- Email: [email protected]
Multiple selfmixing effect in VCSELs with asymmetric external cavity
Optics Communications 260 (2006) 50–56 www.elsevier.com/locate/optcom
Multiple selfmixing effect in VCSELs with asymmetric external cavity Xiang Cheng *, Shulian Zhang The State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China Received 29 June 2005; received in revised form 20 October 2005; accepted 21 October 2005
Abstract Selfmixing fringes shift according to a k/6 optical displacement is observed which is shown to be due to the multiple reflections and the asymmetry of external cavity. A theory of selfmixing taking into account these factors is proposed. The theoretical analysis is seen to yield a simulation in good agreement with the experiment results reported. 2005 Elsevier B.V. All rights reserved. Keywords: Optical selfmixing; Asymmetric cavity; Multiple reflections
1. Introduction It is well known that properties of edge-emitting semiconductor lasers can be significantly affected by the external optical feedback. The light reflected by external mirror causes variations in the threshold gain, output power and spectrum of lasers [1,2]. Since Chung [3] reported that vertical-cavity surface-emitting lasers (VCSELs) had feedback sensitivity comparable to that of conventional edge-emitting lasers, a great deal of interesting has been attracted in the last decades due to the inherent advantages of VCSELs: (1) relative low threshold current; (2) narrow circular profile; (3) single longitudinal mode operation. A number of theories [3–8] have been used to interpret the observed phenomena based on the characteristics of selfmixing effect [4]: the intensity modulation of power and its depth are similar to that produced by a conventional optical interferometer, i.e., a fringe shift according to every half wavelength displacement of the movable external mirror. It was described in the models [1–8] that the effect of selfmixing depended on the amount of light fed back into
the laser cavity and the displacement of reflector which determined the phase of feedback light. Most of them neglected the multiple reflections in the external cavity and the asymmetry of reflector which was not avoidable in the practical device. However the performance of the VCSELs may be significantly influenced by the multiple reflections due to the high reflectivities of their mirrors [9–11]. The purpose of the present work has been to analyze the influence of optical feedback on the laser power modulation taking into account the multiple reflections and the asymmetry of external mirror. In our experiment, not only normal fringes shift according to a k/2 but also the fringes with double and triple frequency are observed. As we know, a fringe shift according to every k/6 in the selfmixing of VCSELs is firstly reported. In order to explain these phenomena, a multi-reflection model is proposed and a coupling parameter is defined. A theoretical analysis and a comparison with the experiment are presented. The experimental results are in good agreement with the theoretical analysis. 2. Experimental setup
*
Corresponding author. Tel.: +86 10 62782859; fax: +86 10 62784503. E-mail address: [email protected] (X. Cheng).
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.051
A block diagram of the experimental arrangement is shown in Fig. 1. The VCSEL is from Honeywell with ultra
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
51
Fig. 1. The experiment setup.
flat window TO-46 style package. At room temperature and free running conditions, the lasing wavelength is 850 nm and the threshold current is Ith = 1.50 mA. Although it was designed to stay single mode over the whole operating current range (1.5–4.0 mA), multimode will appear at very high bias current. In order to avoid any higher transverse mode and bigger beam divergence, it is reasonable to operating the laser at a relative low bias current such as 2.59 mA. The output of the VCSEL is collimated by the microscope objective C. The out beam is split by the beam splitter (BS). One of the laser beams is sent to a movable reflector M. The ratio between the feedback light and the output can be adjusted by a variable attenuator (VA). And a piezoelectric transducer (PZT) which is controlled by the computer through the D/A card is used to push the mirror (M). The other laser beam is sent to a photoelectric detector (D) and the intensity modulation of the VCSEL caused by feedback light is picked up by the computer through the A/D card. 3. Experimental results One dimensional model of the VCSEL, in the presence of feedback, is shown in Fig. 2. The h is the angle between
Fig. 2. One dimensional model of laser. The length of effective VCSEL cavity is denoted by d. The L is that of external cavity (L = 47 mm).
the normal of mirror and the direction of the incident beam. Align the external mirror with the out facet of the laser (h = 0) and detect the output intensity of the laser without VA. A typical self-mixing output waveform, i.e., one fringe shift according to a half wavelength displacement of the external mirror, is got and shown as the lower line in Fig. 3(a). And the upper lines in Fig. 3(a)–(d) are the voltage imposed on the PZT. In our experiment, the PZT is actuated by a triangular voltage ranged from 0 to 450 V and accordingly the maximum displacement of external mirror is about two wavelengths. An asymmetric sawtooth waveform shown as the lower line in Fig. 3b emerges while tilting the reflector with about 0.1 mrad. With the increasing of tilt angle (h), a branch appears in the each feedback fringe shown as the Fig. 3(c). The position and amplitude of the branch is very sensitive to the slight change of the angle (h). A fringe shift according to a k/4 displacement can be easily observed by adjusting the angle of mirror. When the h becomes about 10 mrad and the location of lens makes a change about 0.5 mm (L1 = 8.50 mm), another branch appears in the feedback fringe shown as the Fig. 3(d). Then adjusting the h carefully, a perfect fringe triple of the typical one, i.e., a fringe shift according to every 1/6 wavelength, is acquired as shown in Fig. 4(a). The range of the angle (h), during which the triple fringe appears, is relative to the location of the lens and about 10 mrad is got when L is 8.50 mm. The shape of the triple feedback fringe can also be controlled by adjusting the tilt angle h. Four different fringes are shown in Fig. 4(a)–(d), in which the Fig. 4(a) has a significant value because it indicates the selfmixing effect can be applied to measurement such as length and vibrating with a resolution of one sixth wavelength just counting the fringes. Secondly, we fix the h at the position where the perfect triple fringe appeared. Then adjust the VA from 0 to 3 dB slowly and observe the phenomena. To our surprise, not only the amplitude of fringe is decreased but also the shape of fringe is changed as shown in Fig. 5. The branches in the fringes are not decreased equally. Moreover when VA equals 3 dB, the branches are not obvious any more and the waveform becomes similar to the normal fringe as
52
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
0.1
a
0.05 0 210
220
230
240
250
260
270
0.1
b
Voltage(V)
0.05 0 260
270
280
290
300
310
0.1
c
0.05 0 330
340
350
360
370
380
390
0.15
d
0.1 0.05 170
180
190
200
210
220
230
Time(ms) Fig. 3. Feedback fringes according to different tilt angle of reflector (a) 0 mrad. (b) 0.1 mrad. (c) 1.0 mrad. (d) 10.0 mrad.
0.15
a
0.1 0.05 0 290
300
310
320
330
340
350
0.3
b
0.2
Voltage(V)
0.1 130
140
150
160
170
180
0.3
c 0.2 0.1 0 300
310
320
330
340
350
360
0.3
d
0.2 0.1 0 500
510
520
530
540
550
Time(ms) Fig. 4. An observation of intensity variation vs. PZT voltage as the tilt angle of reflect r is (a) 10.1 mrad; (b) 10.2 mrad; (c) 10.3 mrad; (d) 10.4 mrad.
shown in Fig. 5(c). These phenomena indicate that the branches are the results of the interference among different round trip of reflected light. Otherwise they should be scaled down or up when VA is changed. Finally, we change the position of the lens C. The double and triple fringes also are observed.
4. Theoretical analysis The power modulation mechanism in selfmixing has been assumed to be caused by the variation of the threshold gain of the laser [8]. A number of theories [1–8] have been proposed based on this hypothesis but few of them
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
53
0.15
a
0.1 0.05 0 120
130
140
150
160
170
Voltage(V)
0.15
b
0.1 0.05 0 130
140
150
160
170
180
0.06
c
0.04 0.02 0 150
160
170
180
190
200
Time(ms) Fig. 5. Variation of intensity vs. VA (a) VA = 0 dB. (b) VA = 2 dB. (c) VA = 3 dB.
have taken the multi-feedback into consideration because the external reflection coefficient r3 is usually much less than the reflection coefficient r2 of the laser out facet. Therefore the light amplitude of second and above roundtrip into cavity is so little that it can be neglected [6]. The electric field E(t) inside the cavity undergoing one roundtrip is described [4,5,12] as the following the formula: nd EðtÞ ¼ r1 r2 exp j4pv þ ðg rÞd E0 ðtÞ þ r1 Tr3 c nd þ L þ ðg rÞd E0 ðtÞ; exp j4pv ð1Þ c
efficient varied with the angle and the times of roundtrip. In our experiment, normal fringes, double and even triple fringes can be observed by changing the tilt angle of the reflector and these can not be interpreted in the model which just takes one roundtrip feedback into account. So it is reasonable to vary the equation of E(t) as [12]: nd EðtÞ ¼ r1 r2 exp j4pv þ ðg rÞd E0 ðtÞ c m X C m r1 TRm1 r3 þ
where E0(t) is the initial electric field, r1 represents the amplitude reflection coefficient of the laser rear facet, m is the oscillating frequency of laser, g is the linear gain per unit length in the laser, r accounts for the total loss per unit length within the cavity, and n is the effective refractive index of laser material. The term T = t 0 t is the effective amplitude transmission coefficient, in which t and t 0 are the transmission coefficients through and back the laser out facet, respectively. The d and L are the length of laser cavity and external cavity, respectively. However in the case of VCSEL, light amplitude of second and above roundtrip should not be omitted especially when r3 is not very small or the reflector is tilted within a 20 mrad angle. Richard reported that the light of the second roundtrip can feedback into the cavity while the light of first roundtrip could not enter into the laser cavity in the asymmetric external cavity because of the tilt of reflector [13,14]. When the mirror is tilted by a small angle, the coupling efficient is not constant any more because the centre of spot will move out the active layer. So the coupling
ð2Þ
1
nd þ mL þ ðg rÞd þ dm E0 ðtÞ; exp j4pv c
where Cm is the coupling parameter defined as the power of the reflected light back into cavity divided by that of the total reflected light which arrived at the out facet of laser in the mth roundtrip feedback, R ¼ r3 r02 is the effective amplitude reflection coefficient of light undergoing a roundtrip of external cavity, dm being the additional phase caused by the asymmetrical mirror. According to the oscillation condition and taking the approximation ln(1 + x) = x, the required gain is: m 1 1 k1 X 2mL þ dm ; g ¼ r þ ln C m Rm1 cos 2pm d r1 r2 2d 1 c ð3Þ where k1 ¼
r22 þ
2r2 r3 T Pm 2 2 2 . 1 C m T r3
ð4Þ
54
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
Compared with the required gain without the external cavity, we got: m k1 X 2mL þ dm . Dg ¼ C m Rm1 cos 2pm ð5Þ c 2d 1 According to Eq. (5) between the intensity of laser and the optical gain, the intensity modulation of laser is expressed as: m kk 1 X 2mL m1 þ dm ; CmR cos 2pm ð6Þ DI ¼ kDg ¼ c 2d 1
1
1
0.8
0.8
Coefficient
Coefficient
where k is the modulation coefficient of self-mixing interference. It is clear that the modulation of intensity is not only dependent on the feedback strength and the distance of the external reflector but also on the roundtrip coupling parameter Cm which could be adjusted by tilting mirror or moving the collimating lens. The coupling parameter Cm of the feedback light to the optical field in the laser cavity is obtained by calculating the overlap integral between the fields by means of the 3 · 3 matrix formalism [15] for optical misaligned elements. And the parameter Cm varied with tilt angle at different location of lens is indicated in Fig. 6. The data is calculated under the conditions: foci of lens f = 8.0 mm, L = 47 mm, and the radius of effective medium of Laser is 5 lm. The dashed line represents the coupling parameter of the first roundtrip, the solid line is that of the second, and the doted line is the third.
From Fig. 6, we can find the coupling parameter significantly varied with the tilt angle and the lens displacement. If the light was perfectly collimated, i.e., L1 = 8.00 mm, the feedback light can enter cavity just as the tilt angle within a very narrow range and only the double fringes can be observed as shown in Fig. 6(a). If the distance between the lens and the laser out facet becomes more than the foci of the lens as shown in Fig. 6(c), the coupling parameter of the third round trip will keep constant within a big range of the tilt angle of reflector. Furthermore we calculate the coupling parameter when the displacement of lens is changed as shown in Fig. 7, from which we can easily find that the coupling coefficients of first, second and third roundtrip arrived at the maximum in turn as shown in Fig. 7(a). But when the angle is not zero, the coupling coefficient of first and second roundtrip is no more than 0.2, while the coupling of third roundtrip is nearly 1 within a certain range of displacement of lens. And these account for the phenomena observed in the experiments. According to the formula (6), we simulate the variation of intensity of laser as shown in Fig. 8. Compared Figs. 8(a)–(d) and 4(a)–(d), we can find that the theoretical analysis is in good agreement with the experimental results. The difference among b, c and d is caused by different additional phase dm.
0.6 0.4 0.2 0
0
2
4
6
Angle (rad)
0
8 –4
x 10
1
0.8
0.8
0.4 0.2 0 0
1
2
3
4
5 –3
Angle (rad)
1
0.6
0
b
Coefficient
Coefficient
0.4 0.2
a
c
0.6
x 10
0.6 0.4 0.2
5
10
Angle (rad)
0
15 x 10
–3
d
0
2
4
Angle (rad)
6 x 10
–3
Fig. 6. Coupling parameter of different roundtrip vs. the tilt angle of reflector as the location of the lens is: (a) L1 = 8.00 mm; (b) L1 = 8.25 mm; (c) L1 = 8.50 mm (d) L1 = 8.75 mm.
1
1
0.8
0.8
0.6
0.6
Coefficient
Coefficient
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
0.4 0.2
0.4 0.2
0
0 8
8.5 9 9.5 Displacement(mm)
a
10
1
1
0.8
0.8
0.6 0.4
0
0 8.5 9 9.5 Displacement(mm)
9 9.5 Displacement(mm)
10
8
8.5
9 9.5 Displacement(mm)
10
0.4 0.2
8
8.5
0.6
0.2
c
8 b
Coefficient
Coefficient
55
10 d
Fig. 7. Coupling parameter vs. the displacement of lens as the tilt angle is: (a) 0 mrad; (b) 2 mrad; (c) 4 mrad; (d) 6 mrad.
2
e
1 0
0
10
20
30
40
50
60
0.4
a 0.2 0
0
10
20
30
40
50
60
Voltage(V)
1.0
b 0.5 0
0
10
20
30
40
50
60
1.0
c 0.5 0
0
10
20
30
40
50
60
1.0
d 0.5 0
0
10
20
30
40
50
60
Time(ms) Fig. 8. The simulation of power modulation as R = 0.4 and (a) C1 = C2 = 0, C3 = 1 (b)–(d) C1 = C2 = 0.18C3 = 1 (e) the voltage of PZT.
5. Conclusion We have demonstrated that the tilt of reflector is not an advisable measure to reduce feedback effect on certain con-
dition and it will cause power modulation with a higher frequency. A theory of selfmixing interference of VCSELs is presented taking into account the multiple reflections, the
56
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
position of collimated lens and the asymmetry of the external cavity. In this analysis, the coupling parameters that account for the ratio of the multiple reflections have been introduced. The triple frequency of normal fringes is observed and the potential resolution of selfmixing interference will be tripled, with one fringe produced for every one sixth wavelength instead one per half wavelength. That is to say for a displacement sensor based on fringe counting, the fringe frequency will be one per one sixth wavelength, giving a resolution of 142 nm for a laser diode wavelength of 850 nm. Also we demonstrate that the intensity is the function of the tilt angle of external mirror because it changes both the coupling parameter and the phase condition. So this will present a new way on measuring angle with a high resolution. Acknowledgement This work is supported by the National Nature Science Foundation of China and Tsinghua University.
References [1] A. Olsson, C.L. Tang, IEEE J. Quantum Electron. QE-17 (1981) 1320. [2] L. Goldberg, H.F. Taylor, A. Dandridge, J.F. Weller, R.O. Miles, IEEE T. Microwave Theory Technol. MTT-30 (1982) 401. [3] Y.C. Chung, Y.H. Lee, IEEE Photon. Technol. Lett. 3 (1991) 597. [4] W.M. Wang, K.T.V. Grattan, A.W. Palmer, W.J.O. Boyle, J. Lightwave Technol. 12 (1994) 1577. [5] G. Liu, S. Zhan, J. Zhu, Y. Li, Opt. Commun. 221 (2003) 387. [6] W.M. Wang, W.J.O. Boyle, K.T.V. Grattan, A.W. Palmer, Appl. Opt. 32 (1993) 1551. [7] R. Lang, K. Kobayashi, IEEE J. Quantum Electron. QE-16 (1980) 347. [8] P.J. De Groot, G.M. Gallatin, S.H. Macomber, Appl. Opt. 27 (1988) 4475. [9] J.Y. Law, G.P. Agrawal, IEEE J. Sel. Top. Quantum Electron. 3 (1997) 353. [10] J. Dellunde, A. Valle, L. Pesquera, K.A. Shore, J. Opt. Soc. Am. B 16 (1999) 2131. [11] J. Dellunde, A. Valle, K.A. Shore, J. Opt. Soc. Am. B 13 (1996) 2477. [12] G.H.M. van Tartwijk, D. Lenstra, Quantum Semiclass. Opt. 7 (1995) 87. [13] D.S. Seo, J.D. Park, J.G. Mcinerney, M. Osinski, IEEE J. Quantum Electron. 25 (1989) 2229. [14] R.C. Addy, A.W. Palmer, K.T.V. Grattan, J. Lightwave Technol. 14 (1996) 2672. [15] A.E. Siegman, in: A. Kelly (Ed.), Lasers, University Science, Sausalito, CA, 1986, p. 593.
Multiple selfmixing effect in VCSELs with asymmetric external cavity Xiang Cheng *, Shulian Zhang The State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China Received 29 June 2005; received in revised form 20 October 2005; accepted 21 October 2005
Abstract Selfmixing fringes shift according to a k/6 optical displacement is observed which is shown to be due to the multiple reflections and the asymmetry of external cavity. A theory of selfmixing taking into account these factors is proposed. The theoretical analysis is seen to yield a simulation in good agreement with the experiment results reported. 2005 Elsevier B.V. All rights reserved. Keywords: Optical selfmixing; Asymmetric cavity; Multiple reflections
1. Introduction It is well known that properties of edge-emitting semiconductor lasers can be significantly affected by the external optical feedback. The light reflected by external mirror causes variations in the threshold gain, output power and spectrum of lasers [1,2]. Since Chung [3] reported that vertical-cavity surface-emitting lasers (VCSELs) had feedback sensitivity comparable to that of conventional edge-emitting lasers, a great deal of interesting has been attracted in the last decades due to the inherent advantages of VCSELs: (1) relative low threshold current; (2) narrow circular profile; (3) single longitudinal mode operation. A number of theories [3–8] have been used to interpret the observed phenomena based on the characteristics of selfmixing effect [4]: the intensity modulation of power and its depth are similar to that produced by a conventional optical interferometer, i.e., a fringe shift according to every half wavelength displacement of the movable external mirror. It was described in the models [1–8] that the effect of selfmixing depended on the amount of light fed back into
the laser cavity and the displacement of reflector which determined the phase of feedback light. Most of them neglected the multiple reflections in the external cavity and the asymmetry of reflector which was not avoidable in the practical device. However the performance of the VCSELs may be significantly influenced by the multiple reflections due to the high reflectivities of their mirrors [9–11]. The purpose of the present work has been to analyze the influence of optical feedback on the laser power modulation taking into account the multiple reflections and the asymmetry of external mirror. In our experiment, not only normal fringes shift according to a k/2 but also the fringes with double and triple frequency are observed. As we know, a fringe shift according to every k/6 in the selfmixing of VCSELs is firstly reported. In order to explain these phenomena, a multi-reflection model is proposed and a coupling parameter is defined. A theoretical analysis and a comparison with the experiment are presented. The experimental results are in good agreement with the theoretical analysis. 2. Experimental setup
*
Corresponding author. Tel.: +86 10 62782859; fax: +86 10 62784503. E-mail address: [email protected] (X. Cheng).
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.051
A block diagram of the experimental arrangement is shown in Fig. 1. The VCSEL is from Honeywell with ultra
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
51
Fig. 1. The experiment setup.
flat window TO-46 style package. At room temperature and free running conditions, the lasing wavelength is 850 nm and the threshold current is Ith = 1.50 mA. Although it was designed to stay single mode over the whole operating current range (1.5–4.0 mA), multimode will appear at very high bias current. In order to avoid any higher transverse mode and bigger beam divergence, it is reasonable to operating the laser at a relative low bias current such as 2.59 mA. The output of the VCSEL is collimated by the microscope objective C. The out beam is split by the beam splitter (BS). One of the laser beams is sent to a movable reflector M. The ratio between the feedback light and the output can be adjusted by a variable attenuator (VA). And a piezoelectric transducer (PZT) which is controlled by the computer through the D/A card is used to push the mirror (M). The other laser beam is sent to a photoelectric detector (D) and the intensity modulation of the VCSEL caused by feedback light is picked up by the computer through the A/D card. 3. Experimental results One dimensional model of the VCSEL, in the presence of feedback, is shown in Fig. 2. The h is the angle between
Fig. 2. One dimensional model of laser. The length of effective VCSEL cavity is denoted by d. The L is that of external cavity (L = 47 mm).
the normal of mirror and the direction of the incident beam. Align the external mirror with the out facet of the laser (h = 0) and detect the output intensity of the laser without VA. A typical self-mixing output waveform, i.e., one fringe shift according to a half wavelength displacement of the external mirror, is got and shown as the lower line in Fig. 3(a). And the upper lines in Fig. 3(a)–(d) are the voltage imposed on the PZT. In our experiment, the PZT is actuated by a triangular voltage ranged from 0 to 450 V and accordingly the maximum displacement of external mirror is about two wavelengths. An asymmetric sawtooth waveform shown as the lower line in Fig. 3b emerges while tilting the reflector with about 0.1 mrad. With the increasing of tilt angle (h), a branch appears in the each feedback fringe shown as the Fig. 3(c). The position and amplitude of the branch is very sensitive to the slight change of the angle (h). A fringe shift according to a k/4 displacement can be easily observed by adjusting the angle of mirror. When the h becomes about 10 mrad and the location of lens makes a change about 0.5 mm (L1 = 8.50 mm), another branch appears in the feedback fringe shown as the Fig. 3(d). Then adjusting the h carefully, a perfect fringe triple of the typical one, i.e., a fringe shift according to every 1/6 wavelength, is acquired as shown in Fig. 4(a). The range of the angle (h), during which the triple fringe appears, is relative to the location of the lens and about 10 mrad is got when L is 8.50 mm. The shape of the triple feedback fringe can also be controlled by adjusting the tilt angle h. Four different fringes are shown in Fig. 4(a)–(d), in which the Fig. 4(a) has a significant value because it indicates the selfmixing effect can be applied to measurement such as length and vibrating with a resolution of one sixth wavelength just counting the fringes. Secondly, we fix the h at the position where the perfect triple fringe appeared. Then adjust the VA from 0 to 3 dB slowly and observe the phenomena. To our surprise, not only the amplitude of fringe is decreased but also the shape of fringe is changed as shown in Fig. 5. The branches in the fringes are not decreased equally. Moreover when VA equals 3 dB, the branches are not obvious any more and the waveform becomes similar to the normal fringe as
52
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
0.1
a
0.05 0 210
220
230
240
250
260
270
0.1
b
Voltage(V)
0.05 0 260
270
280
290
300
310
0.1
c
0.05 0 330
340
350
360
370
380
390
0.15
d
0.1 0.05 170
180
190
200
210
220
230
Time(ms) Fig. 3. Feedback fringes according to different tilt angle of reflector (a) 0 mrad. (b) 0.1 mrad. (c) 1.0 mrad. (d) 10.0 mrad.
0.15
a
0.1 0.05 0 290
300
310
320
330
340
350
0.3
b
0.2
Voltage(V)
0.1 130
140
150
160
170
180
0.3
c 0.2 0.1 0 300
310
320
330
340
350
360
0.3
d
0.2 0.1 0 500
510
520
530
540
550
Time(ms) Fig. 4. An observation of intensity variation vs. PZT voltage as the tilt angle of reflect r is (a) 10.1 mrad; (b) 10.2 mrad; (c) 10.3 mrad; (d) 10.4 mrad.
shown in Fig. 5(c). These phenomena indicate that the branches are the results of the interference among different round trip of reflected light. Otherwise they should be scaled down or up when VA is changed. Finally, we change the position of the lens C. The double and triple fringes also are observed.
4. Theoretical analysis The power modulation mechanism in selfmixing has been assumed to be caused by the variation of the threshold gain of the laser [8]. A number of theories [1–8] have been proposed based on this hypothesis but few of them
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
53
0.15
a
0.1 0.05 0 120
130
140
150
160
170
Voltage(V)
0.15
b
0.1 0.05 0 130
140
150
160
170
180
0.06
c
0.04 0.02 0 150
160
170
180
190
200
Time(ms) Fig. 5. Variation of intensity vs. VA (a) VA = 0 dB. (b) VA = 2 dB. (c) VA = 3 dB.
have taken the multi-feedback into consideration because the external reflection coefficient r3 is usually much less than the reflection coefficient r2 of the laser out facet. Therefore the light amplitude of second and above roundtrip into cavity is so little that it can be neglected [6]. The electric field E(t) inside the cavity undergoing one roundtrip is described [4,5,12] as the following the formula: nd EðtÞ ¼ r1 r2 exp j4pv þ ðg rÞd E0 ðtÞ þ r1 Tr3 c nd þ L þ ðg rÞd E0 ðtÞ; exp j4pv ð1Þ c
efficient varied with the angle and the times of roundtrip. In our experiment, normal fringes, double and even triple fringes can be observed by changing the tilt angle of the reflector and these can not be interpreted in the model which just takes one roundtrip feedback into account. So it is reasonable to vary the equation of E(t) as [12]: nd EðtÞ ¼ r1 r2 exp j4pv þ ðg rÞd E0 ðtÞ c m X C m r1 TRm1 r3 þ
where E0(t) is the initial electric field, r1 represents the amplitude reflection coefficient of the laser rear facet, m is the oscillating frequency of laser, g is the linear gain per unit length in the laser, r accounts for the total loss per unit length within the cavity, and n is the effective refractive index of laser material. The term T = t 0 t is the effective amplitude transmission coefficient, in which t and t 0 are the transmission coefficients through and back the laser out facet, respectively. The d and L are the length of laser cavity and external cavity, respectively. However in the case of VCSEL, light amplitude of second and above roundtrip should not be omitted especially when r3 is not very small or the reflector is tilted within a 20 mrad angle. Richard reported that the light of the second roundtrip can feedback into the cavity while the light of first roundtrip could not enter into the laser cavity in the asymmetric external cavity because of the tilt of reflector [13,14]. When the mirror is tilted by a small angle, the coupling efficient is not constant any more because the centre of spot will move out the active layer. So the coupling
ð2Þ
1
nd þ mL þ ðg rÞd þ dm E0 ðtÞ; exp j4pv c
where Cm is the coupling parameter defined as the power of the reflected light back into cavity divided by that of the total reflected light which arrived at the out facet of laser in the mth roundtrip feedback, R ¼ r3 r02 is the effective amplitude reflection coefficient of light undergoing a roundtrip of external cavity, dm being the additional phase caused by the asymmetrical mirror. According to the oscillation condition and taking the approximation ln(1 + x) = x, the required gain is: m 1 1 k1 X 2mL þ dm ; g ¼ r þ ln C m Rm1 cos 2pm d r1 r2 2d 1 c ð3Þ where k1 ¼
r22 þ
2r2 r3 T Pm 2 2 2 . 1 C m T r3
ð4Þ
54
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
Compared with the required gain without the external cavity, we got: m k1 X 2mL þ dm . Dg ¼ C m Rm1 cos 2pm ð5Þ c 2d 1 According to Eq. (5) between the intensity of laser and the optical gain, the intensity modulation of laser is expressed as: m kk 1 X 2mL m1 þ dm ; CmR cos 2pm ð6Þ DI ¼ kDg ¼ c 2d 1
1
1
0.8
0.8
Coefficient
Coefficient
where k is the modulation coefficient of self-mixing interference. It is clear that the modulation of intensity is not only dependent on the feedback strength and the distance of the external reflector but also on the roundtrip coupling parameter Cm which could be adjusted by tilting mirror or moving the collimating lens. The coupling parameter Cm of the feedback light to the optical field in the laser cavity is obtained by calculating the overlap integral between the fields by means of the 3 · 3 matrix formalism [15] for optical misaligned elements. And the parameter Cm varied with tilt angle at different location of lens is indicated in Fig. 6. The data is calculated under the conditions: foci of lens f = 8.0 mm, L = 47 mm, and the radius of effective medium of Laser is 5 lm. The dashed line represents the coupling parameter of the first roundtrip, the solid line is that of the second, and the doted line is the third.
From Fig. 6, we can find the coupling parameter significantly varied with the tilt angle and the lens displacement. If the light was perfectly collimated, i.e., L1 = 8.00 mm, the feedback light can enter cavity just as the tilt angle within a very narrow range and only the double fringes can be observed as shown in Fig. 6(a). If the distance between the lens and the laser out facet becomes more than the foci of the lens as shown in Fig. 6(c), the coupling parameter of the third round trip will keep constant within a big range of the tilt angle of reflector. Furthermore we calculate the coupling parameter when the displacement of lens is changed as shown in Fig. 7, from which we can easily find that the coupling coefficients of first, second and third roundtrip arrived at the maximum in turn as shown in Fig. 7(a). But when the angle is not zero, the coupling coefficient of first and second roundtrip is no more than 0.2, while the coupling of third roundtrip is nearly 1 within a certain range of displacement of lens. And these account for the phenomena observed in the experiments. According to the formula (6), we simulate the variation of intensity of laser as shown in Fig. 8. Compared Figs. 8(a)–(d) and 4(a)–(d), we can find that the theoretical analysis is in good agreement with the experimental results. The difference among b, c and d is caused by different additional phase dm.
0.6 0.4 0.2 0
0
2
4
6
Angle (rad)
0
8 –4
x 10
1
0.8
0.8
0.4 0.2 0 0
1
2
3
4
5 –3
Angle (rad)
1
0.6
0
b
Coefficient
Coefficient
0.4 0.2
a
c
0.6
x 10
0.6 0.4 0.2
5
10
Angle (rad)
0
15 x 10
–3
d
0
2
4
Angle (rad)
6 x 10
–3
Fig. 6. Coupling parameter of different roundtrip vs. the tilt angle of reflector as the location of the lens is: (a) L1 = 8.00 mm; (b) L1 = 8.25 mm; (c) L1 = 8.50 mm (d) L1 = 8.75 mm.
1
1
0.8
0.8
0.6
0.6
Coefficient
Coefficient
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
0.4 0.2
0.4 0.2
0
0 8
8.5 9 9.5 Displacement(mm)
a
10
1
1
0.8
0.8
0.6 0.4
0
0 8.5 9 9.5 Displacement(mm)
9 9.5 Displacement(mm)
10
8
8.5
9 9.5 Displacement(mm)
10
0.4 0.2
8
8.5
0.6
0.2
c
8 b
Coefficient
Coefficient
55
10 d
Fig. 7. Coupling parameter vs. the displacement of lens as the tilt angle is: (a) 0 mrad; (b) 2 mrad; (c) 4 mrad; (d) 6 mrad.
2
e
1 0
0
10
20
30
40
50
60
0.4
a 0.2 0
0
10
20
30
40
50
60
Voltage(V)
1.0
b 0.5 0
0
10
20
30
40
50
60
1.0
c 0.5 0
0
10
20
30
40
50
60
1.0
d 0.5 0
0
10
20
30
40
50
60
Time(ms) Fig. 8. The simulation of power modulation as R = 0.4 and (a) C1 = C2 = 0, C3 = 1 (b)–(d) C1 = C2 = 0.18C3 = 1 (e) the voltage of PZT.
5. Conclusion We have demonstrated that the tilt of reflector is not an advisable measure to reduce feedback effect on certain con-
dition and it will cause power modulation with a higher frequency. A theory of selfmixing interference of VCSELs is presented taking into account the multiple reflections, the
56
X. Cheng, S. Zhang / Optics Communications 260 (2006) 50–56
position of collimated lens and the asymmetry of the external cavity. In this analysis, the coupling parameters that account for the ratio of the multiple reflections have been introduced. The triple frequency of normal fringes is observed and the potential resolution of selfmixing interference will be tripled, with one fringe produced for every one sixth wavelength instead one per half wavelength. That is to say for a displacement sensor based on fringe counting, the fringe frequency will be one per one sixth wavelength, giving a resolution of 142 nm for a laser diode wavelength of 850 nm. Also we demonstrate that the intensity is the function of the tilt angle of external mirror because it changes both the coupling parameter and the phase condition. So this will present a new way on measuring angle with a high resolution. Acknowledgement This work is supported by the National Nature Science Foundation of China and Tsinghua University.
References [1] A. Olsson, C.L. Tang, IEEE J. Quantum Electron. QE-17 (1981) 1320. [2] L. Goldberg, H.F. Taylor, A. Dandridge, J.F. Weller, R.O. Miles, IEEE T. Microwave Theory Technol. MTT-30 (1982) 401. [3] Y.C. Chung, Y.H. Lee, IEEE Photon. Technol. Lett. 3 (1991) 597. [4] W.M. Wang, K.T.V. Grattan, A.W. Palmer, W.J.O. Boyle, J. Lightwave Technol. 12 (1994) 1577. [5] G. Liu, S. Zhan, J. Zhu, Y. Li, Opt. Commun. 221 (2003) 387. [6] W.M. Wang, W.J.O. Boyle, K.T.V. Grattan, A.W. Palmer, Appl. Opt. 32 (1993) 1551. [7] R. Lang, K. Kobayashi, IEEE J. Quantum Electron. QE-16 (1980) 347. [8] P.J. De Groot, G.M. Gallatin, S.H. Macomber, Appl. Opt. 27 (1988) 4475. [9] J.Y. Law, G.P. Agrawal, IEEE J. Sel. Top. Quantum Electron. 3 (1997) 353. [10] J. Dellunde, A. Valle, L. Pesquera, K.A. Shore, J. Opt. Soc. Am. B 16 (1999) 2131. [11] J. Dellunde, A. Valle, K.A. Shore, J. Opt. Soc. Am. B 13 (1996) 2477. [12] G.H.M. van Tartwijk, D. Lenstra, Quantum Semiclass. Opt. 7 (1995) 87. [13] D.S. Seo, J.D. Park, J.G. Mcinerney, M. Osinski, IEEE J. Quantum Electron. 25 (1989) 2229. [14] R.C. Addy, A.W. Palmer, K.T.V. Grattan, J. Lightwave Technol. 14 (1996) 2672. [15] A.E. Siegman, in: A. Kelly (Ed.), Lasers, University Science, Sausalito, CA, 1986, p. 593.