A simple and effective displacement sensor based on external birefringent feedback in Nd:YAG lasers is demonstrated. The measurement is based on the principle that, when linearly polarized light passes through the birefringent external cavity and then is fed back into laser resonator by external object, a phase difference is generated between laser sinusoidal-modulated intensities in the two orthogonal directions. These two sinusoidal intensities with λ/2 period can be subdivided to λ/8 after 4-fold evaluation. Moreover, the directional discrimination can be easily obtained according to the phase relationship between them. The chief advantages of the sensor are that it is compact, small size, flexible, low cost, and robust. Experimental results have shown that the standard deviation of displacement measurement is 0.093μm in a 7mm range and 0.34μm in a 20mm range.
© 2009 Optical Society of America
Laser feedback interferometry (LFI) demonstrates unique features since it was reported in 1960’s to measure changes in the optical path length and the behavior of lasers . Compared with conventional interferometry (Michelson system), these features include not only similar phase sensitivity and modulation depth ratio, but also inherent simplicity (only one optical path), compactness, auto-collimation, robust as well as low cost. Thus, it has been widely studied for displacement measurement [2, 3], velocimetry [4, 5] and imaging [6, 7]. At first, LFI is limited to a resolution of λ/2 based on fringe counting method although it has a wide measurement range. Moreover, it is very hard to judge the movement direction of the external target. Although the sawtooth-like waveform of a laser diode working in the moderate feedback regime [2, 8] can be used to discriminate the direction, the method will not work if the laser does not work in the right feedback regime and the external feedback cavity is not aligned perfectly. The misaligned external cavity  can be used to increase the resolution of LFI, but the measurement range is limited no larger than 1mm. Nowadays, phase measurement technique based on external frequency shifted feedback [10, 11] has improved the resolution of LFI to nanometer level, while, it is at the price of complex structure, high cost, complicated electronics and measurements. Meanwhile, it is so sensitive to the environment that the measurement range and time is limited. Although Wan et al  have reported the quasi-common path LFI to eliminate the environment influence, the range can be no larger than the distance between the reference mirror and target, and also, the measurement speed is limited. In a word, LFI based on fringe counting method has large measurement range, compact structure and low cost, but low resolution and difficulties in direction discrimination. While, the LFI based on phase measurement technique has high resolution, but more complex structure, small measurement range, environment sensitive and high cost. A novel method  to generate two channels of sinusoidal feedback signals with phase difference 90 degree between them has been reported, which is totally different from that produced in conventional interferometer  where the lights do not be fed back into laser cavity. The laser used here is not only a light source but also a self-mixing oscillator combined the lights both in laser cavity and in external feedback cavity. Meanwhile, a simple analysis for the principle of subdivision and directional discrimination is also put forward . In previous researches for external birefringence feedback effects, the total intensity  or the two hopping  polarization states are investigated, but little attention has been paid to the phase relationship of the two orthogonally polarized lights induced by external birefringence element.
The main objective of this paper is to develop an integrated displacement sensor, which is made up of optical source, external feedback mirror, and signal processing. The paper is organized as follows. Section II presents the experiments and the detailed principle of subdivision and directional discrimination of this instrument. The contrast results between this new displacement measurement sensor and dual frequency laser interferometer (Agilent 5529A) is given in Section III and a discussion about the error source of this instrument is proposed in Section IV. Finally, the last section concludes this paper.
2. Experiments and Principle of subdivision and directional discrimination
When the laser is subjected to external birefringent feedback, the output intensity can be expressed as :
where ζ, is optical feedback factor, I0X and I0Y are the laser intensities without optical feedback, φ=4πνlo/c, and δ is the phase retardation of the wave plate inserted in the external cavity.
The experimental setup is just the same as that in Ref 13. When the phase retardation of the wave plate δ = 45°, a phase difference about 90° exists between the two in-quadrature laser intensities, as shown in Fig. 1(a), which is stable and independent of the variation of the external cavity length.
The principle of subdivision and directional discrimination has been simply analyzed in Ref 13, i.e., introducing a threshold intensity to obtain 4-fold subdivision and realize the directional discrimination, which is actualized by voltage comparators. From Fig. 1 (b), a period of optical feedback (λ/2) from A to E is subdivided into four equal parts: AB, BC, CD, and DE. Each zone corresponds to λ/8. Also the movement direction can be discriminated from the sequence of these four zones. Although, these four zones may be not equal exactly, it only brings the error to the last incomplete feedback period which has the zones less than four. The detailed analysis can be seen in last Section.
The detail flow chart of signal processing is as follows (as seen in Fig. 2:
The electronic circuit for signal processing includes a comparator, which reshapes the sinusoidal signals to square ones (named X, Y) at the fast transitions of the photodetected signals, and two monostable triggers capable of generating four pulses (named UPX, DPX, UPY, and DPY). Then these six channels of signals (X, Y, UPX, DPX, UPY, and DPY) are sent to a logical circuit for discriminating the movement direction. An up-down counter with a 6-digit display accumulates the countings which give the displacement in λ/8 units.
The logic discriminating direction circuit consists of AND-OR-INVERT Gates (74LS54) and HEX INVERTERS (74LS04). The forward and backward pulses (PDP, NDP), which represent the information of displacement amount and direction of the external object, will be generated by it when the above six channels of signals are sent to the logic discriminating direction circuit. The logic relation can be expressed as:
When the external feedback mirror moves forward and backward, the output signals are shown in Fig. 3. The PDP end outputs homogeneously four pulses when the external cavity length deceases λ/2, i.e., each pulse corresponding to λ/8. While the NDP end outputs the same four pulses, when the external cavity length increases λ/2. The result of PDP subtracting NDP represents the displacement amount and direction of the external object.
3. Instrument prototype and its performance
Most of the elements in Fig. 4(a) are just the same as those in Ref 13. Some different parts are as follows. The external feedback mirror 5 is adhered with a measuring staff 7, which has a range of about 20mm. The elements 1~7 are packaged by the instrument shell to construct the optical gauge head. The other part including pump source LD, PBS, detectors and signal processing circuits is integrated together to be far away from the gauge head. These two parts are connected with each other by fibers.
Figure 4(b) shows the prototype of this laser feedback measurement instrument.
Then we test the system’s accuracy by contrasting with dual frequency laser interferometer (Agilent 5529A). Due to the use of 4-fold subdivision method, the resolution of the system is 133nm. We use the laser feedback displacement sensor and the interferometer to measure the displacement synchronously with a step of about 200μm. Figure 5 shows the contrasting results between laser feedback system and interferometer (top), and the rms error σ in the displacement (bottom), which we determined by subtracting a least-squares fit from the data measured by our system. The linearity in a 7mm range is 2.57 × 10-5, and the standard deviation is 0.093μm. The linearity in the 20mm range is 1.2 × 10-5, and the standard deviation is 0.34μm. The errors may be caused by the swing of external feedback mirror in the process, or the environmental influences.
The principle of this novel displacement sensor is that the laser intensity changes one period when the external object moves half laser wavelength (seen Fig. 1(a)), which is similar to the traditional laser interferometer and independent of the measurement range. By introducing threshold intensity, a feedback period is subdivided into four equal zones (seen Fig. 1(b)). Thus, these four equal zones appear in sequence when the external object moves half wavelength. According to this principle, this new laser feedback sensor can self-calibrate to wavelength reference for every one period of these four zones without cumulative error.
4.1 value of the wavelength
The principle of this instrument decides that it self-calibrates to wavelength reference. But we do not know the accurate value about this wavelength, because this laser is not a frequency-stabilized one. For estimating the magnitude of frequency drift, which is induced by temperature change, we change the power of the LD from 40mW to 140mW. The frequency drift is founded to be smaller than 2.4GHz. Thus, in common room conditions, the relative error of the wavelength ∆λ/λ will not exceed 8.5 × 10-6. For the range of 20mm, the measurement error is ∆1 = 8.5 × 10-6 × 20mm = 0.17μm.
4.2 displacement amounts smaller than λ/8
In the beginning and the ending of the measurement, the tiny displacement which is smaller the resolution (λ/8) of the instrument, cannot be detected. And these two errors in the beginning and ending of the measurement partially counteract each other. Consequently, the maximum error of this kind will be smaller than one equivalent pulse, i.e., ∆2 = 0.133μm.
4.3 non-uniform four zones
It is very difficult to make the four zones equal exactly in the whole measurement range. But this error only exists in the last incomplete period where there are not four zones. Because this kind of instrument can self-calibrate to the central wavelength as long as the external object moves half wavelength of displacement according to the principle, i.e., every period of four zones strictly corresponding to the λ/2 displacement. Hence, the last incomplete period cannot self-calibrate, which causes the measurement error. Generally, this part of error ∆3 is smaller than λ/8, i.e., ∆3 = 0.133μm.
4.4 temperature influence
The temperature drift is about 1~2°C in common room conditions without constant temperature method. The temperature changes can induce variation of the laser cavity length and the external cavity length, correspondingly the resultant laser frequency drift and faked displacement of external object, respectively. To estimate this influence, a series of zero-point drift experiments has been done. The results show there are less than 4 faked pulses equivalent in 1 hour under common room conditions when the external object keeps still. So, this part of error is ∆4 = 0.532μm
Finally, the combined estimation error ∆ is
In summary, a novel displacement sensor is designed including optical source, external feedback cavity, signal processing circuits based on LD pumped microchip Nd:YAG laser with birefringence external cavity. This new system is compact, robust, low cost, and potential high resolution. The results contrasting with interferometer is: the standard deviation is 0.093μm in a 7mm range and 0.34μm in a 20mm range, and the linearity is 2.57 × 10-5 and 1.2 × 10-5 respectively in the range of 7mm and 20mm. Further experiments indicate that this new sensor has a large measurement range as far as 75mm, which is mainly limited by the misalignment of the external feedback mirror, and taking advantages of retro-reflectors, such as cat’s eye reflector, corner cube prism, the range can be further enlarged. Moreover, the resolution of λ/8 now can also be greatly improved by using of multi-fold subdivision methods adopted in grating encoders. Furthermore, its diode-pumped monolithic version offers a very high potential of cost reduction, small size, compact, and has no problems in manual handling.
The authors acknowledge the support of the National Nature Science Foundation of China (grant 50805084) and China Postdoctoral Science Foundation.
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