## Abstract

The high density intensity fringes and phase behavior in birefringent dual frequency laser with multiple feedback are studied for the first time. It was discovered that the fringes of output intensity are made of bipolar pulses with symmetric external cavity feedback and the fringe density is as high as compared to the conventional feedback. The high density cosine-like fringes are obtained with asymmetric external cavity feedback by adjusting the tilt angle of the feedback mirror and the fringe density is about 22 times higher compared to the conventional feedback. Moreover, there is a phase difference between the two cosine-like fringes and the phase difference is varied with the change of the external cavity length. The experimental results and a theoretical analysis are presented in this work. These results offer a large increase in the resolution for the optical feedback interferometer with the birefringence dual frequency laser.

© 2012 OSA

## 1. Introduction

The output intensity of lasers can be significantly affected by external optical feedback [1, 2]. This phenomenon, also called self-mixing interference, has been widely studied and developed for applications in fields of Doppler velocimetry [3, 4], displacement measurement [5,6], vibrometry [7,8], distance measurement [9,10] and microscopy [11,12], etc.

In previous research, much attention has been paid to the weak feedback effects of lasers with symmetric external cavities [13–15]. It was discovered that the laser intensity modulation induced by the change of an external cavity was similar to the intensity modulation of the two-beam interferometer. One period of laser intensity undulation or a fringe corresponds to λ/2 change of the external cavity length. Recently, in order to increase the fringes density, many researchers have investigated the strong feedback effects of lasers with an asymmetric external cavity [16–18]. Yu [17] reported the double-frequency fringes in a self-mixing system with a laser diode at a high feedback level. Tan [18] discovered the quadruple feedback fringes with an asymmetric external cavity in a single-mode Nd:YAG laser, the fringe density is four times higher compared to the conventional feedback. The resolution of the displacement can be up to λ/8. But so far, little attention has been paid to the multiple feedback in birefringent dual frequency lasers.

In this paper, we demonstrate the high density fringes and phase behavior in birefringent dual frequency lasers with multiple feedback for the first time. At the strong feedback level, the fringes of the output intensity consist of bipolar pulses with a symmetric external cavity feedback. The fringe density is as high as during the conventional feedback. By adjusting the tilt angle of the feedback mirror, the cosine-like high density fringes are obtained. Moreover, there is a phase difference between the two cosine-like fringes and the phase difference is varying with the change of the external cavity length. According to these results, the high density cosine-like fringes in birefringent dual frequency laser offer a potential method to greatly increase the resolution of an optical feedback system.

## 2. Experimental setup

Experiments are carried out on a He-Ne laser operating at 632.8nm. The experimental setup is shown in Fig. 1 .

A plane mirror M_{1} and a concave mirror M_{2} with a radius of 500mm form a half-intra cavity resonator. The corresponding amplitude reflectivities are *r*_{1} = 0.994, *r*_{2} = 0.995. The cavity length of the laser is *L* = 170mm. The lasers capillary T is filled with a Ne^{20}:Ne^{22} = 1:1 and a He:Ne = 7:1 gas mixture to suppress the Lamb dip in the output intensity curve. The inner diameter of the laser capillary is 0.9mm. The concave mirror M_{3} is a feedback mirror with the amplitude reflectivity of *r*_{3} = 0.985. M_{2} and M_{3} form the asymmetric external feedback cavity. The length of the external cavity is *l*. *θ* is the misalignment angle of the feedback mirror M_{3}. The plate Q is a birefringence component made of quartz crystal which splits one frequency into two orthogonally polarized modes (o-light and e-light). The frequency difference is adjusted to 316MHz by changing the angle between the crystalline axis Q and the laser axis. ATT is an attenuator. PBS is a Wollaston prism to separate the two polarized lights which are detected by the photoelectric detectors D_{1} and D_{2}. SP is spectrum analyzer (HP 8590A) that is used to measure the frequency difference of the two orthogonal lights.

## 3. Experimental results

First, in absence of a feedback mirror, the initial output intensities of the two orthogonally polarized lights in the dual frequency laser setup are adjusted to have the same output intensity. Then the feedback loop is closed and the PZT is connected to a triangular wave voltage to make the mirror M_{3} feed the light back into the laser cavity. After that, the feedback level and the misalignment angle of M_{3} are adjusted. The intensity fringes of the conventional weak feedback with symmetric external cavity and the intensity fringes of the strong feedback with symmetric and asymmetric external cavity are shown in Fig. 2
.

Figure 2(a) shows, at the weak feedback level, when *θ* = 0. The output intensities of the two orthogonally polarized modes are both modulated by the change of the external cavity. The modulation depth of the o-light is the same as for the e-light. The feedback fringes are cosine-like and similar to the conventional optical feedback. A fringe corresponds to a half-wavelength variation of the length of the external cavity. Figure 2(b) shows the strong feedback level with *θ* = 0. The feedback fringes of the two orthogonally polarized modes are not cosine-like. They are made of bipolar pulses. The period of the feedback fringes are the same as for the weak feedback. However, at the strong feedback level, when *θ* = 0.9′, the fringes density of the two orthogonally polarized modes are both increased greatly. The intensity modulation curves have envelope fluctuations as shown in Fig. 2(c) or 2(d). From Fig. 2(a) it can be seen that there are four periods on a down-cycle of the PZT voltage and there are about 88 periods in the same range of the PZT voltage as shown in Fig. 2(c), it means that the fringe density of the strong feedback is about 22 times higher compared to the conventional weak feedback. When *θ* = 1.2′, the fringe density is the same as for *θ* = 0.9′, but the latter has the gentle envelope fluctuation curve as shown in Fig. 2(e) or 2(f). With the misalignment angle increasing to 1.8′, the envelope fluctuations disappear and the fringes of the two orthogonally polarized modes tend to be uniform as shown in Fig. 2 (g) or 2(h).

When *θ* = 1.8′, the phase relationship of the two orthogonally polarized lights is studied with different length of the external cavity. When *l* = 110mm, 112mm and 114mm, the experimental results are shown in Fig. 3
. It can be seen that there is a phase difference between two orthogonally polarized lights and the phase difference is varied along with the change of the external cavity length *l*. In Fig. 3 (a), the large amplitude difference of o-light and e-light is used for easy observation.

## 4. Theoretical analysis and simulation

The laser output of single mode laser with asymmetric multiple feedback is given by [19]

For a birefringent dual frequency laser, the two orthogonally polarized modes *I _{o}* and

*I*share the same laser available gain in the laser cavity. From Eq. (1), the laser output of the two orthogonally polarized modes in the dual frequency laser can be expressed as

_{e}*I*

_{o}_{0}and

*I*

_{e}_{0}represent the intensities of o-light and e-light without optical feedback respectively,

*r*

_{2}and

*r*

_{3}are the amplitude reflection coefficients of M

_{2}and M

_{3}.

*K*is a proportionality coefficient, t2 is the transmission coefficient of M2,

*ω*and

_{o}*ω*are the optical angular frequencies,

_{e}*q*is the largest roundtrip times in the external cavity,

*m*represents the beam’s

*m*th roundtrip in the external cavity, also called the feedback order,

*f*is the coupling efficiency of the

_{m}*m*th order feedback beam,

*ν*and

_{o}*ν*are the optical frequencies of o-light and e-light,

_{e}*l*is the length of external cavity.

*L*is the length of the internal cavity,

*c*is the light velocity in vacuum,

*δ*and

_{om}*δ*represent the phase variation of the o-light and e-light after

_{em}*m*round trips induced by the asymmetric external cavity.

From Eq. (2), we can see that the intensity output of two orthogonal polarized lights *I _{o}* and

*I*are both modulated by the change of the length of the external cavity in the birefringent dual frequency laser. At the weak feedback level with symmetric external cavity, the high order feedback can be neglected. Only the first order feedback (

_{e}*q*= 1) beam can reenter into the laser cavity, so Eq. (2) can be written as

Moreover, at the strong feedback level with asymmetric external cavity, there will be multiple feedback beams can reenter into the laser cavity, but not all the high order feedback beams can reenter into the laser cavity. The feedback order *m* that can reenter into the laser cavity (also called effective order) is determined by the length of the external cavity *l* and the parameter of the feedback mirror, including the misalignment angle *θ*, the amplitude reflection coefficient *r*_{3} and the curvature radius *R*. It means that the effective order *m* is the function of the misalignment angle *θ* when the external cavity length and the feedback mirror have been selected. *f _{m}* is the coupling efficiency of the

*m*th order feedback beam, it can be defined as

*d*is the radius of the laser capillary,

*s*is the area of the laser spot which is overlapped with the laser capillary.

Using the ray tracing method, we can calculate the effective order *m* of the feedback beam as well as the corresponding coupling efficiency *f _{m}*. If the length of the external cavity is

*l*= 84mm and the parameters of the feedback mirrors are

*r*

_{3}= 0.90 and

*R*= 500mm. With the misalignment angle of

*θ =*0.9′, the effective feedback orders are the 5th order, the 15th order and the 22nd order and the corresponding coupling efficiencies are

*f*

_{5}= 0.31,

*f*

_{15}= 0.04 and

*f*

_{22}= 0.21. When the misalignment angle is

*θ =*1.2′, the effective feedback orders are still the 5th order, the 15th order and the 22nd order and the corresponding coupling efficiencies are

*f*

_{5}= 0.18,

*f*

_{15}= 0.02 and

*f*

_{22}= 0.30. When the misalignment angle is increasing to

*θ =*1.8′, the effective feedback orders are only the 15th order and the 22nd order and the corresponding coupling efficiencies are

*f*

_{15}= 0.06 and

*f*

_{22}= 0.35, and it is noticeable that

*f*

_{15}is much smaller than

*f*

_{22}. From Eq. (2), the simulation curves of the intensity with different misalignment angles of the feedback mirror are shown in Fig. 4(b).

There are three curves of intensity in Fig. 4(b), and they correspond to different misalignment angles of 0.9′, 1.2′ and 1.8′ respectively. Figure 4(b) shows that the high density fringes are formed due to the multiple high order feedback. The fringe density is about 22 times higher compared to the conventional feedback. When the misalignment angle *θ* increases, the intensity of fringes tend to be uniform and the fringes are cosine-like, that is because the coupling efficiency *f*_{15} is much smaller than *f*_{22}. These considerations agree well with the experimental results shown in Fig. 2(c) to Fig. 2 (h).

When *θ =* 1.8′, *f*_{15} is very small and it may be neglected. The multiple feedback can be simplified as a 22nd order feedback. In this case, from Eq. (2), the phase difference *δ* of the two orthogonally polarized lights is

Equation (7) shows, when the birefringence dual frequency laser is selected, the phase difference of two orthogonally polarized lights is determined by the external cavity length *l*. If Δν = 316MHz, when *l* = 110mm, 112mm and 114mm, the phase differences are 0.02*π*, 0.60*π* and 1.18*π* respectively, which agree well with the experimental results shown in Fig. 3.

## 5. Conclusion

The intensity modulation and phase behavior in birefringent dual frequency laser with multiple feedback are demonstrated. At the strong feedback level, the intensity fringes are made of bipolar pulses with symmetric external cavity feedback. The fringe density is the same as during the conventional feedback. By adjusting the tilt angle of the feedback mirror, the high density cosine-like fringes are obtained and the fringes tend to be uniform along with the increasing of the tilt angle. The fringe density is about 22 times higher compared to the conventional feedback. A λ/44 change of the external cavity length corresponds to one period of intensity modulations respectively. Moreover, there is a phase difference between the two orthogonally polarized fringes with multiple feedback and the phase difference is varied with the changing of the external cavity length. A theoretical analysis based on the compound cavity model of dual frequency lasers and the ray tracing method agrees well with the experimental results. Our results presented in this paper will further improve the design of the dual frequency laser feedback interferometer.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Project no. 60827006).

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