Repetition rate stabilization of the SBS
Q-switched fiber laser by external injection

doi:10.1364/OE.17.003124

« Show journal navigation

Repetition rate stabilization of the SBS Q-switched fiber laser by external injection

Zhengqing Pan, Li Meng, Qing Ye, Haiwen Cai, Zujie Fang, and Ronghui Qu »View Author Affiliations

Optics Express, Vol. 17, Issue 5, pp. 3124-3129 (2009)
http://dx.doi.org/10.1364/OE.17.003124

View Full Text Article

Acrobat PDF (569 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
1. Introduction
2. Numerical analysis
3. Experiment
4. Conclusion
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (11)
Cited By
Figures (5)
Metrics

Abstract

Repetition rate fluctuation is one of the main drawbacks of the low-threshold stimulated Brillouin scattering (SBS) Q-switched fiber laser. A method to stabilize the repetition rate is proposed in this paper by injecting a square-wave modulated light. It is measured experimentally that variance of the repetition rate can be improved from ~20% to ~1% of the period. It is also found that effectiveness of the method depends on modulation frequency and duty cycle of the injection. Its working mechanism is analyzed qualitatively.

1. Introduction

The generation of nanosecond pulse with high peak power is of importance for many applications such as OTDR, lidar, remote sensing and nonlinear optical processes [1–3

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, “Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm,” IEEE Photonics Technol. Lett. 11, 976–978 (1999). [CrossRef]

]. Q-switched fiber lasers have been developed for years to obtain pulsed laser beam with pulse-width ranging from nanoseconds to microseconds. As the pulse duration is mainly dependent on the photon lifetime of resonant cavity, short pulses are hardly achieved in long cavity fiber laser by conventional Q-switched mechanisms. Recently, Q-switched fiber lasers with a Rayleigh-SBS ring mirror successfully generated nanoseconds pulses, much shorter than the round trip time of fiber resonator length [4

S. V. Chernikov, Y. Zhu, and J. R. Taylor, “Supercontinuum self-Q-s witched ytterbium fiber laser,” Opt. Lett. 22, 298–300 (1997). [CrossRef] [PubMed]

, 5

A. A. Fotiadi, O. Deparis, R. Kiyan, S. Chernikov, and A. Ikiades, “Dynamics of passive Q-switching in SBS/Er fiber laser at low pump power,” Proc. SPIE 4354, 125–134 (2001). [CrossRef]

]. The SBS process will supply an intensive and fast transient feedback to generate higher peak power Q-switched pulses much narrower than that by other Q-switches. With assistance of other technologies, such as hybrid Q switching or coherent combination, the peak power of SBS pulse can be raised to a higher level [6

M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, “Nanosecond Pulse Generation in a Passively Q-Switched Yb-Doped Fiber Laser by Cr4+ : YAG Saturable Absorber,” IEEE Photonics Technol. Lett. 18, 764–766 (2006). [CrossRef]

, 7

F. Kong, L. Liu, C. Sanders, Y. C. Chen, and K. K. Lee, “Phase locking of nanosecond pulses in a passively Q-switched two-element fiber laser array,” Appl. Phys. Lett. 90, 151110 (2007). [CrossRef]

]. However, SBS Q-switched pulse has a drawback of repetition rate instability, which comes from the thermal noise incorporated in Brillouin scattering process, and has been observed experimentally [4–7

S. V. Chernikov, Y. Zhu, and J. R. Taylor, “Supercontinuum self-Q-s witched ytterbium fiber laser,” Opt. Lett. 22, 298–300 (1997). [CrossRef] [PubMed]

]. Reference 4 proposed that the enhancement the stabilization of the repetition rate can be achieved by modulating pump power, but did not present experiment data. Pulsed pumping by using a high power laser diode can also serve this purpose and achieved the frequency stability of better than 1% [8

Y. X. Fan, F. Y. Lu, S. L. Hu, K. C. Lu, H. J. Wang, and G. Y. Zhang, “Narrow-linewidth widely tunable hybrid Q-switched double-clad fiber laser,” Opt. Lett. 28, 537–539 (2003). [CrossRef] [PubMed]

]. There was also report of incorporating an active modulator (acousto-optic modulator) into the cavity to stabilize the repetition rate [9

Z. J. Chen, A. B. Grudinin, J. Porta, and J. D. Minelly, “Enhanced Q switching in double-clad fiber lasers,” Opt. Lett. 23, 454–456 (1998). [CrossRef]

In order to keep the compactness and low cost of the laser, we pursue a simple and economical method to stabilize the Q-switched pulse. For the SBS Q-switched laser, the main parameter we can influence is the gain in the cavity. As a kind of gain control technique, slight pump modulation has been successfully used in repetition rate stabilization for saturable absorber passively Q-switched laser [10

N. D. Lai, M. Brunel, F. Bretenaker, and A. L. Floch, “Stabilization of the repetition rate of passively Q-switched diode-pumped solid-state lasers,” Appl. Phys. Lett. 79, 1073–1075 (2001). [CrossRef]

]. However, the method may meet challenge in stabilizing the SBS Q-switched laser whose gain dynamics shows quite stochastic disturbance in every circle. In this letter we propose a novel method of gain control by externally injecting a square-wave modulated light. It is shown experimentally that the repetition rate can be synchronized to the modulation frequency, and the variance of the repetition rate can be improved from ~20% to ~1% of the period. It is also found that effectiveness of the method depends on modulation frequency and duty cycle of the injection. Qualitative analyses and simulation indicate that main origin of the instability is variation of population level in EDF; and cross gain modulation (XGM) induced by the injected light makes the population level starting from a near constant low level in each period, and thus stabilizes the repetition rate of pulse train.

Fig. 1. A schematic diagram of experimental set-up

2. Numerical analysis

The schematic diagram of a SBS Q-switched fiber laser used in this work is shown in Fig. 1, where the solitary SBS Q-switched laser described in previous references is shown inside the dashed square [11

A. A. Fotiadi, P. Mégret, and M. Blondel, “Dynamics of a self-Q-switched fiber laser with a Rayleigh—stimulated Brillouin scattering ring mirror,” Opt. Lett. 29, 1078–1080 (2004). [CrossRef] [PubMed]

], while the square-wave modulated DFB laser is used as the external injection in this work. The SBS pulse generation is known as a result of several dynamic mechanisms including Rayleigh scattering (RS) and cascaded SBS. To understand its mechanism it is necessary to simulate numerically its evolution process by using a set of coupled RS-SBS equations and a two-level model of the active medium.

\frac{n}{c} \frac{\partial E_{k}^{\pm}}{\partial t} \pm \frac{\partial E_{k}^{\pm}}{\partial z} = \frac{g_{SBS}}{2 S} ({ρ_{k - 1}}^{\mp^{*}} {E_{k - 1}}^{\pm} - {ρ_{k}}^{\pm} {E_{k + 1}}^{\pm}) + \frac{1}{2} (σ_{e} N_{2} - σ_{a} N_{1}) E_{k}^{\pm} + η_{k}^{\pm} E_{k}^{\mp}

T_{2} \frac{\partial {ρ_{k}}^{\pm}}{\partial t} + ρ_{k}^{\pm} = {E_{k}}^{\pm} {E_{k + 1}}^{\mp^{*}} + f_{k}^{\pm} (z, t),

\frac{n}{c} \frac{\partial P_{p}}{\partial t} + \frac{\partial P_{p}}{\partial z} = - σ_{p} N_{1} P_{p},

\frac{\partial N_{2}}{\partial t} = \frac{σ_{p}}{Sh υ_{p}} N_{1} P_{p} - (\frac{σ_{e}}{Sh υ_{k}} N_{2} - \frac{σ_{a}}{Sh υ_{k}} N_{1}) [\sum_{k} (E_{k}^{+} E_{k}^{+ *} + E_{k}^{-} E_{k}^{- *}) + E_{inject} E_{inject}^{*}] - \frac{N_{2}}{τ},

\frac{n}{c} \frac{\partial P_{inject}}{\partial t} + \frac{\partial P_{inject}}{\partial t} = (σ_{e} N_{2} - σ_{a} N_{1}) P_{inject}

(1)

Where E ^± _k represent the forward and backward light of k ^th order SBS (k=0,1,2,3), and ρ ^± _k is photo-induced sound wave of k ^th order. P_p is the pump power. N ₁ and N ₂ are the ground and up-level population densities. g_SBS is the SBS gain factor. S is the effective mode area. c is the light velocity in vacuum. h is the Plank’s constant. σ_p and σ_a are the absorption cross sections at pump frequency υ_p and signal frequency υ_k respectively, σ_e is the stimulated emission cross section. T ₂ and τ are the relaxation time of sound wave and N ₂ · η_k (z) is Rayleigh scattering coefficient, and f_k (z,t) is Langevin noise source, which are stochastic process with zero mean 11

]. P_inject is the intensity of external injected light. The injected light will also be amplified, as shown in the last equation, which will increase the depletion of population inversion as reflected in the equation of N ₂.

Fig. 2. Numerically calculated results: (a) Cascaded SBS generation process in corresponding point A and C in Fig .(1); Population inversion dynamics in EDF without(b) and with(c) external injection; (d) Pulse period jittering with and without external injection. Model parameters: FBG’s reflectivity is 99.5% at central frequency with the bandwidth of 30GHz,

The equation set (1) is resolved with Runge–Kutta algorithm with following parameters: P_p = 100mW, N ₁+N ₂ = 6.5 × 10¹⁵mm^-3, g_SBS = 5×10^-8mm/W, S=50 × 10^-6 mm², σ_p=4 × 10^-19 mm², σ_a=3.3×10^-19mm², σ_e=3.8×10^-19mm², τ=10ms, and T ₂ =10ns. η_k (z) and f_k (z,t) are random figures after Ref. 11

. Figure 2(a) shows the calculated pulse generation process. The fiber ring will accumulate photons at its resonance modes; their RS provides backward beam, which will be amplified for the modes coincident with the FBG peak. Since the resonance of ring will effectively narrow the spectral width to meet the Brillouin gain bandwidth [11

], the SBS process will occur under certain light intensity in the ring. The SBS serves as a mirror with relative high reflectivity. The wide bandwidth of FBG allows cascaded SBS generation, which exhausts most of the population inversion. The pulses are attributed to the second order SBS, which comes from the first order SBS traveling leftward in Fig. 1. Figure 2(a) shows the calculated pulse generation process. Since RS-SBS process must have stochastic movement of molecules incorporated, the moment and intensity of the pulses will definitely have stochastic features. Figure 2(b) shows the calculated consequently generated 30 pulses and the corresponding population variation in EDF. To show the instability clearer, 30 pulses are divided into 5 groups and overlapped in the figure with the first pulse of each group synchronized. It is shown that the instabilities include jittering of pulse position, fluctuation of intensity, and fluctuation of inversion after being exhausted. We also found these three parameters have close relationship with each other. The correlation coefficient for peak power and initial inversion is 96%, while it is 98% for initial inversion and period. Therefore, it is conjectured that the fluctuation of initial inversion due to pulse energy variation brings about the time jittering of generated pulse and the stability of pulse period depends on the stability of the initial population level.

It is expected that an external modulated light with proper pulse width can serve the purpose by XGM, and thus synchronize the pulses. Figure 2 (c) shows changes of population inversion in the EDF induced by both SBS Q-switched pulses and injected light with rectangular modulation period of 480μs and duty cycle of 75%. The injected light depletes partly the population and interrupts the process of spontaneous instable increasing; and makes its starting point nearly constant (Δn ₀) by a constant injection level long enough. Then, the onset time of the next generated pulse is limited. Numerical results showed that the mechanism works well and the timing jitter is mitigated from 15μs to 1μs, see Fig. 2 (d).

3. Experiment

The experiment set-up used in this work consists of a 10 meter long erbium-doped fiber, pumped by a 980nm diode laser up-to 130mW, a 2 m long SMF ring resonator with a 10/90 coupler and a fiber Bragg grating with peak reflectivity of 99.5% and bandwidth of 0.3nm and at wavelength of 1554.98nm. The whole length of laser cavity is about 20m. A DFB laser modulated in square wave is used as injection light with adjustable repetition rate and duty cycle, and power around 2mW. The isolators ensure the unidirectional traveling of the light and suppress the Fresnel reflection into the laser cavity from fiber facets.

With the pump power of 100mW, the pulsed output of the solitary SBS Q-switched laser without external injection was measured by an oscilloscope in Digital Phosphor Oscilloscopes (DPO) mode, as shown in Fig. 3(a), where the waveform was synchronized at the middle of the frame. The average repetition rate is 3.2 kHz and the pulse width is quite narrow of about 5ns, as shown in Fig. 3(b). The average power is tested to be 8mW; therefore the pulse energy is at micro Joule level. However, the spacing between adjacent pulses varied greatly, with variance of repetition rate estimated to be near 20% of the period, showing typical characteristics of the SBS self Q-switched fiber laser.

While the DFB laser is operated with modulation frequency of 1 kHz, duty cycle of 70% and injected power of 1mW, clear pulse waveforms can be observed, as shown in Fig. 3(c), indicating that the pulses are synchronized with the injection, with pulse position variance estimated to be less than 1% of the period. Figure 3(d) shows the FBG transmission spectrum and the output SBS spectrum on a linear scale. The central wavelength of FBG locates at 1554.98nm, while the wavelength of the Q-switched pulse is shifted red-forward 0.18nm away (22GHz), which is coincident with frequency shift of the second order SBS. The output line-width was measured about 0.02nm, which might be over estimated because of the resolution of spectrum analyzer. It is also much larger than usual SBS linewidth, which may be attributed to the difference between static process and pulsed dynamic process in this work.

Fig. 3. (a) Typical output of a solitary SBS Q-switched laser; (b) Profile of the generated pulse; (c) Output of the SBS Q-switched laser with external injection; (d) Spectrum of output light and FBG

The injected beam will surely deplete population inversion, and may lower EDF gain at working waveband of Q-switched laser. The average power of the Q-switched pulses was measured to be 2mW under external light injection. Taking the repetition rate into consideration, the pulse energy is 2 μJ, not much difference from that of free running state (~2.5 μJ).

The results verify the proposed stabilization method; but it works under proper parameters, especially the duty cycle of the square wave and modulation frequency. At low modulation frequency and low duty cycle, the depletion of population is not enough, and a second and more pulse will follow the first Q-switched pulse in one period, as shown in Fig. 4(a). It is seen that the onset moment of the first pulse in every period is precisely set, while the second pulse shows a little variations. If the duty cycle and modulation frequency are too high, the gain will be depleted too much, the total gain will be varying around the threshold. Then the laser will generate pulses randomly or even no pulse output. Figure 4 (b) shows experimental measurement of stabilization zones, which is symbolized by the vertical bars.

Fig. 4. (a) Two pulses in one modulation period; (b) Stabilization zone as function of modulation frequency and duty cycle

It should be of interest to point out that although the power of injection light is quite low of only 1mW, it is enough to deplete the gain to suppress SBS evolution, because the depletion is a sustained process, as long as several hundred microseconds. In addition, it is also found experimentally that the wavelength of injected light brings about minor influence to the stabilization. It is because the EDF is mainly a homogeneous broadening medium. So long as the wavelength of injected light is inside the gain spectrum, the XGM will occur. Of cause, the depletion is most sufficient when the injection is at the peak of gain spectrum, i.e. around 1530nm.

Fig. 5. Output of the externally injected SBS Q-switched laser with duty cycle of (a) 70% and (b) 80% and their respective spectrum in (c) and (d) when the injection wavelength is 1554.95nm

When the wavelength of injection falls into the reflection bandwidth of the FBG, much smaller power injection can serve the function of stabilization, as the injection will be amplified twice by reflection and result in sufficient depletion of inversion. In our experiment, the stabilization is achieved using 0.1mW injection power. Figure 5(a) shows the DPO waveform of pulses when 0.1mW 1554.95nm injection is applied with 70% duty cycle and 1 kHz modulation frequency. The width of the pulse bunch in the figure indicates the time jittering, which is 10μs, corresponding to 1% of the modulation period. The pulse is usually generated before the next injection, typically with time slot of 55μs as shown in Fig. 5(a), similar to the case of injection light out of FBG band. However, pulses can also be found shortly after injection turned on when higher injection level, or higher duty cycle or higher frequency is chosen. Figure 5(b) shows the pulse generation with 80% duty cycle modulation, and the same frequency and same injection level as that for 'Fig. 5(a). The Q-switched pulse is generated some micro-seconds after the front edge of injected square wave, in this example. To explain the phenomenon, we investigate the output spectrum of the above two cases with the injection wavelength slightly (0.03nm) shorter than the central wavelength of FBG. In case corresponding to Fig. 5(a), the spectrum was measured with main peak at the same position as the case of injection out of FBG band, as shown in Fig. 5(c), implying that the injection plays a role of gain control as analyzed above. However, the measured output spectrum for case of Fig. 5(b) has an ~0.03nm blue-shifted peak, shown in Fig. 5(d), which is coincident with the 2^nd order SBS of the injected light. It is inferred that under such a condition the injected light plays a seed of the 0^th order lasing beam, which is then quickly transferred into its 2^nd order SBS.

4. Conclusion

In conclusion, a novel method of repetition rate stabilization by external injection of a square wave modulated light is proposed and demonstrated for the SBS Q-switched fiber lasers. It is measured experimentally that variance of the repetition rate can be improved from about 20% without injection to about 1% with injection. It is also found that effectiveness of the method depends on modulation frequency and duty cycle of the injection. Compared with other passive Q-switch, the SBS switch is much fast and very simple. It will be promising in case the drawback of repetition rate instability can be overcome effectively. The results of this paper may be helpful for the purpose; and the stabilization mechanism of XGM by external injection may be also used for other passive Q-switched fiber lasers.

References and links

1.	C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, “Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm,” IEEE Photonics Technol. Lett. 11, 976–978 (1999). [CrossRef]
2.	J. J. Zayhowski, “Passively Q-switched microchip lasers and applications,” Rev. Laser Eng. 26, 841–846 (1998).
3.	P. Petropoulos, H. L. Offerhaus, D. J. Richardson, S. Dhanjal, and N. I. Zheludev, “Passive Q-switching of fiber lasers using a broadband liquefying gallium mirror,” Appl. Phys. Lett. 74, 3619–3621 (1999). [CrossRef]
4.	S. V. Chernikov, Y. Zhu, and J. R. Taylor, “Supercontinuum self-Q-s witched ytterbium fiber laser,” Opt. Lett. 22, 298–300 (1997). [CrossRef] [PubMed]
5.	A. A. Fotiadi, O. Deparis, R. Kiyan, S. Chernikov, and A. Ikiades, “Dynamics of passive Q-switching in SBS/Er fiber laser at low pump power,” Proc. SPIE 4354, 125–134 (2001). [CrossRef]
6.	M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, “Nanosecond Pulse Generation in a Passively Q-Switched Yb-Doped Fiber Laser by Cr4+ : YAG Saturable Absorber,” IEEE Photonics Technol. Lett. 18, 764–766 (2006). [CrossRef]
7.	F. Kong, L. Liu, C. Sanders, Y. C. Chen, and K. K. Lee, “Phase locking of nanosecond pulses in a passively Q-switched two-element fiber laser array,” Appl. Phys. Lett. 90, 151110 (2007). [CrossRef]
8.	Y. X. Fan, F. Y. Lu, S. L. Hu, K. C. Lu, H. J. Wang, and G. Y. Zhang, “Narrow-linewidth widely tunable hybrid Q-switched double-clad fiber laser,” Opt. Lett. 28, 537–539 (2003). [CrossRef] [PubMed]
9.	Z. J. Chen, A. B. Grudinin, J. Porta, and J. D. Minelly, “Enhanced Q switching in double-clad fiber lasers,” Opt. Lett. 23, 454–456 (1998). [CrossRef]
10.	N. D. Lai, M. Brunel, F. Bretenaker, and A. L. Floch, “Stabilization of the repetition rate of passively Q-switched diode-pumped solid-state lasers,” Appl. Phys. Lett. 79, 1073–1075 (2001). [CrossRef]
11.	A. A. Fotiadi, P. Mégret, and M. Blondel, “Dynamics of a self-Q-switched fiber laser with a Rayleigh—stimulated Brillouin scattering ring mirror,” Opt. Lett. 29, 1078–1080 (2004). [CrossRef] [PubMed]

OCIS Codes
(140.3510) Lasers and laser optics : Lasers, fiber
(140.3540) Lasers and laser optics : Lasers, Q-switched
(290.5900) Scattering : Scattering, stimulated Brillouin
(140.3425) Lasers and laser optics : Laser stabilization

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: October 6, 2008
Revised Manuscript: December 29, 2008
Manuscript Accepted: January 24, 2009
Published: February 17, 2009

Citation
Zhengqing Pan, Li Meng, Qing Ye, Haiwen Cai, Zujie Fang, and Ronghui Qu, "Repetition rate stabilization of the SBS Q-switched fiber laser by external injection," Opt. Express 17, 3124-3129 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3124

Sort: Author | Year | Journal | Reset

References

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photonics Technol. Lett. 11, 976-978 (1999). [CrossRef]
J. J. Zayhowski, "Passively Q-switched microchip lasers and applications," Rev. Laser Eng. 26, 841-846 (1998).
P. Petropoulos, H. L. Offerhaus, D. J. Richardson, S. Dhanjal and N. I. Zheludev, "Passive Q-switching of fiber lasers using a broadband liquefying gallium mirror," Appl. Phys. Lett. 74, 3619-3621 (1999). [CrossRef]
S. V. Chernikov, Y. Zhu, and J. R. Taylor, "Supercontinuum self-Q-switched ytterbium fiber laser," Opt. Lett. 22, 298-300 (1997). [CrossRef] [PubMed]
A. A. Fotiadi, O. Deparis, R. Kiyan, S. Chernikov, A. Ikiades, "Dynamics of passive Q-switching in SBS/Er fiber laser at low pump power," Proc. SPIE 4354, 125-134 (2001). [CrossRef]
M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, "Nanosecond Pulse Generation in a Passively Q-Switched Yb-Doped Fiber Laser by Cr4+ : YAG Saturable Absorber," IEEE Photonics Technol. Lett. 18, 764-766 (2006). [CrossRef]
F. Kong, L. Liu, C. Sanders, Y. C. Chen, and K. K. Lee, "Phase locking of nanosecond pulses in a passively Q-switched two-element fiber laser array," Appl. Phys. Lett. 90, 151110 (2007). [CrossRef]
Y. X. Fan, F. Y. Lu, S. L. Hu, K. C. Lu, H. J. Wang, and G. Y. Zhang, "Narrow-linewidth widely tunable hybrid Q-switched double-clad fiber laser," Opt. Lett. 28, 537-539 (2003). [CrossRef] [PubMed]
Z. J. Chen, A. B. Grudinin, J. Porta, and J. D. Minelly, "Enhanced Q switching in double-clad fiber lasers," Opt. Lett. 23, 454-456 (1998). [CrossRef]
N. D. Lai, M. Brunel, F. Bretenaker, and A. L. Floch, "Stabilization of the repetition rate of passively Q-switched diode-pumped solid-state lasers," Appl. Phys. Lett. 79, 1073-1075 (2001). [CrossRef]
A. A. Fotiadi, P. Mégret, and M. Blondel, "Dynamics of a self-Q-switched fiber laser with a Rayleigh-stimulated Brillouin scattering ring mirror," Opt. Lett. 29, 1078-1080 (2004). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper