Controlling the duty cycle of the eigenstates in laser with multiple optical feedback

Zhaoli Zeng; Shulian Zhang; Yidong Tan; Weixin Liu

doi:10.1364/OE.21.019990

1. Introduction

It is well known that a broad variety of dynamics behaviors can occur in lasers subject to optical feedback, such as polarization switching, hysteresis and bistable state [1,2]. In recent years, studies have shown that the optical feedback cannot only generate the polarization flipping but also control the polarization state [3,4]. The reason is that the delayed optical feedback may interact with the polarization competition in laser [5,6]. In previous work, much attention has been paid to the polarization flipping phenomenon in lasers with conventional weak optical feedback [7–12]. Stephan [7,8] experimentally studied the polarization flipping induced by optical feedback from a polarizer external cavity. Floch [9,10] observed the polarization instabilities and presented the polarization control mechanisms in vectorial bistable lasers for one-frequency systems. The application study of polarization hopping in displacement measurement was also demonstrated [11,12], the displacement resolution reached λ/16. Recently, the laser feedback effect induced by multiple optical feedback has attracted considerable interest [13–15]. However, to the best of our knowledge, the polarization dynamics of laser in presence of multiple optical feedback has not been investigated.

In this paper, we demonstrate the polarization behaviors of He-Ne laser with asymmetry feedback cavity. The high-frequency optical fringes with nanoscale resolution are obtained due to the multiple reflections in the asymmetry external cavity. The fringe frequency is dozens of times that of conventional optical feedback. Particularly, the polarization flipping occurred in each optical fringe, and the position of polarization flipping is determined by the residual stress of laser. By applying an appropriate external stress to the laser mirror, the duty cycle of two eigenstates can be changed easily. In addition, the polarization flipping is also observed when the direction of displacement is conversed. These phenomena are interpreted qualitatively by combining the rotation flipping mechanism with compound cavity feedback model. A preliminary application of high-frequency optical fringes in small displacement measurement is conducted.

2. Experimental setup

Experiments are carried out on a single-mode He-Ne laser operating at 632.8nm. The experimental setup is shown in Fig. 1. A plane mirror M₂ and a concave mirror M₁ with a radius of 300mm form a laser resonance cavity. θ is the misalignment angle of the feedback mirror M_f. M₂ and M_f form the asymmetric external feedback cavity in which the laser beam reflects many times before going back to the laser cavity. The laser output of mirror M₁ is split to two parts. One is used to measurement the displacement, and the other is used to measure the frequency difference resulted from the residual stress.

Fig. 1 Experimental setup. M₁, M₂, M_f: mirrors; PZT: piezoceramics; BS: beam splitter; W: Wollaston prism; D_0, D₁, D₂: PIN photoelectric detectors; D: APD photoelectric detector; SP: spectrum analyzer; P: polarizer; OS: oscilloscope; F: forcing device.

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3. Experimental results

In experiment, the phenomenon at weak optical feedback level with isotropic external cavity is studied firstly. In this case, the feedback mirror M_f is a plane mirror with the amplitude reflectivity of r_f = 0.1, and a scanning interferometer is placed in front of polarizer which is used to monitor the laser mode. When the length of feedback cavity is scanned by PZT, the laser intensity (I_t) is modulated by the variation of length of feedback cavity as shown in Fig. 2. From Fig. 2(a), it can be seen that the intensity modulation is similar to cosine function and each modulation period is corresponding to the displacement of λ/2. There is one operating mode in the free spectral range (FSR) which means that the laser is operating at single mode. In addition, Fig. 2(a) shows that there is no polarization flipping at weak feedback level with isotropic external cavity.

Fig. 2 Intensity curves of laser feedback. (a) Conventional feedback; (b) Isotropic strong feedback; (c) on an enlarged time scale of (b).

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However, when the plane feedback mirror M_f is tilted and replaced with a concave mirror with the amplitude reflectivity of r_f = 0.99, the high-frequency optical fringes are obtained due to the multiple feedback in the asymmetry external cavity. The fringe frequency can be 30 times that of conventional weak feedback as shown in Fig. 2(b). The optical resolution of each fringe is up to λ/60. In particular, the polarization flipping in each optical fringe is found in presence of multiple optical feedback as shown in Fig. 2(c).

In Fig. 2(c), the top curve I_t is the total intensity of laser output detected by D₀. I_o and I_e are the intensity curves of two orthogonal polarized lights detected by D₁ and D₂. There are different from that of conventional weak feedback. First, there are dips at B, D and F point of each fringe period AC, CE and EG while the conventional laser feedback curve is similar to cosine as shown in Fig. 2(a). Second, the polarization flips vertically 90° at positions of B, D, or F where the laser intensity transfers from I_o to I_e.

As we known, the causes of polarization flipping mainly included two aspects. One is the external anisotropic in feedback cavity, the other is the internal anisotropic in laser. In our experimental system, there is no any anisotropic element in the external cavity. Hence, the reason of polarization flipping must come from the internal anisotropy of laser, such as the residual stress of laser mirror. Furthermore, the polarization state becomes more sensitive to residual stress at strong optical feedback level than at weak optical feedback level. In order to prove the analysis above, the characteristics of polarization flipping of laser with different residual stress are researched systematically.

The residual stress of laser is changed slightly by applying a force to the laser mirror M₂. Since the stress-birefringence effects, there is a small frequency difference between the orthogonally polarization state. The frequency difference can be measured by using frequency splitting method [16]. So, we can study the relationship between the residual stress and polarization flipping by measuring the frequency difference of two orthogonally polarization state. The experimental results are shown in Fig. 3.

Fig. 3 Waveforms of polarization flipping with different frequency difference (residual stress): (a) Δv = 8MHz; (b) on an enlarged time scale of (a); (c) Δv = 3.5MHz; (d) on an enlarged time scale of (c); (e) Δv = 1.5MHz; (f) on an enlarged time scale of (e).

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Figures 3(a) and 3(b) show, when the residual stress results in frequency difference is 8MHz, there are dips in total intensity modulation curve (I_t). In addition, the polarization flipping occurred in each optical fringe of o-light (I_o) and e-light (I_e), and the position of polarization flipping is near the middle of each fringe. The duty cycle of o-light and e-light is about 2:1. When the residual stress results in frequency difference is 3.5MHz as shown in Figs. 3(c) and 3(d), there are still dips in each fringe period, and the position of polarization flipping moves a little to the right in each fringe. The duty cycle of o-light and e-light in each fringe period is approximately 4:1. In Figs. 3(e) and 3(f), the residual stress result in frequency difference is reduced to 1.5MHz. The dips almost disappeared, and the position of polarization flipping moves to the right edge of each fringe. The duty cycle of o-light and e-light is increased to about 6:1. From Figs. 3(e) and 3(f), it can be seen that, at the up-cycle of PZT voltage, the polarization state of o-light is operating normally and the polarization state of e-light is nearly suppressed. Particularly, Fig. 3(e) shows that the polarization hopping also occurs when the displacement of PZT moves in the opposite direction, which shows that this type of polarization hopping is sensitive to the movement direction of feedback mirror. The mechanism of direction sensitivity will be researched in our future works.

4. Theoretical analysis

When the anisotropy of laser is small, according to Floch rotation flipping mechanism, the flip condition from o-light to e-light can be given by [10]

\dot{ε} = [- \frac{1}{2} \frac{ρ_{+}}{β_{+}} Δ Φ_{o e}] + [\frac{c}{L} \frac{1}{4 α_{+}} \frac{S + 1}{2 S} Δ Φ^{2}_{o e}] > 0

where

ε

is the rotation angle of main axis,

ρ_{+}

is the self-repulsion parameter,

β_{+}

is the self-saturation parameter,

c

is the velocity of light,

L

is the cavity length,

α_{+}

is net gain coefficient,

S = -3 / 20,

Δ Φ_{o e}

is the linear phase anisotropy introduced by residual stress.

Since the stress birefringence effect, there is a frequency difference $Δ ν_{o e}$ between the two orthogonal eigenstates, and $Δ ν_{o e}$ can be written as

Δ ν_{o e} = ν_{o} - ν_{e} = \frac{c}{2 L} \frac{Δ Φ_{o e}}{π}

From Eq. (1), the flip condition can be expressed as

\frac{c}{2 L} \frac{S + 1}{2 S} Δ Φ_{o e} > \frac{γ ξ_{o}}{(2 γ^{2} + ξ_{o}^{2})} [\frac{Z_{i} (ξ_{o})}{Z_{i} (0)} - \frac{1}{η}] F_{o}

where

γ

is the homogeneous width of the transition,

ξ_{o}

is frequency parameter,

Z_{i} (ξ_{o})

is a plasma dispersion function,

η

is relative excitation degree and its about 1.33,

F_{o}

is first-order factor.

From Eq. (2), it can be seen that the polarization flipping is mainly determined by the internal isotropic $Δ Φ_{o e}$ and the first-order factor $F_{o}$ when the frequency parameter $ξ_{o}$ is small. So, Eq. (2) can be simplified as

\begin{array}{l} \frac{c}{2 L} Δ Φ_{o e} < j F_{o} \frac{2 S}{S + 1} \\ j = \frac{γ ξ_{o}}{(2 γ^{2} + ξ_{o}^{2})} [\frac{Z_{i} (ξ_{o})}{Z_{i} (0)} - \frac{1}{η}] \end{array}

In single-mode laser with optical feedback, the first-order factor $F_{o}$ can be written as

F_{o} = \frac{η}{2 π} Δ [0.003 + (0.004 +(1 - R_{e f f}^{o})) / 2]

where

Δ

is the longitudinal mode spacing and is about 1250MHz,

R_{e f f}^{o}

is the effective reflectivity of external feedback cavity.

According compound cavity model with multiple feedback, the effective reflectivity $R_{e f f}^{o}$ may be expressed as [14]

R_{e f f}^{o} = r_{2} [1+ \sum_{m = 1}^{q} t_{2}^{2} r_{2}^{m - 2} r_{3}^{m} p_{m} \exp (m ω_{o} \frac{2 l}{c} + δ_{m})]

where r₂ and r₃ are the amplitude reflection coefficients of M₂ and M_f, l is the length of external cavity, ω_o is the optical angular frequencies, P_m is the coupling efficiency of the mth order feedback beam, δ_m represents the phase variation after m round trips induced by the asymmetric external cavity.

From Eq. (6), we can see that the effective reflectivity $R_{e f f}^{o}$ is modulated by the length of external cavity l. At weak feedback level (q = m = 1), the high-order feedback can be neglected. So, the sine-like feedback fringe with resolution of λ/2 is obtained. However, at strong feedback level, the multiple feedback cannot be neglected. The fringe density will be determined mainly by multiple feedback and the high-frequency optical fringes are observed.

In Eq. (4), the left represents the anisotropic phase of laser, and the right is in relation to the net gain of laser. At weak feedback level, the net gain of laser is not large enough to overcome the mode competition due to the small anisotropic phase of laser. So, there is no polarization flipping accompany with isotropic weak feedback. But, when the feedback level is high, the loss of laser is reduced greatly, and the two orthogonal polarized eigenstates will have enough net gain to overcome the mode competition due to the small anisotropic phase (residual stress). So the polarization flipping occurs when the length of external cavity is modulated and the positions of polarization flipping are mainly determined by $Δ Φ_{o e}$ . Once $Δ Φ_{o e}$ (frequency difference) reduces, the position of flipping moves to the edge of fringe gradually which agrees well with the experimental results showed in Fig. 3.

5. Application in displacement measurement

The experiment result and theory analysis show that the high-frequency optical fringes can be obtained in single-mode laser with multiple optical feedback. Particularly, the polarization flipping occurs in each optical fringe and the duty cycle of two eigenstates can be controlled easily by adjusting the residual stress of laser mirror. These results are useful to develop a high resolution displacement sensor with direction sensitivity.

The preliminary study on the application of high-frequency optical fringes in displacement measurement is conducted. The measurement target (M_f) is driven by PZT with triangular wave voltage. The triangular wave voltage ranges from 0 to 300V and the corresponding displacement of PZT is from 0 to about 1μm. In measurement, the high-frequency optical fringes are first converted to a series of digital pulses. Then the displacement of M_f is measured by counting the number of pulses, here the pulse equivalent is λ/60 (10.5nm). The measurement result is shown in Fig. 4. Figure 4 shows that the displacement of PZT can be measured accurately with high repeatability. The resolution of this method is about 10.5nm. The measurement accuracy is mainly affected by the wavelength shift and the pulse miscount. The pulse miscount generates when the displacement amounts is smaller than one pulse equivalent. The combined estimation error of this system is about 10.7nm. These results indicate that this method has the potential ability to measure small displacement with direction sensitivity and high resolution.

Fig. 4 Displacement measurement results of PZT.

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6. Conclusion

The polarization dynamics of a quasi-isotropic single-mode laser subjected to multiple optical feedback has been demonstrated. When the length of external cavity is modulated, the high-frequency optical fringes with nanoscale resolution are obtained due to the multiple feedback in the asymmetry external cavity. Particularly, the polarization flipping occurs in each optical fringe and the duty cycle of two eigenstates can be controlled easily by adjusting the residual stress of laser mirror. A preliminary study on the application of high-frequency optical fringes in displacement measurement is conducted. The results indicate that it has the potential ability to measure small displacement with direction sensitivity and high resolution.

Acknowledgments

This work was supported by National Natural Science Foundation of China (No.60827006, 60723004) and Scholarship Award for Excellent Doctoral Student granted by Ministry of Education.

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Controlling the duty cycle of the eigenstates in laser with multiple optical feedback

Abstract

1. Introduction

2. Experimental setup

3. Experimental results

4. Theoretical analysis

5. Application in displacement measurement

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (6)

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