Self-mixing interferometry based on sinusoidal phase modulation and integrating-bucket method

Optics Communications 283 (2010) 2186–2192

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Self-mixing interferometry based on sinusoidal phase modulation and integrating-bucket method Dongmei Guo, Ming Wang * School of Physical Science and Technology, Nanjing Normal University, Nanjing 210046, PR China

a r t i c l e

i n f o

Article history: Received 16 September 2009 Received in revised form 22 December 2009 Accepted 11 January 2010

Keywords: Self-mixing interferometry Sinusoidal phase modulation Integrating-bucket

a b s t r a c t In order to realize real-time displacement measurement with high resolution, sinusoidal phase modulation and integrating-bucket method are introduced in the self-mixing interference (SMI) system firstly. The phase of the laser beam is modulated by an electro-optic modulator (EOM) in the external cavity. Theoretical analysis, simulation results and error evaluation are presented. Experimentally, the microdisplacement of a high-precision commercial PZT is reconstructed and the reconstruction accuracy is on the order of nanometers for displacements of a few micrometers. Ó 2010 Published by Elsevier B.V.

1. Introduction Recently, various nondestructive optical measurement methods have been proposed to remotely measure micro-displacement, distance and velocity etc. [1–3]. In these sensors, laser diode self-mixing interference played an important role. Self-mixing system is much simpler than conventional interferometers because many optical elements such as the beam splitter, reference mirror and external photodetector are not required. Based on self-mixing interference, many smart and simple laser sensing systems have been developed. Self-mixing interference was used to measure the displacement with an accuracy of k/2 by counting the interference signal peaks. In order to increase the measurement accuracy beyond k/2, some methods for analysis of SMI signal have been reported. Injection current modulation of the laser diode is the most common modulation method used in SMI system [4]. The main disadvantage of the current modulation is: by varying the injection current, the wavelength of the LD is modulated, but the intensity modulation concurrent with the wavelength modulation leads to measurement errors. External cavity length modulation introduced by PZT is also a very useful method to extract the phase of the SMI signal [5]. But this method does not allow a high modulation frequency due to mechanical constraints, which limit the measurement range. In our recent work, we have proposed a displacement sensor based on SMI system with phase modulation introduced by an electro-optic modulator in the external cavity. Sinusoidal phase

* Corresponding author. Tel./fax: +862583598685. E-mail address: [email protected] (M. Wang). 0030-4018/$ - see front matter Ó 2010 Published by Elsevier B.V. doi:10.1016/j.optcom.2010.01.025

modulation and FFT analysis method have been studied to measure the displacement with great accuracy [6]. But it takes a long time to measure because it is necessary to calculate Fourier transform, filtering and inverse Fourier transform of the SMI signal. This delay time makes it hard to control the micro-displacement of a machine tool with the measured results. Then triangular phase modulation and five-step Schwider–Hariharan algorithm are combined to extract the phase in SMI [7]. The simplicity of this demodulation method makes it possible to realize real-time displacement measurement. Whereas, a triangular signal becomes distorted at high frequency, a sinusoidal signal remains steady, permitting larger measurement range and higher accuracy. So in this paper, sinusoidal phase modulation and integrating-bucket technique are introduced to analysis the SMI signal. During one period of the phase modulation, four integrations of the SMI signal are completed and then the phase of interference can be calculated in a very shot computation time. To the best of our knowledge, it is the first time that sinusoidal phase modulation and integrating-bucket technique is introduced in SMI to improve the measurement accuracy. Experimental results show that using the integrating-bucket method in the sinusoidal phase modulating SMI system, reconstruction accuracy is on the order of nanometers for displacements of a few micrometers. 2. Theoretical analysis 2.1. The theory of self-mixing interference The basic theory of a semiconductor laser SMI can be explained by a three-mirror cavity model as in Fig. 1. M1 and M2 are two

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M1

M2

l

erate optical feedback (C > 1) the interferometric signal waveform becomes sawtooth-like and it exhibits hysteresis. In the following theoretical analysis, only the case when C < 1 is discussed (t  t0). Then the phase of the SMI signal only depends on the external cavity length. The signal peaks appear when the following relationship is satisfied

M3

L

Fig. 1. Schematic diagram of a laser with optical feedback.

D/ ¼ x0 facets of the laser cavity. M3 is the external target. The beam emitted from the laser cavity is reflected by the target, a portion of the laser output back into the laser cavity and mixes with the original light in the laser, forming the self-mixing effect. The theoretical study of the dynamic characteristics of a semiconductor laser with optical feedback has been performed by many researchers. The dynamics of semiconductor laser with optical feedback is described by the rate equations modeled by Lang and Kobayashi [8]. Assuming that laser oscillates in a single longitudinal mode with an angular frequency x0, the complex electric field in the active region is written as E(t) exp[i(x0t + U(t))], where U(t) is the phase change due to the feedback effect. Then, the rate equations for the amplitude and phase of the complex electric field and the carrier density are given by

 d 1 j Eðt  sÞ cos hðtÞ Eð t Þ ¼ g½NðtÞ  N0   1=sp EðtÞ þ dt 2 sin  j Eðt  sÞ d a uðtÞ ¼ g½NðtÞ  N0   1=sp  sin hðtÞ dt 2 sin EðtÞ d NðtÞ  g½NðtÞ  N0 jEðtÞj2 NðtÞ ¼ J  dt ss

ð1Þ

ð2Þ

The feedback parameter j is written by

j ¼ ð1 

r 20 Þr=r 0

ð8Þ

As a consequence, we can only reconstruct the displacement of the external target DL with a resolution of k0/2. Fig. 2 is the simulation results of the SMI signal caused by a sinusoidal motion of the target. Fig. 2(a) simulates the phase variations due to the target movement. Fig. 2(b) is the corresponding SMI signal. Direction of displacement is directly given by the orientation of the sawtooth-like signal. To achieve higher resolution, phase modulation technique is introduced into SMI as below. 2.2. Principle of sinusoidal phase modulation and integrating-bucket technique To achieve higher accuracy, an experimentally controlled additive phase W(t) = asin(2pf(m t + h) is introduced by EOM situated in the external cavity, where a is the modulation depth, fm is the modulation frequency, h is the initial phase of the modulation. Considering that the beam pass through the EOM twice in the external cavity, the modulated interference signal can be written as

IðtÞ ¼ I0 f1 þ m cos ½u þ 2wðtÞg

where g is the linear gain coefficient, N0 is the carrier density at transparency, a is the linewidth enhancement factor, J is the injection current density, sp is the photon life time, ssis the carrier life time, s = 2L/c is the external cavity round-trip time where L external cavity length. sin = 2gl/c is the round-trip time within the laser cavity where l is the internal cavity length and g is the refractive index of the laser cavity. h(t) represents the phase coupling between the original light in the cavity and the delayed light from the external reflector and is given by the following forms:

hðtÞ ¼ x0 t þ /ðtÞ  /ðt  sÞ

2 DL k0 ¼ 2p () DL ¼ c 2

ð3Þ

where r0 and r are the amplitude reflectivities of the laser exit facet and the external reflector, respectively. Since multiple reflections between the laser facet and the external reflector are neglected, Eqs. (1)–(3) are valid for a weak to moderate feedback level. By substituting the steady state solutions into the rate equations, the angular frequency x and the out put power I of the semiconductor laser due to feedback can be expressed by:

x0 s ¼ xs þ C sinðxs þ arctan aÞ

ð4Þ

I ¼ I0 ð1 þ m cos /Þ ¼ I0 ½1 þ m cosðxsÞ

ð5Þ

ð9Þ

Expanding Eq. (9) in a Fourier series, then I(t) can be represented by:

IðtÞ ¼ I0 þ mI0 cos /J 0 ð2aÞ " # 1 X J 2n ð2aÞ cosð2nÞð2pfm t þ hÞ þ mI0 cos / 2 "  mI0 sin / 2

n¼1 1 X

# J ð2nþ1Þ ð2aÞ sinð2n þ 1Þð2pfm t þ hÞ

n¼0

¼ p þ qðtÞ þ rðtÞ

ð10Þ

where J is the Bessel function. Here the integrating-bucket method is introduced to extract the phase U. Within a modulation period T = 1/fm, the time-varying signal I(t) is integrated four times with the integration time T/4 (see Fig. 3), which can be expressed as:

where

  sp m ¼ 2jsp J sp  N0 =sL ss pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ j 1 þ a2 =sin

ð6Þ ð7Þ

For a weak feedback level (C < 1), the variation in the laser frequency caused by optical feedback is very small. The function F(/) is nearly sinusoidal and the signal amplitude increases for increasing level of the optical feedback. At higher feedback level (C  1), the interferometric waveform exhibits a slight distortion. For mod-

Fig. 2. Simulation result of the SMI signal caused by sinusoidal motion of the target.

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R1 ¼ R3 ¼ 2mI0 T sin /

1 X

J 2n1 ð2aÞ½ cosð2n  1Þh

n¼1

 ð1Þn sinð2n  1Þh =ð4n  2Þp 1 X R2 ¼ R4 ¼ 2mI0 T sin / J 2n1 ð2aÞ½ cosð2n  1Þh n¼1  þð1Þn sinð2n  1Þh =ð4n  2Þp

ð14Þ

ð15Þ

Linear combinations of Ci can be formed to have

X ¼ C 1 þ C 3  C 2  C 4 ¼ 8mI0 THX cos /

ð16Þ

Y ¼ C 1 þ C 2  C 3  C 4 ¼ 8mI0 THY sin /

ð17Þ

where

HX ¼ Fig. 3. Illustration of the four-integrating-bucket method used in SMI (a = 1.23, h = 56°).

HY ¼

1 X n¼1 1 X

  J 2n ð2aÞ sin 2nh 1  ð1Þn =4np

ð18Þ

J 2n1 ð2aÞ cosð2n  1Þh=ð4n  2Þp

ð19Þ

n¼1

Ci ¼

Z

iT=4

IðtÞdt ¼ ði1ÞT=4

¼ P i þ Q i þ Ri

Z

Then the phase U can be calculated using the following relationship:

iT=4

½p þ qðtÞ þ rðtÞdt

ði1ÞT=4

i ¼ 1; 2; 3; 4

ð11Þ

/ ¼ arctanðY  HX =X  HY Þ

ð12Þ

Setting a = 1.23 and h = 56°, Hx = HY can be obtained, with the noise based measurement error being minimized [9]. Then the phase U can be expressed by:

where

P1 ¼ P2 ¼ P3 ¼ P4 ¼ ½I0 þ mI0 cos /J 0 ð2aÞT=4 Q 1 ¼ Q 3 ¼ Q 2 ¼ Q 4 1 X   ¼ 2mI0 T cos / J 2n ð2aÞ sin 2nh ð1Þn  1 =4np n¼1

/ ¼ arctanðY=XÞ ð13Þ

ð20Þ

ð21Þ

The phase U obtained using the demodulation technique proposed above is wrapped within the region of p and p. After a

Fig. 4. (a) Simulated micro-displacement of the external target. (b) Extracted phase from the interference signal. (c) Reconstructed displacement. (d) Displacement reconstructed error.

D. Guo, M. Wang / Optics Communications 283 (2010) 2186–2192

phase unwrapping process and following the relationship between the phase U and the length of the external cavity, the movement of the external target can be reconstructed. 2.3. Simulation results and error evaluation In order to evaluate the displacement measurement technique proposed. Simulation results is presented. Assuming that the ac component of the signal detected with a photodiode is given by:

sðtÞ ¼ s0 f1 þ m cos ½uðtÞ þ 2wðtÞg

ð22Þ

Fig. 4(a) is the simulated micro-movement of the external target. The modulation frequency of EOM is 2 kHz. Following the phase detection technique proposed, the phase variation of SMI is extracted from the interference signal as shown in Fig. 4(b). Fig. 4(c) is the reconstructed displacement of external target and Fig. 4(d) is the displacement measurement error. When using the self-mixing interferometer based on sinusoidal phase modulation technique combined with integrating-bucket method for displacement measurement, errors come from mainly following aspects: (1) Errors due to the demodulation algorithm. In order to simulate multiple measurements for a certain displacement, multiple vibration periods is simulated. We use Eq. (23) to obtain the standard deviation of the measured displacement

r¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP un u ðdim  dir Þ2 ti¼1 n1

sðtÞ ¼ s0 cos fu þ 2½wðtÞ þ na ðtÞg

2189

ð25Þ

The standard deviation of the reconstructed displacement when n = 10 versus qa is simulated in Fig. 5. From Fig. 5 we can see that: in general, when the SNR of the modulation signal increase, the standard deviation of the measured results reduce. The SNR of the modulation signal mainly depends on the SNR of high-voltage amplifier which drives EOM. (3) Error induced by noise occurs in the electronic devices ns(t). It is also assumed that the noise ns(t) is Gaussian distribution with zero means and variance r2s . Then the signal to noise (SNR) of the modulation signal is defined by

qs ¼ ðs20 =2Þ=r2s

ð26Þ

The interference signal obtained can be written as:

sðtÞ ¼ s0 f1 þ m cos ½uðtÞ þ 2wðtÞg þ ns ðtÞ

ð27Þ

The standard deviation of the retrieved phase when n = 10 versus qs is simulated in Fig. 6.In order to decrease the measurement error caused by ns(t), high SNR photodetector, amplifier and A/D card are needed during measurement. (4) Error caused by the initial phase error of modulation. In experiment, the initial phase error is unavoidable which will bring measurement error. The simulation result of standard

ð23Þ

where dim is the demodulated result of the peak to peak amplitude of the displacement, dir is the true value of the peak to peak amplitude of the displacement, n is the number of vibration period simulated. The standard deviation r is 0.05 nm in Fig. 4 when n = 10. In Fig. 4(d), only one period vibration is presented because of the size limit of the figure. (2) Error induced by the fluctuations of modulation amplitude in time na(t). It is assumed that the noise na(t) is Gaussian distribution with zero means and variance r2a . Then the signal to noise (SNR) of the modulation signal is defined by

qa ¼ ða2 =2Þ=r2a

ð24Þ

Considering the laser beam pass the EOM twice in the external cavity, the noise should be doubled in the interference signal. Eq. (22) can be written as:

Fig. 5. Standard deviation of the reconstructed displacement versus qa.

Fig. 6. Standard deviation of the reconstructed displacement versus qs.

Fig. 7. Standard deviation of the reconstructed displacement caused by initial phase error of the modulation.

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Eq. (28). The distorted SMI signal is a triangle-like waveform and it can be written as [4]



1 1 I ¼ I0 1 þ m cos / þ cosð3/Þ þ cosð5/Þ þ    9 25

ð28Þ

Fig. 8 simulates the displacement reconstruction error due to the distortion of the SMI signal corresponding to Fig. 4 when n = 10. The standard deviation of the reconstructed displacement in Fig. 8 is 1.34 nm.

Fig. 8. Reconstructed displacement error due to the distortion of SMI signal.

Temperature Controller

LD Driver Lens

Target

PZT

EOM PD

Fig. 10. Measurement result of micro-displacement with frequency 10 Hz, amplitude 600 nm (p-p).

LD EOM Driver

Transimpedance Amplifier

PZT Driver

Signal Generator

A/D

Fig. 9. Experimental setup.

deviation of the reconstructed displacement when n = 10 versus the initial phase error is presented in Fig. 7. Compared with the FFT method, the integrating-bucket method is more sensitive to the initial phase error of the modulation signal. In our measurement, the initial phase error mainly depends on the performance of the A/D card. (5) When the variation in the laser frequency induced by optical feedback cannot be neglected completely, the self-mixing interference signal is not strictly sinusoidal as expressed in

Fig. 11. Measurement result of micro-displacement with frequency 10 Hz, amplitude 1600 nm (p-p).

Table 1 Measurement results of the PZT with sinusoidal micro-displacement (10 Hz). Controlled displacement (nm) (p-p)

Measurement results of four periods (nm) (p-p)

30 100 200 400 600 1000 1600 2000

32.9 107.5 194.6 406.8 605.9 1000.3 1593.3 2000.0

27.4 104.5 196.8 403.3 596.1 999.2 1596.1 2006.9

27.7 108.5 198.6 402.1 600.5 995.5 1595.8 2006.6

30.5 104.9 195.9 404.3 600.8 993.0 1600.0 2006.4

Max error (nm)

Standard deviation (nm)

2.9 8.5 5.4 6.8 6.1 7 6.7 6.9

2.63 7.59 4.40 5.16 4.12 4.83 5.09 6.64

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D. Guo, M. Wang / Optics Communications 283 (2010) 2186–2192 Table 2 Measurement results of the PZT with sinusoidal micro-displacement (p-p: 500 nm). Frequency of movement (Hz)

Measurement results of four periods (nm) (p-p)

10 20 40

489.6 490.3 493.5

495.1 491.9 492.3

495.9 498.5 498.7

Fig. 12. Measurement result of micro-displacement with frequency 8 Hz, amplitude 950 nm (p-p).

3. Experimental setup and results The experimental setup is shown in Fig. 9. It consists of a LD package with a photodetector, an aspherical collimating lens, an EOM (New Focus 4002) and an object target. The external target is fixed on a high-precision commercial PZT (PI, P-841.10) which can obtain a displacement resolution of 0.15 nm based on the close-loop design. The central wavelength k0, maximum output power of the LD are 638 nm and 5 mW, respectively. The temperature of both the laser mount and the lens holder, which are embodied in one piece of aluminum alloy is stabilized at D T < 0.01 C°. And the driving current of LD is stabilized at Di < 4 lA. The angle between the polarization direction of the laser diode and the electro-optically active axis of EOM is 0°. Then EOM can provide pure phase modulation with extremely low amplitude modulation. The interference signal monitored by the PD in the LD

492.9 497.4 493.0

Max error (nm)

Standard deviation (nm)

10.4 9.7 7

8.15 7.50 6.23

Fig. 14. Measurement result of the PZT hysteresis.

package is sent through a transimpedance amplifier and digitized with a 200-kHz, 12-bit analog-to-digital board on a PC bus (National Instrument, NI 6024E). First, the PZT is driven by a sinusoidal signal with frequency 10 Hz. Table 1 list the measurement results of the micro-movement with various amplitude. Fig. 10 is the experimentally reconstructed waveform of the PZT with amplitude 600 nm (p-p) and Fig. 11 is the reconstructed waveform of the PZT with amplitude 1600 nm (p-p). Then, the PZT is driven by a sinusoidal signal with amplitude 500 nm (p-p). Table 2 list the measurement results of the micromovement with various frequency. Again, the PZT is driven by triangular signal with frequency 8 Hz. Fig. 12 is the reconstructed result of the movement with amplitude 950 nm (p-p). The measurement result of the four periods is 951.4 nm, 956.1 nm, 950.4 nm and 957.0 nm, respectively and the standard deviation is 5.43 nm. At last, the high-precision commercial PZT is replaced by a common PZT. A 0–3 V triangular signal with frequency 40 Hz is amplified 40 times and then applied to the PZT. The micro-displacement reconstruction result is shown in Fig. 13. And the corresponding measurement results of PZT hysteresis is shown in Fig. 14. 4. Conclusion

Fig. 13. Micro-displacement reconstruction result of a common PZT.

Integrating-bucket technique is first introduced in the self-mixing interference signal analysis. In our experiment, reconstruction accuracy is on the order of nanometers for displacements of a few micrometers. This will be further improved by carefully refining the system’s electronics and mechanics. The measurement range is limited by both the highest modulation frequency of the EOM and the maximum sampling frequency of the A/D card. And the measurement time delay is short enough to control movement of the object. The proposed method for micro-displacement measurement is effective only for weak feedback, which is the main limitation of the technique.

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Acknowledgements This work was supported by the Natural Science Foundation of China and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (08KJB510008). References [1] S. Merlo, S. Donati, IEEE J. Quant. Electron. 33 (1997) 527.

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