## Abstract

The feedback phenomenon of orthogonally polarized dual frequency laser has not been explained theoretically. This paper gives a model based on Lamb’s semi-classical gas-laser theory for the first time. The intensity reflectivity of the feedback mirror, the polarization characteristics of the dual frequency laser and external cavity length are considered besides the parameters studied before. The intensities of o-light and e-light are tuned by feedback mirror. The intensity alternation, leaning of curves and height difference of the two equal–intensity points etc. are discovered in the region of moderate optical feedback level. The experiments are done and the results are in good agreement with the theoretical model.

©2005 Optical Society of America

## 1. Introduction

Peek[1] first reported that the steady-state intensity of a laser could be modified by introducing coherent optical feedback from an external surface. The physical basis is the interference of the back-reflected field with the standing wave inside the laser resonant cavity. So the feedback effect is also called self-mixing interference. It has been studied hotly in the fields of displacement measurement[2, 3], imaging and vibration analysis[4], and microscope[5].

Theory models have been proposed to explain the feedback effect in a semi-conductor laser. The laser intensity modulated by feedback is sine-like[6] or sawtooth-like[7] with reference to different feedback level. For a gas laser, multiple reflection effect[8] and weak optical feedback[9] have been studied by neglecting many characteristics of a gas laser, such as gain saturation, polarization and mode competition. The simulation curves of these theory models are far different from the experimental curves. Especially, the feedback phenomenon of orthogonally polarized dual frequency laser has not been explained theoretically till now.

In this paper, a theory model based on semi-classical gas-laser theory of Lamb is described to explain the feedback effect of orthogonally polarized dual frequency laser for the first time. The effect of intensity reflectivity of the feedback mirror, the polarization of the dual frequency laser and external cavity length are considered besides the parameters studied before. The intensities of o-light and e-light is tuned by feedback mirror. The intensity alternation, leaning of curves and height difference of the two equal–intensity points etc. are discovered in the region of moderate optical feedback level. The experiments are done to prove the theory predicts and the results are in good agreement with the theoretical model. According to these results, the feedback effects of orthogonally polarized dual frequency laser are promising for application in precision measurement.

## 2. Theoretical model

A novel theory model is presented to explain the modified intensity due to the presence of optical feedback. The internal laser intensity of a two longitudinal mode laser is [10]

$${I}_{e}=\frac{1}{D}\left({\alpha}_{2}{\beta}_{1}-{\alpha}_{1}{\theta}_{21}\right),$$

$$D={\beta}_{1}{\beta}_{2}-{\theta}_{12}{\theta}_{21}$$

With

These equations are suitable for dual frequency laser, too[11]. Where 1/2 means 1 for o-light and 2 for e-light, *I _{o}* and

*I*are the dimensionless intensities of o-light and e-light,

_{e}*α*

_{1/2}is the unsaturated net gain,

*β*

_{1/2}is the saturation parameter,

*θ*

_{12}and

*θ*

_{21}are cross-saturation coefficients,

*α*'

_{1/2}is the small signal gain,

*ν*

_{1/2}is the frequency of the two lights, and

*Q*

_{1/2}is the cavity quality factor. A schematic configuration of a He-Ne laser with the feedback mirror is shown in Fig. 1(a). L is the laser cavity length. The distance between the laser and the external mirror M

_{3}is

*l*.

*R*
_{1} and *R*
_{2} representing the reflectivities of mirror M_{1} and M_{2}, and neglecting all losses other than the transmissions of the laser-end mirrors, we may write[12]

Here *Q*
_{0} is the quality factor of the cavity formed by M_{1} and M_{2}. *λ* = *c* / *ν* is the wavelength. In the presence of an external mirror M_{3}, the feedback beams reenters the laser and superposes with that of the internal laser field. Its phase is determined by the external cavity length *l*. The laser mirror M_{2} and the external mirror M_{3} form an external Fabry-Perot interferometer which now replaces the end mirror M_{2}. The equivalent system is represented by Fig.1(b), in which the effective intensity reflectivity *R _{f}* of the end face of the laser is found to be[1]

Where *δ*
_{1/2} = 4*πl* / *λ*
_{1/2} is the external phase difference between successive reflected beams, and *R*
_{3} is the intensity reflectivity of M_{3}. The effective reflectivity of a laser mirror can thus be considerably changed by applying an external mirror with moderate reflectivity. That is how the intensity of the laser is modulated by feedback effect in our model.

The quality factor *Q*
_{1/2} of the laser cavity with optical feedback can be derived by replacing *R*
_{2} by *R*
_{f1/2} in Eq. (3),

Replacing *Q*
_{0} by *Q*
_{1/2} in Eq. (2) and substituting Eq. (2) in Eq. (1), the modulated intensities of o-light and e-light with feedback effect are

$${I}_{e}={M}_{2}+\frac{c}{8\mathit{DL}}\left(1-{R}_{2}\right)\left(1-{R}_{3}\right)\frac{\left(1+{R}_{2}{R}_{3}\right){N}_{2}+2\sqrt{{R}_{2}{R}_{3}}\left({\theta}_{21}\mathrm{cos}{\delta}_{1}-{\beta}_{1}\mathrm{cos}{\delta}_{2}\right)}{{\left(1+{R}_{2}{R}_{3}\right)}^{3}+2\left(1+{R}_{2}{R}_{3}\right)\left(\mathrm{cos}{\delta}_{2}+\mathrm{cos}{\delta}_{1}\right)+4{R}_{2}{R}_{3}\mathrm{cos}{\delta}_{1}\mathrm{cos}{\delta}_{2}}$$

with

$${M}_{2}={I}_{e0}+\frac{{\alpha}_{2}^{\prime}{\beta}_{1}-{\alpha}_{1}^{\prime}{\theta}_{21}}{D}+\frac{c}{8L}\left(1-{R}_{1}\right){N}_{2},$$

Where *c* is speed of light in vacuum. *I*
_{e0} and *I*
_{o0} are steady state intensities of e-light and o-light. For a certain laser, *N*
_{1}, *N*
_{2} , *M*
_{1} and *M*
_{2} are constant. Consequently, we get the expression of intensities as function of *δ*
_{1/2}(or *l*) for reasonable values of *R*
_{1}, *R*
_{2} and *R*
_{3}. Here

With the frequency difference

Because the two frequencies (o-light and e-light) are orthogonally polarized, we use W.M.Doyle’s expressions[13] for *α*'_{1/2} , *β*
_{1/2} , *θ*
_{1/2} and *θ*
_{21} in our model which extend the semi-classical gas-laser theory of Lamb to describe the behavior of a gas laser having generalized polarization characteristics.

The computational intensities curve as a function of external cavity length *l* for gas lasers with ∆*ν* = 240*MHz*, *R*
_{1} = *R*
_{2} = 0.995, *R*
_{3} = 0.4 and L=170mm are shown in Fig. 2, in which the variation range of *l* is 2 *μm* and *α*
_{1} = 2.38×10^{6} , *α*
_{2} = 2.41×10^{6} , *β*
_{1} = 7.44×10^{5} , *β*
_{2} = 7.5×10^{5} , *θ*
_{12} = 5.35×10^{5} , *θ*
_{2l} = 5.39×10^{5} . According to the Eq. (1), Eq. (7) and Eq. (8), the parameters’ values are *D* = 2.7×l0^{11} , *N*
_{1} = -2.15×10^{5} , *N*
_{2} = -2.06×10^{5}, *M*
_{1} = -2.38×10^{11}, *M*
_{2} = -2.27×l0^{11}.

From Fig. 2 we can see that the output intensities of o-light and e-light change periodically as the external mirror moves along the laser’s axial direction. When one light reaches its peak, the other reaches its valley. There is obvious humpback on the opposite side of the two curves and both curves are leaning. The two equal–intensity points in one period have height difference. These phenomena are very different from the intensity tuning curve of weak optical feedback (sine-like or sawtooth-like) or normal dual frequency laser.

## 3. Experimental setup

In order to prove the theoretical model, an experimental setup is shown in Fig. 3. The operating wavelength of the half-intracavity He-Ne laser is 632.8nm. T is the discharge tube filled with *Ne*
^{20} : *Ne*
_{22} = 1:1 gas mixture to suppress the Lamb dip in the output intensity curve. The ration of gaseous pressure in laser is *He* : *Ne* = 7:1.

A plane mirror M_{1} and a concave mirror M_{2} which has a radius of 1m form the laser cavity. Their reflectivities are *R*
_{1} = *R*
_{2} = 0.995 and L=170mm M_{3} is external mirror with a reflectivity of *R*
_{3} = 0.4 that reflects light back into the internal cavity. M_{2} and M_{3} together form the feedback external cavity, whose length is *l*=95mm. The plate Q is a birefringence component made of quartz crystal by which a frequency is split into two orthogonally polarized frequencies (o-light and e-light). The frequency difference of the two frequencies can be changed from 40MHz to one longitudinal mode spacing by changing the angle *θ* between the crystalline axis of Q and the laser axis. D_{1}, D_{2} are photoelectric detectors that is used to detect the output intensities of two orthogonally polarized lights, respectively. The computer’s function is to get the signal from A/D and control the input of D/A. The laser modes are observed by F-P and OS. A voltage given by a D/A card and amplified by a PI amplifier is used to drive the PZT. The voltage of the D/A card changes about 1V and the PZT changes a distance of *λ* / 2.

## 4. Experimental results

The experimental curves for ∆*ν* = 240*MHz* are shown in Fig. 4. Compared Fig. 4 with Fig. 2, we can see that the experimental waveforms have the same characteristics as the theoretical simulation curves. The modulation intensities of o-light and e-light change periodically and inversely. The humpback exists and the curves are leaning which are in good agreement with the simulated curves. The equal-intensity point appears when o-light and e-light have the same gain in the equivalent system shown in Fig. 1(b). The two equal–intensity points in one period have height difference. Consequently, the theory predict is observed in the experiment.

## 5. Conclusions

Our theoretical model based on Lamb semi-classical gas-laser theory gives a new idea to explain many phenomena of a dual frequency laser with moderate optical feedback regime. The theoretical analysis and comparison with the experimental results on the feedback effect of orthogonally polarized dual frequency laser are presented in this paper and they are in good agreement with each other. The output intensities of the two orthogonally polarized frequencies change periodically as the external mirror moves along the laser’s axial direction. There is obvious humpback on the opposite side of the two curves and both curves are leaning. This is very different from the intensity tuning curve of weak optical feedback or common dual frequency laser. The equal-intensity point appears when o-light and e-light have the same gain in the equivalent system. There is height difference between the two equal-intensity points in one period. When one light reaches its peak, the other reaches its valley. In other words, the external cavity length *l* changes every *λ* / 2, the laser intensity produce a fringe for o-light and e-light, respectively. This result can be used to measure displacement of external mirror M_{3} and our results presented in this paper will also advance the research of dual frequency self-mixing interferometer.

## Acknowledgments

This work was supported by the project 60438010 Nature Science Foundation of China.

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