## Abstract

The self-mixing interference in birefringent dual frequency laser is systematically studied for the first time. The output intensities of two orthogonal modes are both modulated by external cavity length, and their phase relationship is experimentally and theoretically demonstrated. When frequency difference is greater than line width of homogeneous broadening gain curve, the phase relationship is determined by phase difference of two modes. If the frequency difference is smaller than the line width, modes competion will play an important role. Our results can advance the research of self-mixing interferometer of orthogonally polarized dual frequency laser.

© 2004 Optical Society of America

## 1. Introduction

Self-mixing interference in lasers, also called external optical feedback, can dramatically affect the output intensity, coherence and stability of lasers [1–3]. It has been widely used in the fields of displacement [4], vibration [5] and Doppler velocity [6] measurement because of the inherent simplicity and compactness as well as self-alignment [7, 8]. The theory models have been proposed to explain the effects of optical feedback [8, 9]. However, in former researches, little attention has been paid to the optical feedback of orthogonally polarized dual frequency lasers.

There are three kinds of dual frequency He-Ne lasers. The first one is the Zeeman dual frequency laser including longitudinal and transverse Zeeman dual frequency lasers. Both two kinds of Zeeman lasers cannot output a frequency difference over 3 MHz [10]. The second one is birefringent dual frequency laser which outputs a frequency difference at least 40 MHz [11–13]. In order to make up the blank frequency difference from 3 MHz to 40 MHz, the third kind dual frequency laser, called Zeeman-birefringent dual frequency laser [14], is invented. According to the frequency difference, the dual frequency laser is divided into three cases: small frequency difference ranging from 0–3MHz, medium frequency difference ranging from 3–40 MHz and large frequency difference ranging from 40 MHz to infinity.

In this paper, the self-mixing interference in birefringent dual frequency He-Ne laser is systematically studied for the first time. With the optical feedback loop on, the intensities of two orthogonal polarized lights are both modulated by the length of external cavity, and *λ*/2 change of the external cavity length makes the two intensities both undulate one period. The phase relationship of the two intensity modulation curves is determined by frequency difference between the two orthogonally polarized lights, initial length of external cavity and modes competion. When the frequency difference is smaller than the line width of homogeneous broadening gain curve, the phase relationship mainly depends on modes competion. However, if the frequency difference is greater than the line width, the phase relationship is determined by the frequency difference and initial length of external cavity. According to these results, the self-mixing interference effects of orthogonally polarized dual frequency laser are promising for application in precision measurement.

## 2. Experimental setup

Experiments are carried out on a half-intracavity He-Ne laser operating at 632.8nm. The experimental setup is shown in Fig. 1. The ration of gaseous pressure in laser is He:Ne=7:1 and Ne^{20}:Ne^{22}=1:1.

M_{1} and M_{2} are laser mirrors with the amplitude reflectivity of r_{1}=0.999 and r_{2}=0.994, respectively, and the distance *L* between them is 135mm. QC and the half-intracavity laser form an orthogonally polarized dual frequency laser. Due to the birefringent effect of QC, a single mode of the laser can be split into two orthogonally polarized modes, which are called o-light and e-light. When the angle between the crystalline axis of QC and the laser axis is changed, the frequency difference between the two orthogonally polarized modes can be adjusted. M_{E} is external reflective mirror with the amplitude reflectivity of r_{3}=0.2, used to reflect beams back into laser. M_{E} and M_{2} together form the feedback external cavity, whose length is *l*. The photodetectors, D_{1} and D_{2} can be used to detect the output intensities of two orthogonally polarized lights, respectively. The laser modes are observed by SI. SP is used to measure the frequency difference of the two orthogonal modes.

## 3. Theoretical analyses

Since the external reflection coefficient r_{3} is much smaller than the reflection coefficient r_{2}, the multiple reflection effect within the external cavity can be neglected. In the presence of optical feedback, the light beams can be divided into two parts. The first one travels within the internal cavity, while the second one travels in the external cavity and then couples into the internal cavity. These two parts of electric fields superpose in the internal cavity and construct the self-mixing interference. The oscillating condition of a dual frequency laser with optical feedback can be given by [15–17]

$${r}_{1}{r}_{\mathit{eff}}^{e}\mathrm{exp}\left[\left({g}_{e}-{\alpha}_{e}\right)L\right]\mathrm{exp}\left(i{\omega}_{e}{\tau}_{c}\right)=1$$

where ${r}_{\mathit{\text{eff}}}^{o}$
and ${r}_{\mathit{\text{eff}}}^{e}$
represent the complex effective field amplitude reflectivities of o-light and e-light due to the laser coupling mirror M_{2} and external mirror M_{E} respectively, *g*_{o}
and *g*_{e}
are the laser linear gains, *α*_{o}
and *α*_{e}
are the internal losses, *ω*_{o}
and *ω*_{e}
are the optical angular frequencies of o-light and e-light, *τ*_{c}
=*2L/c* is the laser beam round-trip time in internal cavity, and *c* is light velocity in vacuum. Since r_{3} is very small and the external mirror is perfectly aligned, we can get ${r}_{\mathit{\text{eff}}}^{o}$
=*r*
_{2}[1+*ζ* exp(*iω*_{o}*τ*)] and ${r}_{\mathit{\text{eff}}}^{e}$
=*r*
_{2}[1+*ζ* exp(*iω*_{e}*τ*)], where *τ*=*2l/c* represents the laser beam round-trip time in external cavity, and *ζ*=(1-${r}_{2}^{2}$)*r*
_{3}/*r*
_{2} is optical feedback factor. The threshold gains change *Δg*_{o}
=*g*_{o}*-g*_{o0}
and Δge=ge-ge0 due to the optical feedback, where go0 and ge0 are the threshold gains of laser without optical feedback, can be obtained from Eq. (1). To make the result independent of internal cavity length *L*, we define normalized threshold gains change *ΔG*_{o}
=*Δg*_{o}*L* and *ΔG*_{e}*=Δg*_{e}*L*. Because *ζ*≪1, Δ*G*_{o}
=-ln(|${r}_{\mathit{\text{eff}}}^{o}$
|/*r*
_{2})≈*η*_{o}
cos(*ω*_{o}*τ*) and Δ*G*_{e}
≈*η*_{e}
cos(*ω*_{e}
*τ*) where *η*_{o}
and *η*_{e}
are intensity optical feedback factors, which can be obtained from Eq. (1). Because variations of laser intensities are proportional to *ΔG*_{o}
and *ΔG*_{e}
, the output intensities of two orthogonally polarized lights with optical feedback can be written as:

$${I}_{e}={I}_{e0}+{\epsilon}_{e}{\eta}_{e}\mathrm{cos}\left({\omega}_{e}\tau \right),$$

where *I*_{o0}
and *I*_{e0}
are intensities of two orthogonally polarized lights without optical feedback, *ε*_{o}
and *ε*_{e}
are constants. Equation (2) shows that intensities of two orthogonally polarized lights are both modulated when the feedback loop is on, and are similar to sine waves. For convenience, we rewrite the Eq. (2) as:

$${I}_{e}={I}_{e0}+{\epsilon}_{e}{\eta}_{e}\mathrm{cos}\left(\frac{4\pi}{c}{\nu}_{e}l\right)$$

where *ν*_{o}
and *ν*_{e}
are the optical frequencies of o-light and e-light. From Eq. (3), we can find that both of the two intensities vary one period when the length of external cavity changes λ/2. However, there is a phase difference *δ* between *I*_{o}
and *I*_{e}
,

where Δ*ν* is frequency difference between the two orthogonal polarized lights, and Λ=1100*MHz* is longitudinal mode spacing of laser. From Eq. (4), when laser is fixed, *δ* is determined by and is proportional to the length of external cavity and frequency difference. However, if Δ*ν* is smaller than the line width of homogeneous broadening gain curve (about 100–300MHz), the hole-burnings of the two orthogonal modes will cross. Consequently, the modes competion must be considered. The phase relationship between *I*_{o}
and *I*_{e}
does not only depend on Eq. (4), and a phenomenon that increase of one mode intensity will be accompanied decrease of the other mode will be observed. If Δ*ν* is greater than the line width, the modes competion can be neglected, and the phase relationship between *I*_{o}
and *I*_{e}
is determined only by Eq. (4).

As we all know, in the presence of the optical feedback, the laser frequency will be shifted. Since the reflectivity of laser cavity mirror (M_{2}) is nearly 1 and |*r*
_{3}|≪|*r*
_{2}| in our experiments, the frequency shifts caused by optical feedback are very small [8, 18], and the frequency difference variation induced by optical feedback can be neglected.

## 4. Experimental results and discussion

In absence of optical feedback, the initial output intensities of o-light and e-light in dual frequency laser are made even. Different initial lengths of external cavity and frequency differences are considered in experiments. When the initial length of external cavity is 67.5mm, i.e., *l*=*L/2*, the intensity modulation curves of two orthogonally polarized lights with different frequency differences are shown in Fig. 2.

Figure 2 shows, in the presence of optical feedback, the intensities of two orthogonal polarized lights are both modulated by the length of external cavity. They look like the form of sine wave and have a phase difference between them. From Eq. (4), the phase differences of Figs. 2(a)–2(f) are 0.06*π*, 0.14*π*, 0.254*π*, 0.5*π*, 0.664*π* and *π* respectively. But, Fig.2 (a) and (b) disagree with the results of Eq. (3), and the phase differences between *I*_{o}
and *I*_{e}
shown in Figs. 2(a) and 2(b) are far greater than the values calculated from Eq. (4). In Fig. 2(a) and 2(b), since the frequency difference of two orthogonal modes is smaller than the line width of homogeneous broadening gain curve, the hole-burnings of the two orthogonal modes cross. Due to the existence of competion between two modes, increase of one mode intensity will lead to decrease of the other mode. In this case, the phase relationship of *I*_{o}
and *I*_{e}
mainly depends on modes competion. Once the frequency difference of two orthogonal modes is greater than the line width of homogeneous broadening gain curve, the phase relationship of *I*_{o}
and *I*_{e}
will be determined by phase difference given by Eq. (4), as shown by Figs. 2(c)–(f), which agree with the calculated results by Eq. (3).

If initial length of external cavity is 135mm, i.e., *l*=*L*, the intensity modulation curves of the two orthogonal polarized lights with different frequency differences are shown by Fig. 3.

Form Eq. (4), the phase differences of Figs. 3(a)–3(f) are 0.127π, 0.272π, 0.509π, π, 1.33π and 2π respectively. Figures 3(a)–3(f) show, if the frequency difference of two orthogonal modes is smaller than the line width of homogeneous broadening gain curve, the phase relationship between *I*_{o}
and *I*_{e}
will be mainly determined by modes competion. Otherwise, the phase relationship is determined by phase difference given by Eq. (4).

Let initial length of external cavity be 270mm, i.e., l=2L, the intensity modulation curves of two orthogonal polarized lights with different frequency differences are shown by Fig. 4.

From Eq. (4), the phase differences of Figs. 4(a)–4(f) are 0.254*π*, 0.545*π*, 1.02*π*, 2*π*, 2.66*π* and 4*π* respectively. The figure also show, when the frequency difference of two orthogonal modes is smaller than the line width of homogeneous broadening gain curve, the phase relationship of *I*_{o}
and *I*_{e}
is mainly determined by modes competion. Otherwise, the phase relationship is determined by phase difference given by Eq. (4).

Figures 2–4 indicate, if Δ*ν* is small, the phase relationship of *I*_{o}
and *I*_{e}
disagree with the calculated results from Eq. (3), due to the existence of modes competion. In this case, considering Fig. 2(a) and 2(b)–Fig. 4(a) and 4(b), we can find, for a certain Δ*ν*, the phase difference between two modes can finely change the phase relationship of *I*_{o}
and *I*_{e}
. However, it is difficult to accurately calculate the influence of modes competion on the phase relationship of *I*_{o}
and *I*_{e}
. More experiments, which are not presented here, have proven that the line width of homogeneous broadening gain curve of He-Ne laser used by our experiments is about 200MHz. As long as Δ*ν* is greater than 200MHz, the phase relationship between *I*_{o}
and *I*_{e}
agrees well with the calculated result by Eq. (3), so the modes competion can be neglected. For example, when Δ*ν*=550*MHz*, from Eq. (4), the initial external cavity lengths of 67.5mm, 135mm, and 270mm are corresponding to the phase differences of 0.5*π, π*, and 2*π* respectively. Comparing the theoretical results with experimental results shown by Fig. 2(d)–Fig. 4(d), we can find that they are in good agreement. Therefore, the phase relationship between *I*_{o}
and *I*_{e}
can be controlled easily. Meanwhile, by spectrometer, we also find that the frequency difference is almost unchanged in the presence of optical feedback.

## 5. Conclusions

A self-mixing interference of orthogonally polarized dual frequency laser is proposed and demonstrated. The intensity variations of two orthogonal polarized modes and their phase relationship are presented. In the weak optical feedback regime, intensity modulation curves of two modes are both similar to sine waves, and λ/2 change of the external cavity length corresponds one period of intensity undulations respectively. If the frequency difference between two modes is smaller than the line width of homogeneous broadening gain curve of laser, the phase relationship of two intensity modulation curves will mainly depend on modes competion. Otherwise, the phase relationship is determined by phase difference of two modes. Using these characteristics, we can easily adjust the phase relationship between two output intensities. Our results presented in this paper will advance the research of dual frequency self-mixing interferometer, and can also be applied to other kinds of lasers.

## Acknowledgments

This work was supported by the Nature Science Foundation of China and Tsinghua University.

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