INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS
J. Opt. A: Pure Appl. Opt. 4 (2002) S400–S405
PII: S1464-4258(02)36470-5
Digital synthetic-heterodyne interferometric demodulation Michael J Connelly Department of Electronic and Computer Engineering, University of Limerick, Limerick, Republic of Ireland E-mail:
[email protected]
Received 5 April 2002, in final form 16 September 2002 Published 4 November 2002 Online at stacks.iop.org/JOptA/4/S400 Abstract Synthetic-heterodyne demodulation is a very useful technique for signal detection in interferometric sensors. The demodulation process is usually accomplished using analogue circuits. Improved functionality can be obtained by using a digital signal processor. In this paper, an expression is derived for the sensor sensitivity where both laser phase noise and signal acquisition quantization noise are considered. The demodulation technique requires modulation of the laser frequency, usually accomplished by modulation of the laser current. An expression is derived for the second-harmonic distortion caused by the laser power modulation. The detection scheme was implemented on a digital signal processor and used to detect dynamic pressure signals with a bandwidth of 550 Hz. Keywords: Optical fibre sensor, synthetic-heterodyne, interferometer, digital
signal processing (Some figures in this article are in colour only in the electronic version)
1. Introduction
2. Theory
Interferometric optical sensors can be used to detect a variety of stimuli including strain, temperature and pressure [1]. The basic principle is that an applied stimulus causes a phase shift between two light beams. Both beams are detected simultaneously and converted to an electrical signal, which is then processed to obtain a signal proportional to the stimulus. Because the detection process is non-linear, relatively complex techniques must be applied to obtain a linear relationship between an output electrical signal and the induced phase shift. Common techniques include pseudo-heterodyne [2], passive and active homodyne [3] and synthetic-heterodyne [4]. In this paper, we describe a digital implementation of the synthetic-heterodyne technique and use it to detect a phase signal from an external cavity with includes a mirror mounted on an accelerometer to simulate a phase disturbance [5]. We also derive new equations for the sensor sensitivity and secondharmonic distortion (HD2) taking into account laser phase and signal quantization noise and laser power modulation. The particular application of the sensor in this study is to detect dynamic pressure waves such as sound.
A schematic diagram of the sensor system under consideration in this paper is shown in figure 1. The output from a frequencymodulated laser diode is transmitted down an optical fibre to an external cavity. The fibre end is partially reflective. The output light from the fibre is collimated by a lens and travels through air for a distance L and then reflected by a flexible membrane, with reflectivity R2 1. As the fibre end reflectivity R1 1, the cavity can be modelled as a twobeam interferometer. If the membrane is displaced, the cavity length is changed leading to a phase shift between the two reflected light beams. The reflected light beams travel back down the fibre and are detected by a photodiode. The detected photocurrent is converted to a voltage by a resistive load, filtered, amplified and passed through an analogue-to-digital converter (ADC). The digital signal is then digitally processed using synthetic-heterodyne demodulation to obtain a signal proportional to the induced dynamic phase shift. This signal can be further processed, displayed or converted to an analogue signal by passing it through a digital-to-analogue converter (DAC) and a low-pass filter. The detected photocurrent can be
1464-4258/02/060400+06$30.00 © 2002 IOP Publishing Ltd Printed in the UK
S400
Digital synthetic-heterodyne interferometric demodulation
in-phase local oscillators at angular frequencies ω0 and 2ω0 and low-pass filtered (detection bandwidth B) to give
written as I (t) = A{1 + V cos[θ (t) + φ(t)]}
(1)
S1 = AV J1 (C) sin φ(t)
where A = R p Pin γ (R1 + R2 )
(2)
where R p is the photodiode responsivity, Pin the laser output power and γ the fraction of power from the extrinsic cavity coupler directed to the photodiode. The visibility V is given by √ R1 R2 . (3) V = R1 + R2 The visibility is maximized if R1 = R2 . It is assumed that the reflected light beams have identical polarizations. θ (t) is the phase difference between the reflected beams due to the optical carrier frequency and φ(t) the externally induced phase shift. In the synthetic-heterodyne technique θ (t) is modulated. This modulation can be achieved in many ways but one of the most convenient is to modulate the laser frequency by amplitude modulating the laser drive current. The laser frequency modulation is converted to phase modulation by the non-zero cavity path difference. The instantaneous laser frequency is ν = ν0 +
dν i (t) dI
(4)
where ν0 is the unmodulated laser frequency and i (t) the laser modulation current. The laser phase is then t ν(t) dt θ (t) = 2π 0 dν t i (t) dt . (5) = 2π ν0 t + dI 0 If sinusoidal modulation is applied to the laser current then i (t) = Im cos(ω0 t)
(6)
where Im and ω0 are the laser modulation current amplitude and angular frequency respectively. Hence (5) becomes dν Im sin(ω0 t) . (7) θ (t) = 2π ν0 t + dI ω0 Hence θ (t) ≡ θ (t + τ ) − θ (t) = 2πν0 τ + C cos(ω0 t + θ0 )
(8)
where τ = 2L/c is the cavity round-trip time. For ω0 τ < 1 we can make the approximation C ≈ 2π
dν Im τ dI
(9)
θ0 ≈ 0. Equation (1) can now be written as I (t) = A{1 + V cos[C cos(ω0 t) + φ(t)]}.
S2 = AV J2 (C) cos φ(t)
(10)
Equation (10) can be expanded into a series of sidebands that contain the signal of interest φ(t). In synthetic-heterodyne demodulation, the first two sidebands of (10) are mixed with
(11)
where J1 and J2 are Bessel functions of the first kind of order one and two respectively. Taking the time derivatives of S1 and S2 gives ˙ cos φ(t) S3 = AV J1 (C)φ(t) ˙ sin φ(t). S4 = − AV J2 (C)φ(t)
(12)
Multiplying S1 by S4 and S2 by S3 gives ˙ sin2 φ(t) S5 = −(AV )2 J1 (C)J2 (C)φ(t) ˙ cos2 φ(t). S6 = (AV )2 J1 (C)J2 (C)φ(t)
(13)
Subtracting S5 from S6 gives ˙ S0 = (AV )2 J1 (C)J2 (C)φ(t).
(14)
S0 can be integrated to obtain a signal proportional to φ(t). S0 depends on the product J1 (C)J2 (C), which reaches its maximum value of 0.22 when C = 2.37. This condition dictates the magnitude of the modulation current used for a given laser and cavity length.
3. Sensitivity 3.1. Output noise power There are a number of noise sources that limit the sensor sensitivity. In practice receiver dark current noise and signal shot noise are negligible in comparison with other noise sources. The dominant noise sources are receiver thermal noise, laser phase noise induced intensity noise and quantization noise, which arises when digital signal processing is used. The thermal noise current spectral density (A2 Hz−1 ) is 4kT F G th = (15) RL where k is the Boltzmann constant, T temperature, R L the photodiode load resistance and F the receiver postamplifier noise figure. An expression for the noise spectral density of the light intensity from a two-beam interferometer in the case of a monomode laser source is derived in [6]. Using this expression, it can be easily shown for the setup of figure 1 that the photocurrent noise spectral density due to the laser phase noise is given by G p = 8γ 2 R1 R2 τc f (τ/τc )Pin2 R 2p
(16)
where τc is the laser coherence time and f (τ/τc ) is the coherence function [6] given by τ f (τ/τc ) = 1 − exp(−τ/τc ) 1 + (17) τc f (τ/τc ) can be approximated by 2 1 τ , f (τ/τc ) ≈ 2 τc 1,
τ τc
(18)
τ τc S401
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dc bias Bias-tee Modulation current i (t ) = I m cos ω 0t DFB laser
Isolator Splitter Optical fibre
Extrinsic cavity R2 R1
Pressure wave
Lens
I(t) Optical receiver
Digital signal processing
Flexible L mirror Output signal proportional to dynamic pressure
Figure 1. Dynamic pressure sensor system using synthetic heterodyne demodulation. FIR low-pass filter
X Amplifier From optical receiver
Anti-aliasing filter
Cos(w0t) ADC Cos(2w0t) FIR low-pass filter
X
d dt
X
Amplifier
+ d dt
Analog output to low-pass filter
Integrator
DAC
X Reset switch
Range check
0
Figure 2. Synthetic-heterodyne demodulation digital implementation.
τc is related to the laser spectral linewidth ν by τc =
1 . ν
3.2. Output signal power (19)
The quantization noise current spectral density is given by
2 V A– D /2 Q (20) Gq = 6Ts R 2L where V A– D and Q are the voltage range and number of bits of the digital signal processor input ADC. Ts is the sampling time (inverse of the sampling frequency). In (20) it is assumed that unity strength sampling pulses are used in the ADC quantizer. The total current noise spectral density η (single-sided) at the ADC input is simply the sum of the individual uncorrelated noise spectral densities, i.e. η = G th + G p + G q .
(21)
η can be considered to be white noise over the bandwidth of the detection process. It is an elementary but tedious process, using techniques described in [7], to calculate the output noise power N after synthetic-heterodyne demodulation. We simply state the result here: N= S402
[AV J2 (C)]2 η B . 2
(22)
If the input phase signal φ(t) is sinusoidal with amplitude D and angular frequency ωs , then the detected signal power, after integration of S0 , is given by S=
(AV )4 [D J1 (C)J2 (C)]2 . 2
(23)
The sensor signal-to-noise ratio (SNR), from (22) and (23) is given by [AV J1 (C)D]2 S = . (24) SNR ≡ N ηB The sensor sensitivity can be defined as the phase amplitude Dmin required for an SNR of unity. From (24) we get √ η (25) rad Hz−1/2 . Dmin = AV J1 (C) If laser phase noise induced intensity noise is dominant, then Dmin is proportional to the cavity length.
4. Second-harmonic distortion The laser power varies as the current is modulated. This can cause phase errors [8]. In synthetic-heterodyne demodulation,
Digital synthetic-heterodyne interferometric demodulation
S2
2.4 kHz local oscillator
S3
4.8 kHz local oscillator
S4 Input to upper filter
S1 x S4 Input to lower filter
S2 x S3
S1
So
Figure 3. Internal signals on the DSP card. The vertical axes are normalized to the maximum signal amplitudes for clarity. The lengths of the horizontal time axes are 20 ms.
Acquired optical receiver signal
Detected phase signal at 100 Hz
Figure 4. Optical receiver output and detected phase signals. The ramping of the phase signal is due to the laser power modulation. The vertical axes are normalized to the maximum signal amplitudes for clarity. The lengths of the left and right horizontal time axes are 7 and 29 ms respectively.
the laser power variation causes terms proportional to cos φ(t) and sin φ(t) to be present in S1 and S2 respectively and ˙ sin φ(t) and φ(t) ˙ cos φ(t) to also terms proportional to φ(t) be present in S3 and S4 respectively. An advantage of the synthetic-heterodyne technique is that these potentially degrading terms cancel out when S0 is calculated. There is also a constant term in S1 , caused by the laser power modulation. If this effect is included in the analysis then (14) becomes ˙ S0 = (AV )2 J1 (C)J2 (C)φ(t) 1−
α sin φ(t) 2V J1 (C)
(26)
where α=
d Pin Im . dI Pin
(27)
If φ(t) = D cos(ωt) is small compared to unity then (26) becomes S0 = −(AV )2 J1 (C)J2 (C)ω D sin(ωt) α × 1− D cos(ωt) 2V J1 (C) = − (AV )2 J1 (C)J2 (C)ω D α × sin(ωt) − [1 + sin(2ωt )] . 4V J1 (C)
(28) S403
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If S0 is integrated (to retrieve φ(t)) then the constant term in (28) will give rise to a signal that increases linearly with time. There is also a second harmonic present in (28), which may interfere with other signals within the demodulation scheme detection bandwidth. The HD2 can be defined as the ratio between the amplitudes of the fundamental and its second harmonic, i.e. αD dB. (29) 2HD = 20 log 8V J1 (C)
5. Digital implementation and experiment The basic experimental setup is shown in figure 1. A block diagram of the synthetic-heterodyne demodulation technique as implemented on a digital signal processor (dSPACE DS1102) is shown in figure 2. The output signal from the optical receiver (R L = 1 k ) is passed through a dc blocking capacitor, anti-aliasing filter (bandwidth equal to ω0 /π + B), amplified (so the receiver output signal range is equal to the ADC voltage range) and converted to a digital signal. The digital signal is then processed using synthetic-heterodyne demodulation. The DSP card has a 16-bit ADC and operates at a sampling frequency of 12 kHz. The digital filters are finiteimpulse response (FIR) filters. Differentiation is approximated by ( fn − f n−1 ) d f (t) (30) ≈ dt Ts where f n and f n−1 are samples of a function f at times t and t + Ts . Integration is approximated by f (t) dt ≈ Ts
n
fj.
(31)
written that allows the internal DSP signals to be monitored. Some of these signals are shown in figure 3. The optical receiver output signal and resulting detected phase signal at 100 Hz are shown in figure 4. The phase signal is as expected, a sinusoid at 100 Hz. The waveform exhibits sharp jumps that are due to quantization effects inherent in the sampling process. These can be removed if the signal is converted to an analogue signal and passed through a smoothing low-pass filter. The waveform also has a component that increases slowly with time. As described above, this is caused by the laser power modulation. In this case phase resets were required approximately every 5 s, i.e. the reset waveform is a triangular function with a period of 5 s. This means that the lowest frequency that can be detected by the sensor is approximately 1 Hz (assuming frequency components above the fifth harmonic of the reset waveform are negligible). Other frequencies, within the detection bandwidth of 550 Hz, were successfully detected. The thermal noise limited sensitivity is 0.2 µrad Hz−1/2 . If a typical laser linewidth of 10 MHz is assumed, the theoretical sensitivity (in the absence of quantization noise) from (25) is 7.8 µrad Hz−1/2 . The quantization noice introduced by the 16-bit ADC is for the dominant noise source, leading to a theoretical sensitivity of 0.49 rad Hz−1/2 . This theoretical sensitivity was not confirmed experimentally in the prototype system. The quantization noise can be reduced to less than the laser phase noise induced intensity noise by the use of a 32-bit ADC. There is some distortion of the output signal; however, it was difficult to determine if this is due to additional reflections from the cavity, the laser power modulation or the accelerometer.
6. Conclusions
j =0
Due to the laser power modulation, there is a dc component in the signal prior to integration. This will eventually cause an overflow error in the processor. To avoid this problem, the integrator output is reset to zero if it exceeds its allowed range. If required, the demodulated phase signal can be further processed or converted to an analogue signal using a DAC and smoothing low-pass filter. A prototype system was constructed using a temperaturestabilized DFB laser operating at a bias current of 90 mA, with an output power of 1 mW. The side-mode rejection ratio of the laser was greater than 35 dB and so is a monomode source. At the bias point chosen, dν/dI = 850 MHz mA−1 . A 10 cm long external cavity was used. An external low-reflectivity mirror mounted on an accelerometer was used to simulate a flexible reflective membrane. For evaluation purposes, the accelerometer was driven by a sinusoidal voltage at 100 Hz to simulate a cavity displacement. The laser modulation current had a frequency of 2.4 kHz. The current amplitude of 3.6 mA was chosen to give the optimum value of C (2.37) for the particular cavity length used. The fibre uncoated cleaved end reflectivity is 0.04. The mirror has a reflectivity approximately equal to 0.1. γ = 1. A low-noise optical receiver (R p = 0.9 A W−1 ) was used to detect the return signal from the external cavity. This signal was then passed through an anti-aliasing filter, dc blocking capacitor and amplified before acquisition and digital signal processing. Software was S404
A prototype of a digital implementation of the syntheticheterodyne technique, with a detection bandwidth of 550 Hz, has been demonstrated. Important practical limitations, including laser phase noise, signal quantization noise, laser power modulation and second harmonic distortion were analysed.
Acknowledgment This work was carried out while the author was a visiting researcher at the European Commission Joint Research Centre (JRC), 21020 Ispra (VA), Italy. The author thanks Dr Maurice Whelan of the JRC for assistance with the experimental set-up and useful discussions.
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Digital synthetic-heterodyne interferometric demodulation
[5] Connelly M J and Whelan M 2001 Analysis of digital synthetic-heterodyne detection of a fibre cavity sensor Proc. 3rd Topical Meeting on Optoelectronic Distance Measurement and Applications, ODIMAP III (Pavia, Italy) pp 331–6 [6] Petermann K and Weidel E 1981 Semiconductor laser noise in an interferometer system IEEE J. Quantum Electron.
17 1251–6 [7] Taub H and Schilling D L 1987 Principles of Communication Systems (New York: McGraw-Hill) [8] Onodera R, Ishii Y, Ohde N, Takahashi Y and Yoshino T 1995 Effect of laser-diode power change on optical heterodyne interferometry IEEE J. Lightwave Technol. 13 675–81
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