The self-mixing interference effects with a folding feedback cavity in a Zeeman-birefringence dual frequency laser have been investigated theoretically and experimentally. The fringe frequency of the self-mixing interference system can be doubled due to the hollow cube corner prism, with which a folding cavity is formed. The intensities of the two frequencies are changed periodically in the modulation of the external cavity length. When the phase difference between the two frequencies equals π/2, the intensity modulation curves can be divided into four zones with equal width in a period. Each zone corresponds to one polarization state. Based on the experimental results, a novel displacement sensor with a high resolution of λ/16, as well as functions of direction discrimination, is discussed.
©2006 Optical Society of America
Much attention [1–2] has been drawn to the fact that laser behaviors can be significantly affected by the external optical feedback, that is, feedback of a portion of the laser output into the laser cavity from a reflecting external surface. The phenomenon of optical feedback is also called self-mixing interference. Compared with conventional interference system, self-mixing interference system does not only have similar phase sensitivity and modulation depth ratio, but also inherent simplicity, compactness and robustness, as well as self-aligning capability. Thus, much theoretical and experimental research based on self-mixing interference has been done in the fields of Doppler velocity measurement [3–5], distance and displacement [6–8], imaging and vibration .
However, a fringe shift of the laser output intensity corresponds to only a half wavelength displacement of a feedback mirror in the above self-mixing interference system, which critically restricts the system’s resolution from being further advanced. Many efforts have been made to advance the resolution of self-mixing interference systems. Addy et al.  reported that the fringe frequency of a self-mixing interferometric sensor system can be doubled because of the misalignment of the external reflector. However, the frequency-doubled fringe is observed to be independent from the external cavity length on a scale of approximately 1 mm only, and it cannot be used for a displacement measurement greater than 1 mm. Liu et al.  also presented a way to double the fringe frequency by putting a quartz crystal plate in the external cavity at a specific angle between the crystalline axis and the laser beam. It relies upon the angle so strictly that it is hard to get the doubled fringe. Moreover, the intensity modulation depth of the doubled fringe is much shallower than that of the normal fringes.
This paper presents a simple but effective method to double the fringe frequency using a folding feedback cavity, which is formed by a hollow cube corner prism (HCCP). It is well known that the HCCP has a property of retro-direction reflection. The output light passed through the HCCP is parallel to the incident light, and then it is reflected back into the laser by a mirror. The HCCP and the mirror compose a folding feedback cavity. The twice passing in the external cavity doubles the fringe frequency. Although the cube corner prisms were used in Michelson interferometers many years ago, the special use in the self-mixing interferometer system of dual frequency laser is still supposed to be the first time.
There are three kinds of dual frequency He-Ne lasers. The first one is the Zeeman dual frequency laser with frequency difference no larger than 3 MHz. The second one is the birefringent dual frequency laser, and the frequency difference is at least 40 MHz. The Zeeman-birefringent dual frequency laser  used in our experiments makes up the blank of frequency difference ranging from 3 to 40 MHz. With the optical feedback loop on, the intensities of the two frequencies are changed periodically. The fringe frequencies of the two modes are both doubled. So the fringe shift will correspond to λ/4. When the phase difference between the two frequencies equals π/2, the intensity modulation curves can be divided into four zones with equal width in a period. Each zone corresponds to one polarization state. Based on the experimental results, a novel displacement sensor is discussed.
2. Experimental setup
The experimental setup is schematically shown in Fig. 1. A Zeeman-birefringent dual frequency laser is used. QC is a uniaxial quartz crystal inside the laser cavity. Due to the birefringent effect of QC, a single mode of the laser can be split into two orthogonally polarized modes that are called o-light and e-light. θ is the angle between the crystalline axis of QC and the laser axis, which determines the frequency difference between the two modes. B presents the homogeneous magnetic field.
The internal cavity consists of a concave mirror M1 and a plane mirror M2. The internal cavity length is 165 mm. The whole right surface of M2 is coated with high reflectivity film and half of its left surface is coated with low reflectivity film whose reflectivity is 18%, while the other half of the left surface is antireflection coated, as shown in Fig. 2(a). Here M2 is used as both the internal cavity mirror and the external feedback mirror. HCCP is a hollow cube corner prism constructed to have three reflectors with high reflecting layers at incident angle 54.7°. The three reflectors are arranged as three adjacent faces of a cube, as shown in Fig. 2(b). Compared with common cube corner prism, the HCCP has less absorbing loss. P is a piezoelectric transducer (PZT) that drives the HCCP. BS is a beam splitter that is used to divide the taillights into two parts. One part is divided by a Wollaston prism W into two orthogonally polarized lights - o-light and e-light, and their intensities are detected by photo detectors D1 and D2, respectively; and the other part is detected by D3.
3. Theoretical analysis
Various theories based on the model of equivalent cavity of Fabry-Perot interferometer have been developed and used primarily for explaining the intensity modulation characteristics in a single mode laser. In our study here a simple theory is presented to relate the dual frequency laser with a folding feedback cavity.
In the presence of optical feedback, the light beam can be divided into two parts. The first one travels within the internal cavity, while the other travels in the external cavity and then couples into the internal cavity. These two parts superpose in the internal cavity and construct the self-mixing interference. After the two trips, the initial electric vector E 0 (t) in the laser cavity turns into final electric vector E(t) and these trips can be expressed as
where r 1 is the amplitude reflection coefficient of M1 and r 2 , r 3 are that of M2 and an assumed plane mirror M3 replacing the HCCP, respectively. t 2 is the amplitude transmission coefficient of M2. ξ, is the feedback coupling coefficient of M2 when the beam travels from the external cavity to the internal cavity. L and l are the laser cavity length and external cavity length, respectively. c is the speed of light in vacuum, g is the linear gain per unit length due to the simulated emission inside the laser cavity in the presence of feedback. n is the effective refractive index of the laser cavity material. Δl presents the displacement of the external feedback mirror of M3 in the absence of HCCP.
For E(t) = E 0 (t) when the laser comes to balance, Eq. (1) can be expressed as
Define α = t 2 r 3 ξ/r 2 , δl =4πvl/c , φ = 4πv∙Δl/c and φ is the variation of the external phase. Solving Eq. (2) and we get
For a He-Ne laser, t 2 is very small. ζ is nearly the same to t 2. This results that α = t 2 r 3 ξ/r 2 is very small too. So α is much smaller than 1.
It is known that (x 2 + y 2)½ ≈ x when x ≫ y. With this approximation, Eq. (3) can be simplified. We can obtain
Considering ln(1 + z) ≈ z when z is very small, Eq. (5) is solved to be
In the absence of M3, there is no feedback and α = 0 , so the linear gain per unit length in the absence of feedback is
The excess required gain Δg in the presence of optical feedback can be expressed as
Since the laser intensity can be expressed as 
where K is a constant and I 0 is the initial laser intensity without optical feedback.
When the HCCP is placed in the external cavity, the displacement Δl of the HCCP will bring the displacement of the external feedback cavity about 2Δl. It is because of the HCCP causing the feedback cavity folded that the displacement variation of the external cavity doubled. So the variation of the external phase φH can be expressed as
And the laser output intensity with optical feedback in the present of HCCP can be expressed as
In Fig. 3 and also in the following other figures, the vertical axis represents the laser intensity and the horizontal axis represents the voltage of the D/A card. The PZT changes a displacement of λ/2 when the voltage of the D/A card changes 500 mV. The simulation results of the laser output intensity without and with HCCP are shown as dot-pointed curve I 1 in Fig. 3(a), and circle-pointed curve I 2 in Fig. 3(b), respectively. The theoretical analysis shows that the fringe frequency can be doubled by the addition of a hollow cube corner prism in the external cavity.
Because the two orthogonally polarized lights share the same laser available gain in the laser cavity, the sum of the gain of o-light and e-light without optical feedback is the same as that with optical feedback. Thus, the excess required gain Δgo of o-light and Δge of e-light can be expressed as
With Eq. (14), we can express the laser intensity of o-light Io and that of e-light Ie in the presence of hollow cube corner prism, respectively as
where I 0o and I 0e are the laser intensities of o-light and e-light without optical feedback.
And the phase difference δ between Io and Ie can be expressed as 
where Δv is the frequency difference between the two orthogonal polarized lights.
Based on Eq. (15) and Eq. (16), when the phase difference and the initial intensities of o-light and e-light are equal without optical feedback, the simulated laser intensity variations of o-light and e-light with optical feedback are shown in Fig. 4.
As Fig. 4(a) shows, in the presence of optical feedback, the intensities of the two orthogonal polarized lights are both modulated by the variation of the external cavity length. The laser intensities of o-light and e-light change crossly and periodically. Both of the lights are doubled, and a period corresponds to a λ/4 variation of the external cavity length.
If a suitable threshold intensity is assumed and the intensity lower than the threshold can be ignored, as shown in Fig. 4(b), a period can be divided into four ranges as marked in Fig. 4(b) by A, B, C, D, and E. From A to B, only o-light oscillates. From B to C, both o-light and e-light oscillate. And only e-light oscillate from C to D. From D to E, neither o-light nor e-light exists. Thus a period of λ/4 can be subdivided into four polarization states and the resolution of a self-mixing sensing system can be further increased.
In addition, the direction of the sensing system can be judged with these polarization states. As we know, the o-light and e-light have orthogonal linear polarization and the four states appear in turn. If the HCCP moves toward the mirror M2 to decrease the external cavity length, the appearance order of the four polarization states will be o-light (A to B), o-light and e-light (B to C), e-light (C to D), no light (D to E), o-light (E to F), and so on. On the contrary, if the HCCP moves away from the mirror M2 to increase the external cavity length, the appearance order of the four polarization states will be o-light (F to E), no light (E to D), e-light (D to C), o-light and e-light (C to B), o-light (B to A), and so on. So, by analyzing the order of the appearance of the four polarization states of laser output, the direction of the displacement can be observed.
4. Experimental results and discussion
At first, let’s consider a conventional optical feedback system. The HCCP is replaced by a feedback mirror in Fig. 1. PZT is used to drive the feedback mirror to change the length of feedback cavity. The laser output intensity detected by D3 is shown in Fig. 5(a). Then, let’s consider the optical feedback with a folding feedback cavity as shown in Fig. 1. The output light passing through HCCP is parallel to the incident light, and then it reflects back to the laser by the mirror M2. The HCCP and the mirror compose a folding feedback cavity. The twice passing in the external cavity doubles the fringe frequency as shown in Fig. 5(b).
Comparing Fig. 5(b) with Fig. 5(a), the fringe frequency of the self-mixing interference system can be doubled with the folding feedback cavity, which is predicted by theory above. By using the folding cavity, the resolution of a sensing system can be doubled, besides which the type of the laser is on longer a restriction.
At last, the intensity modulation curves of the two orthogonally polarized lights are observed. The initial output intensities of o-light and e-light are made even by changing the internal cavity length before the optical feedback loop is connected. θ can be adjusted to change the frequency difference of the dual frequency laser. When the frequency difference is equal to 28 MHz, the detected intensities of o-light and e-light are shown in Fig. 6.
Comparing Fig. 6 with Fig. 4(a), we can find that they are nearly the same. The intensity characteristics of the two lights have been analyzed in the theoretical analysis part. So what the theory predicts in Fig. 4(a) can be seen in the experimental result as shown in Fig. 6. When the HCCP is moved along laser axis, the intensities of both the two lights change periodically. An increase of o-light intensity always accompanies a decrease of e-light intensity. The frequencies of the two modulation curves are also doubled.
From above, we can see that the experimental results are in good agreement with the theoretical analysis.
Figure 6 suggests that the resolution of a self-mixing sensing system based on dual frequency laser can be further increased. If an appropriate threshold intensity for the output light is given by a signal processing circuits, the remaining intensity curves in each period shows four different polarization zones: o-light, o-light and e-light, e-light, and no light, as shown in Fig. 4(b). So a period of λ/4 with a folding feedback cavity can be subdivided into four polarization states. The resolution of λ/16 can be obtained.
Of course, it is difficult to make these four zones be exactly equal to each other, although the width of these zones can be adjusted by the frequency difference. However, the bandwidth of each period, such as the four zones from A to D, is fixed, which is equal λ/4. Therefore, the HCCP is moved by λ/4 if the four zones appear once in turn, which is always tenable. As a result, the sensor based on this principle can calibrate itself once every period and there is no error accumulation during the measurement. In other words, it has the function of self-calibration.
Furthermore, the increase or decrease of feedback length can be discriminated by the different appearing sequences of the four zones. For example, if a polarized mode is located at point C in Fig. 4(b), it will enter the zone of both o-light and e-light oscillate (BC) with the feedback length increasing, or the zone of only e-light oscillate (CD) with the feedback length decreasing. Using two electronic comparators after the detectors to transform each zone into an electronic pulse and, respectively, sending the pulses to the non-inverse or the inverse input port of a reversible counter, a displacement with direction, finally, can be obtained by counting the quantity of pulses.
In a conventional optical feedback system, the feedback mirror should be adjusted exactly to ensure the light can be fed back into the laser cavity, which is restricted in real application. But in our system, M2 is used as both internal cavity mirror and the external feedback mirror. The output light passing through HCCP is parallel to the incident light. It does not require to adjust HCCP exactly, which makes the system has high noise immunity.
So a novel displacement sensor with a high resolution of λ/16, as well as functions of self-calibration, direction discrimination and high stabilization, becomes real.
The self-mixing interference effects with a folding feedback cavity in a Zeeman-birefringence dual frequency laser have been investigated. The doubled fringe of the self-mixing interference system with a folding feedback cavity has been demonstrated theoretically and experimentally. The intensities of the e-light and o-light are modulated periodically. An increase of o-light intensity always accompanies a decrease of e-light intensity. Based on the intensity modulation curves of the two lights, a novel displacement sensor has been presented.
This research was supported by the National Nature Science Foundation of China grant 60438010.
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