LASER DISPLACEMENT SENSOR using Self - Mixing Effect

Transcription

1 LASER DISPLACEMENT SENSOR using Self - Mixing Effect A report submitted to the department of Electrical Engineering, as a part of Guided Research in Electrical Engineering leading to the degree of Bachelor of Science in Electrical Engineering and Computer Science. Prepared by, ANKUR AGRAWAL Fall 2004 Supervised by : Dr. Dietmar Knipp, Professor of Electrical Engineering. School of Engineering and Science International University Bremen, Germany. 1

2 TABLE OF CONTENTS ACKNOWLEDGEMENTS Abstract INTRODUCTION Motivation to this work Laser Displacement Sensor. 7 2 THE SELF MIXING EFFECT Definition of Self-Mixing and a short history of optical feedback Operating Principle Theory of Self-Mixing Effect DISPLACEMENT SENSING Displacement Sensing using Self-Mixing Effect Relative Displacement Measurement using Self-Mixing Effect Absolute Displacement Measurement using Self-Mixing Effect EXPERIMENTS Experimental Setup Results and Discussion SUMMARY Tasks Completed Further Work REFERENCES APPENDIX

3 ACKNOWLEDGEMENTS I am very thankful to all the people who are involved in this project directly or indirectly. It was a great opportunity for me to work on this topic, and learn from my experiences. I am obliged to Dr. Dietmar Knipp for providing me this opportunity and guiding me all through the way until the completion of the project. I am thankful to him for enlightening me with his immense knowledge in the field and discussing the key points of the project. I would also like thank my family for their help and support during the project. Finally I would like to thank all my friends, Ajay Singh, Ajay Chauhan, Sudharsan Ganesan, and Hrishikesh Venkatraman for their hidden support towards development of this project. 3

4 Agrawal, Ankur, Laser Displacement Sensor using Self Mixing Effect Department of Electrical Engineering, International University Bremen, Bremen 28759, Germany. Guided Research, Fall Abstract This report describes a new technique for displacement sensing for the development of displacement sensors using the self-mixing effect in a diode laser. The main advantage of this technique for laser sensor development is the development of compact, low cost, precise displacement sensors using the existing laser diode design. The main advantages of using self-mixing effect for displacement sensing, is that the measurement set up is extremely compact consisting of the laser diode only. The use of fewer components decreases the price of the device. The accuracy of measurement is of the order of half wavelength of the light emitted. These benefits make this technique interesting for development of low cost displacement sensors. In this report, different methods for displacement measurement using self-mixing have been described. The first method considers relative displacement measurement, while the second method considers absolute target distance measurement. Many experiments and results have been discussed and a proof of the concept has been established, that self mixing is an effective tool for displacement sensing, and low cost displacement sensors can be developed using on this concept. Keywords: self-mixing effect, relative displacement measurement, absolute displacement measurement, and proof of concept. 4

5 1 INTRODUCTION The demand for low cost displacement sensors is increasing day by day [1-3]. The technology advances in the field of Laser Sensors in the last few decades has improved greatly, and this has led to some optical sensors already being routinely deployed in displacement sensing, range finding, velocity measurement, 3-D imaging, optical microscopes and various medical applications [1-6]. The development of an optical microphone, based on laser light intensity modulation, is already being researched by researchers all over the world [1]. All the above mentioned developments demand for low cost, displacement sensors. In this report it shall be proved, that self-mixing effect can be used for displacement sensing, and that this concept can be applied to development of low cost, and precise optical displacement sensors using the existing laser diode. The main advantages of using self-mixing Effect, based displacement sensors compared to conventional displacement sensors is that the measurement set up is simple, because basically only one optical component, the laser diode, is needed. The use of fewer components decreases the price of the device, thus making it inexpensive to use. Moreover, an interferometer can be implemented in a small size and it is easy to control because only one optical axis has to be adjusted. In addition, an accuracy, which corresponds to half of the wavelength of the light source, can be achieved. These benefits make self mixing based displacement sensing techniques interesting for development of low cost laser displacement sensors. However, before the application of this technique to development of displacement sensors, the primary concern is if self-mixing effect is a reliable approach for displacement sensing?? This question shall be answered in the next sections of this report. First, the motivation behind choosing self-mixing effect for displacement sensing shall be discussed followed by an introduction into classification of laser displacement sensors. This shall be followed by the theory of self-mixing effect and the different techniques for displacement sensing using self-mixing effect shall be discussed. A number of experiments performed, and the results obtained shall be discussed in the following section which shall be compared with the expected theoretical results to 5

6 establish the reliability of self-mixing effect for displacement sensing. A summary of the tasks accomplished and further work possible shall be provided at the end of the report. 1.1 Motivation to this work The approach of using Laser for displacement sensing is a well-established technique, which is widely used in the industrial and laboratory environments to measure displacement and applications such as mechanical metrology, machine-tool control, profilometry and vibrometry have flourished [4]. However, the usual techniques, mentioned above rely on the use an external interferometer, i.e. an optical transducer made up of lens, prisms and mirrors, which is read-out using laser light. These types of sensors are costly, complex and comparatively, not very easy to use. The use of Self Mixing effect for displacement sensing, which makes the technique described in this report, new has been chosen from the observed phenomena that optical feedback in semiconductor lasers significantly affects the operating behavior of the laser [3]. The back reflected Laser light when self mixed, or made to interfere with the light already present in the Laser cavity, causes fluctuations in the optical output power which can be observed using a monitor photodiode [4-6]. The availability of a monitor diode in the package of commercially available Laser diodes makes this technique ideal for the development of a displacement sensor. (Fig. 1.a) (Fig. 1. b) Fig. 1. Laser Diode (TO Package). Microscopic view (1.a) (Source: 6

7 As seen from Fig. 1, that presence of a monitor photodiode in the package of already available Laser diode reduces the number of components, and hence no other component in addition to the Laser diode is needed. This reduces the cost of the sensor and also makes it compact and also easy to operate. This makes the development of precise, low cost displacement sensors possible. 1.2 Laser Displacement Sensor A Laser Displacement sensor uses Laser Light modulation to sense displacement. It behaves as a Light modulating acoustic sensor without any direct conversion of mechanical energy (or sound energy in case of an Optical Microphone, discussed later) into Light Energy. The energy is supplied by the light but the displacement (signal) information is supplied by the acoustic signal. Light has three properties, Intensity (or amplitude), Phase (or frequency) and Polarization which can be modulated [1]. Hence, the laser displacement sensors are based on the following three approaches. 1. The Laser Light Intensity Modulation Technique. 2. The Polarization Modulation Technique. 3. The Phase Modulation Technique. The Laser Light Intensity Modulation Technique implies selective removal of a part of the energy of the signal from the optical path before it is monitored by a photodiode after being reflected from a target, whose displacement is to be sensed. The reflected light has energy less than the transmitted light, where the energy is shed mechanically due to the displacement of the target. This energy difference is used to sense displacement. This is one of the most common principles of operation of an Optical Microphone. The energy of the reflected light in constantly monitored and by its rate of the change the frequency of the sound wave can be extracted and the amount of light deflected will correspond to the energy of the sound wave. 7

8 Fig. 2. Optical Microphone based on Intensity Modulation (Source: Phone-or Company) Fig. 2. above shows an Optical Fiber Based Microphone manufactured by the Phone- Or[2] Company in Israel. The response of this microphone, currently offered by them, is not linear and a distortion of the signal towards the ends of its range is observed, as seen from the datasheets available at the company s website. However there is a large volume of world-wide research activity going on to improve the performance of the Optical Microphone. Implementing the optical microphone as a sound sensor in many applications is very useful since the devices benefit from its electromagnetic insusceptibility. The Laser Light Polarization Modulation Technique implies modulating the polarization of the laser light, which is eventually reduced to intensity modulation, for a direct response from the photodiode. Polarization Modulated Displacement sensors require a polarizer (analyzer) or birefrin gent element to perform this reduction. However, this scheme is rarely used in development of displacement sensors today [1] and hence we shall not discuss this approach further. The Laser Light Phase Modulation Technique implies, modulation of the phase (or frequency) of the laser light. This technique further branches into two approaches [1]. The first approach relies on use of grating type displacement sensors, which refers to a periodic structure intended to alter the transmission or reflection of light through a medium. The direction where constructive interference occurs due to multiple discontinuities of the grating serves as the exit path for light. The magnitude and 8

9 direction of the reflection paths depends on the periodicity of the optical grating, the wavelength and the phase of the laser light. The second approach is the Interferometric Phase Modulation Technique which relies on the use of Interferometers. This is the approach where we shall mainly concentrate our self. It operates by both changing the physical length or the refractive index of an optical test path and recombining the result from a reference path. The total phase φ of a wave with wavenumber k along an optical path of length L and refractive index n is φ = 2 π L 2 π nl k. L = = = k λ λ o Where, λo and ko are the free space wavelength and wavenumber, respectively [1]. The phase change due to a perturbation of the environment is o nl (1) φ = ko ( nl ) = ko( n L + L n) (2) It can be seen from equation (2) that whenever there is a displacement of the target, the length of the optical path, L changes, and hence the phase (or frequency) of the wave changes, since the refractive index remains constant independent of the variations in L. The Interferometric Phase Modulated displacement sensors require an interferometer to reduce the phase modulation to intensity modulation, to which the photodetector directly responds [1]. In the classical technique using the well-known Michelson and Mach Zehnder interferometers, the interferometer is an optical transducer made up of lens, prisms and mirrors, which is read-out using the laser light. In the approach chosen by us, a fraction of the light is back reflected or backscattered by a remote target and is allowed to re-enter the laser cavity. This modulates both the amplitude and the frequency of the lasing field. This approach is called as the self-mixing, feedback or induced-modulation interferometry, because a part of the fedback light is self-mixed or is made to interfere, with the light in the laser cavity. The laser source here (consisting of a laser diode and a photo diode), acts as a sensitive detector for the path length 2ks (where k = 2π/λ, λ being the wavelength of light and s is the target distance) which is the roundtrip distance traveled by the light to the target and back [4-9],[11-17]. The interference inside the laser cavity, due to self mixing causes a fluctuation in the optical output power. These fluctuations in the output power appear in the form of peaks, 9

10 which are then detected by the monitor photodiode, and hence, the displacement is sensed. The displacement sensing and the different approaches to displacement sensing using self mixing effect is discussed later in detail. The principle of self mixing was first demonstrated by Rudd using gas lasers to detect the Doppler shift caused by a moving remote reflector [4-7], [18-21]. The first complete self-mixing interferometer/vibrometer and use of Laser Diode as source/detector was reported by the Turning point experiments by Donati [4]. Many sensing applications for displacement and absolute distance measurements, based on the self-mixing effect in low-cost commercial Fabry Perot (FP) Laser Diodes have appeared in the scientific literatures since 1986 [4]. The main advantages of the displacement sensing using self-mixing scheme [4] are: (1) An optical interferometer external to the source is not needed; this results in a very simple, part-count-saving and compact set-up. (2) An external photo detector is not required, because the sensing of the signal is done by the monitor photodiode contained in the package of the Laser Diode. (3) The scheme has a high sensitivity, being a sort of coherent detection that easily attains the quantum detection regime (i.e. sub-nm sensitivity in path length is possible); (4) It can operate successfully on rough diffusive surfaces. (5) The laser beam carries the information and hence it can be picked up even at remote target locations. (6) The whole scheme is cost effective and easy to implement. 10

11 2. THE SELF MIXING EFFECT 2.1 Definition of self-mixing and a short history of optical feedback A self-mixing effect in a laser diode, occurs when a fraction of the emitted laser light, backreflected or backscattered from a remote target, re-enters into the laser cavity, and interacts with the original laser light, to generate a modulation of both the amplitude and the frequency of the lasing field. This causes fluctuations in optical output power, monitored by a photodetector placed on opposite side of laser cavity than primary light output. History of the self-mixing effect goes back to 1960 s when the laser was invented. The optical feedback in semiconductor lasers was earlier considered undesirable [3-11] as it significantly affects the operating behavior of the laser. The intensity modulation of the self-mixing effect was modeled as variable losses in a laser cavity and it was used only for refractive index monitoring [20-22]. However, the later experiments and the theoretical analyses have shown that optical feedback is useful in purposes such as mode selectivity and line width reduction [3-7], and that self-mixing finds a lot of applications in displacement sensing, distance measurement, range finding, velocity measurement, and many other fields. It was noticed that the fringe shift caused by an external reflector corresponds to the optical displacement of λ/2, where λ is the operating wavelength of the laser. In addition, the intensity modulation was noticed to be comparable to conventional interferometers. Based on these two features, the phenomenon was given the name self-mixing effect. Sometimes the names autodyne effect or self-modulation are also used [20-22]. 2.2 Operating Principle The principle of operation of self-mixing can be explained by considering the Electric field phasors of the Lasing field, denoted by Eo and the Electric field phasor of backrelected light, denoted by Er. The setup consists of the classic three mirror Fabry- Perot cavity model [3-4], [23] as shown in Fig. 3. The Laser diode is separated from the remote target by a distance s, and the internal monitoring photodiode is integrated into 11

12 the packaging of the laser diodes to observe the changes in the optical power in the laser with self-mixing. Fig. 3. Conventional Self-Mixing configuration using a Laser Diode. [4] The power emitted by the laser is denoted by P o and the power of light which is reflected back from the remote target is denoted by Pr, and the power attenuation of the external cavity is denoted by A. The reflected power is always less than the power emitted, since Pr = Po/A, and A is always greater than 1. A simple interpretation for the process of injection detection process is that when the small backreflected field phasor Er having phase φ(t) = 2ks(t), (where k = 2π/λ and s(t) is the distance of the remote target) re-enters the laser cavity and adds to the lasing field phasor Eo, then the lasing field amplitude and frequency are modulated by the term φ equal to 2ks. Hence, the Frequency Modulated term is sine(2ks) and the Amplitude Modulated term is cosine(2ks). From the two quadrature signals, the interferometric phase φ (which is equal to 2ks) can be conveniently retrieved, and hence the target displacement is measured [4]. This detection scheme, which resembles homodyning at radio frequencies applies conveniently to gas lasers. However, this explanation does not apply to single mode Fabry Perot laser diodes because the frequency modulation term cannot be detected by heterodyning, due to their large linewidth and because the intrinsic active medium of the semiconductor is non-linear in nature and it couples the refractive index and the optical gain to the injected carrier density, which makes the amplitude modulation term different from the cosine function. A complete analysis of the Laser Diode with optical feedback can be performed by using the equations first derived by Lang and Kobayashi [23]. This is discussed in detail in the next section. 12

13 2.3 Theory of Self - Mixing Effect The following theory of self-mixing is considered for a single mode semiconductor laser. The phenomenon can be modeled as three mirror Fabry-Perot cavity model [3-16], [19-22]. The schematic arrangement showing how light is reflected by a remote target to reenter the laser cavity, is shown in fig. 4. M1 and M2 represent the two mirrors forming the laser cavity of length L and refractive index µe, having reflection coefficients Γ 1 and Γ 2s respectively. The remote target is modeled as a third external mirror placed along the optical axis at a distance Lext from mirror M2, and having a reflection coefficient Γ 2ext. An internal monitor photodiode is located in the package behind the laser cavity to monitor the power fluctuations within it. Fig. 4. Three-mirror Fabry Perot model for self-mixing effect in a single mode semiconductor laser diode. [3] To simplify the figure, mirror M2 is combined with the target to form a single mirror having a complex reflection coefficient, given by equation (3). [3], [24].This combines the laser cavity and the external cavity to form a compound cavity. [24] Γ 2 ( 1 Γ2s ) Γ exp( c ) 2new ( ν ) = Γ2 S + 2ext j2πν τ ext (3) Where, νc is the optical emission frequency of the laser, and τext=2lext/c is the round trip propagation time through the external cavity (which is assumed to be air, and c is the speed of light). 13

14 As observed from the equation (3), the phase of reflectivity of the new complex mirror depends on the propagation delay, and hence the target distance. Therefore we can see that many properties of the light emitted from the Laser, are related to the target distance. Among these properties are the light output frequency, and the threshold gain. The laser threshold gain is defined as the minimum gain needed by the optical cavity to sustain lasing. The reflected light which re-enters the laser cavity is delayed, and hence it interferes with the light present inside in the cavity, affecting these threshold properties of t he laser. Therefore, any variation in the external reflectance, due to self-mixing will cause threshold gain to fluctuate. The variation of the threshold gain, g, of the laser with weak external optical feedback (κext<<1) is shown in equation (4). g = g c g th = κ L ext cos( 2 πν c τ ext ) (4) Where, gth is the threshold gain without optical feedback, gc is the threshold gain with feedback and κext is the coupling coefficient to the external cavity, which varies between zero and one. It indicates the quantity of light being coupled outside of the laser cavity. It is given by equation (5) κ ext = Γ Γ 2 s 2 ( 1 Γ 2 s ) 2 ext (5) The variation in threshold gain ( g) is minimum when the reflected light and light inside the laser cavity are in phase. This happens when the phase for the reflected light φext=2πντext, is an integer multiple of 2π, which occurs when the target distance is an integer multiple of λ/2, where λ is the wavelength of the light emitted by the laser. A similar explanation can be provided for variation in the emission frequency of the laser. Again due to the optical feedback, there is a variation in the external reflectance, which affects the frequency of emitted light. The fluctuation of the emission frequency, 14

15 which is related to the excess phase of the laser cavity (denoted by φl), is given by equation (6) τ ext 2 φl = 2πτext( ν c νth) + κ ext 1+ α sin( 2πνcτ ext + arctan( α) ) (6) where φl is the excess phase or τlround trip phase change for the compound cavity compared to 2πm, νth is the emission frequency without optical feedback, νc is the emission frequency with feedback, τl is the round trip delay in the laser cavity (τl=2lµeg/c, where µeg is the effective group refractive index.) α is called as the linewidth enhancement factor. It is related to the amount of stimulated emission and it is the ratio of the real to imaginary parts of the variation in the complex refractive index. Its value typically ranges between 3 and 7. [3] Now, from the theory of lasers, we also know that both the phase and the amplitude conditions must be fulfilled, for successful operation of the laser. The phase condition states that the round trip phase of the compound cavity must be an integer multiple of 2π. This shown in equation (4) 2 βl + φext = 2πm (7) Where, β is the phase constant of the optical wave (where β=2πνµe/c), m is an integer (m = 0, 1, 2.) and φext is the phase term of the external cavity. The amplitude condition states that the gain of the compound cavity must exceed the cavity losses to produce leasing. This is shown in equation (8) ( g c α s ) L Γ 1 Γ 2 s exp = 1 (8) Where, gc is threshold gain for the compound cavity and αs is the cavity loss factor [24]. By applying the phase condition, given by equation (7) to the excess phase φl given in equation (6), we can set φl to zero and the possible emission frequencies, with feedback can then be numerically solved from equation (6) and after calculation of ν, the gain gc can be solved from equation (4). [24] The strength of the external feedback is also a very important consideration in selfmixing interferometry. The feedback parameter of self-mixing effect is directly C fb τ τ L κ ext = ext α

16 proportional to the amount of light, re-entering the laser cavity. [3] Its value is given by equation (9). (9) Under weak feedback conditions (Cfb < 1) only one solution for emission frequency exists, with φl = 0, otherwise many solutions exist. Further, when Cfb > 1, the operation of the laser is no longer stable, leading to increased noise and mode hopping. [24]. Hence, Cfb < 1 shall always be considered for single mode operating laser. When the parameter Cfb << 1, power fluctuations at the laser output are sinusoidal, while as Cfb increases, the fluctuations become more saw-tooth-like. This saw-tooth signal can be used to detect the direction of a moving target, because the self-mixing signal changes its inclination in respect to the phase of the external target. [24] This shall be discussed in detail later in the report. The output of the laser is measured in terms of its optical output power. The power fluctuations, measured by the internal monitor photodiode are related to the variations in the threshold gain. For small variations of the threshold gain (i.e. g/g th << 1), the fluctuation in threshold gain can be also be related to the threshold current flowing through the laser. The gain in the diode laser is produced by driving a high current density into the active area. [25] Since the external optical feedback causes gain variation, hence the threshold current also varies. This change in the threshold current causes a change in the actual Laser Diode carrier density. [4] All the above statements are combined in equation (10). [3] g = g c g th T saγ ev I th (10) Where Ith is the variation in the threshold current ( Ith = Ic Ith), Ic is the laser diode threshold current with feedback, Ith is the laser diode threshold current without feedback, e is the elementary charge of an electron, Ts is the spontaneous recombination rate, V is the active volume of the laser cavity, a is the constant of proportionality used in the linearlised dependence of the threshold gain gth as a function of the carrier density n in 16

17 the laser diode depending on the threshold gain carrier density characteristics, and Γ is the mode confinement factor of the semiconductor. The optical output power of the laser can be related to the operating current when the laser is operating well above the threshold. This is given by equation (11) [3] (11) P = η ( Iop Ith ) Where, η is the conversion efficiency between optical output power and drive current in milliwatts/milliamperes. Now, by combining equations (11) and (10), we get equation (12), given below. P = η ( I op [ I th + gev T saγ ]) (12) Now, by substituting the value of g from equation (4) in (12) we get equation (13) P κ ext qv = η [ Iop + cos( 2πν cτ ext ) Ith ] LT saγ (13) We can see from Equation (13) that the optical output power of a laser has a sinusoidal relation with the target distance (since, τext = 2Lext/c) and the frequency of light emitted by the laser. Hence when we vary Lext linearly, Power varies periodically, with frequency ντext. This also corresponds to the mode hops occurring every λth/2 of the target displacement accompanied by a 2π round trip phase shift. This periodic fluctuation of the output optical power is sensed by the monitor photodiode. Therefore Self Mixing effect can be used to sense displacement of the target. The displacement sensing using Self Mixing Effect is discussed in detail in the next section of the report. 17

18 3. DISPLACEMENT SENSING 3.1 Displacement sensing using Self-Mixing Effect Self-Mixing Effect can be recognized as a special tool for performing a special type of displacement detection. This application of self-mixing, which is also commonly referred to as injection-detection, [10] is the optical counterpart of the superheterodyne detection, a well known technique in radio waves. Displacement sensing using self-mixing is a special type of coherent detection scheme, [10] which has the advantage of always working at the quantum limit of the incoming signal as well as of being eventually homodyne (the self-mixing scheme) or heterodyne (the synchronization scheme). In both cases, self-mixing can be used to measure the phase (related to optical pathlength) or the amplitude (related to the suffered attenuation) of the incoming signal. This, fact about self-mixing is most used in displacement sensing. As, seen from the equations derived in previous section, that due to self mixing effect in a semiconductor laser, a linear variation of target distance, causes power to fluctuate periodically, and the frequency of fluctuation is same as the resonant modes, which occurs at every λ/2 of the target displacement, where λ is the wavelength of light emitted by the laser. Using this property of self-mixing in lasers, we can employ many techniques for displacement sensing. However, we shall mainly concentrate on the following two approaches for displacement measurement. 1. Relative Displacement Measurement. 18

19 2. Absolute Displacement Measurement. Relative Displacement measurement is the classical technique for displacement sensing. I.e. here we displace the target, or in other words we vary the target distance. This varying of target distance then affects the threshold conditions and hence causes the output optical power to fluctuate. This fluctuating power signal, which appears in the form of a saw-tooth like waveform, is monitored by the photodiode and is used to determine the nature and magnitude of the displacement, and hence reconstruct the displacement waveform. The distance between the fluctuating peaks in the power waveform corresponds to a target displacement of a half-wavelength. Hence by counting the number of peaks for one half period of vibration of the signal we obtain the information about displacement of the target in one direction. Since the displacement sensed, is relative to a mean position, this method is called relative displacement measurement. This approach is discussed in detail in the next section. Absolute Displacement Measurement is a slightly different approach. In this case we determine the distance of the target from the laser, and hence we do not move the target. We know that by changing the frequency of light emitted, phase shifts of 2π will occur at every resonant mode of the external cavity in conjunction with the power fluctuations with frequency of ντext. The frequency is varied by modulating the injection current through the laser, which fluctuates the output power and the resonant frequency is obtained from the analysis of the output power waveform. Hence by determining the frequency at resonance, the value of Lext is calculated. This approach is also discussed in detail in the next section. Since the absolute distance of the target is determined, within acceptable error limits, this approach is called as the absolute displacement measurement. 3.2 Relative Displacement Measurement using Self-Mixing Effect In this approach we drive the laser diode with a constant dc injection current. Both the output optical power and the output frequency will remain constant, until there is no target present, or until there is no self mixing. But with the introduction of self-mixing, as 19

20 discussed in the previous section, as we move the target, a periodic output optical power fluctuations are observed, which are progressively distorted from a sinusoidal signal until it resembles a saw-tooth like waveform, with increasing feedback, as more and more light re-enters the laser cavity, in the week feedback regime (0.1 < Cfb < 1) [4]. This saw tooth like waveform exhibits fast switchings with hysteresis and these switchings occur periodically, with a period Lext = λ/2, and are upward and downward for, decreasing and increasing Lext, respectively, where λ is the wavelength of light emitted by the laser without feedback. [9]. This can also be derived from the theory of selfmixing discussed in the previous section. By looking at the equation (13), we can observe that the phase term of the optical output power with feedback is 2πνcτext and with linear variation in target distance (Lext) the power fluctuates sinusoidally with round trip phase shift of 2π. [17] Τhis is shown mathematically in equation (16). c λ πν ext = 2π Lext = = 2ν th 2 ( 2 cτ ) Where λ is the wavelength corresponding to the emission frequency without feedback. Hence we (14) see that the spacing between the adjacent peaks in the power waveform corresponds to a target displacement of half wavelength of the emitted light. Therefore, it is easy to recover variation in target distance Lext in steps of λ/2 without ambiguity. It must be noted that the transitions are independent of the signal slope, so that no constraint is imposed on the rate of rise of displacement function, up to the limit of the switching time. Fig. 5 below shows the theoretical diagram of the saw-tooth like output optical power with fast switchings and hysterisis produced by a sinusoidal motion of the target. Directional discrimination of this self-mixing interference waveform is obtained by changing the motion direction of the target showing up and down switchings as the target approaches or moves away from the laser. [10] 20

21 λ / 2 Fig. 5 Optical output power variations caused by a sinusoidal motion of the target. [10] In fig. 5 above, D represents the sinusoidal displacement of the target versus time, and Pc-Ps represents the difference in output optical power with and without feedback, versus time. The next step is the accurate reconstruction of the displacement waveform from the obtained output power signal. In classical interferometry, the reconstruction of the displacement waveform accurately, without any directional ambiguity would require two interferometric channels, i.e. two signals in quadrature or a displacement signal in presence of a carrier signal. But, in our case since we have only one single interferometric channel available, provided by direct detection of the output power of a semiconductor laser in presence of optical feedback the reconstruction of displacement signal without directional ambiguity becomes a challenging task. However the missing information is provided by the direction of transition of the saw-tooth waveform, and hence the signal is reconstructed. The displacement direction is recovered by the slope of the saw-tooth like signal and the direction of the transitions, which carries the information if whether the displacement signal is increasing or decreasing. Before describing the reconstruction process in detail, It shall be worthy to derrive a mathematical expression, which shall be used later to reconstruct the displacement waveform. 21

22 By substituting equations (6) in equation (9) and by setting the excess phase to zero we obtain equation (15) Cfb ν th = νc + sin ext + 2πτext ( 2πνcτ arctan( α) ) Now, using the expression derived for output power previously, and assuming that κ does not depend on the length of the external cavity and neglecting all second order contributions, the variation in the output optical power due to feedback with respect to the power without feedback can be shown by equation (16). [9] P = P max cos ( 2πνcτ ) Where P is the optical output power variations due to feedback with respect to the unperturbed laser. Now, by combing equations (15) and (16), the optical power variations with feedback as a function of the external cavity round trip phase is obtained and this is given in equation (17) when 0 < 2πνcτext < π and by equation (18) when -π < 2πνcτext < 0. [9] ext (15) (16) P C P P 2 2πτextνth = arccos( ) +.[ α( ) + 1 ( ) ] + 2πm P max 2 1+ α P max P max (17) Where, 0 < 2πνcτext < π and m = 0, 1, 2. 2πτ ext P C P P 2 νth = arccos( ) +.[ α( ) 1 ( ) ] + 2π ( m + 1) P max 2 1+ α P max P max (18) Where, -π < 2πνcτext < 0 and m = 0, 1, 2. T he equations (17) and (18) are the stationary solutions and provide output power for different values of the external cavity length. Since we now vary the external cavity length in a quasi-stationary way by applying the displacement signal, the equations (19) and (20) are obtained using F(t) = P(t)/ Pmax, using which the displacement signal is reconstructed. P(t) represents the variation of P over time. 1 C s( t) = ( ){ φ o + arccos( F( t)) +.[ α( F( t)) + 1 F 2 ( t) ] + 2πm} 2k 2 1+α (19) 22

23 For, df ds ( ).( ) < 0 dt dt 1 s( t) = ( ){ φo arccos( F( t)) + 2k C 2 1+ α.[ α( F( t)) 1 F 2 ( t) ] + 2π ( m + 1)} For, df ds ( ).( ) > 0 dt dt (20) Where k = 2π/λ and m = 0, 1, 2, S(t) is the displacement signal which is superimposed on a quiescent distance Lext and which is reconstructed using the equations given above. The value of m is increased or decreased by 1 if ds/dt is positive or negative respectively, and m is updated every two zero crossings of F(t). φo is the initial phase (2kLext) having modulus 2π. Fig. 6 shows the displacement waveform reconstruction process for the optical output power variation of a laser diode with feedback observed due to a target moving sinusoidally. 23

24 λ 2 Fig. 6. Reconstruction of displacement waveform for Relative displacement measurement. In Fig. 6 above v(t) represents the sinusoidal waveform which is used to modulate the target distance. F(t) is the distorted cosinusoidal signal corresponding to ratio of variation in optical output power due to feedback with respect to unperturbed laser, to the maximum variation in power. S(t) is the reconstructed displacement signal corresponding to the optical power waveform. For reconstruction, we shall assume that displacement is zero at zero time. As we know that the spacing between the neighboring peaks corresponds to λ/2 of the target displacement, the motion of the target is then obtained by just adding these λ/2 displacements, with the proper sign. The peak to peak amplitude of the sinusoidal displacement is given by multiplying the number of peaks observed during the half-period of the vibration with λ/2. The sign of the derivative of displacement waveform (ds/dt) is obtained from the inspection of F(t) waveform. The displacement signal, s(t) increases when F(t) exhibits 24

25 downward transitions, from positive to negative. And vice versa when F(t) exhibits upward transitions from negative to positive, then the displacement signal, s(t) decreases. Also, by looking at the points in F(t) where two consecutive transitions (from positive to negative and then vice versa) exhibits similar slopes, we can discern where the signal is stationary. Hence this reconstruction routine repeats itself in intervals of transitions from positive to negative or vice versa. These above rules can be implemented to reconstruct the displacement waveform from F(t) with the help of equations (19) and (20) on a computer, to give a resolution of λ/2, [17] as shown in Fig. 6. However, with good signal processing and with the choice of good algorithms a resolution up to λ/12 can also be achieved. [6]. 3.3 Absolute Displacement Measurement using Self-Mixing Effect In this approach, we shall measure the distance of the target. From the analysis if equation (13) it is already known that the power of the laser with feedback is related to the emission frequency and that the power exhibits a sinusoidal relationship with the target distance Lext. Since we shall measure the target distance, hence we shall keep it constant and modulate is the other factor (viz. emission frequency) to obtain the fluctuations in the output power. These output power fluctuations, which appear with a frequency of ντext, due to the variation in emission frequency, also correspond to the resonant modes of the external cavity, where a phase shift of 2π occurs. Hence, a phase shift of 2π due to variation in frequency occurs when, 2 πm = 2πντ ext (21) Where, m corresponds to the resonant mode, such that m = 0, 1, It can be seen from equation (21) that distinct resonant modes develop in the laser cavity and the frequency of the laser is forced to the closest one of these modes. Now, at the fist mode of resonance, when m = 1, the resonant frequency (known as fundamental mode frequency, denoted by νo) can be defined as the difference between 25

26 frequencies of any two adjacent resonant modes. Hence, by substituting m = 1 and expanding τext (τext = 2Lext/c) in equation (21), we get equation (22), which gives us the expression for the absolute distance of the target from the laser. L ext c = 2ν o Hence we see that by determining the resonant frequency it is possible to obtain the absolute distance of the target. We shall now describe the process by which we shall vary the emission frequency which will then cause power fluctuations, which shall be used to determine this resonant frequency. It has been found by researchers [3-4] that by modulating the current through the laser with a triangular wave the optical frequency of the laser beam is varied proportionally to the current flowing in the laser. This current modulation also effects the excess phase of the laser, described by equation (6) as the emission frequency of the laser without feedback (νth), changes with current. Therefore an additional parameter is introduced in the expression for excess phase to indicate the amount of optical frequency change with the injected current. It is known as the frequency modulation coefficient, denoted by Ω, and expressed in GigaHertz/milliAmperes. [3-4] Thus with this new parameter, equation (6) now becomes equation (23) given below. (22) ( 2πνcτ arctan( )) φ L = 2πτext( νc ( νth + IΩ)) + Cfbsin + α ext (23) Hence there is also a corresponding change in the expression for the output optical power, given by equation (13) and the new expression for optical output power with feedback includes a new current modulation term I, [3] and is described by equation (24) given below. P κ = η [ Iop + I + cos( 2πν cτext )] LT saγ ext qv (24) 26

27 The plot of the output optical power of a laser diode with feedback and injection current modulated with a triangular wave, is a triangular wave with small steps. These steps correspond to the periodic power fluctuations caused by the different resonant modes occurring in the laser cavity. Fig. 7 below shows a theoretical curve produced in Matlab, to demonstrate the output power waveform with small steps due to current modulation of 2milliAmperes peak to peak with a triangular wave for absolute displacement measurement using self-mixing effect. Fig. 7 Graph of Injection current over time with the corresponding fluctuation of Power over time. Simulation. In Fig. 7 above, Delta I represents the triangular waveform of the injection current modulation, which produces the corresponding triangular power waveform with the small periodic steps, represented by Delta P, as would be monitored by the photodiode in the real experimental setup. These fluctuations represent the resonant modes in the laser cavity and the average spacing between these steps (fluctuations) is proportional to the spacing between the resonant modes. [3-4]. Hence by finding the average distance 27

28 between these steps, once can determine the resonant frequency, which is then used to determine the target distance according to equation (22). To make the power fluctuations more distinct, so that the average spacing between them can be determined more accurately, we take a derivate of the power waveform, and we get a series of sharp peaks, where each peak corresponds to a fluctuation in the power waveform, and the average spacing between the fluctuations is simply the average spacing between the peaks. The derivative of the power waveform is shown in Fig. 8 ψ Fig.8 Graph of derivative of power fluctuations over time. - Simulation In Fig. 8 above ψ represents the distance between two peaks and correspondinglyψ avg is used to denote the average distance between the peaks. Since the spacing between the resonant modes is proportional to the average distance between the peaks, the fundamental frequency is therefore calculated using this average distance. This is shown in equation (25) [3], below. ν o = 4 ipkfω ψ avg (25) 28

29 Where, ipk is the peak amplitude of the injection current, f is the modulation frequency, and 4ipkf is the slope of the current signal. Substituting equation (25) in equation (22) we get equation (26), as given below. L ext = c 8ipkfΩ ψ avg (26) Hence, the absolute distance to the target from the laser, is determined using equation (26). Hence absolute displacement measurement is performed using the information (average distance between the peaks) obtained due to self-mixing effect in a laser diode. 4. EXPERIMENTS 4.1 Experimental Setup 29

30 The main aim of the project was to prove if self-mixing effect can be used as a tool for displacement sensing. This proof of concept was required to be established, so that further experiments of displacement sensing can be performed and the research can be extended to various important applications like the optical microphone. Many experiments were conducted to observe the self-mixing effect in semiconductor lasers, with and without displacement of the target, to study the process of displacement sensing using self mixing effect, and the results were studied and noted. The results and findings of the experiments are discussed later in the next section. Let us first discuss the steps of the experiments. Fig. 9 below shows the block diagram of the experimental setup used during the experiments. Fig. 9 Block diagram of the used experimental set-up. As we can observe from the block diagram in Fig. 9 above, the experimental setup is extremely simple and compact, consisting of a laser diode and photodiode in a single package, a reflecting target (the piezo actuator with the mirror), and some other instruments to support and enable their operation during the course of the experiments. The piezo actuator used, simulates the modulation of a target membrane, and a mirror is attached at the end of the piezo actuator facing the laser light beam, to reflect the light back into the laser cavity. The piezo controller is used to control the piezo actuator, and hence, the movement of the target. A laser driver is used to drive the laser at the desired current levels. This in-turn also controls the intensity and the frequency of the light emitted by the laser. The signal generator is used to produce a test signal (or waveform), which is applied to the piezo controller as the displacement signal incase of relative 30

31 displacement measurement experiments, or is applied to the laser driver (shown by the dashed line) to modulate the laser injection current in case of absolute displacement measurement experiments. A transimpedence amplifier is connected at the output of the monitor photodiode, to convert the low magnitude current produced by the photodiode into a measurable voltage which is then analyzed by the oscilloscope connected to the transimpedence amplifier. The entire setup is mounted on an optical table, before the experiments are performed. All experiments were performed in the conditions of weak feedback regime (i.e. Cfb < 1). Many experiments were conducted, to study the effect of self-mixing on the output of the laser with moving as well as stationery targets, and to perform a study of both approaches for displacement sensing, namely relative displacement sensing and absolute displacement sensing, and the corresponding output optical power waveforms were observed. It was noted that there are many factors controlling the output of lasers in addition to the ones mentioned in the theoretical analysis. The most important ones among them are optical alignment of the laser and the target reflecting the laser light, and the experimental conditions like stability of the optical table used (due to the effect of vibrations). A systematic experimental study of the two displacement sensing approaches was performed and the data collected was compared with the expected theoretical results, to establish a proof of the concept that self-mixing effect can be used to perform displacement sensing. In the first study of absolute displacement measurement the laser driver was driven by a triangular waveform generated by the signal generator, to produce a current variation of 2 milliamperes peak to peak and the corresponding variations in the photodiode current was observed. The results were similar to the results of simulated experiment, already discussed in the previous section; hence we will not discuss it again. In the second study of relative displacement measurement, the piezo actuator (target) was modulated by a sinusoidal wave of 6 volts peak to peak biased at +2 volts above ground level, generated by the signal generator. Many sets of observations were noted by varying the signal voltage and the biasing levels. The laser was constantly driven at a current of 87 milliamperes, and the output current of the photodiode was observed in the oscilloscope, with and without a filtering of the output signal at 10 Hertz bandwidth limit. 31

32 To limit the noise in the output signal, an assembly of a band pass filter was also tried, using the design described in [3]. However, it was not used in our final experiments and hence we will not discuss it further. The obtained results and screenshots from the oscilloscope, from the experiment were carefully observed, and are discussed in the next section. 4.2 Results and Discussion The experiments for relative displacement measurement were conducted as described in the previous section and the screenshots obtained from the oscilloscope are shown in Fig. 10. below. Fig 10a. Yellow trace: displacement signal applied to Piezo Actuator. (5 Volts/div) (Target modulated with a 6 Volts p-p sinusoidal signal at +2Volts bias). Blue trace: Output signal of Photodiode. (50 millivolts/div) 32

33 Fig 10b. Yellow trace: displacement signal applied to Piezo Actuator (5 volts/div) (Target modulated with a 4 Volts p-p sinusoidal signal at +1Volts bias). Blue trace: Output signal of Photodiode. (50 millivolts/div) In the screenshots shown in Fig. 10a. and 10b., the yellow signal represents the displacement signal applied to the Piezo actuator, generated by the signal generator. The Blue signal represents the output of the photodiode via the transimpedence amplifier. As we observe the output of the photodiode in the screenshots, we can see a series of fluctuating peaks which represents the power fluctuations. The presence of spikes in the output signal proves the success of the experiment. These fluctuations or spikes represent the mode hops which occur at every λ/2 of the target displacement accompanied by a 2π phase shift, as discussed in the theory or relative displacement measurement. Hence the spacing between two adjacent spikes corresponds to λ/2 displacement of the target. The information about the target displacement is calculated by simply counting the number of peaks for one half period of vibration of the signal and then multiplying by λ/2. This gives the information about the peak to peak amplitude of the sinusoidal displacement. Using this information, the sinusoidal displacement waveform can be reconstructed, as explained in the theory. Hence the presence of spikes in the output 33

34 signal of the photodiode gives us information about the displacement of the target and this establishes a proof of the concept that self-mixing effect can be used for displacement sensing. The first sixty samples of the data collected for the output signal from the oscilloscope, corresponding to the screenshot seen in Fig. 10a, have been reproduced below in table 1. The output of photodiode is further plotted in Fig 11, to show the spikes more clearly. Table 1: Data showing output signal corresponding to target modulation by 6 volts p-p sinusoidal signal biased at +2 volts above ground level, versus time. Time (sec) Applied Displacement Signal (volts) Photodiode Output Signal (volts)

35 Time (sec) Applied Displacement Signal (volts) Photodiode Output Signal (volts) The data provided above is taken for the following settings of the oscilloscope. Scale: Displacement signal 5 Offset: Displacement signal -0.2 Scale: Photodiode output 0.05 Offset: Photodiode output Scale: Time

36 The output of the photodiode from table 1 is plotted below in Fig. 11, below, to show the presence of the spikes in the output more clearly. Fig. 11 Graph showing the plot of photodiode output versus time Fig. 11 shows the plot of the photodiode output against time. As observed in the graph, the presence of spikes or fluctuations clearly agrees to the theoretical explanation for the relative displacement measurement using self-mixing. This proves the success of the experiment and establishes the concept of using self mixing for displacement sensing and hence it is verified that self-mixing is a reliable approach for displacement sensing. The plot for data collected for the output signal for the applied sinusoidal displacement signal at 4 volts peak to peak with +1 volts biasing above ground level shown in screenshot in Fig 10b gives similar results to the plot in Fig 11, and has been provided in the appendix attached at the end of the repot. 36

37 5. SUMMARY It has been demonstrated that Self-Mixing Effect can be used as a reliable tool for displacement measurement. It has been clearly established from the analysis of the results obtained from experiments performed, that self-mixing effect can be used to perform displacement sensing. This was the main aim of the project and this has been fulfilled to the fullest satisfaction. It can be concluded that self mixing effect as a technique for displacement sensing, can be used as a reliable approach for the development of low cost, extremely compact, precise laser displacement sensors. During the course of the project to prove the concept of applicability of self mixing for displacement sensing, several tasks have been accomplished. These have been discussed in the next section. 5.1 Tasks Completed 1. Studying the two approaches of displacement sensing namely relative displacement sensing and absolute displacement sensing using the self-mixing effect. Various experiments and theoretical analysis was performed to study the two displacement measurement approaches. Simulations were prepared in Matlab and the results of the experiments were compared with those of the theoretical simulations prepared. The results have been documented and serve as the foundation for our conclusions. 2. Identified that self-mixing is a reliable approach for displacement sensing. The study and results of experiments of relative displacement measurement shows that self-mixing effect can be used for relative displacement sensing. Similarly the study of absolute displacement measurement shows that self mixing effect can be used to perform absolute target distance measurements. 3. Building the experimental setup. The experimental setup was designed, built and tested, and there experiments were performed there on. There were many constraints with respect to the physical setup of the experiment. Optical alignment of the laser and the target was a major challenge to the experimental setup. It 37

38 plays a very important role in the output signal, and hence accurate optical alignment is very important. Other factors which affected the output included stability of the optical table, due to the vibrations. 4. Established a proof of the concept that Self-Mixing effect can be used for displacement sensing, and this technique can hence be applied to development of low cost displacement sensors. 5.2 Further Work The research can be extended to the development of precise low cost displacement sensors for various displacement sensing purposes. The best example of development of such a sensor is the development of optical microphones. Other applications of these displacement sensors based on self-mixing shall be in biomedical optics for LDF, blood pressure pulse registration and measurement of skin vibration due to MMG. [25] Recently, displacement sensors based on self-mixing technique have also been introduced to everyday practical applications. It is reported to have been used in optical touch sensitive interfaces where two self-mixing interferometers measure the movement of the fingertip. The movement of the finger is used to control the optical scroll device. [31], [25-28]. Fig. 12 below shows the use of such a sensor for scrolling function in Phillips mobile phone handset. Fig. 12 Application of displacement sensors based on self mixing effect in Phillips Mobile phone handsets for scrolling function. 38