) Swept sine cancellation behavior at null (V A 2 = 0.000 V).

Fig. 16
Fig. 16

Noise intermodulation suppression performance of the ratio-only noise canceller of Fig. 14. This measurement was made with the apparatus of Fig. 15 and arranged so that the Pockels cell and Glan–Taylor combination produced a harmonic-rich 5-kHz common-mode modulation of approximately 30% p.-p. A liquid-crystal ferroelectric modulator was put in the comparison beam to provide a differential signal. The liquid-crystal modulator was driven with a small amplitude sine wave at 50 kHz to produce a sinusoidal modulation 0.3% p.-p. The result is a 50-kHz differential optical modulation that has large modulation sidebands at harmonics of 5 kHz (upper trace: comparison beam blocked, incandescent lamp supplying i comp). When the comparison beam was unblocked (lower trace: same i comp), the strong 5-kHz intermodulation peaks essentially disappeared.

Fig. 17
Fig. 17

On-chip temperature stabilization of Q 1/Q 2, which are diagonally opposite sections of a MAT-04. Of the other two sections, Q 5 is a temperature sensor and Q 4 a heater. The design temperature is 330 K with a reference voltage of approximately 5 V.

Fig. 18
Fig. 18

Drift performance of the temperature-controlled ratio-only noise canceller of Figs. 17 and 19, run in a small uninsulated metal box with windowless photodiodes to control étalon fringes in the setup used in Fig. 9. The signal and comparison beam powers were 1.25 and 2.50 mW, respectively. The Hamamatsu S1722-01 photodiodes were inadequately passivated for windowless operation, as shown by the strong popcorn noise, which was not observed with the photodiode windows intact. Scale: 10-4 extinction/div vertical, 200 s/div horizontal.

Fig. 19
Fig. 19

Schematic of the differential and high-dynamic range noise canceller. A second signal photodiode D 3 and transistor Q 4 are added to Fig. 3. Photocurrent i sig1 is made slightly (perhaps a few percent) larger than i sig2. Most of the photocurrent flows from D 1 to D 3, leaving all of the desired signal, the shot noise, and a reduced dc current. The differential pair operates at lower current, reducing the effects of r b noise, i C 2 shot noise, and r E nonlinearity, so the limiting SNR closely approaches the signal-to-shot-noise ratio of the two strong beams.

Fig. 20
Fig. 20

Cancellation performance of the differential circuit of Fig. 19 at 50 kHz. The data are the ratios of the residual 50 kHz at the output of A 1 with the canceler operating to the total 50-kHz power in both beams combined (i sig1 = 1.48 mA, i sig2 = 1.36 mA). The cancellation performance is nearly independent of the comparison beam power, in contrast to the situation of Fig. 8, even though the total signal photocurrent is twice as great as that of the worst performing trace of Fig. 8.

Fig. 21
Fig. 21

Noise floor of the differential noise canceller of Fig. 19 (i sig1 = 1.83 mA, i sig2 = 1.77 mA, i comp = 0.15 mA). The total signal beam power was 14 mW. The available signal from A 1 is 18.6 V, or +25.4 dBV. The measured 3–100-kHz noise voltage spectral density is -134.4 dBV rms/Hz or -134.6 dBV rms after correction for the 30-nV/√Hz instrument noise. The 1-Hz SNR is thus 160.0 dB.

Fig. 22
Fig. 22

Simplified noise model of a differential BJT pair.

Equations (22)

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in=2eiB,
1SNRout=1SNRsignal+1SNRcomp,
iC2iC1=expeΔVBEkT.
ΔVBE=-kTe lnicompisignal-1.
VNA1=R14eBisignal,
VNA2=1γkTe4eisignal11-isignalicomp,  =2kTγeisignal1+expeγVA2kT,
fc=e2 πkTisignal1+expeΔVBE/kTγRfRC.
hfe=iCiB,
hFE=iCiB.
Amin=1hFE1-hFEhfe.
Amin=iC2icomp-iC2icompiC2icomp.
AminekTisample-isignalisample×rE1isample-isignal-rE2isignal.
in22=2 eiC21-NN+1iC1iC1+iC2.
ΔVcomp=rE2iC2-rE1iC1,
ΔVcomp=-iC1rE2+rE1+icomprE2,
Amin=2 sinπΔzfmodc.
iC2iC1=expqΔVBEkTexpqkTiC1rE1-iC2rE2.
AminqkTicomp-isignalicomprE1icomp-isignal-rE2isignal.
iN2=νN1+νN2re1+re2+iNE1/re21/re1+1/re2,
νN2=2 qiCre2=2 k2T2qiC
in22=2 qiC2
in22=2 qiC21-NN+1iC1iC1+iC2.

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