Optimal power split ratio for autobalanced photodetection Chia-Yu Chang1 and Jow-Tsong Shy1,2,* 1
Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan 2
Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan *Corresponding author: [email protected]
Received 7 October 2013; revised 14 December 2013; accepted 14 December 2013; posted 16 December 2013 (Doc. ID 198921); published 14 January 2014
The noise suppression of the autobalanced photoreceiver devised by Hobbs [Proc. SPIE 1376, 216 (1990)] had been determined to depend on the photocurrent ratio of reference beam to signal beam under the condition of constant signal beam photocurrent, and the best noise cancellation was suggested at a ratio close to 2. But in most applications, the available optical power has a limit. Therefore, to optimize the sensitivity of measurements, we should consider how to allocate the beam power in the case of fixed total optical power. In this paper, we measure the air Faraday rotation at different azimuth angles of beam polarization, which correspond to different photocurrent ratios. The signal-to-noise ratio at each photocurrent ratio is determined, and the best sensitivity appears at the photocurrent ratio of 1. This best sensitivity achieved is 3.02 × 10−8 rad Hz−1∕2 , which is about 1.3 times the shot noise limit. Our results are useful for sensitive optical measurements with the autobalanced photoreceiver. © 2014 Optical Society of America OCIS codes: (120.1880) Detection; (120.5410) Polarimetry; (120.7000) Transmission. http://dx.doi.org/10.1364/AO.53.000347
1. Introduction
Differential detection is commonly used to detect a relative amplitude change between two signal channels. The advantage of differential detection is removing the common mode noise by subtraction. In photonics, the differential detection scheme has been applied to measure various small effects for a long time. For example, it had been used in the measurement of Faraday effect [1], Faraday rotation spectroscopy [2], and absorption spectroscopy [3]. In these experiments, the original light beam is split into two beams, usually named the signal beam and reference beam. In the meanwhile, a small relative power difference between these two is induced by the sample under test, and a pair of photodetectors is used to measure each beam. Ideally the power of the two beams should be matched so that the excess noise 1559-128X/14/030347-04$15.00/0 © 2014 Optical Society of America
can be removed as completely as possible and small effects can be sensed. In actual implementation, there are tiny differences between the two photodetectors. Hence, to have better noise cancellation, it is the photocurrents in the two channels that must be maintained in balance. The performance of differential photodetection for measuring small effects has been proven to be very sensitive. However, balancing photocurrents is critical, and the noise subtraction is not always perfect. Hobbs devised an autobalanced circuitry, linking its differential output to a negative feedback to automatically compensate for photocurrent unbalance [4]. With this technology, balancing the photocurrents is not critical. This invention was demonstrated primarily for gas absorption spectroscopy, and the noise floor can achieve about 3 dB above the shot noise of the signal beam [5]. Hobbs had investigated the noise suppression performance in which the original beam was amplitude modulated by a few parts in 103 , which was treated as noise 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
347
at a specific frequency, before splitting [6]. Then it was separated into the signal beam and comparison beam (also called reference beam). The signal beam power was kept constant, while the comparison beam power was set at a specific photocurrent ratio. Both beams impinged on the autobalanced photoreceivers. By continuously sweeping the modulation frequency, the noise cancellation behavior in the frequency domain was determined. This experiment investigated the noise cancellation behavior under a constant signal beam power, and the noise suppression had been found to depend on the photocurrent ratio of reference beam to signal beam. Later, the ratio was suggested as being close to 2 to get the best noise cancellation [7]. In our previous air Faraday rotation measurement we set the ratio at 1.77 [8], and in Faraday rotation spectroscopy for oxygen detection by Brumfield and Wysocki the ratio was set from 1.6 to 2 [9]. However, in the measurement scheme with two beams there are other concerns: total optical power has a limit, and the larger signal beam power usually acquires larger signal size. Therefore, to better utilize the autobalanced photoreceiver, we should consider how to properly allocate the beam power to achieve the highest sensitivity. The measurement of Faraday rotation is an ideal example for studying this question. The Faraday effect is a phenomenon occurring when linearly polarized light propagates through a material on which a longitudinal magnetic field is applied. Because of the magnetic field-induced circular birefringence, the plane of beam polarization rotates in the material. The angle of rotation Δθ is proportional to the magnetic induction B and the optical path length d, that is, Δθ VBd, where the Verdet constant V is the characteristic rotation of the material per unit field strength and per unit length. The Verdet constants of gases are smaller than those of liquids and solids by three orders of magnitude. We have been able to measure the tiny air Faraday effect with autobalanced photodetection [8]. In this paper, we report and discuss the measurement of air Faraday rotation at different azimuth angles of polarization which correspond to different power split ratios of reference beam to signal beam. 2. Experimental Setup
The experimental setup is depicted in Fig. 1. The setup is the same as our previous setup for air Faraday measurement [8], except the light source is replaced by a two-mode stabilized 632.8 nm He–Ne laser. After the optical isolator, the beam is linearly polarized, and its power is 0.42 mW. A current source drives the solenoid with a 10 kHz AC current which induces a longitudinal AC magnetic field with an amplitude of 14.72 G. The autobalanced photoreceiver Nirvana (2007, New Focus) used in this work is based on Hobbs’s circuit. The 10 kHz is set so as to be higher than the cutoff frequency of the feedback loop 348
APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014
He-Ne Laser
Optical Isolator
M
REF
SIG
Autobalanced photoreceiver
M
PBS
HWP Lens
Current Source
Solenoid
Ref
M Sig
Lock-in Amplifier Out
Computer
Fig. 1. Experimental setup. Here, M: mirror; HWP: half-wave plate; PBS: polarizing beam splitter.
in the circuit. In general, the photocurrent ratio of I REF to I SIG is suggested to be close to 2 [10]. The linearly polarized beam gets a small AC polarization rotation Δθ after going through the solenoid due to the air Faraday effect. The half-wave plate (HWP) is used to adjust the polarization angle of the beam incidence onto the polarizing beam splitter (PBS) to be θ with respect to that of the signal beam. The angle θ corresponds to a specific power splitting ratio. The autobalanced output signal is sent to a lock-in amplifier. The average and standard deviation of the lock-in output in 100 s are calculated as signal root-mean-square (RMS) value and noise, and their quotient is the signal-to-noise ratio (SNR). In the present setup, with the incident beam power being PI onto the PBS, the signal beam power PSIG and reference beam power PREF are as follows: PSIG θ Δθ PI cos2 θ Δθ ≅ PI cos2 θ − PI sin 2θΔθ;
(1a)
PREF θ Δθ PI sin2 θ Δθ ≅ PI sin2 θ PI sin 2θΔθ:
(1b)
The signal (SIG) and reference (REF) detectors are assumed to induce photocurrents with the same responsivity so that the two photocurrents can be proportional to two power expressions in Eqs. (1a) and (1b) directly. The autobalanced circuit uses a feedback loop to automatically balance the two photocurrents within the cutoff frequency. In this operation, the first terms in the photocurrent expressions, I SIG and I REF , are maintained in balance by an internally adjusted gain g, such that PI cos2 θ − g · PI sin2 θ 0; g
cos2 θ : sin2 θ
(2)
(3)
The common mode noise in the two photocurrents is subtracted in the operation. Due to our setup where Δθ changes with frequency higher than the cutoff frequency, the second terms result in the differential signal: ΔV AB ∝ −V TOTAL sin 2θ1 gΔθ:
(4)
Here, V TOTAL is the corresponding voltage proportional to the incident beam power PI, which is the sum of PSIG and PREF . We can express sin 2θ × 1 g in the formula (4) in the form of the photocurrent ratio r being equal to I REF ∕I SIG . With the replacement, the autobalanced output signal has the following relation: ΔV AB ∝ −
V TOTAL p Δθ: r
(5)
3. Results and Discussion
In our measurement the ratio I REF ∕I SIG is varied from 0.71 to 5.23. The measured RMS values of ΔV AB , noise, and SNR against photocurrent ratio are shown in Fig. 2. The RMS values of ΔV AB display dependence on r−1∕2 , which agrees well with formula (5). The noise is suppressed to a minimum at the photocurrent ratio of 1. Regarding photocurrent ratios lower than 1, the noise sharply increases. For photocurrent ratios larger than 1 the noise increases a little and then decreases slightly. The signal and noise behaviors demonstrate that the circuit can adjust the internal gain based on Eq. (2) throughout the range of measurement; however, the adjustment could be unstable so as to cause more unbalanced common mode noise and signal fluctuation when I REF is lower than I SIG . These relative behaviors cause the SNR to reach the maximum at the photocurrent ratio of 1. Regarding ratios lower than 1, the SNR drops quickly. At a photocurrent ratio of 0.71 it has fallen below half of its peak. But for ratios larger than 1, the SNR gradually descends to a relatively steady level. In the range from 1 to 5.23, it remains above half of the peak. This indicates the autobalanced circuit cancels noise well 14
7 Signal
Noise
12
6
10
5
8
4
6
3
4
2
2
1
0
0
1
2
3
4
5
6
Voltage (1×10-6 volt)
SNR
SNR
0
Photocurrent ratio IREF/I SIG Fig. 2. Measured RMS values of autobalanced signal ΔV AB , noise, and SNR versus photocurrent ratio IREF ∕ISIG .
in the range where I REF is larger than I SIG , which agrees with Hobbs’s analysis [6]. The experiment shows that for the highest sensitive polarization measurement, the optimized optical setup should be at the photocurrent ratio of 1. The sensitivity at the ratio of 1 is estimated to be more 50% higher than that of 2. For example, we set the ratio at 1.77 in our previous air Faraday rotation measurement, and a sensitivity of 2.99 × 10−8 rad Hz−1∕2 was achieved [8]. The sensitivity could reach 2.06 × 10−8 rad Hz−1∕2 if we set the ratio to 1. Particularly, in the present experiment, we use a He–Ne laser with 3 times lower power, and the best sensitivity at the ratio of 1 is 3.02 × 10−8 rad Hz−1∕2 , which is about 1.3 times the shot noise limit. For another example, the sensitivity could be improved by more than 40% in Faraday rotation spectroscopy for oxygen detection by Brumfield and Wysocki [9]. Next, for absorption detection with the autobalanced photodetection, only the signal beam passes through the sample cell and acquires a transmission change from 1 to (1 − Δt). Due to the action of the feedback loop, the common mode noise is subtracted, and the variation caused by absorption with frequency higher than the cutoff frequency remains. The differential signal is ΔV AB ∝ −V SIG Δt:
(6)
Thus the Nirvana’s autobalanced output signal has the relation as follows: ΔV AB ∝ −
V TOTAL Δt: 1r
(7)
The formula (7) implies that the relative sizes of minimum detectable Δt can be found by multiplying the size of minimum measurable ΔV AB by (1 r) at a specific r. We have measured the noise levels at different photocurrent ratios. These noise levels can be used as the size of minimum measurable ΔV AB in formula (7). These products of corresponding noise levels and (1 r) can be indices of minimum detectable changes, and the inverses of these products can be indices of SNR for an absorption measurement at different photocurrent ratios. This relative SNR trend is plotted in Fig. 3 with its value at the photocurrent ratio of 1 being normalized to 1. Figure 3 shows that for the highest sensitive absorption measurement, the optimized optical setup should be also at the photocurrent ratio of 1. Finally, in respect to the two power expressions shown in Eq. (1), various laser wavelengths correspond to various levels of responsivity, which is the proportional constant of beam power to photocurrent. Therefore, changing the laser wavelength or power used for measurement will cause the SIG and REF photocurrents to alter together with the same proportion, and normally the signal and excess noise are also altered in the same way. Hence, the optimal optical setup should be unaffected. 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
349
Estimated relative SNR
1.2
for absorption measurement should be also at the ratio of 1. These results are useful to set up autobalanced photodetection with high sensitivity.
1 0.8
This work is supported by the National Science Council of Taiwan.
0.6 0.4
References 0.2 0
0
1
2
3
4
5
6
Photocurrent ratio IREF/I SIG Fig. 3. Estimated relative SNR for absorption measurement versus photocurrent ratio IREF ∕I SIG .
4. Conclusions
We have presented a polarization experiment to optimize the sensitivity in measurements with autobalanced photodetection. In this experiment, the autobalanced photoreceiver is found to run noise cancellation well in the range where the photocurrent ratios of reference beam to signal beam are larger than 1. The best sensitivity for the polarization measurement appears at the ratio of 1. The sensitivity at the ratio of 1 can be more 50% higher than that of 2. Particularly, we use an He–Ne laser and achieve the best sensitivity of 3.02 × 10−8 rad Hz−1∕2, which is about 1.3 times the shot noise limit. The estimated relative SNR of absorption measurement is also deduced by using measured noise levels. The curve of the relative SNR shows that the best sensitivity
350
APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014
1. L. R. Ingersoll and W. L. James, “A sensitive photoelectric method for measuring the Faraday effect,” Rev. Sci. Instrum. 24, 23–25 (1953). 2. H. Adams, D. Reinert, P. Kalkert, and W. Urban, “A differential detection scheme for Faraday rotation spectroscopy with a color center laser,” Appl. Phys. B 34, 179–185 (1984). 3. C. B. Carlisle and D. E. Cooper, “Tunable-diode-laser frequency-modulation spectroscopy using balanced homodyne detection,” Opt. Lett. 14, 1306–1308 (1989). 4. P. C. D. Hobbs, “Shot noise limited optical measurements at baseband with noisy lasers,” Proc. SPIE 1376, 216–221 (1990). 5. K. L. Haller and P. C. D. Hobbs, “Double beam laser absorption spectroscopy: shot noise-limited performance at baseband with a novel electronic noise canceller,” Proc. SPIE 1435, 298–309 (1991). 6. P. C. D. Hobbs, “Ultrasensitive laser measurements without tears,” Appl. Opt. 36, 903–920 (1997). 7. X. Wang, M. Jefferson, P. C. D. Hobbs, W. P. Risk, B. E. Feller, R. D. Miller, and A. Knoesen, “Shot-noise limited detection for surface plasmon sensing,” Opt. Express 19, 107–117 (2011). 8. C. Y. Chang, L. Wang, J. T. Shy, C. E. Lin, and C. Chou, “Sensitive Faraday rotation measurement with auto-balanced photodetection,” Rev. Sci. Instrum. 82, 063112 (2011). 9. B. Brumfield and G. Wysocki, “Faraday rotation spectroscopy based on permanent magnets for sensitive detection of oxygen at atmospheric conditions,” Opt. Express 20, 29727–29742 (2012). 10. “Nirvana Auto-Balanced Photoreceivers: Model 2007 & 2017 User’s Manual,” New Focus, 2002, p. 12.
Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan 2
Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan *Corresponding author: [email protected]
Received 7 October 2013; revised 14 December 2013; accepted 14 December 2013; posted 16 December 2013 (Doc. ID 198921); published 14 January 2014
The noise suppression of the autobalanced photoreceiver devised by Hobbs [Proc. SPIE 1376, 216 (1990)] had been determined to depend on the photocurrent ratio of reference beam to signal beam under the condition of constant signal beam photocurrent, and the best noise cancellation was suggested at a ratio close to 2. But in most applications, the available optical power has a limit. Therefore, to optimize the sensitivity of measurements, we should consider how to allocate the beam power in the case of fixed total optical power. In this paper, we measure the air Faraday rotation at different azimuth angles of beam polarization, which correspond to different photocurrent ratios. The signal-to-noise ratio at each photocurrent ratio is determined, and the best sensitivity appears at the photocurrent ratio of 1. This best sensitivity achieved is 3.02 × 10−8 rad Hz−1∕2 , which is about 1.3 times the shot noise limit. Our results are useful for sensitive optical measurements with the autobalanced photoreceiver. © 2014 Optical Society of America OCIS codes: (120.1880) Detection; (120.5410) Polarimetry; (120.7000) Transmission. http://dx.doi.org/10.1364/AO.53.000347
1. Introduction
Differential detection is commonly used to detect a relative amplitude change between two signal channels. The advantage of differential detection is removing the common mode noise by subtraction. In photonics, the differential detection scheme has been applied to measure various small effects for a long time. For example, it had been used in the measurement of Faraday effect [1], Faraday rotation spectroscopy [2], and absorption spectroscopy [3]. In these experiments, the original light beam is split into two beams, usually named the signal beam and reference beam. In the meanwhile, a small relative power difference between these two is induced by the sample under test, and a pair of photodetectors is used to measure each beam. Ideally the power of the two beams should be matched so that the excess noise 1559-128X/14/030347-04$15.00/0 © 2014 Optical Society of America
can be removed as completely as possible and small effects can be sensed. In actual implementation, there are tiny differences between the two photodetectors. Hence, to have better noise cancellation, it is the photocurrents in the two channels that must be maintained in balance. The performance of differential photodetection for measuring small effects has been proven to be very sensitive. However, balancing photocurrents is critical, and the noise subtraction is not always perfect. Hobbs devised an autobalanced circuitry, linking its differential output to a negative feedback to automatically compensate for photocurrent unbalance [4]. With this technology, balancing the photocurrents is not critical. This invention was demonstrated primarily for gas absorption spectroscopy, and the noise floor can achieve about 3 dB above the shot noise of the signal beam [5]. Hobbs had investigated the noise suppression performance in which the original beam was amplitude modulated by a few parts in 103 , which was treated as noise 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
347
at a specific frequency, before splitting [6]. Then it was separated into the signal beam and comparison beam (also called reference beam). The signal beam power was kept constant, while the comparison beam power was set at a specific photocurrent ratio. Both beams impinged on the autobalanced photoreceivers. By continuously sweeping the modulation frequency, the noise cancellation behavior in the frequency domain was determined. This experiment investigated the noise cancellation behavior under a constant signal beam power, and the noise suppression had been found to depend on the photocurrent ratio of reference beam to signal beam. Later, the ratio was suggested as being close to 2 to get the best noise cancellation [7]. In our previous air Faraday rotation measurement we set the ratio at 1.77 [8], and in Faraday rotation spectroscopy for oxygen detection by Brumfield and Wysocki the ratio was set from 1.6 to 2 [9]. However, in the measurement scheme with two beams there are other concerns: total optical power has a limit, and the larger signal beam power usually acquires larger signal size. Therefore, to better utilize the autobalanced photoreceiver, we should consider how to properly allocate the beam power to achieve the highest sensitivity. The measurement of Faraday rotation is an ideal example for studying this question. The Faraday effect is a phenomenon occurring when linearly polarized light propagates through a material on which a longitudinal magnetic field is applied. Because of the magnetic field-induced circular birefringence, the plane of beam polarization rotates in the material. The angle of rotation Δθ is proportional to the magnetic induction B and the optical path length d, that is, Δθ VBd, where the Verdet constant V is the characteristic rotation of the material per unit field strength and per unit length. The Verdet constants of gases are smaller than those of liquids and solids by three orders of magnitude. We have been able to measure the tiny air Faraday effect with autobalanced photodetection [8]. In this paper, we report and discuss the measurement of air Faraday rotation at different azimuth angles of polarization which correspond to different power split ratios of reference beam to signal beam. 2. Experimental Setup
The experimental setup is depicted in Fig. 1. The setup is the same as our previous setup for air Faraday measurement [8], except the light source is replaced by a two-mode stabilized 632.8 nm He–Ne laser. After the optical isolator, the beam is linearly polarized, and its power is 0.42 mW. A current source drives the solenoid with a 10 kHz AC current which induces a longitudinal AC magnetic field with an amplitude of 14.72 G. The autobalanced photoreceiver Nirvana (2007, New Focus) used in this work is based on Hobbs’s circuit. The 10 kHz is set so as to be higher than the cutoff frequency of the feedback loop 348
APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014
He-Ne Laser
Optical Isolator
M
REF
SIG
Autobalanced photoreceiver
M
PBS
HWP Lens
Current Source
Solenoid
Ref
M Sig
Lock-in Amplifier Out
Computer
Fig. 1. Experimental setup. Here, M: mirror; HWP: half-wave plate; PBS: polarizing beam splitter.
in the circuit. In general, the photocurrent ratio of I REF to I SIG is suggested to be close to 2 [10]. The linearly polarized beam gets a small AC polarization rotation Δθ after going through the solenoid due to the air Faraday effect. The half-wave plate (HWP) is used to adjust the polarization angle of the beam incidence onto the polarizing beam splitter (PBS) to be θ with respect to that of the signal beam. The angle θ corresponds to a specific power splitting ratio. The autobalanced output signal is sent to a lock-in amplifier. The average and standard deviation of the lock-in output in 100 s are calculated as signal root-mean-square (RMS) value and noise, and their quotient is the signal-to-noise ratio (SNR). In the present setup, with the incident beam power being PI onto the PBS, the signal beam power PSIG and reference beam power PREF are as follows: PSIG θ Δθ PI cos2 θ Δθ ≅ PI cos2 θ − PI sin 2θΔθ;
(1a)
PREF θ Δθ PI sin2 θ Δθ ≅ PI sin2 θ PI sin 2θΔθ:
(1b)
The signal (SIG) and reference (REF) detectors are assumed to induce photocurrents with the same responsivity so that the two photocurrents can be proportional to two power expressions in Eqs. (1a) and (1b) directly. The autobalanced circuit uses a feedback loop to automatically balance the two photocurrents within the cutoff frequency. In this operation, the first terms in the photocurrent expressions, I SIG and I REF , are maintained in balance by an internally adjusted gain g, such that PI cos2 θ − g · PI sin2 θ 0; g
cos2 θ : sin2 θ
(2)
(3)
The common mode noise in the two photocurrents is subtracted in the operation. Due to our setup where Δθ changes with frequency higher than the cutoff frequency, the second terms result in the differential signal: ΔV AB ∝ −V TOTAL sin 2θ1 gΔθ:
(4)
Here, V TOTAL is the corresponding voltage proportional to the incident beam power PI, which is the sum of PSIG and PREF . We can express sin 2θ × 1 g in the formula (4) in the form of the photocurrent ratio r being equal to I REF ∕I SIG . With the replacement, the autobalanced output signal has the following relation: ΔV AB ∝ −
V TOTAL p Δθ: r
(5)
3. Results and Discussion
In our measurement the ratio I REF ∕I SIG is varied from 0.71 to 5.23. The measured RMS values of ΔV AB , noise, and SNR against photocurrent ratio are shown in Fig. 2. The RMS values of ΔV AB display dependence on r−1∕2 , which agrees well with formula (5). The noise is suppressed to a minimum at the photocurrent ratio of 1. Regarding photocurrent ratios lower than 1, the noise sharply increases. For photocurrent ratios larger than 1 the noise increases a little and then decreases slightly. The signal and noise behaviors demonstrate that the circuit can adjust the internal gain based on Eq. (2) throughout the range of measurement; however, the adjustment could be unstable so as to cause more unbalanced common mode noise and signal fluctuation when I REF is lower than I SIG . These relative behaviors cause the SNR to reach the maximum at the photocurrent ratio of 1. Regarding ratios lower than 1, the SNR drops quickly. At a photocurrent ratio of 0.71 it has fallen below half of its peak. But for ratios larger than 1, the SNR gradually descends to a relatively steady level. In the range from 1 to 5.23, it remains above half of the peak. This indicates the autobalanced circuit cancels noise well 14
7 Signal
Noise
12
6
10
5
8
4
6
3
4
2
2
1
0
0
1
2
3
4
5
6
Voltage (1×10-6 volt)
SNR
SNR
0
Photocurrent ratio IREF/I SIG Fig. 2. Measured RMS values of autobalanced signal ΔV AB , noise, and SNR versus photocurrent ratio IREF ∕ISIG .
in the range where I REF is larger than I SIG , which agrees with Hobbs’s analysis [6]. The experiment shows that for the highest sensitive polarization measurement, the optimized optical setup should be at the photocurrent ratio of 1. The sensitivity at the ratio of 1 is estimated to be more 50% higher than that of 2. For example, we set the ratio at 1.77 in our previous air Faraday rotation measurement, and a sensitivity of 2.99 × 10−8 rad Hz−1∕2 was achieved [8]. The sensitivity could reach 2.06 × 10−8 rad Hz−1∕2 if we set the ratio to 1. Particularly, in the present experiment, we use a He–Ne laser with 3 times lower power, and the best sensitivity at the ratio of 1 is 3.02 × 10−8 rad Hz−1∕2 , which is about 1.3 times the shot noise limit. For another example, the sensitivity could be improved by more than 40% in Faraday rotation spectroscopy for oxygen detection by Brumfield and Wysocki [9]. Next, for absorption detection with the autobalanced photodetection, only the signal beam passes through the sample cell and acquires a transmission change from 1 to (1 − Δt). Due to the action of the feedback loop, the common mode noise is subtracted, and the variation caused by absorption with frequency higher than the cutoff frequency remains. The differential signal is ΔV AB ∝ −V SIG Δt:
(6)
Thus the Nirvana’s autobalanced output signal has the relation as follows: ΔV AB ∝ −
V TOTAL Δt: 1r
(7)
The formula (7) implies that the relative sizes of minimum detectable Δt can be found by multiplying the size of minimum measurable ΔV AB by (1 r) at a specific r. We have measured the noise levels at different photocurrent ratios. These noise levels can be used as the size of minimum measurable ΔV AB in formula (7). These products of corresponding noise levels and (1 r) can be indices of minimum detectable changes, and the inverses of these products can be indices of SNR for an absorption measurement at different photocurrent ratios. This relative SNR trend is plotted in Fig. 3 with its value at the photocurrent ratio of 1 being normalized to 1. Figure 3 shows that for the highest sensitive absorption measurement, the optimized optical setup should be also at the photocurrent ratio of 1. Finally, in respect to the two power expressions shown in Eq. (1), various laser wavelengths correspond to various levels of responsivity, which is the proportional constant of beam power to photocurrent. Therefore, changing the laser wavelength or power used for measurement will cause the SIG and REF photocurrents to alter together with the same proportion, and normally the signal and excess noise are also altered in the same way. Hence, the optimal optical setup should be unaffected. 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
349
Estimated relative SNR
1.2
for absorption measurement should be also at the ratio of 1. These results are useful to set up autobalanced photodetection with high sensitivity.
1 0.8
This work is supported by the National Science Council of Taiwan.
0.6 0.4
References 0.2 0
0
1
2
3
4
5
6
Photocurrent ratio IREF/I SIG Fig. 3. Estimated relative SNR for absorption measurement versus photocurrent ratio IREF ∕I SIG .
4. Conclusions
We have presented a polarization experiment to optimize the sensitivity in measurements with autobalanced photodetection. In this experiment, the autobalanced photoreceiver is found to run noise cancellation well in the range where the photocurrent ratios of reference beam to signal beam are larger than 1. The best sensitivity for the polarization measurement appears at the ratio of 1. The sensitivity at the ratio of 1 can be more 50% higher than that of 2. Particularly, we use an He–Ne laser and achieve the best sensitivity of 3.02 × 10−8 rad Hz−1∕2, which is about 1.3 times the shot noise limit. The estimated relative SNR of absorption measurement is also deduced by using measured noise levels. The curve of the relative SNR shows that the best sensitivity
350
APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014
1. L. R. Ingersoll and W. L. James, “A sensitive photoelectric method for measuring the Faraday effect,” Rev. Sci. Instrum. 24, 23–25 (1953). 2. H. Adams, D. Reinert, P. Kalkert, and W. Urban, “A differential detection scheme for Faraday rotation spectroscopy with a color center laser,” Appl. Phys. B 34, 179–185 (1984). 3. C. B. Carlisle and D. E. Cooper, “Tunable-diode-laser frequency-modulation spectroscopy using balanced homodyne detection,” Opt. Lett. 14, 1306–1308 (1989). 4. P. C. D. Hobbs, “Shot noise limited optical measurements at baseband with noisy lasers,” Proc. SPIE 1376, 216–221 (1990). 5. K. L. Haller and P. C. D. Hobbs, “Double beam laser absorption spectroscopy: shot noise-limited performance at baseband with a novel electronic noise canceller,” Proc. SPIE 1435, 298–309 (1991). 6. P. C. D. Hobbs, “Ultrasensitive laser measurements without tears,” Appl. Opt. 36, 903–920 (1997). 7. X. Wang, M. Jefferson, P. C. D. Hobbs, W. P. Risk, B. E. Feller, R. D. Miller, and A. Knoesen, “Shot-noise limited detection for surface plasmon sensing,” Opt. Express 19, 107–117 (2011). 8. C. Y. Chang, L. Wang, J. T. Shy, C. E. Lin, and C. Chou, “Sensitive Faraday rotation measurement with auto-balanced photodetection,” Rev. Sci. Instrum. 82, 063112 (2011). 9. B. Brumfield and G. Wysocki, “Faraday rotation spectroscopy based on permanent magnets for sensitive detection of oxygen at atmospheric conditions,” Opt. Express 20, 29727–29742 (2012). 10. “Nirvana Auto-Balanced Photoreceivers: Model 2007 & 2017 User’s Manual,” New Focus, 2002, p. 12.
