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Suppression of first order stimulated Raman scattering in erbium-doped fiber laser based LIDAR transmitters through induced bending loss
Optics Communications 250 (2005) 403–410 www.elsevier.com/locate/optcom
Suppression of first order stimulated Raman scattering in erbium-doped fiber laser based LIDAR transmitters through induced bending loss Peter D. Dragic
*
Department of Electrical and Computer Engineering, University of Illinois, 1308 W Main Street, Urbana, IL 61801, USA Received 30 November 2004; accepted 16 February 2005
Abstract A model is presented for suppression of first-order stimulated Raman scattering (SRS) in pulsed Er-doped fiber amplifiers with wavelength-selective induced bending loss. The model estimates the effectiveness of SRS suppression through fiber coiling in a pulsed amplifier, considering both the single- and dual-clad fiber cases. The basic approach is to fit analytical expressions to the evolution of a quasi-CW signal in the amplifier. Then, assuming that SRS does not deplete the signal, exact analytical expressions for the propagation of the Stokes wave are determined, from which the SRS threshold is estimated. Finally, example calculations are presented, and the model is used to determine a fiber geometry for the suppression of SRS through fiber coiling. 2005 Elsevier B.V. All rights reserved. PACS: 42.65.D; 42.55.W; 42.81; 42.68.W; 42.79.Q Keywords: Stimulated Scattering, Raman; Fiber lasers; Fiber Optics; LIDAR
1. Introduction Fiber lasers have been attracting increasing attention as transmitter sources for light detection and ranging (LIDAR) systems. This is attributable
*
Tel.: +1 217 333 8399; fax: +1 217 333 4303. E-mail address: [email protected].
to several qualities that potentially make fibers superior to the lasers traditionally used for this application. In contrast to these more traditional lasers, fiber lasers are compact, efficient, lightweight, and can be air-cooled. These characteristics, along with their high beam quality and low pointing jitter make them especially desirable for space-based applications. It is useful to note that fiber lasers in LIDAR applications are typically
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.02.048
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P.D. Dragic / Optics Communications 250 (2005) 403–410
master oscillator-power amplifier (MOPA) lasers [1,2]. Unfortunately, fiber lasers suffer from a number of undesirable parasitic phenomena that have precluded a far reach to some of the most important LIDAR applications. One of these parasitics, stimulated Brillouin scattering (SBS), limits the power per unit bandwidth available in very narrow linewidth systems [3–5]. Another, amplified spontaneous emission (ASE) is a problem in pulsed systems as it depopulates the upper state between pulses. This leads to a continuous-wave (CW) signal component that appears as ÔbackgroundÕ in the received signal, substantially degrading system signal-to-noise ratio (SNR) when the atmospheric returns are very small. This leads to the required use of an optical chopper at the receiver. ASE also drains energy from the upper state, thereby substantially limiting laser performance. Here, it is assumed that the circumvention of both SBS [6–9] and ASE [10–13] shall be independent of the results of this paper. A third parasitic phenomenon is stimulated Raman scattering (SRS). SRS is an interaction between the optical field and optical phonons that tends to Stokes-shift the LIDAR signal [3,4,14–16]. This effect is undesirable where strict wavelength control is mandatory, such as in spectroscopic systems (e.g., differential absorption LIDAR (DIAL) and resonance fluorescence applications). The result is that the peak transmitter output power is limited by the SRS threshold, limiting available system SNR. Here, a model is presented for first-order SRS management and weakening in pulsed Er-doped fiber amplifiers (fiber with gain) incorporating wavelength-selective induced bending loss [17,18]. The microbending of optical fiber has already been shown to be an effective way to inflict mode-selective loss to induce single mode (SM) operation in a multimode (MM) fiber [19]. Bending loss can also be used to suppress emission near 1064 nm in Nd-doped fibers to allow for laser operation near 930 nm for the water vapor application [20]. The coiling technique has also recently been used to suppress higher order Stokes components in the Raman interaction
[21]. Our modeling results indicate that this coiling technique can also be applied to the management and reduction of the effects of first-order SRS in Er-doped fiber in order to increase overall system power and thus SNR. This model is not meant as an attempt to precisely predict laser performance. Several factors, such as fiber imperfection and finite upper state lifetime, can alter the results presented here. The model is meant to estimate the effectiveness of SRS suppression by fiber coiling in a pulsed amplifier. Our basic approach is to fit analytical expressions to the evolution of a quasi-CW signal in the amplifier. We have chosen a co-propagating signal/pump configuration. Then, assuming that SRS does not deplete the signal, we can determine exact analytical expressions for the propagation of the Stokes wave, from which the SRS threshold can be estimated.
2. Model We begin with the quasi-CW [4] equation governing the evolution of the Stokes signal Ps(z) d P p ðzÞ þ as P s ðzÞ ¼ gR P s ðzÞ p ; ð1Þ dz Aeff where as is the induced (bending) attenuation coefficient at the Stokes wavelength, gR is the Raman gain coefficient, and Aeff is the effective mode area of the pump signal. The p refers to the pump wavelength (15XX nm, different from the amplifier pump wavelength of 976 nm) and s to the Stokes signal (15XX nm + Dk). Here, it is reasonably assumed that the Stokes wavelength falls outside the Er gain bandwidth, since the peak Raman wavelength shift (Dk) in silica fibers is 100 nm at 1550 nm. A non-depleted evolution of the pump is also assumed, which is governed only by the net gain provided by the fiber amplifier. In the case of a deeply saturated amplifier, the simple model of exponential signal growth is not appropriate. We will therefore subsequently present analytical expressions that approximate pump evolution for the cases of single- and dual-clad fiber. This leads to exact expressions for the evolution of the Stokes
P.D. Dragic / Optics Communications 250 (2005) 403–410
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signal that can be used to estimate amplifier performance. Finally, the analytical expression for bendinduced as [1/meters] is provided in [17] for an optical mode as pffiffiffi 4Dw3 p 1 a exp 3aV 2 R as ¼ ð2Þ 1=2 ; 4 Aseff V2 w wR þ a 2Dw
over the fiber length. Then, the effective input power can be found from
where a is the radius of the fiber core, Aseff is the effective mode area of the Stokes wave, R is the fiber bending radius, is the index difference, D is the fiber V-number (normalized frequency), and w is approximated in [22] by the following in a step index fiber within the range 1.5 < V < 2.5
where Dmfwhm is the width of the Raman gain spectrum. With this boundary condition, we determine that the solution to (5) is gR P p ðLÞ P s ðzÞ ¼ hms Beff exp as z 2Apeff gR P p ðLÞ coshðcðz Lfit ÞÞ ln exp : 2Apeff c coshðcLfit Þ
w ¼ 1:1428V 0:9960:
ð3Þ
P s ð0Þ ¼ hms Beff ;
ð7Þ
where Beff is the effective bandwidth (Hz) and is estimated in our case by [3] pffiffiffi p Dmfwhm Beff ¼ ; ð8Þ 2 ðP p ðLÞgR =cp Apeff Þ1=2
ð9Þ 2.1. Single-clad fibers As will be demonstrated, a good model for the pump in a saturated single-clad fiber is the general hyperbolic tangent function P p ðzÞ ¼
P p ðLÞ ð1 þ tanhðcðz Lfit ÞÞÞ; 2
ð4Þ
where c and Lfit are amplifier fitting parameters. In this analysis, Pp(L) is the maximum amplifier output power for a given amplifier pump (976 nm) power, with fiber length L selected such that the output power is maximized. Inserting (4) into (1), we obtain
d P p ðLÞ þ as P s ðzÞ ¼ gR P s ðzÞ p ð1 þ tanhðcðz Lfit ÞÞÞ: dz 2Aeff ð5Þ
The general solution to (5) is given by gR P p ðLÞ P s ðzÞ ¼ C exp as ðz Lfit Þ 2Apeff gR P p ðLÞ lnðcoshðcðz Lfit ÞÞÞ ; ð6Þ exp 2Apeff c where C is a constant. To determine C, we assume a known effective signal input Ps(0). Smith [3] showed that if this input at z = 0 is taken to be one photon per mode, then this input is equivalent to a sum of all spontaneous emission integrated
2.2. Dual-clad fibers A good model for the pump in a saturated dualclad fiber is the following exponential function: P p ðzÞ ¼ Dfit P p ðLÞð1 expðczÞÞ;
ð10Þ
where Pp(L) is the defined as before, and c and Dfit are fitting parameters for amplifier performance. Inserting (10) into (1), we obtain d P p ðLÞ þ as P s ðzÞ ¼ gR P s ðzÞDfit p ð1 expðczÞÞ: dz Aeff ð11Þ The general solution to (11) is given by gR Dfit P p ðLÞ P s ðzÞ ¼ C exp as z Apeff g Dfit P p ðLÞ expðczÞ ; exp R p Aeff c
ð12Þ
where C is again a constant. Invoking the boundary condition Ps(0) again, we achieve the following solution to (11): g Dfit P p ðLÞ P s ðzÞ ¼ hms Beff exp as R p z Aeff g Dfit P p ðLÞ ðexpðczÞ 1Þ : ð13Þ exp R p Aeff c
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3. Raman gain coefficient The Raman gain coefficient gR has an inverse dependence on pump wavelength. Therefore, we can estimate the Raman gain coefficient near 1550 nm after [3,4,14–16]. Beyond about 1650 nm (400 cm1) the gain coefficient decreases and wavelength increases, both decreasing the Raman contribution and increasing the bending loss. Thus, these long wavelengths are neglected (see Fig. 1).
4. Example calculation: typical single-clad fiber We begin with a model for the output of a single mode, single-clad fiber. We will consider results in the range of 0.5–4 W of optical pump power injected into the fiber. In the former case, near 500 mW, fiber coupled semiconductor edge-emitters [23,24] can be applied. In the latter, for higher-power operation, Yb-doped fiber lasers near 980 nm [25,26] can be employed. Second, we arbitrarily define a ÔtypicalÕ system that operates at 10 kHz with 50 ns pulse width, corresponding to a 15 km atmosphere with 7.5 m range resolution. This provides a duty cycle of 1/2000. We assume a ÔtypicalÕ erbium-doped single mode single-clad fiber with a core diameter of 4.4 lm, NA of 0.16, and doping concentration of 1000 ppm wt. The cut-off wavelength of this fiber is then 920 nm. We assume that the amplifier operates near 1550 nm so that the signal absorp-
tion coefficient is about 1.25 · 1025 m2 and the emission cross-section is about 2.5 · 1025 m2 [27]. We also assume that the lifetime of the erbium 4I13/2 upper state is 10 ms. With the assumption that the pump absorption cross-section is about 1 · 1025 m2, we invoke the amplifier models found in [28] for 1 mW of average injected signal power, and arrive at the results for the average signal power evolution as a function of length shown in Fig. 2. Pump power is shown to the right. Fig. 2 also shows the results of the best fit for the single-clad fiber amplifier models. The fit parameters are Lfit = 8.8 m and c = 0.26. We see that the fit is very good further along in the fiber, but breaks down in the first few meters of the fiber. However, since the power levels in the first few meters of fiber are well below the SRS threshold, this should not significantly affect the calculations. Furthermore, the fit is also very good in the whole range 500 mW–4 W of pump power, but degrades below 500 mW. We assume a fiber length of 20 m. as · L for this 20-m fiber is plotted in Fig. 3 for several fiber bend radii. To estimate the SRS threshold, we follow Smith [3] and assume that the critical output power occurs when Ps(L) = Pp(L). Using Eq. (2) and solving Eq. (9) for Pcrit = Ps(L) = Pp(L), we arrive at the results shown in Fig. 4 for the SRS threshold as a function of wavelength for the same fiber bend radii in Fig. 3. We have also assumed that Dmfwhm = 8 THz. Since the peak Raman gain in this case is around 1650 nm, the lowest critical power is at this wavelength.
Amp Model Pump = 4W
Fit Pump = 3W
Pump = 2W
Pump = 1W
Fig. 1. 1550 nm-Pumped Raman gain spectrum after [3,4,14– 16].
Fig. 2. Modeling results for a typical single-clad fiber along with the tanh fit.
P.D. Dragic / Optics Communications 250 (2005) 403–410
0.020m
0.022m
0.025m 5m
Fig. 3. as · L vs. wavelength for the single-clad fiber for several bending radii. Bend radii are provided in the plot.
0.020m 0.022m 0.025m 5m
Fig. 4. Critical peak output power (kW) vs. wavelength for various fiber bending radii.
A drawback to bending is that background attenuation at the signal wavelength increases (bending loss), resulting in the need for a greater amount of pump power to achieve the output powers shown in Fig. 2. The excess bending loss (ap · L) at 1550 nm for bend radii 5, 0.025, 0.022, and 0.020 m are 0, 0.099, 0.372, and 1.02, respectively. However, we see from Fig. 4 that through coiling such a fiber, the SRS threshold can be increased from about 1.35 to 2.7 kW (1.35 W average, 135 lJ/pulse), or about a factor of 2. Even with the increased signal loss, a twofold increase in the SRS threshold can be very desirable for increased system power.
407
1000 ppm wt Yb and Er doping, respectively. This ratio (20:1) was found to maximize the energy transfer efficiency between the Yb and Er atoms [29]. Additionally, we assume a core diameter of 12 lm, cladding diameter of 400 lm, core NA of 0.06, and a pumping wavelength of 915 nm. We choose a pumping wavelength off of the peak absorption coefficient so that a longer fiber can be used such that more Erbium atoms are available for greater pulse energies. We assume an Yb absorption coefficient of about 7.5 · 1025 m2 [30] and an energy transfer efficiency of 20% [29,31]. Finally, we assume a 100 ns pulse width with 5 kHz repetition rate (30 km atmosphere and 15 m range resolution), giving a duty cycle of 1/2000. Fig. 5 shows the amplifier modeling results for 10 mW of average input signal power along with the fit. The fit parameters are Dfit = 1.04 and c = 0.120. We see that in the range of pumping powers of interest (40–100 W), the numerical fit is very good over the whole fiber length. ap · L is plotted in Fig. 6 for a 30-m length of this fiber. Again, there is an increased background loss at the signal wavelength and ap · L is found to be 0, 0.229, 0.53, and 0.931 for bending radii of 5, 0.34, 0.31, and 0.29 m, respectively, at 1550 nm. Using the same definition for the critical SRS output power, we arrive at the results shown in Fig. 7. For the two extreme cases in Fig. 5, R = 5 and 0.29 m, the available peak output power only increases from about 5.3–8.0 kW (4.0 W average power, 0.8 mJ/pulse), or an increase of about 50%. This is meager at best, and probably does not justify the excess signal loss due to bending.
Amp Model
Pump = 100W
Fit Pump = 80W Pump = 60W Pump = 40W
5. Example calculation: typical dual-clad fiber For the dual-clad fiber example, we assume an Yb–Er co-doped fiber. We assume 20,000 and
Fig. 5. Modeling results for a typical dual-clad fiber along with the fit.
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P.D. Dragic / Optics Communications 250 (2005) 403–410
Index of Refraction
Yb/Er Doping Profile
0.29m 0.31m
0.34m
Fig. 6. as · L vs. wavelength for the dual-clad fiber for several bending radii.
0.29m 0.31m 0.34m 5m
Fig. 7. Critical peak output power (kW) vs. wavelength for various fiber bending radii.
6. Fiber optimization In order for the bending loss technique to be useful enough to suppress SRS, the loss at the Stokes wavelength should be maximized while minimizing the loss at the desired signal wavelength. It turns out that the loss contrast between these two wavelengths increases as the fiber V-number decreases. As a result, we may design a fiber geometry that can optimally take advantage of this loss, such as the proposed cross-sectional geometry shown in Fig. 8, where both the core radius and NA are kept small. In the figure, R1 is the core radius, R2 is the doping radius, and R3 is the inner cladding radius. With the addition of a separate doping layer, both the overlap with the pump in the inner cladding and pulse energy can be increased while retaining a small core diameter for a small fiber V-number. However, proper dopants should be used to match the index of the doped layer to that of the inner cladding.
R3
r
Fig. 8. Yb/Er doping and index of refraction profiles for the suppression of SRS. R1 is the core radius, R2 is the doping radius, and R3 is the inner cladding radius.
Fig. 9 shows the plot of the normalized fundamental mode (intensity) for two optical fibers, both with NAs of 0.06, but with different core diameters; 7.5 and 12 lm. The plot is along the radial direction from the center of the fiber. Both fibers have NAs of 0.06. We see that these modes have, for our purposes here, very similar modefield diameters. From inspection of Fig. 9 and noting the wide tails associated with the 7.5-lm fiber, we choose R2 = 10 lm, and retain the same doping concentrations as in Part V. R3 is chosen to be 200 lm, as in Part 5. The V-number of the 7.5-lm fiber is 0.91 at 1550 nm. This falls outside the bounds of the expression for w in Eq. (3). In the case of small V, Marcuse [32] provides a better approximation for w. This is given by 1 J 0 ðV Þ w ¼ 1:122 exp ; ð14Þ V J 1 ðV Þ where the JÕs represent the Bessel functions. Using this expression, we can plot as · L for the fiber of Fig. 8. The results are shown in Fig. 10 for 20-m-long fibers with (1) 7.5-, (2) 8.0-, and (3) 1 Relative Power
5m
R1 R2
0.8 0.6 0.4 7.5 micron
0.2 12 micron
0
0
5
10 15 Distance (microns)
20
Fig. 9. Plots of the fundamental optical modes for fibers with 7.5- and 12-lm cores.
P.D. Dragic / Optics Communications 250 (2005) 403–410
1
2
3
Fig. 10. as · L vs. wavelength for (1) 7.5-lm diameter, 8.85 m bend radius; (2) 8-lm core, 4.45 m bend radius; (3) 10-lm core, 0.87 m bend radius.
10.0-lm core diameters and 20-lm doping layers. The bending radii were chosen such that as · L = 1 at 1550 nm. These radii are (1) 8.85 m, (2) 4.45 m, and (3) 0.87 m, respectively. Invoking the amplifier models for a 40–100 W pumping range and retaining all other remaining parameters from Part 5 (doping concentrations, cross-sections, energy transfer efficiency, upper state lifetime, input signal power, repetition rate and pulse width), the amplifier fit parameters were found to be Dfit = 1.02 and c = 0.32. The core size in the range of 7.5–10 lm had little influence on the fit parameters. Thus, the same fit parameters were used for all three cases. With these results, the critical output powers vs. wavelength for these cases are shown in Fig. 11. The critical peak output power of a 7.5-lm-core fiber with as = 0 is 6.6 kW (660 lJ/pulse, case 4).
1
2
3
409
Thus, if large bending radii can be tolerated, the critical output power can be increased to nearly 30 kW (15 W average power, 3 mJ/pulse). However, it is interesting to note that SRS will first occur in these fibers at wavelengths shorter than 1650 nm, such as near 1628 nm for case 1. Bending radii on the order of several meters may not seem practical. However, in a setting where the laser is in a lidar facility, it would be possible to coil the fiber loosely around a room, such as Ôthe telescope roomÕ. Furthermore, if the doped dual-clad fiber is used as the delivery fiber, or if a similar fiber is produced without the rare earth for delivery purposes, the fiber can be carefully strung as it leads to the telescope in a monostatic system. Alternatively, if the fiber is used for delivery in straight-line segments, these large bending radii could possibly be achieved if the fiber were contained in something similar to a stretched-out coiled phone cord with a larger radius, whereby a helical path is generated.
7. Conclusion We have presented a model for the management and suppression of SRS in pulsed Er-doped fiber amplifiers (fiber with gain) with wavelength-selective induced bending loss. Our modeling results indicate that the coiling technique can be applied to the management and substantial reduction of the effects of first-order SRS in Er-doped fiber based systems, with the right waveguide design. The optimal fiber for the suppression of SRS has a low V-number (<1.0) and requires a large minimum bending radius. However, if this bending radius is tolerable in a system, substantial power increases can be realized before the onset of SRS. Furthermore, by coupling bending with fiber grating technologies, the SRS limitation of fiber laser technologies can be greatly diminished.
4
References Fig. 11. Critical peak output power (kW) vs. wavelength for cases 1, 2, and 3. Case 4 is the example of a fiber with no bending loss.
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[2] C. Alegria, Y. Jeong, C. Codemard, J.K. Sahu, J.A. Alvarez-Chavez, L. Fu, M. Ibsen, J. Nilsson, IEEE Photon. Technol. Lett. 16 (8) (2004) 1825. [3] R.G. Smith, Appl. Opt. 11 (11) (1972) 2489. [4] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 1995. [5] C.N. Parnell, P.St.J. Russell, T.P. Newson, J. Opt. Soc. Am. B 10 (4) (1993) 684. [6] L. Eskildsen, P.B. Hansen, U. Koren, B.I. Miller, M.G. Young, K.F. Dreyer, Electron. Lett. 32 (15) (1996) 1387. [7] C.A.S. deOliveira, C.K. Jen, A. Shang, C. Saravanos, J. Opt. Soc. Am. B 10 (6) (1993) 969. [8] M. Ohashi, M. Tateda, IEEE J. Lightwave Technol. 11 (12) (1993) 1941. [9] K. Shiraki, M. Ohashi, M. Tateda, Electron. Lett. 31 (8) (1995) 668. [10] M.A. Arbore, Y. Zhou, G. Keaton, T.J. Kane, Proc. SPIE 4989 (2003) 47. [11] Y.H. Chen, L.G. Sheu, C.C. Yang, I.W. Wei, W.C. Lin, Y.Z. Lee, C.H. Hsia, Proc. SPIE 3420 (1998) 188. [12] J. Nilsson, J.D. Minelly, R. Paschotta, A.C. Tropper, D.C. Hanna, Opt. Lett. 23 (5) (1998) 355. [13] P.F. Wysocki, in: M.J.F. Digonnet (Ed.), Rare Earth Doped Fiber Lasers and Amplifiers, Marcel Dekker, New York, 2001, p. 583. [14] R.H. Stolen, E.P. Ippen, A.R. Tynes, Appl. Phys. Lett. 20 (2) (1972) 62. [15] R.H. Stolen, E.P. Ippen, Appl. Phys. Lett. 22 (6) (1973) 276. [16] R.H. Stolen, Proc. IEEE 68 (10) (1980) 1232. [17] J-I. Sakai, T. Kimura, Appl. Opt. 17 (10) (1978) 1499. [18] J-I. Sakai, Appl. Opt. 18 (7) (1979) 951. [19] J.P. Koplow, D.A.V. Kliner, L. Goldberg, Opt. Lett. 25 (7) (2000) 442.
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Suppression of first order stimulated Raman scattering in erbium-doped fiber laser based LIDAR transmitters through induced bending loss Peter D. Dragic
*
Department of Electrical and Computer Engineering, University of Illinois, 1308 W Main Street, Urbana, IL 61801, USA Received 30 November 2004; accepted 16 February 2005
Abstract A model is presented for suppression of first-order stimulated Raman scattering (SRS) in pulsed Er-doped fiber amplifiers with wavelength-selective induced bending loss. The model estimates the effectiveness of SRS suppression through fiber coiling in a pulsed amplifier, considering both the single- and dual-clad fiber cases. The basic approach is to fit analytical expressions to the evolution of a quasi-CW signal in the amplifier. Then, assuming that SRS does not deplete the signal, exact analytical expressions for the propagation of the Stokes wave are determined, from which the SRS threshold is estimated. Finally, example calculations are presented, and the model is used to determine a fiber geometry for the suppression of SRS through fiber coiling. 2005 Elsevier B.V. All rights reserved. PACS: 42.65.D; 42.55.W; 42.81; 42.68.W; 42.79.Q Keywords: Stimulated Scattering, Raman; Fiber lasers; Fiber Optics; LIDAR
1. Introduction Fiber lasers have been attracting increasing attention as transmitter sources for light detection and ranging (LIDAR) systems. This is attributable
*
Tel.: +1 217 333 8399; fax: +1 217 333 4303. E-mail address: [email protected].
to several qualities that potentially make fibers superior to the lasers traditionally used for this application. In contrast to these more traditional lasers, fiber lasers are compact, efficient, lightweight, and can be air-cooled. These characteristics, along with their high beam quality and low pointing jitter make them especially desirable for space-based applications. It is useful to note that fiber lasers in LIDAR applications are typically
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.02.048
404
P.D. Dragic / Optics Communications 250 (2005) 403–410
master oscillator-power amplifier (MOPA) lasers [1,2]. Unfortunately, fiber lasers suffer from a number of undesirable parasitic phenomena that have precluded a far reach to some of the most important LIDAR applications. One of these parasitics, stimulated Brillouin scattering (SBS), limits the power per unit bandwidth available in very narrow linewidth systems [3–5]. Another, amplified spontaneous emission (ASE) is a problem in pulsed systems as it depopulates the upper state between pulses. This leads to a continuous-wave (CW) signal component that appears as ÔbackgroundÕ in the received signal, substantially degrading system signal-to-noise ratio (SNR) when the atmospheric returns are very small. This leads to the required use of an optical chopper at the receiver. ASE also drains energy from the upper state, thereby substantially limiting laser performance. Here, it is assumed that the circumvention of both SBS [6–9] and ASE [10–13] shall be independent of the results of this paper. A third parasitic phenomenon is stimulated Raman scattering (SRS). SRS is an interaction between the optical field and optical phonons that tends to Stokes-shift the LIDAR signal [3,4,14–16]. This effect is undesirable where strict wavelength control is mandatory, such as in spectroscopic systems (e.g., differential absorption LIDAR (DIAL) and resonance fluorescence applications). The result is that the peak transmitter output power is limited by the SRS threshold, limiting available system SNR. Here, a model is presented for first-order SRS management and weakening in pulsed Er-doped fiber amplifiers (fiber with gain) incorporating wavelength-selective induced bending loss [17,18]. The microbending of optical fiber has already been shown to be an effective way to inflict mode-selective loss to induce single mode (SM) operation in a multimode (MM) fiber [19]. Bending loss can also be used to suppress emission near 1064 nm in Nd-doped fibers to allow for laser operation near 930 nm for the water vapor application [20]. The coiling technique has also recently been used to suppress higher order Stokes components in the Raman interaction
[21]. Our modeling results indicate that this coiling technique can also be applied to the management and reduction of the effects of first-order SRS in Er-doped fiber in order to increase overall system power and thus SNR. This model is not meant as an attempt to precisely predict laser performance. Several factors, such as fiber imperfection and finite upper state lifetime, can alter the results presented here. The model is meant to estimate the effectiveness of SRS suppression by fiber coiling in a pulsed amplifier. Our basic approach is to fit analytical expressions to the evolution of a quasi-CW signal in the amplifier. We have chosen a co-propagating signal/pump configuration. Then, assuming that SRS does not deplete the signal, we can determine exact analytical expressions for the propagation of the Stokes wave, from which the SRS threshold can be estimated.
2. Model We begin with the quasi-CW [4] equation governing the evolution of the Stokes signal Ps(z) d P p ðzÞ þ as P s ðzÞ ¼ gR P s ðzÞ p ; ð1Þ dz Aeff where as is the induced (bending) attenuation coefficient at the Stokes wavelength, gR is the Raman gain coefficient, and Aeff is the effective mode area of the pump signal. The p refers to the pump wavelength (15XX nm, different from the amplifier pump wavelength of 976 nm) and s to the Stokes signal (15XX nm + Dk). Here, it is reasonably assumed that the Stokes wavelength falls outside the Er gain bandwidth, since the peak Raman wavelength shift (Dk) in silica fibers is 100 nm at 1550 nm. A non-depleted evolution of the pump is also assumed, which is governed only by the net gain provided by the fiber amplifier. In the case of a deeply saturated amplifier, the simple model of exponential signal growth is not appropriate. We will therefore subsequently present analytical expressions that approximate pump evolution for the cases of single- and dual-clad fiber. This leads to exact expressions for the evolution of the Stokes
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signal that can be used to estimate amplifier performance. Finally, the analytical expression for bendinduced as [1/meters] is provided in [17] for an optical mode as pffiffiffi 4Dw3 p 1 a exp 3aV 2 R as ¼ ð2Þ 1=2 ; 4 Aseff V2 w wR þ a 2Dw
over the fiber length. Then, the effective input power can be found from
where a is the radius of the fiber core, Aseff is the effective mode area of the Stokes wave, R is the fiber bending radius, is the index difference, D is the fiber V-number (normalized frequency), and w is approximated in [22] by the following in a step index fiber within the range 1.5 < V < 2.5
where Dmfwhm is the width of the Raman gain spectrum. With this boundary condition, we determine that the solution to (5) is gR P p ðLÞ P s ðzÞ ¼ hms Beff exp as z 2Apeff gR P p ðLÞ coshðcðz Lfit ÞÞ ln exp : 2Apeff c coshðcLfit Þ
w ¼ 1:1428V 0:9960:
ð3Þ
P s ð0Þ ¼ hms Beff ;
ð7Þ
where Beff is the effective bandwidth (Hz) and is estimated in our case by [3] pffiffiffi p Dmfwhm Beff ¼ ; ð8Þ 2 ðP p ðLÞgR =cp Apeff Þ1=2
ð9Þ 2.1. Single-clad fibers As will be demonstrated, a good model for the pump in a saturated single-clad fiber is the general hyperbolic tangent function P p ðzÞ ¼
P p ðLÞ ð1 þ tanhðcðz Lfit ÞÞÞ; 2
ð4Þ
where c and Lfit are amplifier fitting parameters. In this analysis, Pp(L) is the maximum amplifier output power for a given amplifier pump (976 nm) power, with fiber length L selected such that the output power is maximized. Inserting (4) into (1), we obtain
d P p ðLÞ þ as P s ðzÞ ¼ gR P s ðzÞ p ð1 þ tanhðcðz Lfit ÞÞÞ: dz 2Aeff ð5Þ
The general solution to (5) is given by gR P p ðLÞ P s ðzÞ ¼ C exp as ðz Lfit Þ 2Apeff gR P p ðLÞ lnðcoshðcðz Lfit ÞÞÞ ; ð6Þ exp 2Apeff c where C is a constant. To determine C, we assume a known effective signal input Ps(0). Smith [3] showed that if this input at z = 0 is taken to be one photon per mode, then this input is equivalent to a sum of all spontaneous emission integrated
2.2. Dual-clad fibers A good model for the pump in a saturated dualclad fiber is the following exponential function: P p ðzÞ ¼ Dfit P p ðLÞð1 expðczÞÞ;
ð10Þ
where Pp(L) is the defined as before, and c and Dfit are fitting parameters for amplifier performance. Inserting (10) into (1), we obtain d P p ðLÞ þ as P s ðzÞ ¼ gR P s ðzÞDfit p ð1 expðczÞÞ: dz Aeff ð11Þ The general solution to (11) is given by gR Dfit P p ðLÞ P s ðzÞ ¼ C exp as z Apeff g Dfit P p ðLÞ expðczÞ ; exp R p Aeff c
ð12Þ
where C is again a constant. Invoking the boundary condition Ps(0) again, we achieve the following solution to (11): g Dfit P p ðLÞ P s ðzÞ ¼ hms Beff exp as R p z Aeff g Dfit P p ðLÞ ðexpðczÞ 1Þ : ð13Þ exp R p Aeff c
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3. Raman gain coefficient The Raman gain coefficient gR has an inverse dependence on pump wavelength. Therefore, we can estimate the Raman gain coefficient near 1550 nm after [3,4,14–16]. Beyond about 1650 nm (400 cm1) the gain coefficient decreases and wavelength increases, both decreasing the Raman contribution and increasing the bending loss. Thus, these long wavelengths are neglected (see Fig. 1).
4. Example calculation: typical single-clad fiber We begin with a model for the output of a single mode, single-clad fiber. We will consider results in the range of 0.5–4 W of optical pump power injected into the fiber. In the former case, near 500 mW, fiber coupled semiconductor edge-emitters [23,24] can be applied. In the latter, for higher-power operation, Yb-doped fiber lasers near 980 nm [25,26] can be employed. Second, we arbitrarily define a ÔtypicalÕ system that operates at 10 kHz with 50 ns pulse width, corresponding to a 15 km atmosphere with 7.5 m range resolution. This provides a duty cycle of 1/2000. We assume a ÔtypicalÕ erbium-doped single mode single-clad fiber with a core diameter of 4.4 lm, NA of 0.16, and doping concentration of 1000 ppm wt. The cut-off wavelength of this fiber is then 920 nm. We assume that the amplifier operates near 1550 nm so that the signal absorp-
tion coefficient is about 1.25 · 1025 m2 and the emission cross-section is about 2.5 · 1025 m2 [27]. We also assume that the lifetime of the erbium 4I13/2 upper state is 10 ms. With the assumption that the pump absorption cross-section is about 1 · 1025 m2, we invoke the amplifier models found in [28] for 1 mW of average injected signal power, and arrive at the results for the average signal power evolution as a function of length shown in Fig. 2. Pump power is shown to the right. Fig. 2 also shows the results of the best fit for the single-clad fiber amplifier models. The fit parameters are Lfit = 8.8 m and c = 0.26. We see that the fit is very good further along in the fiber, but breaks down in the first few meters of the fiber. However, since the power levels in the first few meters of fiber are well below the SRS threshold, this should not significantly affect the calculations. Furthermore, the fit is also very good in the whole range 500 mW–4 W of pump power, but degrades below 500 mW. We assume a fiber length of 20 m. as · L for this 20-m fiber is plotted in Fig. 3 for several fiber bend radii. To estimate the SRS threshold, we follow Smith [3] and assume that the critical output power occurs when Ps(L) = Pp(L). Using Eq. (2) and solving Eq. (9) for Pcrit = Ps(L) = Pp(L), we arrive at the results shown in Fig. 4 for the SRS threshold as a function of wavelength for the same fiber bend radii in Fig. 3. We have also assumed that Dmfwhm = 8 THz. Since the peak Raman gain in this case is around 1650 nm, the lowest critical power is at this wavelength.
Amp Model Pump = 4W
Fit Pump = 3W
Pump = 2W
Pump = 1W
Fig. 1. 1550 nm-Pumped Raman gain spectrum after [3,4,14– 16].
Fig. 2. Modeling results for a typical single-clad fiber along with the tanh fit.
P.D. Dragic / Optics Communications 250 (2005) 403–410
0.020m
0.022m
0.025m 5m
Fig. 3. as · L vs. wavelength for the single-clad fiber for several bending radii. Bend radii are provided in the plot.
0.020m 0.022m 0.025m 5m
Fig. 4. Critical peak output power (kW) vs. wavelength for various fiber bending radii.
A drawback to bending is that background attenuation at the signal wavelength increases (bending loss), resulting in the need for a greater amount of pump power to achieve the output powers shown in Fig. 2. The excess bending loss (ap · L) at 1550 nm for bend radii 5, 0.025, 0.022, and 0.020 m are 0, 0.099, 0.372, and 1.02, respectively. However, we see from Fig. 4 that through coiling such a fiber, the SRS threshold can be increased from about 1.35 to 2.7 kW (1.35 W average, 135 lJ/pulse), or about a factor of 2. Even with the increased signal loss, a twofold increase in the SRS threshold can be very desirable for increased system power.
407
1000 ppm wt Yb and Er doping, respectively. This ratio (20:1) was found to maximize the energy transfer efficiency between the Yb and Er atoms [29]. Additionally, we assume a core diameter of 12 lm, cladding diameter of 400 lm, core NA of 0.06, and a pumping wavelength of 915 nm. We choose a pumping wavelength off of the peak absorption coefficient so that a longer fiber can be used such that more Erbium atoms are available for greater pulse energies. We assume an Yb absorption coefficient of about 7.5 · 1025 m2 [30] and an energy transfer efficiency of 20% [29,31]. Finally, we assume a 100 ns pulse width with 5 kHz repetition rate (30 km atmosphere and 15 m range resolution), giving a duty cycle of 1/2000. Fig. 5 shows the amplifier modeling results for 10 mW of average input signal power along with the fit. The fit parameters are Dfit = 1.04 and c = 0.120. We see that in the range of pumping powers of interest (40–100 W), the numerical fit is very good over the whole fiber length. ap · L is plotted in Fig. 6 for a 30-m length of this fiber. Again, there is an increased background loss at the signal wavelength and ap · L is found to be 0, 0.229, 0.53, and 0.931 for bending radii of 5, 0.34, 0.31, and 0.29 m, respectively, at 1550 nm. Using the same definition for the critical SRS output power, we arrive at the results shown in Fig. 7. For the two extreme cases in Fig. 5, R = 5 and 0.29 m, the available peak output power only increases from about 5.3–8.0 kW (4.0 W average power, 0.8 mJ/pulse), or an increase of about 50%. This is meager at best, and probably does not justify the excess signal loss due to bending.
Amp Model
Pump = 100W
Fit Pump = 80W Pump = 60W Pump = 40W
5. Example calculation: typical dual-clad fiber For the dual-clad fiber example, we assume an Yb–Er co-doped fiber. We assume 20,000 and
Fig. 5. Modeling results for a typical dual-clad fiber along with the fit.
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P.D. Dragic / Optics Communications 250 (2005) 403–410
Index of Refraction
Yb/Er Doping Profile
0.29m 0.31m
0.34m
Fig. 6. as · L vs. wavelength for the dual-clad fiber for several bending radii.
0.29m 0.31m 0.34m 5m
Fig. 7. Critical peak output power (kW) vs. wavelength for various fiber bending radii.
6. Fiber optimization In order for the bending loss technique to be useful enough to suppress SRS, the loss at the Stokes wavelength should be maximized while minimizing the loss at the desired signal wavelength. It turns out that the loss contrast between these two wavelengths increases as the fiber V-number decreases. As a result, we may design a fiber geometry that can optimally take advantage of this loss, such as the proposed cross-sectional geometry shown in Fig. 8, where both the core radius and NA are kept small. In the figure, R1 is the core radius, R2 is the doping radius, and R3 is the inner cladding radius. With the addition of a separate doping layer, both the overlap with the pump in the inner cladding and pulse energy can be increased while retaining a small core diameter for a small fiber V-number. However, proper dopants should be used to match the index of the doped layer to that of the inner cladding.
R3
r
Fig. 8. Yb/Er doping and index of refraction profiles for the suppression of SRS. R1 is the core radius, R2 is the doping radius, and R3 is the inner cladding radius.
Fig. 9 shows the plot of the normalized fundamental mode (intensity) for two optical fibers, both with NAs of 0.06, but with different core diameters; 7.5 and 12 lm. The plot is along the radial direction from the center of the fiber. Both fibers have NAs of 0.06. We see that these modes have, for our purposes here, very similar modefield diameters. From inspection of Fig. 9 and noting the wide tails associated with the 7.5-lm fiber, we choose R2 = 10 lm, and retain the same doping concentrations as in Part V. R3 is chosen to be 200 lm, as in Part 5. The V-number of the 7.5-lm fiber is 0.91 at 1550 nm. This falls outside the bounds of the expression for w in Eq. (3). In the case of small V, Marcuse [32] provides a better approximation for w. This is given by 1 J 0 ðV Þ w ¼ 1:122 exp ; ð14Þ V J 1 ðV Þ where the JÕs represent the Bessel functions. Using this expression, we can plot as · L for the fiber of Fig. 8. The results are shown in Fig. 10 for 20-m-long fibers with (1) 7.5-, (2) 8.0-, and (3) 1 Relative Power
5m
R1 R2
0.8 0.6 0.4 7.5 micron
0.2 12 micron
0
0
5
10 15 Distance (microns)
20
Fig. 9. Plots of the fundamental optical modes for fibers with 7.5- and 12-lm cores.
P.D. Dragic / Optics Communications 250 (2005) 403–410
1
2
3
Fig. 10. as · L vs. wavelength for (1) 7.5-lm diameter, 8.85 m bend radius; (2) 8-lm core, 4.45 m bend radius; (3) 10-lm core, 0.87 m bend radius.
10.0-lm core diameters and 20-lm doping layers. The bending radii were chosen such that as · L = 1 at 1550 nm. These radii are (1) 8.85 m, (2) 4.45 m, and (3) 0.87 m, respectively. Invoking the amplifier models for a 40–100 W pumping range and retaining all other remaining parameters from Part 5 (doping concentrations, cross-sections, energy transfer efficiency, upper state lifetime, input signal power, repetition rate and pulse width), the amplifier fit parameters were found to be Dfit = 1.02 and c = 0.32. The core size in the range of 7.5–10 lm had little influence on the fit parameters. Thus, the same fit parameters were used for all three cases. With these results, the critical output powers vs. wavelength for these cases are shown in Fig. 11. The critical peak output power of a 7.5-lm-core fiber with as = 0 is 6.6 kW (660 lJ/pulse, case 4).
1
2
3
409
Thus, if large bending radii can be tolerated, the critical output power can be increased to nearly 30 kW (15 W average power, 3 mJ/pulse). However, it is interesting to note that SRS will first occur in these fibers at wavelengths shorter than 1650 nm, such as near 1628 nm for case 1. Bending radii on the order of several meters may not seem practical. However, in a setting where the laser is in a lidar facility, it would be possible to coil the fiber loosely around a room, such as Ôthe telescope roomÕ. Furthermore, if the doped dual-clad fiber is used as the delivery fiber, or if a similar fiber is produced without the rare earth for delivery purposes, the fiber can be carefully strung as it leads to the telescope in a monostatic system. Alternatively, if the fiber is used for delivery in straight-line segments, these large bending radii could possibly be achieved if the fiber were contained in something similar to a stretched-out coiled phone cord with a larger radius, whereby a helical path is generated.
7. Conclusion We have presented a model for the management and suppression of SRS in pulsed Er-doped fiber amplifiers (fiber with gain) with wavelength-selective induced bending loss. Our modeling results indicate that the coiling technique can be applied to the management and substantial reduction of the effects of first-order SRS in Er-doped fiber based systems, with the right waveguide design. The optimal fiber for the suppression of SRS has a low V-number (<1.0) and requires a large minimum bending radius. However, if this bending radius is tolerable in a system, substantial power increases can be realized before the onset of SRS. Furthermore, by coupling bending with fiber grating technologies, the SRS limitation of fiber laser technologies can be greatly diminished.
4
References Fig. 11. Critical peak output power (kW) vs. wavelength for cases 1, 2, and 3. Case 4 is the example of a fiber with no bending loss.
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