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The new phenomena of orthogonally polarized lights in laser feedback
15 December 2001
Optics Communications 200 (2001) 303±307 www.elsevier.com/locate/optcom
The new phenomena of orthogonally polarized lights in laser feedback Lu Li *, Shulian Zhang, Shiqun Li, Ping Xue State Key Laboratory of Precision Measurement Technology, Tsinghua University, 100084 Beijing, China Department of Precision Instruments, Tsinghua University, 100084 Beijing, China Department of Physics, Tsinghua University, 100084 Beijing, China Received 11 September 2000; received in revised form 11 December 2000; accepted 11 September 2001
Abstract Some new phenomena of orthogonally polarized lights are observed in the laser-feedback system and in the new system that combines the frequency splitting technique with laser feedback. Results show that there are obvious mode competition between orthogonally polarized lasers throughout the laser-feedback process. This mode competition shows randomness. Further analysis shows that this mode competition between orthogonally polarized lights can effectively increase the laser intensity changing rate relative to the piezoelectric transducer voltage, which provide a new way to improve the axial resolution of a laser-feedback microscope. Ó 2001 Published by Elsevier Science B.V. Keywords: Laser-feedback; Frequency splitting; Polarized lights
The steady-state intensity I0 of a laser can be modi®ed by introducing coherent optical feedback from an external surface. The physical basis is the interference of the backre¯ected ®eld with the standing wave inside the laser resonant cavity [1]. The internal laser intensity I0 of a two-mirror laser is [4]
frequency of the laser radiation, and Q0 is the passive resonation quality. Using R1 and R2 representing the re¯ectivities of mirror M1 and M2 (Fig. 1), and neglecting all losses other than those due to the transmission of the laser-end mirrors, we may write [4]
a a0 I0 b
Q0
pm=Q0 ; b
1
where a is the unsaturated net gain, b is the saturation parameter, a0 is the small signal gain, m is the *
Corresponding author. Address: 2701 Ridge Road #101, Berkeley, CA 94709, USA. Tel.: +1-510-841-2696. E-mail address: [email protected] (L. Li).
1
4pL=k : R1 1 R 2
2
The mirror M2 and the external mirror M3 form an external Fabry±Perot interferometer which now replaced the end mirror M2 of the simple twomirror laser resonator. The intensity re¯ectivity R of the external interferometer which varies strongly with the spacing l aects the resonation Q and by that the laser intensity. R is found to be [4]
0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 5 6 7 - X
304
L. Li et al. / Optics Communications 200 (2001) 303±307
Fig. 1. Arrangement of the laser mirrors M1 and M2 and the external mirror M3 .
R R2 1
R2 f1 1 R3 p =1 R2 R3 2 R2 R3 cos dg;
3
where d 4pl=k is the external phase dierence between successive re¯ected beams, apart from a noninteresting constant phase. Using the R replacing the R2 in Eq. (2), we get the value of Q. And then Q replacing Q0 in Eq. (1), we can drive an expression here for the quasistatic change in the internal laser intensity in terms of the re¯ectivities R3 and the external phase dierence d [4]: I I0 c=4Lb 1
R2 f1 1 R3 p =1 R2 R3 2 R2 R3 cos dg:
4
Laser-feedback eect has been used widely in microscope, for imaging nanometer-scale microfabricated structures and for picometer and nanometer motional analysis of piezoelectric devices, aiding not only in the development of more precise methods of microfabrication, but also in the rapidly growing ®eld of acoustic sensors and other related piezoelectric devices [1±3]. Now a laserfeedback microscope has been constructed capable
of proving an axial resolution of a target surface topography of 1 nm and a lateral resolution of 200 nm [2]. We start our experiments based on the physical theories above. For comparison, we repeat the laser-feedback experiment using the conventional method ®rst. Fig. 2 shows the conventional laser-feedback loop. The 0.6328 lm He±Ne laser used has an half-intracavity. The discharge tube T is ®lled with 20 Ne:22 Ne 1:1 gas mixture to suppress the Lamb dip in the output power curve. The cavity consists of a plane mirror M2 and a concave mirror M1 which has a radius of 1 m and is mounted on a piezoelectric transducer (PZT) P0 . Their re¯ectivities are 0.986 and 0.999 respectively. The length of the oscillating cavity is 195 mm, thus the distance between adjacent modes is 769 MHz. M3 is the external mirror that re¯ects laser back into the internal cavity with a re¯ectivity of 0.5. P is also a PZT which drives the mirror M3 . D is a photoelectric detector for measuring the laser intensity. C is a signal processing circuit. F±P is a Fabry±Perot scanning interferometer. First we detect the output intensity I0 in the absence of mirror M3 . In this situation, we adjust M1 by the voltage on the PZT P0 to force the laser into single mode status. This mode is TEM00 almost in the center of the gain curve. Then we connect the feedback loop to detect the curve of the output intensity Ifeedback versus the position of M3 , referring to Eq. (4). The detected curves are shown in Fig. 3.
Fig. 2. The conventional experimental setup of laser feedback.
L. Li et al. / Optics Communications 200 (2001) 303±307
Fig. 3. Output intensity versus the position of M3 using conventional laser feedback.
In Fig. 3, the vertical axis represents the laser intensity with 0.01 mW/400 units. The horizontal axis represents the position of M3 with 0.3 lm/40 V. The laser intensity has a DC noise of about 50 units and a AC noise of about 5 units. The PZT has a nonlinearity of about 10%. Using the F±P scanning interferometer set behind the mirror M3 , we ®nd that once the feedback loop is connected, the mode status changes from a single mode into three modes. Q modulation is a simple explanation for this phenomenon. If we use Eqs. (2) and (3) and substituting the experimental parameters: R1 0:999, R2 0:986, R3 0:5, L 0:195 m, k 0:6328 lm, d 0±2p, we can see that the value of Q is changed greatly relative to Q0 . And in a big portion of a cycle (M3 has a displacement of k=2 or the voltage on the PZT is changed 40 V or so), Q is much greater than Q0 .
305
pm=Q represents the loss of the laser cavity. A high loss is induced because the value of Q0 is rather low. In this condition, only the centered mode has a positive value after subtracting the loss pm=Q from the gain a0 , referring to Eq. (1), and thus only one mode exists. When Q is changed bigger after connecting the feedback loop, the loss is changed lower. And as a result, the single mode status of the laser is changed into three modes status. In order to study the characteristics of orthogonally polarized lights in laser feedback, we start our second experiment by adding a Wollaston prism before the photoelectric detector to separate the orthogonally polarized lights (we call them Alight(s) and B-light(s)). Two detectors are used to receive the lights separately. The new apparatus is shown in Fig. 4. In this experiment, S is a Wollaston prism that separates A-light(s) and B-light(s) spatially. D1 and D2 are two photoelectric detectors. The status of the laser in the absence of the mirror M3 is the same as the experiment above with single mode (I0A or I0B ) which has an output intensity of about 800 units. Because the DC noise is 50 units, either I0A or I0B must be below that value. After the feedback loop is connected, we get the curve of Alight(s) output intensity IA versus the position of M3 and the curve of B-light(s) output intensity IB versus the position of M3 . We show them in Fig. 5. From Fig. 5 we can see that when the intensity of B-light(s) output intensity is maximum the Alight(s) output intensity almost is minimum, and when B-light(s) output intensity is minimum the A-light(s) output intensity reaches the maximum, vice versa. It shows the characteristics of mode
Fig. 4. The experimental setup after separating the orthogonally polarized lights.
306
L. Li et al. / Optics Communications 200 (2001) 303±307
Fig. 5. A-light(s) output intensity and B-light(s) output intensity versus the position of M3 (the others are the same as those in Fig. 3).
competition between orthogonally polarized lights. In the absence of M3 , the single mode located at the center position of the laser gain line is a Blight. But Fig. 5 shows that B-light(s) has zero intensity in a part of the PZT scan period (PZT is changed 40 V or so when M3 has a displacement of k=2). This phenomenon also results from the mode competition between orthogonally polarized lights. As we can also see in Fig. 5, although there is only a B-light in the absence of M3 , after the feedback loop is connected, the average laser intensity of A-light(s) is obviously bigger than that of B-light(s). Furthermore, we ®nd that both Alight(s) and B-light(s) could have higher average
Fig. 6. The detected curves showing the randomness of mode competition between orthogonally polarized lights in a given cycle (the others are the same as those in Fig. 3).
laser intensity in a given cycle. This randomness happens in each cycle as shown in Fig. 6. By combining the laser feedback with the frequency splitting technique [5], we construct a new experimental system shown in Fig. 7. In this experiment, Q is a uniaxial quartz crystal. The optical rotational eect of the quartz plate has been considered. We use the uniaxial quartz crystal to make every mode split into o-light and elight, and thus getting a frequency splitting of 117 MHz. Except for the single mode splits into o-light and e-light with equal intensity (I0o I0e ), the others are all the same as the former experiments in the absence of M3 . The detected curves are shown in Fig. 8.
Fig. 7. The experimental setup of combining the laser feedback with frequency splitting.
L. Li et al. / Optics Communications 200 (2001) 303±307
307
Fig. 8. o-Lights output intensity and e-lights output intensity versus the position of M3 (the others are the same as those in Fig. 3).
Comparing Fig. 8 with the former curves, we note the obvious symmetry of Fig. 8. That is because the frequency splitting technique makes the relation between orthogonally polarized lights much more symmetric than the former one. In the former case, the adjacent A-light mode and B-light mode has a large frequency separation of about 769 MHz. The large frequency separation between the A-light and the B-light makes the gain characteristics very dierent for the two. While in Fig. 8, the o-light and e-light split from the same mode has a relatively much shorter frequency separation of 117 MHz, the two lasers have almost the same characteristics. Thus the characteristics of the all o-lights and that of the all e-lights are almost the same. Another point in this experiment is that, in the process of moving the external mirror, using the F±P scanning interferometer behind the mirror M3 , we are able to see six modes. Three of them are o-lights, and the other three are e-lights. We see that the three o-lights would increase and decrease almost synchronously. And the three elights also have this synchronism. It shows that the coupling eect between the modes whose polarization directions are parallel to each other would be stronger than that between the modes whose polarization directions are orthogonal to each other.
At last we ®nd that our new methods can be used to improve the axial resolution of LFM. This can be proved by comparing the results shown in Figs. 3, 5, and 8. We can calculate their ratios of the laser intensity variation dividing the PZT voltage variation when the voltage of PZT is changed around 90 V or so. The value of the ratio restricts the resolution using the laser-feedback microscope. Respectively, the three ratios are 59, 173, and 117 units/V from Figs. 3, 5, and 8. The lowest ratio is obtained from the curve that is detected by the conventional method in a laser-feedback microscope. The other two ratios are almost 2±3 times of that quantitatively. From the comparison above, we conclude that the new methods provide us a new way to improve the axial resolution of a laserfeedback microscope.
References [1] A. Bearden, M.P. O'Neill, L.C. Osborne, T.L. Wong, Opt. Lett. 18 (1993) 238. [2] T.L. Wong, S.L. Sabato, A. Bearden, J. Microsc. 177 (1995) 162. [3] W.M. Wang, W.J.O. Boyle, K.T.V. Grattan, A.W. Palmer, Appl. Opt. 32 (1993) 1551. [4] Th.H. Peek, P.T. Bolwin, C.Th.J. Alkemade, Am. J. Phys. 35 (1967) 820. [5] S. Yang, S. Zhang, Opt. Commun. 68 (1) (1988) 55.
Optics Communications 200 (2001) 303±307 www.elsevier.com/locate/optcom
The new phenomena of orthogonally polarized lights in laser feedback Lu Li *, Shulian Zhang, Shiqun Li, Ping Xue State Key Laboratory of Precision Measurement Technology, Tsinghua University, 100084 Beijing, China Department of Precision Instruments, Tsinghua University, 100084 Beijing, China Department of Physics, Tsinghua University, 100084 Beijing, China Received 11 September 2000; received in revised form 11 December 2000; accepted 11 September 2001
Abstract Some new phenomena of orthogonally polarized lights are observed in the laser-feedback system and in the new system that combines the frequency splitting technique with laser feedback. Results show that there are obvious mode competition between orthogonally polarized lasers throughout the laser-feedback process. This mode competition shows randomness. Further analysis shows that this mode competition between orthogonally polarized lights can effectively increase the laser intensity changing rate relative to the piezoelectric transducer voltage, which provide a new way to improve the axial resolution of a laser-feedback microscope. Ó 2001 Published by Elsevier Science B.V. Keywords: Laser-feedback; Frequency splitting; Polarized lights
The steady-state intensity I0 of a laser can be modi®ed by introducing coherent optical feedback from an external surface. The physical basis is the interference of the backre¯ected ®eld with the standing wave inside the laser resonant cavity [1]. The internal laser intensity I0 of a two-mirror laser is [4]
frequency of the laser radiation, and Q0 is the passive resonation quality. Using R1 and R2 representing the re¯ectivities of mirror M1 and M2 (Fig. 1), and neglecting all losses other than those due to the transmission of the laser-end mirrors, we may write [4]
a a0 I0 b
Q0
pm=Q0 ; b
1
where a is the unsaturated net gain, b is the saturation parameter, a0 is the small signal gain, m is the *
Corresponding author. Address: 2701 Ridge Road #101, Berkeley, CA 94709, USA. Tel.: +1-510-841-2696. E-mail address: [email protected] (L. Li).
1
4pL=k : R1 1 R 2
2
The mirror M2 and the external mirror M3 form an external Fabry±Perot interferometer which now replaced the end mirror M2 of the simple twomirror laser resonator. The intensity re¯ectivity R of the external interferometer which varies strongly with the spacing l aects the resonation Q and by that the laser intensity. R is found to be [4]
0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 5 6 7 - X
304
L. Li et al. / Optics Communications 200 (2001) 303±307
Fig. 1. Arrangement of the laser mirrors M1 and M2 and the external mirror M3 .
R R2 1
R2 f1 1 R3 p =1 R2 R3 2 R2 R3 cos dg;
3
where d 4pl=k is the external phase dierence between successive re¯ected beams, apart from a noninteresting constant phase. Using the R replacing the R2 in Eq. (2), we get the value of Q. And then Q replacing Q0 in Eq. (1), we can drive an expression here for the quasistatic change in the internal laser intensity in terms of the re¯ectivities R3 and the external phase dierence d [4]: I I0 c=4Lb 1
R2 f1 1 R3 p =1 R2 R3 2 R2 R3 cos dg:
4
Laser-feedback eect has been used widely in microscope, for imaging nanometer-scale microfabricated structures and for picometer and nanometer motional analysis of piezoelectric devices, aiding not only in the development of more precise methods of microfabrication, but also in the rapidly growing ®eld of acoustic sensors and other related piezoelectric devices [1±3]. Now a laserfeedback microscope has been constructed capable
of proving an axial resolution of a target surface topography of 1 nm and a lateral resolution of 200 nm [2]. We start our experiments based on the physical theories above. For comparison, we repeat the laser-feedback experiment using the conventional method ®rst. Fig. 2 shows the conventional laser-feedback loop. The 0.6328 lm He±Ne laser used has an half-intracavity. The discharge tube T is ®lled with 20 Ne:22 Ne 1:1 gas mixture to suppress the Lamb dip in the output power curve. The cavity consists of a plane mirror M2 and a concave mirror M1 which has a radius of 1 m and is mounted on a piezoelectric transducer (PZT) P0 . Their re¯ectivities are 0.986 and 0.999 respectively. The length of the oscillating cavity is 195 mm, thus the distance between adjacent modes is 769 MHz. M3 is the external mirror that re¯ects laser back into the internal cavity with a re¯ectivity of 0.5. P is also a PZT which drives the mirror M3 . D is a photoelectric detector for measuring the laser intensity. C is a signal processing circuit. F±P is a Fabry±Perot scanning interferometer. First we detect the output intensity I0 in the absence of mirror M3 . In this situation, we adjust M1 by the voltage on the PZT P0 to force the laser into single mode status. This mode is TEM00 almost in the center of the gain curve. Then we connect the feedback loop to detect the curve of the output intensity Ifeedback versus the position of M3 , referring to Eq. (4). The detected curves are shown in Fig. 3.
Fig. 2. The conventional experimental setup of laser feedback.
L. Li et al. / Optics Communications 200 (2001) 303±307
Fig. 3. Output intensity versus the position of M3 using conventional laser feedback.
In Fig. 3, the vertical axis represents the laser intensity with 0.01 mW/400 units. The horizontal axis represents the position of M3 with 0.3 lm/40 V. The laser intensity has a DC noise of about 50 units and a AC noise of about 5 units. The PZT has a nonlinearity of about 10%. Using the F±P scanning interferometer set behind the mirror M3 , we ®nd that once the feedback loop is connected, the mode status changes from a single mode into three modes. Q modulation is a simple explanation for this phenomenon. If we use Eqs. (2) and (3) and substituting the experimental parameters: R1 0:999, R2 0:986, R3 0:5, L 0:195 m, k 0:6328 lm, d 0±2p, we can see that the value of Q is changed greatly relative to Q0 . And in a big portion of a cycle (M3 has a displacement of k=2 or the voltage on the PZT is changed 40 V or so), Q is much greater than Q0 .
305
pm=Q represents the loss of the laser cavity. A high loss is induced because the value of Q0 is rather low. In this condition, only the centered mode has a positive value after subtracting the loss pm=Q from the gain a0 , referring to Eq. (1), and thus only one mode exists. When Q is changed bigger after connecting the feedback loop, the loss is changed lower. And as a result, the single mode status of the laser is changed into three modes status. In order to study the characteristics of orthogonally polarized lights in laser feedback, we start our second experiment by adding a Wollaston prism before the photoelectric detector to separate the orthogonally polarized lights (we call them Alight(s) and B-light(s)). Two detectors are used to receive the lights separately. The new apparatus is shown in Fig. 4. In this experiment, S is a Wollaston prism that separates A-light(s) and B-light(s) spatially. D1 and D2 are two photoelectric detectors. The status of the laser in the absence of the mirror M3 is the same as the experiment above with single mode (I0A or I0B ) which has an output intensity of about 800 units. Because the DC noise is 50 units, either I0A or I0B must be below that value. After the feedback loop is connected, we get the curve of Alight(s) output intensity IA versus the position of M3 and the curve of B-light(s) output intensity IB versus the position of M3 . We show them in Fig. 5. From Fig. 5 we can see that when the intensity of B-light(s) output intensity is maximum the Alight(s) output intensity almost is minimum, and when B-light(s) output intensity is minimum the A-light(s) output intensity reaches the maximum, vice versa. It shows the characteristics of mode
Fig. 4. The experimental setup after separating the orthogonally polarized lights.
306
L. Li et al. / Optics Communications 200 (2001) 303±307
Fig. 5. A-light(s) output intensity and B-light(s) output intensity versus the position of M3 (the others are the same as those in Fig. 3).
competition between orthogonally polarized lights. In the absence of M3 , the single mode located at the center position of the laser gain line is a Blight. But Fig. 5 shows that B-light(s) has zero intensity in a part of the PZT scan period (PZT is changed 40 V or so when M3 has a displacement of k=2). This phenomenon also results from the mode competition between orthogonally polarized lights. As we can also see in Fig. 5, although there is only a B-light in the absence of M3 , after the feedback loop is connected, the average laser intensity of A-light(s) is obviously bigger than that of B-light(s). Furthermore, we ®nd that both Alight(s) and B-light(s) could have higher average
Fig. 6. The detected curves showing the randomness of mode competition between orthogonally polarized lights in a given cycle (the others are the same as those in Fig. 3).
laser intensity in a given cycle. This randomness happens in each cycle as shown in Fig. 6. By combining the laser feedback with the frequency splitting technique [5], we construct a new experimental system shown in Fig. 7. In this experiment, Q is a uniaxial quartz crystal. The optical rotational eect of the quartz plate has been considered. We use the uniaxial quartz crystal to make every mode split into o-light and elight, and thus getting a frequency splitting of 117 MHz. Except for the single mode splits into o-light and e-light with equal intensity (I0o I0e ), the others are all the same as the former experiments in the absence of M3 . The detected curves are shown in Fig. 8.
Fig. 7. The experimental setup of combining the laser feedback with frequency splitting.
L. Li et al. / Optics Communications 200 (2001) 303±307
307
Fig. 8. o-Lights output intensity and e-lights output intensity versus the position of M3 (the others are the same as those in Fig. 3).
Comparing Fig. 8 with the former curves, we note the obvious symmetry of Fig. 8. That is because the frequency splitting technique makes the relation between orthogonally polarized lights much more symmetric than the former one. In the former case, the adjacent A-light mode and B-light mode has a large frequency separation of about 769 MHz. The large frequency separation between the A-light and the B-light makes the gain characteristics very dierent for the two. While in Fig. 8, the o-light and e-light split from the same mode has a relatively much shorter frequency separation of 117 MHz, the two lasers have almost the same characteristics. Thus the characteristics of the all o-lights and that of the all e-lights are almost the same. Another point in this experiment is that, in the process of moving the external mirror, using the F±P scanning interferometer behind the mirror M3 , we are able to see six modes. Three of them are o-lights, and the other three are e-lights. We see that the three o-lights would increase and decrease almost synchronously. And the three elights also have this synchronism. It shows that the coupling eect between the modes whose polarization directions are parallel to each other would be stronger than that between the modes whose polarization directions are orthogonal to each other.
At last we ®nd that our new methods can be used to improve the axial resolution of LFM. This can be proved by comparing the results shown in Figs. 3, 5, and 8. We can calculate their ratios of the laser intensity variation dividing the PZT voltage variation when the voltage of PZT is changed around 90 V or so. The value of the ratio restricts the resolution using the laser-feedback microscope. Respectively, the three ratios are 59, 173, and 117 units/V from Figs. 3, 5, and 8. The lowest ratio is obtained from the curve that is detected by the conventional method in a laser-feedback microscope. The other two ratios are almost 2±3 times of that quantitatively. From the comparison above, we conclude that the new methods provide us a new way to improve the axial resolution of a laserfeedback microscope.
References [1] A. Bearden, M.P. O'Neill, L.C. Osborne, T.L. Wong, Opt. Lett. 18 (1993) 238. [2] T.L. Wong, S.L. Sabato, A. Bearden, J. Microsc. 177 (1995) 162. [3] W.M. Wang, W.J.O. Boyle, K.T.V. Grattan, A.W. Palmer, Appl. Opt. 32 (1993) 1551. [4] Th.H. Peek, P.T. Bolwin, C.Th.J. Alkemade, Am. J. Phys. 35 (1967) 820. [5] S. Yang, S. Zhang, Opt. Commun. 68 (1) (1988) 55.