## Abstract

The polarization dynamics of laser subjected to weak optical feedback from birefringence external cavity are studied theoretically and experimentally. It is found that polarization flipping with hysteresis is induced by birefringence feedback, and the intensities of two eigenstates are both modulated by external cavity length. The variations of hysteresis loop and duty ratios of two eigenstates in one period of intensity modulation with phase differences of birefringence element in external cavity are observed. When the phase difference is *π*/2, the two eigenstates will equally alternatively oscillate, and the width of hysteresis loop is the smallest.

©2005 Optical Society of America

## 1. Introduction

Single-mode He-Ne laser usually emits linearly polarized light, but its polarization may abruptly flip between two eigenstates [1]. Floch and co-workers [2,3] observed the polarization flipping and hysteresis effect by changing the anisotropy values of laser intracavity. Stephan and co-workers [4,5] experimentally and theoretically studied the polarization switching induced by optical feedback from a polarizer external cavity. Although the polarization flipping and hysteresis effect were observed, they could not modify the size of hysteresis loop, and only one polarization intensity can be modulated. Recently, the polarization switching induced by optical feedback has attracted considerable interest [6–9]. However, to the best of our knowledge, the polarization control by changing the anisotropic values of external feedback cavity has not been investigated.

In this letter, we demonstrate the influence of the optical feedback from birefringence external cavity on the laser polarization states and output intensity. When the length of external cavity is changed, the polarization flipping with hysteresis between two eigenstates will occur, and the intensities of the two eigenstates are both modulated. The width of hysteresis loop decreases when the phase difference of birefringence element in external cavity increases, and the duty ratios of the two eigenstates in one period of laser intensity modulation also vary with the phase difference. When the phase difference is π/2, the width of hysteresis loop is the smallest, and the duty ratios are equal. In this case, the two eigenstates will equally alternatively oscillate, and the square wave oscillation can be observed. Each polarization switching corresponds to λ/4 change of the external cavity length.

## 2. Experimental setup

Experiments are carried out on a single mode, linearly polarized He-Ne laser with natural anisotropy. The wavelength λ is 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 reflectivities of R_{1}=99.8% and R_{2}=98.8%, respectively, and the distance *L* between them is 150mm. M_{E} is external mirror with reflectivity of R_{3}=10%, used to reflect laser beams back into the laser. M_{E}, together with M_{2} and G can form a birefringence external cavity. The length of external cavity *l* is 100mm. D_{1} is used to detect the laser intensity. D_{2} is used to detect the variations of laser polarization state. Due to the stress birefringence effect, when a force is applied on G, the two optical axes of G are parallel to the two principal stress directions, and the force-induced birefringence phase difference is proportional to the magnitude of force. According to the coordinates system shown in Fig. 1, the two optical axes of G are along y-axis and x-axis, respectively.

## 3. Experimental results

In our experiments, the initial polarization direction of laser is parallel to y-axis. The force-induced birefringence phase difference *δ* can be given by *δ*=8*λF*/*πDf*
_{0}, where *D* is the diameter of *G*, *f _{0}* is the fringe value of the optical materials, and

*F*is the force applied on G. Therefore, the different phase differences between the two principal optical axes of G can be obtained by changing the magnitudes of force. When the length of external cavity is scanned by PZT, the intensity modulation curves can be obtained, and different phase differences of G correspond to different intensity modulation curves as shown in Fig. 2.

In Figs. 2(a)–2(d), the solid lines are the intensity modulation curves that PZT voltage is increased, i.e., M_{E} moves toward the laser and the length of external cavity is decreased. The dash lines represent the intensity modulation curves that PZT voltage is decreased, i.e., M_{E} moves away from the laser, and the length of external cavity is increased.

From the Fig. 2, we can find that there are dips on the intensity modulation curves, which is different from the conventional optical feedback intensity curve whose profile is similar to sine wave. If observe the laser output through a polarizer, we can find that the polarization of laser hops at the dip points, and the intensities of two eigenstates are both modulated by the length of external cavity. When PZT voltage is increased, the position of polarization flipping from y-polarization (P_{y}) to x-polarization (P_{x}) is at point D. If PZT voltage is decreased, the position of polarization flipping from P_{x} to P_{y} is at point C. From Figs. 2(a)–2(d), we can find, if the phase difference of G is changed, the position of polarization flipping is also different. This indicates that the duty ratios of the two eigenstates in a period of intensity modulation curve vary with the phase difference. The relationship between the duty ratios of the two eigenstates and the phase difference are shown in Fig. 3(a). Meanwhile, for a certain phase difference of G, when the moving direction of feedback mirror (M_{E}) is different, the polarization flipping points C and D are not superposition. This indicates the hysteresis effect of polarization flipping. When the phase difference is changed, the width of hysteresis loop is also changed. Using this result, we can control the polarization switching outside the laser. As known, the length variation of external cavity is proportional to the voltage applied on PZT, so the voltage increments on PZT can be used to represent the width of hysteresis loop shown by the space between point C and D. The relationship curve of the hysteresis loop width and the phase difference is shown in Fig. 3(b). When *δ*=π/2, the width of hysteresis loop is the smallest.

When *δ*=π/2, the curve of intensity modulation is similar to the full wave rectification of sine wave, as shown in Fig. 2(d). Observing the output intensity through a polarizer, we can find that the duty ratios are nearly equal and the profile of intensity curve is similar to a square wave due to the existence of the laser initial intensity. Because *λ*/2 change of the external cavity length corresponds to one period of intensity modulation, in this case, each polarization switching will correspond to *λ*/4 change of the external cavity length.

## 4. Theoretical analyses

The eigenstates polarizations depend on the active medium, on a linear phase anisotropy and on a loss anisotropy. For smaller phase anisotropy of intracavity, the polarization flipping conforms to the rotation mechanism [2]. Let initial polarization direction of laser be parallel to y-axis. When the following inequality is satisfied [3], the polarization flipping from P_{y} to P_{x} will occur

where *α* is laser net gain, *β* and *θ* are self and cross saturation coefficient, *ρ* is self pushing coefficient, ΔΦ* _{xy}* is the phase anisotropy in internal cavity,

*t*and

_{x}*t*represent the transmission coefficients of P

_{y}_{x}and P

_{y}respectively. The first term of Eq. (1) represents the effect of the active medium, the second term represents the effect of the phase anisotropy of intracavity and the third term represents the effect of the loss anisotropy.

In the presence of optical feedback, due to *R*
_{3}≪*R*
_{2}, according to the model [10] of equivalent cavity of Fabry-Perot interferometer, when the length of external cavity changes, the equivalent mirror reflectivities along y-axis and x-axis can be given by

$${R}_{y-x}={R}_{2},$$

where *φf*=4*πl*/*λ* represents the phase of external cavity. Due to *R _{y-y}*≠

*R*, the two eigenstates of one laser mode will subject to different losses. Substitute Eq. (2) into Eq. (1), we can get the condition of polarization flipping from P

_{y-x}_{y}to P

_{x}

where κ=(*R*
_{3}/*R*
_{2})^{1/2}(1-*R*
_{2}). Because the frequency shift caused by optical feedback and the intracavity anisotropy are very small [11], if we neglect saturation effects, Eqs. (2) and (3) show that the azimuth of polarization will be along the larger reflectivity axis, and the right term in Eq. (3) can be assumed as zero. In this case, the light will be polarized along y-axis (cos*φ _{>}*0) or x-axis (cos

*φ*<0). When polarization direction of laser is along y-axis, the intensity variation can be obtained by ΔI

_{f}_{y}=

*η*cos

*φ*[8], where

_{f}*η*represents optical feedback factor. Similarly, when the polarization direction of laser is parallel to x-axis, the equivalent mirror reflectivities along x-axis and y-axis are given by

$${R}_{x-y}={R}_{2},$$

where *δ* is the phase difference between two principal optical axes of G. The P_{x} to P_{y} flip condition is similar to Eq. (1), and only the signs of the first and third terms are changed. The condition of polarization flipping from P_{x} to P_{y} can be written as

The light will be polarized along x-axis (cos(*φf*-2*δ*)>0) or y-axis (cos(*φ _{f}*-2

*δ*)<0). If polarization direction of laser is along x-axis, the intensity variation is ΔI

_{x}=

*η*cos(

*φ*-2

_{f}*δ*). When the length of external cavity is changed, the dependence of laser intensity and polarization flipping on

*δ*can be illustrated in Fig. 4. The horizontal dot lines in Fig. 4 represent the right-hand sides of Eqs. (3) and (5), which are nearly equal to zero.

Firstly, we decrease the length of external cavity. Because the initial polarization direction of laser is P_{y}, the intensity variation is ΔI_{y} shown by solid lines in Fig. 4. In Figs. 4(a)–4(c), when starts from point A to the right and reaches point D, the condition of polarization flipping from P_{y} to P_{x} is satisfied from Eq. (3). The polarization direction of laser jumps from P_{y} to P_{x}, i.e., from point D to point E, and the intensity turns into ΔI_{x} shown by dash lines. At point F, the polarization should jump from P_{x} to P_{y} from Eq. (5). However, due to *R _{x-x}*>

*R*, P

_{y-y}_{y}will subject to more losses and be suppressed. The polarization still remains P

_{x}. Once reaches point H, due to

*R*>

_{y-y}*R*, from Eq. (5), the polarization will jump back to P

_{x-x}_{y}. The intensity becomes ΔI

_{y}again till point I, and then begins another period. The trace of intensity modulation within a period is $\overline{\mathrm{ADDE}}\stackrel{\cdots}{\text{E}}\stackrel{\cdots}{\text{F}}\stackrel{\cdots}{\text{H}}\overline{\mathrm{HI}}$ shown in Figs. 4(a)–4(c). Then, we increase the length of external cavity. For conveniently, we start from point F to the left. Because the polarization direction of laser at point F is P

_{x}, the intensity variation is ΔI

_{x}shown by dash lines. In Figs. 4(a)–4(c), when reaches point C, from Eq. (5), the polarization will jump from P

_{x}to P

_{y}, i.e., from point C to point B. At point J, the polarization will jump back to P

_{x}. The intensity becomes ΔI

_{x}again till point K, and then begins another period. The trace of intensity modulation within a period is $\stackrel{\cdots}{F}\stackrel{\cdots}{C}\stackrel{\cdots}{C}\stackrel{\cdots}{B}\overline{\mathrm{BAJ}}\stackrel{\cdots}{J}\stackrel{\cdots}{K}$. If δ=π/2, Fig. 4(d) shows, the length of external cavity whether is increased or is decreased, the positions of polarization flipping are both at point B. The intensity modulation curve is $\overline{\mathrm{AB}}\stackrel{\cdots}{B}\stackrel{\cdots}{C}$ or $\overline{\mathrm{CB}}\stackrel{\cdots}{B}\stackrel{\cdots}{A}$, which is full wave rectification of sine wave. From the conditions of polarization flipping Eqs. (3) and (5), the duty ratios of the two eigenstates can be written by D

_{y}=2(

*π*-

*δ*) and

*D*=2

_{x}*δ*. The normalized curves that the duty ratios vary with

*δ*are shown in Fig. 5(a).

The stabilities Eqs. (3) and (5) can also be used to explain the hysteresis loop apparent in Fig. 2 and the fact that we have observed a decrease in the size of the loop with increasing *δ*. The relationship between the width of hysteresis loop and the phase difference of birefringence element can be given by *W _{H}*=

*π*-2

*δ*. The curve that the width of hysteresis loop varies with

*δ*is shown in Fig. 5(b).

Above analyses show that the intensities of the two eigenstates are both modulated by the length of external cavity, and there is a hysteresis loop between P_{y}→P_{x} and P_{x}→P_{y}. The width of hysteresis loop CD shown by lower traces in Figs. 4(a)–4(d) decreases with increasing the phase difference of birefringence element. If *δ*=*π*/2, the width of hysteresis loop is the smallest, and nearly equal to zero. Meanwhile, in a period of laser intensity modulation, the duty ratios of two eigenstates also vary with the value of phase difference. The greater phase difference the smaller difference of duty ratios. When *δ*=*π*/2, the duty ratios of two eigenstates are equal. The theoretical analyses are in good agreement with the experimental results.

## 5. Conclusions

We have demonstrated the polarization control outside the laser. By adjusting the phase difference of birefringence element in the external feedback cavity, we can change the width of hysteresis loop and the duty ratios of two eigenstates. When *δ*=*π*/2, the duty ratios are equal, and intensity curve is similar to the full wave rectification of sine wave. If we observe the laser intensity through a polarizer, the square wave can be output. In this case, the width of hysteresis loop is the smallest, and each polarization switching corresponds to *λ*/4 change of the external cavity length. Our results are promising for applications in optical switching, optical bistability, and precision measurements of some physical quantities.

## Acknowledgments

The Nature Science Foundation of China supported this work.

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