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External optical feedback effects in a frequency locking dual frequency laser
Optics Communications 271 (2007) 492–498 www.elsevier.com/locate/optcom
External optical feedback effects in a frequency locking dual frequency laser Wei Mao *, Shulian Zhang, Yidong Tan, Weixin Liu The State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China Received 14 June 2006; received in revised form 11 September 2006; accepted 16 October 2006
Abstract We present a systematic study of external optical feedback effects on the output characteristics of the two orthogonally polarized modes in a frequency locking dual frequency laser. Accompanied by the change of the external cavity length, the length of the resonance cavity is changed very slowly to set the oscillating modes sweeping across the laser gain curve. Comprehensive results are obtained and a strong mode competition is observed. Particularly, when the laser runs in the middle region of the gain curve, which mode can oscillate is determined by the movement direction of the external feedback mirror. Based on the phenomenon, a displacement sensor with directional discrimination is proposed and its capability is also discussed. 2006 Elsevier B.V. All rights reserved. Keywords: Optical feedback; Frequency locking; Mode competition; Displacement sensor; Directional discrimination
1. Introduction The optical feedback phenomenon in a laser occurs when a portion of the laser output is reflected back into its resonator by a moving reflector or object, which was first reported by King and Steward [1]. It was found that the laser intensity modulation caused by an external optical feedback was similar to that produced by a conventional optical interferometer. Thus, optical feedback is also called self-mixing interference. The phenomena have been widely studied in gas lasers [1–3], semiconductor lasers [4–6] and microchip solid lasers [7,8]. Recently, much attention has been paid to the optical feedback of orthogonally polarized dual frequency He– Ne lasers [9–11]. Although the size of a He–Ne laser is much bigger than a semiconductor laser, the characteristic of a He–Ne laser beam is quite better and the focus lens and precise temperature controller are not needed. Particularly, more characteristics [12–16] of optical feedback can *
Corresponding author. Tel.: +86 10627 82838; fax: +86 10627 84503. E-mail address: [email protected] (W. Mao).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.035
be obtained because of the interaction between the two orthogonally polarized lights. In previous works, we have acquired some results about optical feedback regimes [12], strong optical feedback [13], polarized optical feedback [15] and polarization hopping in optical feedback [16], which are based on dual frequency lasers. But the optical feedback effects in a frequency locking dual frequency laser are not studied. Especially, the output characteristics of a laser running on different positions of the laser gain curve are not systematically studied. Compared with a conventional interferometer, the selfmixing interference system has not only similar phase sensitivity and modulation depth ratio, but also inherent simplicity, compactness, robustness as well as cheap. Thus, a series of application researches [14,17] carried out mainly in the measurement fields of displacement. However, these applications have some insufficiencies. First, it is hard to judge the direction of the displacement. Although the sawtooth-like waveform of a laser diode working in the moderate feedback regime [6,17] can be used to discriminate the direction, the method will not work if the laser does not work in the right feedback regime and the external
W. Mao et al. / Optics Communications 271 (2007) 492–498
feedback cavity is not aligned perfectly. And it will bring on an acceptable measurement error. Second, these systems have strict demands of the object to be measured, such as the optical feedback levels, the alignment, and so on. Thus, these factors restrict the measurement range of the optical feedback system. In this paper, we systematically study the external optical feedback effects on the output characteristics of the two orthogonally polarized modes in a frequency locking dual frequency laser, which runs on different positions of the laser gain curve. According to the experimental results, the laser gain curve can be divided into three regions with optical feedback. When the laser runs in both sideward regions of the laser gain curve, only one mode can oscillate and the other mode is always extinct. However, when the laser runs in the middle region of the gain curve, mode of which can oscillate between the two orthogonally polarized modes is determined by the movement direction of the external feedback mirror. It means that the intensity modulation curves of optical feedback contain the direction information of the external feedback object. We carry out some related experiments to confirm the phenomenon. Based on the phenomenon, a displacement sensor with directional discrimination is proposed. And the sensor’s capability is also discussed. 2. Experimental setup Experiments are performed with a half-intracavity frequency locking dual frequency He–Ne laser operating at 632.8 nm. The experimental setup is schematically shown in Fig. 1. The laser internal cavity consists of a plane mirror M2 and a concave mirror M1, which has a radius of 1 m. Their reflectivities are R2 = 0.990 and R1 = 0.995, respectively. The discharge tube T is filled with He:Ne = 7:1 and 20 Ne:22Ne = 1:1 gas mixture to suppress the Lamb dip in the output power curve. WQC is a wedged quartz crystal with the wedge angle h being 1.5, whose both sides are coated with antireflecting films. M3 is the external feedback mirror with a reflectivity of R3 = 0.35. The laser internal cavity length L is 155 mm and the external cavity length l
Fig. 1. Experimental setup. M1, M2: mirrors; T: discharge tube; WQC: wedged quartz crystal; M3: external feedback mirror; W: Wollaston prism; PZT1, PZT2: piezoelectric transducers; D1, D2: photoelectric detectors.
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is 215 mm. PZT1 and PZT2 are two piezoelectric transducers that drive the mirrors to modify the laser internal cavity length and the external cavity length, respectively. Increasing the voltage of PZT1 or PZT2 will result in a decrease of the internal cavity length or the external cavity length. On the contrary, decreasing the PZT voltage will result in an increase of the cavity lengths L or l. The tail lights are divided by a Wollaston prism W into two orthogonally polarized lights, o-light and e-light. The two polarized lights are detected by photoelectric detectors D1 and D2. The signals are gathered by the A/D card and then sent to the computer.
3. Experimental results At first, we do not introduce optical feedback, so the external feedback mirror M3 is removed from Fig. 1. By driving PZT1 to modulate the length of the internal cavity, the laser frequencies sweep across the laser gain curve orderly. When mirror M2 moves a displacement of half wavelength of the laser, the laser frequencies vary by one longitudinal mode interval. A period of the intensity tuning curve can be acquired as shown in Fig. 2. In Fig. 2 and also in the following other figures, the open-circle curve is for Io and the solid-circle curve is for Ie. Io and Ie represent the intensities of the o-light and the e-light respectively. The curve with triangular symbols is for V1, which represents the driving signal applied on the PZT1. From Fig. 2, we can see that the two split modes nearly have no co-existing region. When the length of the internal cavity is decreasing, only o-light oscillates at first. Suddenly, o-light extinguishes and e-light begins to oscillate at the right region of the laser gain curve. It is because that the frequency difference is only 25 MHz caused by the WQC, which is smaller than the locking threshold about 40 MHz [18]. However, in the center of the curve there is
Fig. 2. Intensity tuning curve of the frequency locking dual frequency laser without optical feedback.
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still a very small region, where both lights can co-exist but not stably. Strong mode competition exists between the two split modes. So only one mode can oscillate and the laser is a so-called frequency locking dual frequency laser. Then, we fix the external feedback mirror M3 and make the feedback cavity aligned perfectly. We drive mirror M3 to modulate the external cavity length with a triangle wave whose period is about 1 s, and at the same time we drive mirror M2 with an increasing voltage to modify the internal cavity length. It takes 130 s when M2 moves half the laser wavelength. The modulation frequency of the internal cavity length is far lower than that of the external feedback cavity length. So the variation of the internal cavity length is nearly of no effect to the optical feedback. We can obtain the optical feedback characteristics of the laser running on different positions of the laser gain curve. The experimental results are shown in Fig. 3 when M2 moves half the laser wavelength during the optical feedback. The vertical axis represents the laser intensity and the horizontal axis represents the time. The o-light’s intensity curve is shifted up along the vertical axis by 3.5 V to discriminate it from the intensity curve of the e-light. The square curve is for V2, which represents the driving signal applied on the PZT2. According to the experimental results in Fig. 3, the laser gain curve can be divided into three regions with optical feedback. When the laser runs on the both sideward regions A and C of the laser gain curve, only one mode can oscillate and the other mode is always extinct. Whereas when the laser runs in the middle region B of the gain curve, both modes can oscillate. In order to study the optical feedback effects detailedly, we modify the internal cav-
ity length to set the laser operating on the three regions, respectively. The experimental results are shown in Fig. 4, which relate to the three regions. In region A, only o-light oscillates. Although the threshold gain of the laser is modulated due to optical feedback, strong mode competition still exists between the two split modes and e-light has not enough gain to oscillate. With the internal cavity length decreasing, the average intensity of o-light increases and the modulation depth of o-light caused by optical feedback also increases, as shown in Figs. 3 and 4a. In region B, both o-light and e-light can oscillate. However, they oscillate by turns, not synchronously. Which mode can oscillate is determined by the movement direction of the external cavity mirror. When the voltage of PZT2 increases, that is the feedback mirror M3 moves toward the laser, only e-light oscillates. When the voltage of PZT2 decreases, o-light oscillates, while e-light is extinct. The oscillating mode is changed with the alteration of the movement direction of the external feedback mirror. Accidentally, when the oscillating mode’s intensity is near its minimum, the other mode may oscillate transitorily with a very low intensity. A strong mode competition can be observed. With the internal cavity length decreasing, the average intensity of o-light or e-light increases and then decreases in some sort, and modulation depth does not seem to vary very much, as shown in Figs. 3 and 4b. Compared with the modulation depth of o-light in region A, the modulation depth of o-light or e-light is much deeper in region B. In region C, only e-light oscillates. In Fig. 4c, it is shown that when the external feedback mirror M3 moves a dis-
Fig. 3. Intensity modulation characteristics with optical feedback in the frequency locking dual frequency laser during the laser internal cavity being tuned.
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Fig. 4. An observation of the intensities of the two modes in different regions of the laser gain curve. (a) In region A, (b) In region B and (c) In region C.
placement of half wavelength of the laser, a period fringe of e-light is produced, but o-light is of zero intensity all the time no matter whether V2 is increasing or decreasing. With the internal cavity length decreasing in Fig. 3, the average intensity of e-light decreases and the modulation depth of e-light caused by optical feedback also decreases. The above experimental results show that when the frequency locking dual frequency laser runs in the region B, we can acquire the intensity modulation curves of optical feedback that contain the direction information of the external feedback object. Thus, based on the phenomenon a displacement sensor with directional discrimination can be easily realized. In order to confirm the phenomenon, some related experiments are carried out. We consider four main factors which may affect the results. The first one is the external cavity length l. we have done a lot of experiments at different external cavity lengths, such as l = 100 mm, l = 155 mm, l = 280 mm, and so on. The results are nearly the same as shown in Fig. 4b. When the voltage of PZT2 increases, only e-light oscillates, whereas when the voltage of PZT2 decreases, o-light oscillates and e-light is extinct. The second factor is the feedback level of the external cavity. We use a mirror whose reflectivity is as high as 0.88 as the external feedback mirror. When the laser runs in region B, the experimental result is shown in Fig. 5. From Fig. 5, we can see that the intensity modulation curves also contain the direction information of the exter-
Fig. 5. An observation of the intensities of the two modes at a high feedback level when the laser runs in region B.
nal feedback object at high feedback levels. Compared with Fig. 4b, the modulation depth of o-light or e-light is much deeper in Fig. 5. The third factor is the misalignment of the external feedback mirror. We adjust the mirror M3 to destroy the alignment of the external cavity in a small degree, so as to ensure that most of the light can be fed back into the internal cavity. A small portion of the light may be unable to be fed back into the laser internal cavity because of the misalignment. It will decrease the reflectivity of the feedback mirror indirectly, and reduce the feedback level. The exper-
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imental results are similar to that in Figs. 4b and 5. So it is non-sensitive to the feedback levels. The fourth factor is the dithering of the laser cavity mirror M2. When the laser runs in region B at the time of 50 s, we change the voltage V1 of M2 to make it move towards the contrary direction. The experimental result is shown in Fig. 6. From Fig. 6, we can see that although the movement direction of M2 is changed, the output characteristics of o-light and e-light remain unchanged. In region B, when the voltage of PZT2 increases, only e-light oscillates, whereas when the voltage of PZT2 decreases, o-light oscillates and e-light is extinct, no matter the movement direction of M2 is changed or not. These experimental results prove that the intensity modulation characteristics of o-light and e-light in region B are of nonsensitivity to the factors, such as the external cavity length, the feedback level, the misalignment of the external feedback mirror and the movement direction of the laser cavity mirrors. The mode that can oscillate is determined strongly by the movement direction of the external cavity mirror. It is of great importance for directional discrimination in the application of measurement.
oscillate only when both the corresponding net gain coefficient ai and the effective gain coefficient a0i are positive a0i ¼ ai aj
hij bj
ði; j ¼ 1; 2; i 6¼ jÞ;
ð1Þ
where hij and bj are the coefficients which represent the cross-saturation and self-saturation of the ith split mode. In Ref. [18], the evolution curves of ai and a0i with the frequency differences are given. We can see that when the frequency difference is 25 MHz, the effective gain coefficient a0i is about 0.6 · 105 Hz, although the corresponding net gain coefficient ai is positive. Thus, one frequency is locked similar to the experimental results in Fig. 2. In the following part, we will analyze the intensity characteristics of the frequency locking dual frequency laser with optical feedback. 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 feedback 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 single mode helium neon laser with optical feedback can be given by [19]
4. Theoretical analysis
r1 reff exp½ði2pmsL Þ þ 2ðg aÞL ¼ 1;
Because of birefringence effects of the wedged quartz crystal in the internal cavity, using the frequency split technology [9], one mode is split into two orthogonally polarized frequencies with the frequency difference being 25 MHz. According to Lamb’s analysis, a split mode will
where r1 is the amplitude reflection coefficient of M1, reff represents the complex effective field amplitude reflection coefficient due to the laser coupling mirror M2 and the external feedback mirror M3, m is the laser frequency, sL = 2L/c is the laser beam round-trip time in the internal
Fig. 6. An observation of the intensities of the two modes when the movement direction of M2 is changed.
ð2Þ
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cavity, c is the speed of light in vacuum, g is the laser linear gain per unit length, and a is the internal loss. Based on the three-cavity-mirror model [6,20], we can get reff ¼ r2 ½1 þ f expði2pmsl Þ;
ð3Þ
where r2, r3 are the amplitude reflection coefficient of M2 and M3, sl = 2l/c is the laser beam round-trip time in the external feedback cavity, and f ¼ ð1 r22 Þr3 =r2 is the optical feedback factor. From Eqs. (2) and (3), the threshold gain change DG can be expressed as 1 f cos ð2pmsl Þ; ð4Þ 2L where g0 is the threshold gain of laser without optical feedback. Because variations of laser intensity is proportional to DG, the laser intensity with optical feedback can be written as h i m I ¼ I 0 1 þ f cos ð2pmsl Þ ; ð5Þ 2L where I0 is the intensity of the laser without optical feedback, m is a proportionality coefficient that associated with laser gain and the optical feedback factor f. Thus, for a frequency locking dual frequency laser used in the paper, the intensity of o-light Io and e-light Ie with optical feedback can be expressed as h i m I o or e ¼ I 0o or e 1 þ f cos ð2pmo or e sl Þ ; ð6Þ 2L
DG ¼ g g0 ¼
where I0o and I0e are the intensities of the o-light and the elight without optical feedback, mo and me are the frequencies of the o-light and the e-light, respectively. When the internal cavity length decreases, the mode whose frequency is larger than the other will first enter into the laser gain curve and it will be in an absolutely leading situation in optical feedback. Thus, o-light first oscillates and e-light has not enough gain to oscillate due to strong mode competition in region A. In this region, with o-mode sweeping to the middle region of the laser gain curve, the available gain increases and the average intensity of o-light I0o also increases without optical feedback. From Eq. (6), we can see that the modulation depth of o-light, which m can be expressed as I 0o 2L f, will increase with the decreasing of the internal cavity length in region A. Similarly, in region C with e-light sweeping from the middle region to the sideward region of the laser gain curve, the available gain and the average intensity of e-light I0e decreases, which results in the decrease of the modulation depth of e-light. In region B, the laser runs on the middle region of the laser gain curve. The available gain in this region is bigger than that in region A or C. So the o-light or the e-light have the largest modulation depth. The experimental results are in good agreement with the theoretical analysis.
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In the above several experiments, we can find out that when the laser runs in the middle region of the gain curve, the mode of which can oscillate between the o-light and the e-light is determined by the movement direction of the external feedback mirror. 5. Discussion In this paper, the output characteristics of the two orthogonally polarized modes subjected to optical feedback in a frequency locking dual frequency laser are systematically studied during the laser internal cavity when tuned. From the experiments, we can see that the larger the initial intensity of one light without optical feedback, the deeper the modulation depth of the light with optical feedback in regions A and C. The feature can be used to improve the optical feedback system’s sensitivity for displacement by adjusting the laser internal cavity length to let the laser run towards the middle region of the laser gain curve. In Figs. 4a and c, all the curves are symmetrical and the movement direction of the external feedback mirror is not easy to be discriminated. However, in Fig. 4b, the intensity modulation curves contain the direction information of the external feedback mirror and have the largest modulation depths. Sufficient experimental results have proven that when the frequency locking dual frequency laser runs in the middle region of the laser gain curve with optical feedback, the movement direction of the external feedback mirror can be discriminated by the oscillating mode. A displacement sensor based on the phenomenon has some vital capabilities in actual application. Firstly, the system has the capability of directional discrimination. A displacement sensor that cannot discriminate the displacement direction or has poor capability of directional discrimination, cannot be called a sensor. In the sensor proposed in this paper, o-light oscillating represents the external feedback mirror moving away from the laser, and e-light oscillating represents the feedback mirror moving toward the laser. Using a polarized beam splitter to divide the two orthogonally polarized lights, and detecting the intensities of the two lights separately, we can easily obtain the displacement information of the external feedback mirror in the two contrary directions. Secondly, the system has little confines about the object to be measured and thus no additional attachments are needed. From Fig. 5 we can see the system is nonsensitive to the feedback levels. So a target mirror or an attenuator, which is usually used to adjust the feedback level of the system, does not need. The system is much simple and compact. The range of the object can be measured has been broadened. Thirdly, the measurement range of the system will be very large. In part 3, we have done several experiments to research the phenomenon at different external cavity lengths. The system can work well. Moreover, when the external feedback cavity is not aligned perfectly, we also
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can obtain the displacement information of the external object. It ensures that the large measurement range of the system is possible. Lastly, the system is robust. From Fig. 3 we can see that region B is almost two times larger than region A or region C. And from Fig. 6, we can find out that the dithering of the laser cavity mirror has little effect to the system. So, when the laser runs on region B, the system has strong capability against the variation of the laser cavity length caused by vibration, temperature variation, and so on. Furthermore, the system has a low cost but a high quality. Acknowledgements This work is supported by the National Nature Science Foundation of China and Tsinghua University. The authors also wish to thank the reviewers and editors for their valuable recommendations on the manuscript. References [1] P.G.R. King, G.J. Steward, New Sci. 17 (1963) 180.
[2] B. Ovryn, J.H. Andrews, Opt. Lett. 23 (1998) 1078. [3] M.J. Rudd, J. Phys. E. 1 (1968) 723. [4] R. Lang, K. Kobayashi, IEEE J. Quantum Electron. QE-16 (1980) 347. [5] P.J. De. Groot, G.M. Gallatin, S.H. Macomber, Appl. Opt. 27 (1988) 4475. [6] W.M. Wang, K.T.V. Grattan, A.W. Palmer, W.J.O. Boyle, J. Lightwave Technol. 12 (1994) 1577. [7] E. Lacot, R. Day, F. Stoeckel, Opt. Lett. 24 (1999) 744. [8] X.J. Wan, S.L. Zhang, G. Liu, L.G. Fei, Opt. Eng. 44 (2005) 104204. [9] S.L. Zhang, H. Guo, K.L. Li, Y.M. Han, Opt. Laser Eng. 23 (1995) 1. [10] W. Holzapfel, W. Settgast, Appl. Phys. B 49 (1989) 69. [11] W. Mao, S.L. Zhang, L.G. Fei, Chin. Phys. 15 (2006) 2036. [12] W. Mao, S.L. Zhang, L.Q. Zhang, J. Zhu, Y. Li, Chin. Phys. Lett. 23 (2006) 1188. [13] W. Mao, S.L. Zhang, Chin. Phys. 15 (2006) 340. [14] W. Mao, S.L. Zhang, L. Cui, Y.D. Tan, Opt. Exp. 14 (2006) 182. [15] G. Liu, S.L. Zhang, T. Xu, J. Zhu, Y. Li, Opt. Commun. 241 (2004) 159. [16] L.G. Fei, S.L. Zhang, X.B. Zong, Opt. Commun. 246 (2005) 505. [17] G. Giuliani, M. Norgia, S. Donati, T. Bosch, J. Opt. A Pure Appl. Opt. 4 (2002) S283. [18] X.B. Zong, W.X. Liu, S.L. Zhang, Chin. Phys. Lett. 22 (2005) 1906. [19] W. Mao, S.L. Zhang, Appl. Opt. 45 (2006) 7723. [20] W.M. Wang, W.J.O. Boyle, K.T.V. Grattan, A.W. Palmer, Appl. Opt. 32 (1993) 1551.
External optical feedback effects in a frequency locking dual frequency laser Wei Mao *, Shulian Zhang, Yidong Tan, Weixin Liu The State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China Received 14 June 2006; received in revised form 11 September 2006; accepted 16 October 2006
Abstract We present a systematic study of external optical feedback effects on the output characteristics of the two orthogonally polarized modes in a frequency locking dual frequency laser. Accompanied by the change of the external cavity length, the length of the resonance cavity is changed very slowly to set the oscillating modes sweeping across the laser gain curve. Comprehensive results are obtained and a strong mode competition is observed. Particularly, when the laser runs in the middle region of the gain curve, which mode can oscillate is determined by the movement direction of the external feedback mirror. Based on the phenomenon, a displacement sensor with directional discrimination is proposed and its capability is also discussed. 2006 Elsevier B.V. All rights reserved. Keywords: Optical feedback; Frequency locking; Mode competition; Displacement sensor; Directional discrimination
1. Introduction The optical feedback phenomenon in a laser occurs when a portion of the laser output is reflected back into its resonator by a moving reflector or object, which was first reported by King and Steward [1]. It was found that the laser intensity modulation caused by an external optical feedback was similar to that produced by a conventional optical interferometer. Thus, optical feedback is also called self-mixing interference. The phenomena have been widely studied in gas lasers [1–3], semiconductor lasers [4–6] and microchip solid lasers [7,8]. Recently, much attention has been paid to the optical feedback of orthogonally polarized dual frequency He– Ne lasers [9–11]. Although the size of a He–Ne laser is much bigger than a semiconductor laser, the characteristic of a He–Ne laser beam is quite better and the focus lens and precise temperature controller are not needed. Particularly, more characteristics [12–16] of optical feedback can *
Corresponding author. Tel.: +86 10627 82838; fax: +86 10627 84503. E-mail address: [email protected] (W. Mao).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.035
be obtained because of the interaction between the two orthogonally polarized lights. In previous works, we have acquired some results about optical feedback regimes [12], strong optical feedback [13], polarized optical feedback [15] and polarization hopping in optical feedback [16], which are based on dual frequency lasers. But the optical feedback effects in a frequency locking dual frequency laser are not studied. Especially, the output characteristics of a laser running on different positions of the laser gain curve are not systematically studied. Compared with a conventional interferometer, the selfmixing interference system has not only similar phase sensitivity and modulation depth ratio, but also inherent simplicity, compactness, robustness as well as cheap. Thus, a series of application researches [14,17] carried out mainly in the measurement fields of displacement. However, these applications have some insufficiencies. First, it is hard to judge the direction of the displacement. Although the sawtooth-like waveform of a laser diode working in the moderate feedback regime [6,17] can be used to discriminate the direction, the method will not work if the laser does not work in the right feedback regime and the external
W. Mao et al. / Optics Communications 271 (2007) 492–498
feedback cavity is not aligned perfectly. And it will bring on an acceptable measurement error. Second, these systems have strict demands of the object to be measured, such as the optical feedback levels, the alignment, and so on. Thus, these factors restrict the measurement range of the optical feedback system. In this paper, we systematically study the external optical feedback effects on the output characteristics of the two orthogonally polarized modes in a frequency locking dual frequency laser, which runs on different positions of the laser gain curve. According to the experimental results, the laser gain curve can be divided into three regions with optical feedback. When the laser runs in both sideward regions of the laser gain curve, only one mode can oscillate and the other mode is always extinct. However, when the laser runs in the middle region of the gain curve, mode of which can oscillate between the two orthogonally polarized modes is determined by the movement direction of the external feedback mirror. It means that the intensity modulation curves of optical feedback contain the direction information of the external feedback object. We carry out some related experiments to confirm the phenomenon. Based on the phenomenon, a displacement sensor with directional discrimination is proposed. And the sensor’s capability is also discussed. 2. Experimental setup Experiments are performed with a half-intracavity frequency locking dual frequency He–Ne laser operating at 632.8 nm. The experimental setup is schematically shown in Fig. 1. The laser internal cavity consists of a plane mirror M2 and a concave mirror M1, which has a radius of 1 m. Their reflectivities are R2 = 0.990 and R1 = 0.995, respectively. The discharge tube T is filled with He:Ne = 7:1 and 20 Ne:22Ne = 1:1 gas mixture to suppress the Lamb dip in the output power curve. WQC is a wedged quartz crystal with the wedge angle h being 1.5, whose both sides are coated with antireflecting films. M3 is the external feedback mirror with a reflectivity of R3 = 0.35. The laser internal cavity length L is 155 mm and the external cavity length l
Fig. 1. Experimental setup. M1, M2: mirrors; T: discharge tube; WQC: wedged quartz crystal; M3: external feedback mirror; W: Wollaston prism; PZT1, PZT2: piezoelectric transducers; D1, D2: photoelectric detectors.
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is 215 mm. PZT1 and PZT2 are two piezoelectric transducers that drive the mirrors to modify the laser internal cavity length and the external cavity length, respectively. Increasing the voltage of PZT1 or PZT2 will result in a decrease of the internal cavity length or the external cavity length. On the contrary, decreasing the PZT voltage will result in an increase of the cavity lengths L or l. The tail lights are divided by a Wollaston prism W into two orthogonally polarized lights, o-light and e-light. The two polarized lights are detected by photoelectric detectors D1 and D2. The signals are gathered by the A/D card and then sent to the computer.
3. Experimental results At first, we do not introduce optical feedback, so the external feedback mirror M3 is removed from Fig. 1. By driving PZT1 to modulate the length of the internal cavity, the laser frequencies sweep across the laser gain curve orderly. When mirror M2 moves a displacement of half wavelength of the laser, the laser frequencies vary by one longitudinal mode interval. A period of the intensity tuning curve can be acquired as shown in Fig. 2. In Fig. 2 and also in the following other figures, the open-circle curve is for Io and the solid-circle curve is for Ie. Io and Ie represent the intensities of the o-light and the e-light respectively. The curve with triangular symbols is for V1, which represents the driving signal applied on the PZT1. From Fig. 2, we can see that the two split modes nearly have no co-existing region. When the length of the internal cavity is decreasing, only o-light oscillates at first. Suddenly, o-light extinguishes and e-light begins to oscillate at the right region of the laser gain curve. It is because that the frequency difference is only 25 MHz caused by the WQC, which is smaller than the locking threshold about 40 MHz [18]. However, in the center of the curve there is
Fig. 2. Intensity tuning curve of the frequency locking dual frequency laser without optical feedback.
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still a very small region, where both lights can co-exist but not stably. Strong mode competition exists between the two split modes. So only one mode can oscillate and the laser is a so-called frequency locking dual frequency laser. Then, we fix the external feedback mirror M3 and make the feedback cavity aligned perfectly. We drive mirror M3 to modulate the external cavity length with a triangle wave whose period is about 1 s, and at the same time we drive mirror M2 with an increasing voltage to modify the internal cavity length. It takes 130 s when M2 moves half the laser wavelength. The modulation frequency of the internal cavity length is far lower than that of the external feedback cavity length. So the variation of the internal cavity length is nearly of no effect to the optical feedback. We can obtain the optical feedback characteristics of the laser running on different positions of the laser gain curve. The experimental results are shown in Fig. 3 when M2 moves half the laser wavelength during the optical feedback. The vertical axis represents the laser intensity and the horizontal axis represents the time. The o-light’s intensity curve is shifted up along the vertical axis by 3.5 V to discriminate it from the intensity curve of the e-light. The square curve is for V2, which represents the driving signal applied on the PZT2. According to the experimental results in Fig. 3, the laser gain curve can be divided into three regions with optical feedback. When the laser runs on the both sideward regions A and C of the laser gain curve, only one mode can oscillate and the other mode is always extinct. Whereas when the laser runs in the middle region B of the gain curve, both modes can oscillate. In order to study the optical feedback effects detailedly, we modify the internal cav-
ity length to set the laser operating on the three regions, respectively. The experimental results are shown in Fig. 4, which relate to the three regions. In region A, only o-light oscillates. Although the threshold gain of the laser is modulated due to optical feedback, strong mode competition still exists between the two split modes and e-light has not enough gain to oscillate. With the internal cavity length decreasing, the average intensity of o-light increases and the modulation depth of o-light caused by optical feedback also increases, as shown in Figs. 3 and 4a. In region B, both o-light and e-light can oscillate. However, they oscillate by turns, not synchronously. Which mode can oscillate is determined by the movement direction of the external cavity mirror. When the voltage of PZT2 increases, that is the feedback mirror M3 moves toward the laser, only e-light oscillates. When the voltage of PZT2 decreases, o-light oscillates, while e-light is extinct. The oscillating mode is changed with the alteration of the movement direction of the external feedback mirror. Accidentally, when the oscillating mode’s intensity is near its minimum, the other mode may oscillate transitorily with a very low intensity. A strong mode competition can be observed. With the internal cavity length decreasing, the average intensity of o-light or e-light increases and then decreases in some sort, and modulation depth does not seem to vary very much, as shown in Figs. 3 and 4b. Compared with the modulation depth of o-light in region A, the modulation depth of o-light or e-light is much deeper in region B. In region C, only e-light oscillates. In Fig. 4c, it is shown that when the external feedback mirror M3 moves a dis-
Fig. 3. Intensity modulation characteristics with optical feedback in the frequency locking dual frequency laser during the laser internal cavity being tuned.
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Fig. 4. An observation of the intensities of the two modes in different regions of the laser gain curve. (a) In region A, (b) In region B and (c) In region C.
placement of half wavelength of the laser, a period fringe of e-light is produced, but o-light is of zero intensity all the time no matter whether V2 is increasing or decreasing. With the internal cavity length decreasing in Fig. 3, the average intensity of e-light decreases and the modulation depth of e-light caused by optical feedback also decreases. The above experimental results show that when the frequency locking dual frequency laser runs in the region B, we can acquire the intensity modulation curves of optical feedback that contain the direction information of the external feedback object. Thus, based on the phenomenon a displacement sensor with directional discrimination can be easily realized. In order to confirm the phenomenon, some related experiments are carried out. We consider four main factors which may affect the results. The first one is the external cavity length l. we have done a lot of experiments at different external cavity lengths, such as l = 100 mm, l = 155 mm, l = 280 mm, and so on. The results are nearly the same as shown in Fig. 4b. When the voltage of PZT2 increases, only e-light oscillates, whereas when the voltage of PZT2 decreases, o-light oscillates and e-light is extinct. The second factor is the feedback level of the external cavity. We use a mirror whose reflectivity is as high as 0.88 as the external feedback mirror. When the laser runs in region B, the experimental result is shown in Fig. 5. From Fig. 5, we can see that the intensity modulation curves also contain the direction information of the exter-
Fig. 5. An observation of the intensities of the two modes at a high feedback level when the laser runs in region B.
nal feedback object at high feedback levels. Compared with Fig. 4b, the modulation depth of o-light or e-light is much deeper in Fig. 5. The third factor is the misalignment of the external feedback mirror. We adjust the mirror M3 to destroy the alignment of the external cavity in a small degree, so as to ensure that most of the light can be fed back into the internal cavity. A small portion of the light may be unable to be fed back into the laser internal cavity because of the misalignment. It will decrease the reflectivity of the feedback mirror indirectly, and reduce the feedback level. The exper-
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imental results are similar to that in Figs. 4b and 5. So it is non-sensitive to the feedback levels. The fourth factor is the dithering of the laser cavity mirror M2. When the laser runs in region B at the time of 50 s, we change the voltage V1 of M2 to make it move towards the contrary direction. The experimental result is shown in Fig. 6. From Fig. 6, we can see that although the movement direction of M2 is changed, the output characteristics of o-light and e-light remain unchanged. In region B, when the voltage of PZT2 increases, only e-light oscillates, whereas when the voltage of PZT2 decreases, o-light oscillates and e-light is extinct, no matter the movement direction of M2 is changed or not. These experimental results prove that the intensity modulation characteristics of o-light and e-light in region B are of nonsensitivity to the factors, such as the external cavity length, the feedback level, the misalignment of the external feedback mirror and the movement direction of the laser cavity mirrors. The mode that can oscillate is determined strongly by the movement direction of the external cavity mirror. It is of great importance for directional discrimination in the application of measurement.
oscillate only when both the corresponding net gain coefficient ai and the effective gain coefficient a0i are positive a0i ¼ ai aj
hij bj
ði; j ¼ 1; 2; i 6¼ jÞ;
ð1Þ
where hij and bj are the coefficients which represent the cross-saturation and self-saturation of the ith split mode. In Ref. [18], the evolution curves of ai and a0i with the frequency differences are given. We can see that when the frequency difference is 25 MHz, the effective gain coefficient a0i is about 0.6 · 105 Hz, although the corresponding net gain coefficient ai is positive. Thus, one frequency is locked similar to the experimental results in Fig. 2. In the following part, we will analyze the intensity characteristics of the frequency locking dual frequency laser with optical feedback. 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 feedback 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 single mode helium neon laser with optical feedback can be given by [19]
4. Theoretical analysis
r1 reff exp½ði2pmsL Þ þ 2ðg aÞL ¼ 1;
Because of birefringence effects of the wedged quartz crystal in the internal cavity, using the frequency split technology [9], one mode is split into two orthogonally polarized frequencies with the frequency difference being 25 MHz. According to Lamb’s analysis, a split mode will
where r1 is the amplitude reflection coefficient of M1, reff represents the complex effective field amplitude reflection coefficient due to the laser coupling mirror M2 and the external feedback mirror M3, m is the laser frequency, sL = 2L/c is the laser beam round-trip time in the internal
Fig. 6. An observation of the intensities of the two modes when the movement direction of M2 is changed.
ð2Þ
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cavity, c is the speed of light in vacuum, g is the laser linear gain per unit length, and a is the internal loss. Based on the three-cavity-mirror model [6,20], we can get reff ¼ r2 ½1 þ f expði2pmsl Þ;
ð3Þ
where r2, r3 are the amplitude reflection coefficient of M2 and M3, sl = 2l/c is the laser beam round-trip time in the external feedback cavity, and f ¼ ð1 r22 Þr3 =r2 is the optical feedback factor. From Eqs. (2) and (3), the threshold gain change DG can be expressed as 1 f cos ð2pmsl Þ; ð4Þ 2L where g0 is the threshold gain of laser without optical feedback. Because variations of laser intensity is proportional to DG, the laser intensity with optical feedback can be written as h i m I ¼ I 0 1 þ f cos ð2pmsl Þ ; ð5Þ 2L where I0 is the intensity of the laser without optical feedback, m is a proportionality coefficient that associated with laser gain and the optical feedback factor f. Thus, for a frequency locking dual frequency laser used in the paper, the intensity of o-light Io and e-light Ie with optical feedback can be expressed as h i m I o or e ¼ I 0o or e 1 þ f cos ð2pmo or e sl Þ ; ð6Þ 2L
DG ¼ g g0 ¼
where I0o and I0e are the intensities of the o-light and the elight without optical feedback, mo and me are the frequencies of the o-light and the e-light, respectively. When the internal cavity length decreases, the mode whose frequency is larger than the other will first enter into the laser gain curve and it will be in an absolutely leading situation in optical feedback. Thus, o-light first oscillates and e-light has not enough gain to oscillate due to strong mode competition in region A. In this region, with o-mode sweeping to the middle region of the laser gain curve, the available gain increases and the average intensity of o-light I0o also increases without optical feedback. From Eq. (6), we can see that the modulation depth of o-light, which m can be expressed as I 0o 2L f, will increase with the decreasing of the internal cavity length in region A. Similarly, in region C with e-light sweeping from the middle region to the sideward region of the laser gain curve, the available gain and the average intensity of e-light I0e decreases, which results in the decrease of the modulation depth of e-light. In region B, the laser runs on the middle region of the laser gain curve. The available gain in this region is bigger than that in region A or C. So the o-light or the e-light have the largest modulation depth. The experimental results are in good agreement with the theoretical analysis.
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In the above several experiments, we can find out that when the laser runs in the middle region of the gain curve, the mode of which can oscillate between the o-light and the e-light is determined by the movement direction of the external feedback mirror. 5. Discussion In this paper, the output characteristics of the two orthogonally polarized modes subjected to optical feedback in a frequency locking dual frequency laser are systematically studied during the laser internal cavity when tuned. From the experiments, we can see that the larger the initial intensity of one light without optical feedback, the deeper the modulation depth of the light with optical feedback in regions A and C. The feature can be used to improve the optical feedback system’s sensitivity for displacement by adjusting the laser internal cavity length to let the laser run towards the middle region of the laser gain curve. In Figs. 4a and c, all the curves are symmetrical and the movement direction of the external feedback mirror is not easy to be discriminated. However, in Fig. 4b, the intensity modulation curves contain the direction information of the external feedback mirror and have the largest modulation depths. Sufficient experimental results have proven that when the frequency locking dual frequency laser runs in the middle region of the laser gain curve with optical feedback, the movement direction of the external feedback mirror can be discriminated by the oscillating mode. A displacement sensor based on the phenomenon has some vital capabilities in actual application. Firstly, the system has the capability of directional discrimination. A displacement sensor that cannot discriminate the displacement direction or has poor capability of directional discrimination, cannot be called a sensor. In the sensor proposed in this paper, o-light oscillating represents the external feedback mirror moving away from the laser, and e-light oscillating represents the feedback mirror moving toward the laser. Using a polarized beam splitter to divide the two orthogonally polarized lights, and detecting the intensities of the two lights separately, we can easily obtain the displacement information of the external feedback mirror in the two contrary directions. Secondly, the system has little confines about the object to be measured and thus no additional attachments are needed. From Fig. 5 we can see the system is nonsensitive to the feedback levels. So a target mirror or an attenuator, which is usually used to adjust the feedback level of the system, does not need. The system is much simple and compact. The range of the object can be measured has been broadened. Thirdly, the measurement range of the system will be very large. In part 3, we have done several experiments to research the phenomenon at different external cavity lengths. The system can work well. Moreover, when the external feedback cavity is not aligned perfectly, we also
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can obtain the displacement information of the external object. It ensures that the large measurement range of the system is possible. Lastly, the system is robust. From Fig. 3 we can see that region B is almost two times larger than region A or region C. And from Fig. 6, we can find out that the dithering of the laser cavity mirror has little effect to the system. So, when the laser runs on region B, the system has strong capability against the variation of the laser cavity length caused by vibration, temperature variation, and so on. Furthermore, the system has a low cost but a high quality. Acknowledgements This work is supported by the National Nature Science Foundation of China and Tsinghua University. The authors also wish to thank the reviewers and editors for their valuable recommendations on the manuscript. References [1] P.G.R. King, G.J. Steward, New Sci. 17 (1963) 180.
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