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Multiwavelength fiber laser with an intracavity polarizer
Optics Communications 253 (2005) 352–361 www.elsevier.com/locate/optcom
Multiwavelength fiber laser with an intracavity polarizer Thierry Chartier a
a,*
, Adrian Mihaescu a, Gilles Martel b, Ammar Hideur b, Franc¸ois Sanchez c
Laboratoire d’Optronique, UMR CNRS 6082 FOTON, ENSSAT, 6 rue de Ke´rampont, 22305 Lannion Cedex, France b Groupe d’Optique et d’Optronique, CORIA UMR CNRS 6614, Universite´ de Rouen, Avenue de lÕuniversite´, 76801 Saint Etienne du Rouvray Cedex, France c Laboratoire POMA UMR CNRS 6136, Universite´ d’Angers, 2 Bd. Lavoisier, 49045 Angers Cedex 01, France Received 22 September 2004; received in revised form 21 April 2005; accepted 27 April 2005
Abstract We present the experiment that allows multiwavelength operation and tunability of a neodymium-doped fiber laser with an intracavity polarizer. Then, we present the experimental method to extract the Jones matrix of the fiber as a function of the wavelength. Hence, using the round-trip operator method, we compare both experimental and theoretical wavelength selection and find a excellent agreement. 2005 Elsevier B.V. All rights reserved. PACS: 42.55.Wd; 42.81.Gs Keywords: Fiber lasers; Birefringence; Polarization
1. Introduction Multiwavelength fiber lasers have extensively been studied in recent years [1–10]. They are of great interest for various applications such as wavelength-division-multiplexed communication systems, fiber sensors or instrument testing. Many approaches have been proposed to obtain *
Corresponding author. Tel.: +33 296469144; fax: +33 296370199. E-mail address: [email protected] (T. Chartier).
multiwavelength operation from rare-earth-doped fiber lasers. Most of these methods include intracavity components such as Fabry–Perot etalon [1], Mach–Zhender fiber interferometer [2], fiber Bragg gratings [3], fiber loop mirrors [4], acousto-optic frequency-shifter [5] or intracavity polarizer [6–10]. The latter method is very attractive because of its simplicity. It consists to insert a polarizer in the laser cavity containing the birefringent gain fiber. Because of the wavelength dependence of the phase shift induced by fiber birefringence, wavelength selection occurs, leading
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.04.077
T. Chartier et al. / Optics Communications 253 (2005) 352–361
to multiwavelength operation. For example, 24-line multiwavelength operation over 17-nm spectral range has been demonstrated with an erbium-doped fiber laser using this technique [6]. Assuming that the fiber of length L can be modeled by a linear retardation plate of birefringence B, the spacing Dk between each line can simply be calculated from [7–9,12] Dk ¼
k20 ; 2BL
353
ous work [11], this method does not need the use of the PSP and is applied to a multiwavelength fiber laser. In Section 2, we present the experiment on the tunability of a multiwavelength neodymiumdoped fiber laser with an intracavity polarizer. In Section 3, the experimental method to extract the Jones matrix of the fiber is presented. Section 4 is devoted to the theoretical prediction of the tunability and comparison with experiment.
ð1Þ
where k0 is the central wavelength of the spectrum. The advantage of this method is also to provide tunability of the spectral lines by rotation of the polarizer or some intracavity polarization controllers [7–11]. However, tunability of the laser lines is more difficult to model. For example, the previous theoretical approach which consists to assume that the laser cavity contains only both a linear retardation plate and a polarizer fails to describe the tunability when the polarizer rotates [11,12]. The complicated nature of fiber birefringence imposes a more realistic model. In [11], Friedman et al. use the principal states of polarization (PSP) to describe the tunability of a single-wavelength erbium-doped fiber laser with an intracavity polarizer. The PSP, initially introduced to characterize polarization mode dipersion in fiberbased transmission systems [13], are extracted using Jones matrix eigenanalysis [14]. The aim of our paper is to present a novel approach to describe the tunability of a multiwavelength fiber laser with an intracavity polarizer. The principle is to use an experimental method to extract the Jones matrix of the birefringent gain fiber as a function of the wavelength and to apply the round-trip operator of the cold cavity to explain wavelength selection. In contrast with previ-
2. Experiment on the laser 2.1. Description of the laser The set-up of the Nd-doped fiber laser with an intracavity polarizer is shown in Fig. 1. We use a 500-ppm-Nd-doped fiber as the gain medium. The Ge-doped-silica core has a diameter of 2.7 lm, the numerical aperture is 0.24 leading to a cutoff wavelength of 850 nm. The fiber is 20-m-long and is wound on a drum. The intrinsic birefringence B of the fiber has been evaluated using the magneto-optic method and is of the order of 105 [16]. The laser diode operates at 810 nm and is focused in the fiber through the microscope objective O1. Its maximum output power is 150 mW. The microscope objective O2 collimates the output beam. The cavity is composed of two mirrors M1 and M2. Mirror M1 has high reflectivity around 1080 nm and high transmission at 810 nm. Mirror M2 is the output coupler with a reflectivity of 80% around 1080 nm. The bulk rotatable polarizer has an extinction ratio greater than 30 dB around 1080 nm. The orientation of its transmission axis with respect to the laboratory frame of reference is h. The output signal is detected with an optical spectrum analyzer (resolution of 0.01
Nd-doped fiber
y
polarizer (θ)
θ
Laser diode O1 M1
O2 M2
Fig. 1. Schematic representation of the laser.
x
z
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T. Chartier et al. / Optics Communications 253 (2005) 352–361
nm). The threshold of the laser is about 10 mW of pump power. In the following, measurements will be made at 40 mW of pump power for which the laser delivers 0.5 mW. 2.2. Results In the absence of polarizer in the cavity, the laser operates along two linear and orthogonal polarizations [15]. The output spectrum of the laser, is given in Fig. 2(a). The spectral width is about 12 nm (3000 GHz) around 1086 nm. Since the free spectral range of the cavity is about 5 MHz, the spectrum contains several hundred of thousands longitudinal modes of the cavity. No wavelength selection occurs. When the polarizer is present in the cavity and oriented along h, the output signal is polarized
(a)
No polarizer
(b)
θ = 0˚
(c)
θ = 50˚
(e)
1078
2.3. Discussion
θ = 35˚
(d)
θ = 70˚
1080
1082
1084
1086
1088
1090
Wavelength (nm)
Fig. 2. Spectra versus h.
1092
along h and its spectrum reduces only into peaks. Each peak of this multiwavelength-spectrum can be tuned by rotating the polarizer. Figs. 2(b)–(e) give examples of spectra for different values of the polarizer angle h. In order to map the tunability of the spectrum, we recorded spectra for h varying from 0 to 180 by step of 5. For each spectrum, we measured the central wavelength of each peak. Fig. 3 reports the evolution of peak-wavelengths as a function of h. We note the periodicity of 90. We also note that for h around the particular values of 0 and 90, the periodicity in the spectrum is 3 nm, in good agreement with Eq. (1), while for h around 45, the periodicity is 1.5 nm. Moreover, Fig. 3 shows this unexpected phenomenon of splitting of the peak wavelengths around 20 and 70. Note that near threshold, the spectrum is narrower and does not exhibit a sufficient number of peaks to relate accurately this periodical phenomenon. Far above threshold the number of peaks does not increase significantly to justify a pumping at maximum pump power.
1094
As previously mentioned in the introduction, wavelength selection in a cavity containing both a birefringent media and a polarizer is well understood [7–9,12]. Let us recall the basic principle. In a laser cavity with no polarization-dependent loss (no polarizer for example), the output polarization state is fixed by the resonance condition stipulating that a polarization state must remain unchanged after one round-trip in the cavity. If the cavity contains only one linear birefringent element, this condition is satisfied for two polarization states aligned with the eigenaxes of the birefringent element, whatever the wavelength. If a polarizer is now inserted in this cavity, one single polarization state is fixed by the polarizer. Laser operation occurs for polarizations that remain aligned with the polarizer axis after one round-trip. A polarization state that enters the birefringent media with a certain angle with respect to its eigenaxes experiences a wavelength-dependent round-trip phase shift equal to D/ = 4pBL/k0. To satisfy the resonance condition, the lasing polarizations are the ones
T. Chartier et al. / Optics Communications 253 (2005) 352–361
355
1094
Peak wavelength (nm)
1092 1090 1088 1086 1084 1082 1080 0
30
60
90
120
150
180
Angle θ (deg.) Fig. 3. Evolution of the peak-wavelengths versus h.
for which the phase shift is a multiple of 2p. This explains wavelength selection in the spectrum of a fiber laser and spacing Dk given by Eq. (1). However, this simple interpretation fails to explain the three following experimental observations. First, if the polarizer is aligned with the birefringence axis, no wavelength-selection should occur. This point is not verified by the experiment. Second, the spacing Dk should be independent of the angle h. In the experiment Dk varies from 1.5 to 3 nm. Third, no tuning of the lasing wavelengths should be observed when the polarizer is rotated but rather a change in the loss of the lasing polarizations [11,12]. However, tuning (and splitting) has been observed in the experiment. The reason of this mismatching between theory and experiment is due to the complicated nature of fiber birefringence that cannot be described as a simple linear retardation plate. We propose in the following a more complete description of the fiber birefringence that allows to predict the experimentally observed laser spectrum.
3. Experiment on the fiber 3.1. Model for the fiber In [16], we showed that a fiber of length L can be described by the Jones matrix M of an elliptical
birefringent plate, i.e. a birefringent medium whose eigenpolarizations are elliptical. If M is written in an appropriate basis of linear polarizations (X, Y), M depends only on two parameters a and b a b M¼ ð2Þ b a with L b L þ i sin ; 2 2 a L b ¼ sin ; 2 a ¼ cos
ð3Þ ð4Þ
and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a2 þ b 2 .
ð5Þ
Parameters a and b are related, respectively, to the circular and linear birefringence of the fiber, corresponds to the total (elliptical) birefringence. All of these parameters are expressed in radians/meter. The basis (X, Y) has a particular signification: polarizations X and Y coincide with the azimuth of the elliptical eigenpolarizations of the fiber. Note that if (X, Y) makes an angle /0 with respect to the laboratory basis (x, y), matrix of the fiber writes, in the laboratory basis ML ¼ Rð/0 ÞMRð/0 Þ;
ð6Þ
where R(/) is the rotation matrix of an angle /
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T. Chartier et al. / Optics Communications 253 (2005) 352–361
Rð/Þ ¼
cos /
sin /
sin /
cos /
.
ð7Þ
In summary, in the laboratory basis, three parameters define the Jones matrix ML of the fiber of length L: a, b and /0. 3.2. Extracting parameters In [16], we demonstrated the propriety that, for any elliptical birefringent, there always exists an input linear polarization (oriented along /in) that exits the fiber linearly polarized (along /out). The angle /0 is the bisector of the angle between /in and /out and is given by /0 ¼
/in þ /out . 2
ð8Þ
A method to extract a and b knowing /0 has been proposed in [16]. In this paper a new method is presented. It simply consists to launch in the fiber a linear polarization oriented at 45 with respect to /0 and to analyse the Stokes parameters of the corresponding output state of polarization. In the fiber basis (X, Y), this input state, oriented at 45 of the basis axes, writes 1 1 Ein ¼ pffiffiffi . ð9Þ 2 1 The corresponding output state is calculated as follows: Eout ¼ MEin ;
ð10Þ
where M is given by Eqs. (2)–(5). Using Appendix A to calculate the Stokes parameters of Eout, we find:
S 0 ¼ 1; a S 1 ¼ sin L; S 2 ¼ cos L; b S 3 ¼ sin L.
ð11Þ ð12Þ ð13Þ ð14Þ
We see that simple relations link Stokes parameters to fiber parameters. From Eqs. (12)–(14), we easily find: 1 ¼ ½arccos S 2 2kp; L S1 a ¼ 2 ; S 1 þ S 23 S3 b ¼ 2 . S 1 þ S 23
ð15Þ ð16Þ ð17Þ
3.3. Experimental set-up We present in Fig. 4 the experimental set-up to measure the Jones matrix of the fiber as a function of the wavelength in the range 1080– 1094 nm. The tunable laser source is a linearly polarized ytterbium-doped double-clad fiber laser similar to that described in [17] but delivering here only few mW. The quarter-wave plate at 1080 nm transforms the linear polarization into a circular one. The polarizer allows to inject the appropriate linear polarization in the fiber. This association of a quarter-wave plate and a polarizer is convenient to control the input state of polarization since it is wavelength independent. The output state of polarization is analyzed with a polarization analyzer (Thorlabs polarimeter). We performed measurements in
Fig. 4. Experimental set-up.
T. Chartier et al. / Optics Communications 253 (2005) 352–361 100
φin (deg.)
50
φout (deg.)
(a)
0 -50 -100
70
(b)
60 50 40 30 20 60
φ0 (deg.)
the range 1080–1094 nm, with a step of 0.25 nm. The procedure is the following. First, by rotating the polarizer, we find the input linear polarization that exits the fiber linearly polarized. This gives both angles /in and /out and we calculate /0 according to (8). Then, we inject in the fiber the linear polarization oriented at 45 with respect to /0 and measure the Stokes parameters s1, s2 and s3 of the output polarization with the polarimeter in the laboratory basis (x, y). A simple transformation of s1, s2 and s3 will give S1, S2 and S3 in the fiber basis. In order to use these results to explain experimental tunability of the fiber laser, the very important point here was not to move the fiber between both experiments on the laser and on the fiber.
357
(c)
40 20 0 -20 -40 0.5
(d)
3.4. Results S1
0.0
-0.5
0.5
(e)
S2
0.0 -0.5 -1.0 1.0
(f)
0.5
S3
Fig. 5 represent evolution of the six parameters /in, /out, /0, S1, S2 and S3 as a function of the wavelength. First of all, let us point out the periodicity of 3 nm for each parameter. Secondly, it seems that each parameter obeys a simple law. Dots represent experimental data while lines are fitting functions. Fig. 5(a) represents the measured values of /in together with a fit by linear functions of slope equal to 60/nm. Fig. 5(b) gives the measured values of /out and the corresponding fit by the sine curve of equation: /out = 45 + 22.5 sin (2pk/3 + p/3). Wavelength k is expressed in nm. Fig. 5(c) is deduced from data of both Figs. 5(a) and (b) according to Eq. (8). Figs. 5(d)–(f) represent the normalized Stokes parameters S1, S2 and S3 of the output polarization state in the basis (X, Y) of the fiber. We used Appendix B with the experimental values of /0 to change the basis of the Stokes parameters. In Fig. 5(d), S1 is fitted by S1 = 0.3 + 0.6 sin (2pk/ 3 + 5p/6) while in Fig. 5(e) S2 is fitted by S2 = 0.4 + 0.6 sin (2pk/3 + p/6). Fig. 5(f) represents S3 with its fitting func-tion deduced from S1 and S2 according to: S 20 ¼ S 21 þ S 22 þ S 23 with S0 = 1. Using experimental data of Fig. 5, we can extract parameters a, b and according to Eqs. (15)–(17). Corresponding fitting function for each parameter can also be calculated from Eqs. (15)–(17) using the fitting functions given
0.0 -0.5 -1.0 1080
1082
1084
1086
1088
1090
1092
1094
Wavelength (nm)
Fig. 5. Evolution of measured parameters versus wavelength.
above. Fig. 6 represents the extracted values of aL, bL and L. In Eqs. (15)–(17), the signs + or have been chosen in order to avoid discontinuities in the evolution of parameters. We set the value of k = 0, this leads to oscillations of L around p. In reality L, which corresponds to the total phase shift, is greater. For example, with a birefringence of 105 and a fiber length of L = 20, L should be of the order of 185 · 2p rad. However, the value of k has no real importance for our study since appears in Eqs. (2)–(5) as
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T. Chartier et al. / Optics Communications 253 (2005) 352–361 M1
αL, βL, εL (rad.)
3π/2
MLT(λ)
M2
PT(θ)
y
π π/2
A
θ
z
0 −π/2
x
αL βL εL
−π
ML (λ)
−3π/2 1080
P(θ)
Fig. 7. Schematic representation of the cavity. 1082
1084
1086
1088
1090
1092
1094
Wavelength (nm)
Fig. 6. Extracted values of aL, bL and L and their corresponding fitting function.
the argument of trigonometric functions or through ratios a/ or b/ which are independent of k (see Eqs. (16) and (17)). Figs. 6 and 5(c) show that both ellipticity and azimuth of fiber birefringence evolves versus wavelength. The fiber birefringence is generally elliptical but for some particular wavelengths (1080, 1083, 1086 nm, etc.) the fiber birefringence is linear (a = 0) and the value of the phase shift L is such that the fiber acts like a quarter-wave plate (L = p/2 or 3p/2). At this stage no explanation is given to describe wavelength evolution of the fiber birefringence. This point should be the subject of further work. 4. Theory 4.1. The round-trip operator The round-trip operator method describes, in term of Jones matrices, the resonance condition stipulating that a state of polarization must remains unchanged after one round-trip in the laser cavity. This method has already been successfully used to describe polarization properties of lowpower CW fiber lasers [18,19]. The laser cavity of Fig. 1 can be resumed to the schematic representation of Fig. 7 if we are only interested in polarization effects. We have assumed that only two anisotropic elements are present in the cavity: the fiber and the polarizer. Let ML
and P(h) be, respectively, the Jones matrices of the fiber and the polarizer in the laboratory basis (x, y) for light propagating from left to right (forward direction). Matrix ML is given by Eq. (6) and P(h) by cos2 h sin h cos h PðhÞ ¼ . ð18Þ sin h cos h cos2 h The Jones matrices of these components when light propagates in the backward direction are the transpose matrices MTL and PT ðhÞ [18]. We immediately note that PT(h) = P(h). The roundtrip operator C at point A is the Jones matrix experienced by a polarization starting from point A and propagating through the cavity C ¼ PðhÞML MTL PðhÞ.
ð19Þ
Using Eq. (6) for ML, we find C ¼ PðhÞRð/0 ÞMMT Rð/0 ÞPðhÞ;
ð20Þ
where M is the Jones matrix of the fiber in its own basis in the forward direction and is given by Eqs. (2)–(5). Eigenvectors and eigenvalues of C represent, respectively, the eigenpolarizations of the laser at point A and the transmission of the eigenvectors through the cavity. Using experimental results of Section 3, we can, at this stage, calculate C for each angle h and each wavelength in the range 1080–1094 nm. Moreover, using the fitting functions found for a, b, and /0 we can simulate a continuous evolution of C versus h and k.
T. Chartier et al. / Optics Communications 253 (2005) 352–361
4.2. Simulations It is obvious that one eigenvalue of C is always null, corresponding to the eigenvector oriented at 90 of the transmission axis of the polarizer. Let c be the non-zero eigenvalue corresponding to the polarization oriented along the polarizer axis. Modulus |c| corresponds to the transmission of this polarization after one round-trip through the cavity. Fig. 8 represents the wavelengthdependence of |c| for three particular values of h. We note a strong modulation of |c|. Longitudinal modes for which the transmission coefficient |c| is below a certain value cannot satisfy the oscillation condition: gain = loss. Consequently, the lasing wavelengths observed in the experimental spectra correspond to the maximums of |c|. Note that the round-trip operator analysis without the intra-cavity polarizer does not give any wavelength selection but a constant value equal to unity for both eigenvalues of C. We have verified that wavelength-selection always occurs for any values of h. We have also verified that, for a simply linear birefringence model for the fiber, no tuning occurs. Tunability of the spectrum is also visible in Fig. 8. In order to compare both experimental
1.0
0.8
1092 1090 1088 1086 1084 1082 1080 0
30
60
90
120
150
180
Angle θ (deg.)
Fig. 9. Theoretical (lines) and experimental (crosses) evolution of the laser spectrum peak-wavelengths versus h.
and theoretical tunability, we performed simulation of |c| from h=0 to 180 by step of 1. Then we reported wavelengths corresponding to the maximums of |c| as a function of h. Results are shown in Fig. 9 together with the experimental data of Fig. 3. We first note the good adequacy between theoretical and experimental data. In particular, splitting of the peak wavelengths is clearly visible. However, some discrepancies occur. In particular, a red-shift of theoretical prediction and the presence of additional wavelength around h = 0, h = 90 and h = 180. The first one may be due to a drift of fiber parameters between both experiment on the laser and on the fiber, due for example to temperature changes. The second one may be attributed to the mismatch between experimental data of Fig. 5 and corresponding fitting functions. However, to the authors knowledge, Fig. 9 represents the first direct comparison between experimental and theoretical tunability of a multiwavelength fiber laser with an intracavity polarizer.
|γ |
0.6
1094
Peak wavelength (nm)
Note that the round-trip operator method is not valid for high-power lasers or ultrashort pulsed lasers because of the intensity dependance of birefringence.
359
0.4
0.0 1080
5. Conclusion
20˚ 40˚ 60˚
0.2
1082
1084
1086
1088
1090
1092
1094
Wavelength (nm)
Fig. 8. Wavelength-dependence of |c| for three particular values of h.
We have presented multiwavelength operation of a Nd-doped fiber laser with an intracavity polarizer. We have shown tunability and splitting of the lasing peaks. In order to explain the experimental observations, we have presented a novel
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T. Chartier et al. / Optics Communications 253 (2005) 352–361
approach based on the Jones matrix analysis of fiber birefringence and on the derivation of the round-trip operator of the cavity. Excellent agreement between theory and experiment validates the principle of the method. Let us point out that this approach is not limited by the spectral range of the laser and can be applied to other types of multiwavelength fiber lasers based on polarization selection. Possible extension of this work concerns the comprehension of the wavelength evolution of fiber birefringence.
Acknowledgements The authors thank Pr. Pierre Pellat-Finet (LAUBS, Lorient, France) for the idea to inject a polarization oriented at 45, Dr. Se´bastien Coe¨tmellec (CORIA, Saint-Etienne du Rouvray, France) for helpfull discussions and Pr. Pascal Besnard (UMR FOTON, Lannion, France) for his comments on the manuscript.
Appendix A. Relations between Jones vectors, Stokes parameters and ellipticity and azimuth In the most general case, the Jones vector V representing an elliptical state of polarization S, has two complex coordinates and can be written as ! Ax expðiux Þ ax þ ibx V¼ . ðA:1Þ ¼ Ay expðiuy Þ ay þ iby The Stokes parameters of S write: s0 ¼ A2x þ A2y ¼ a2x þ b2x þ a2y þ b2y ; s1 ¼
A2x
A2y
¼
a2x
þ
b2x
a2y
b2y ;
ðA:2Þ ðA:3Þ
s2 ¼ 2Ax Ay cosðuy ux Þ ¼ 2ðax ay þ bx by Þ;
ðA:4Þ
s3 ¼ 2Ax Ay sinðuy ux Þ ¼ 2ðax by ay bx Þ.
ðA:5Þ
The ellipticity v and the azimuth / of S can be expressed as following: s3 sin 2v ¼ ; ðA:6Þ s0 s2 ðA:7Þ tan 2/ ¼ . s1
Appendix B. Change of basis for the Stokes parameters Let s0, s1, s2 and s3 be the Stokes parameters of a polarization state S in a basis (x, y) of linear polarizations. As a function of ellipticity v and azimut / of S, they write: s0 ¼ 1; s1 ¼ s0 cos 2v cos 2/;
ðB:1Þ ðB:2Þ
s2 ¼ s0 cos 2v sin 2/;
ðB:3Þ
s3 ¼ s0 sin 2v.
ðB:4Þ
Let us now consider a new basis (X, Y) of linear polarizations that make an angle /0 with respect to the basis (x, y). In this new basis the Stokes parameters of S write: S 0 ¼ 1;
ðB:5Þ
S 1 ¼ S 0 cos 2v cos 2ð/ /0 Þ ¼ s1 cos 2/0 þ s2 sin 2/0 ;
ðB:6Þ
S 2 ¼ S 0 cos 2v sin 2ð/ /0 Þ ¼ s2 cos 2/0 s1 sin 2/0 ;
ðB:7Þ
S 3 ¼ S 0 sin 2v ¼ s3 .
ðB:8Þ
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Multiwavelength fiber laser with an intracavity polarizer Thierry Chartier a
a,*
, Adrian Mihaescu a, Gilles Martel b, Ammar Hideur b, Franc¸ois Sanchez c
Laboratoire d’Optronique, UMR CNRS 6082 FOTON, ENSSAT, 6 rue de Ke´rampont, 22305 Lannion Cedex, France b Groupe d’Optique et d’Optronique, CORIA UMR CNRS 6614, Universite´ de Rouen, Avenue de lÕuniversite´, 76801 Saint Etienne du Rouvray Cedex, France c Laboratoire POMA UMR CNRS 6136, Universite´ d’Angers, 2 Bd. Lavoisier, 49045 Angers Cedex 01, France Received 22 September 2004; received in revised form 21 April 2005; accepted 27 April 2005
Abstract We present the experiment that allows multiwavelength operation and tunability of a neodymium-doped fiber laser with an intracavity polarizer. Then, we present the experimental method to extract the Jones matrix of the fiber as a function of the wavelength. Hence, using the round-trip operator method, we compare both experimental and theoretical wavelength selection and find a excellent agreement. 2005 Elsevier B.V. All rights reserved. PACS: 42.55.Wd; 42.81.Gs Keywords: Fiber lasers; Birefringence; Polarization
1. Introduction Multiwavelength fiber lasers have extensively been studied in recent years [1–10]. They are of great interest for various applications such as wavelength-division-multiplexed communication systems, fiber sensors or instrument testing. Many approaches have been proposed to obtain *
Corresponding author. Tel.: +33 296469144; fax: +33 296370199. E-mail address: [email protected] (T. Chartier).
multiwavelength operation from rare-earth-doped fiber lasers. Most of these methods include intracavity components such as Fabry–Perot etalon [1], Mach–Zhender fiber interferometer [2], fiber Bragg gratings [3], fiber loop mirrors [4], acousto-optic frequency-shifter [5] or intracavity polarizer [6–10]. The latter method is very attractive because of its simplicity. It consists to insert a polarizer in the laser cavity containing the birefringent gain fiber. Because of the wavelength dependence of the phase shift induced by fiber birefringence, wavelength selection occurs, leading
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.04.077
T. Chartier et al. / Optics Communications 253 (2005) 352–361
to multiwavelength operation. For example, 24-line multiwavelength operation over 17-nm spectral range has been demonstrated with an erbium-doped fiber laser using this technique [6]. Assuming that the fiber of length L can be modeled by a linear retardation plate of birefringence B, the spacing Dk between each line can simply be calculated from [7–9,12] Dk ¼
k20 ; 2BL
353
ous work [11], this method does not need the use of the PSP and is applied to a multiwavelength fiber laser. In Section 2, we present the experiment on the tunability of a multiwavelength neodymiumdoped fiber laser with an intracavity polarizer. In Section 3, the experimental method to extract the Jones matrix of the fiber is presented. Section 4 is devoted to the theoretical prediction of the tunability and comparison with experiment.
ð1Þ
where k0 is the central wavelength of the spectrum. The advantage of this method is also to provide tunability of the spectral lines by rotation of the polarizer or some intracavity polarization controllers [7–11]. However, tunability of the laser lines is more difficult to model. For example, the previous theoretical approach which consists to assume that the laser cavity contains only both a linear retardation plate and a polarizer fails to describe the tunability when the polarizer rotates [11,12]. The complicated nature of fiber birefringence imposes a more realistic model. In [11], Friedman et al. use the principal states of polarization (PSP) to describe the tunability of a single-wavelength erbium-doped fiber laser with an intracavity polarizer. The PSP, initially introduced to characterize polarization mode dipersion in fiberbased transmission systems [13], are extracted using Jones matrix eigenanalysis [14]. The aim of our paper is to present a novel approach to describe the tunability of a multiwavelength fiber laser with an intracavity polarizer. The principle is to use an experimental method to extract the Jones matrix of the birefringent gain fiber as a function of the wavelength and to apply the round-trip operator of the cold cavity to explain wavelength selection. In contrast with previ-
2. Experiment on the laser 2.1. Description of the laser The set-up of the Nd-doped fiber laser with an intracavity polarizer is shown in Fig. 1. We use a 500-ppm-Nd-doped fiber as the gain medium. The Ge-doped-silica core has a diameter of 2.7 lm, the numerical aperture is 0.24 leading to a cutoff wavelength of 850 nm. The fiber is 20-m-long and is wound on a drum. The intrinsic birefringence B of the fiber has been evaluated using the magneto-optic method and is of the order of 105 [16]. The laser diode operates at 810 nm and is focused in the fiber through the microscope objective O1. Its maximum output power is 150 mW. The microscope objective O2 collimates the output beam. The cavity is composed of two mirrors M1 and M2. Mirror M1 has high reflectivity around 1080 nm and high transmission at 810 nm. Mirror M2 is the output coupler with a reflectivity of 80% around 1080 nm. The bulk rotatable polarizer has an extinction ratio greater than 30 dB around 1080 nm. The orientation of its transmission axis with respect to the laboratory frame of reference is h. The output signal is detected with an optical spectrum analyzer (resolution of 0.01
Nd-doped fiber
y
polarizer (θ)
θ
Laser diode O1 M1
O2 M2
Fig. 1. Schematic representation of the laser.
x
z
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T. Chartier et al. / Optics Communications 253 (2005) 352–361
nm). The threshold of the laser is about 10 mW of pump power. In the following, measurements will be made at 40 mW of pump power for which the laser delivers 0.5 mW. 2.2. Results In the absence of polarizer in the cavity, the laser operates along two linear and orthogonal polarizations [15]. The output spectrum of the laser, is given in Fig. 2(a). The spectral width is about 12 nm (3000 GHz) around 1086 nm. Since the free spectral range of the cavity is about 5 MHz, the spectrum contains several hundred of thousands longitudinal modes of the cavity. No wavelength selection occurs. When the polarizer is present in the cavity and oriented along h, the output signal is polarized
(a)
No polarizer
(b)
θ = 0˚
(c)
θ = 50˚
(e)
1078
2.3. Discussion
θ = 35˚
(d)
θ = 70˚
1080
1082
1084
1086
1088
1090
Wavelength (nm)
Fig. 2. Spectra versus h.
1092
along h and its spectrum reduces only into peaks. Each peak of this multiwavelength-spectrum can be tuned by rotating the polarizer. Figs. 2(b)–(e) give examples of spectra for different values of the polarizer angle h. In order to map the tunability of the spectrum, we recorded spectra for h varying from 0 to 180 by step of 5. For each spectrum, we measured the central wavelength of each peak. Fig. 3 reports the evolution of peak-wavelengths as a function of h. We note the periodicity of 90. We also note that for h around the particular values of 0 and 90, the periodicity in the spectrum is 3 nm, in good agreement with Eq. (1), while for h around 45, the periodicity is 1.5 nm. Moreover, Fig. 3 shows this unexpected phenomenon of splitting of the peak wavelengths around 20 and 70. Note that near threshold, the spectrum is narrower and does not exhibit a sufficient number of peaks to relate accurately this periodical phenomenon. Far above threshold the number of peaks does not increase significantly to justify a pumping at maximum pump power.
1094
As previously mentioned in the introduction, wavelength selection in a cavity containing both a birefringent media and a polarizer is well understood [7–9,12]. Let us recall the basic principle. In a laser cavity with no polarization-dependent loss (no polarizer for example), the output polarization state is fixed by the resonance condition stipulating that a polarization state must remain unchanged after one round-trip in the cavity. If the cavity contains only one linear birefringent element, this condition is satisfied for two polarization states aligned with the eigenaxes of the birefringent element, whatever the wavelength. If a polarizer is now inserted in this cavity, one single polarization state is fixed by the polarizer. Laser operation occurs for polarizations that remain aligned with the polarizer axis after one round-trip. A polarization state that enters the birefringent media with a certain angle with respect to its eigenaxes experiences a wavelength-dependent round-trip phase shift equal to D/ = 4pBL/k0. To satisfy the resonance condition, the lasing polarizations are the ones
T. Chartier et al. / Optics Communications 253 (2005) 352–361
355
1094
Peak wavelength (nm)
1092 1090 1088 1086 1084 1082 1080 0
30
60
90
120
150
180
Angle θ (deg.) Fig. 3. Evolution of the peak-wavelengths versus h.
for which the phase shift is a multiple of 2p. This explains wavelength selection in the spectrum of a fiber laser and spacing Dk given by Eq. (1). However, this simple interpretation fails to explain the three following experimental observations. First, if the polarizer is aligned with the birefringence axis, no wavelength-selection should occur. This point is not verified by the experiment. Second, the spacing Dk should be independent of the angle h. In the experiment Dk varies from 1.5 to 3 nm. Third, no tuning of the lasing wavelengths should be observed when the polarizer is rotated but rather a change in the loss of the lasing polarizations [11,12]. However, tuning (and splitting) has been observed in the experiment. The reason of this mismatching between theory and experiment is due to the complicated nature of fiber birefringence that cannot be described as a simple linear retardation plate. We propose in the following a more complete description of the fiber birefringence that allows to predict the experimentally observed laser spectrum.
3. Experiment on the fiber 3.1. Model for the fiber In [16], we showed that a fiber of length L can be described by the Jones matrix M of an elliptical
birefringent plate, i.e. a birefringent medium whose eigenpolarizations are elliptical. If M is written in an appropriate basis of linear polarizations (X, Y), M depends only on two parameters a and b a b M¼ ð2Þ b a with L b L þ i sin ; 2 2 a L b ¼ sin ; 2 a ¼ cos
ð3Þ ð4Þ
and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a2 þ b 2 .
ð5Þ
Parameters a and b are related, respectively, to the circular and linear birefringence of the fiber, corresponds to the total (elliptical) birefringence. All of these parameters are expressed in radians/meter. The basis (X, Y) has a particular signification: polarizations X and Y coincide with the azimuth of the elliptical eigenpolarizations of the fiber. Note that if (X, Y) makes an angle /0 with respect to the laboratory basis (x, y), matrix of the fiber writes, in the laboratory basis ML ¼ Rð/0 ÞMRð/0 Þ;
ð6Þ
where R(/) is the rotation matrix of an angle /
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T. Chartier et al. / Optics Communications 253 (2005) 352–361
Rð/Þ ¼
cos /
sin /
sin /
cos /
.
ð7Þ
In summary, in the laboratory basis, three parameters define the Jones matrix ML of the fiber of length L: a, b and /0. 3.2. Extracting parameters In [16], we demonstrated the propriety that, for any elliptical birefringent, there always exists an input linear polarization (oriented along /in) that exits the fiber linearly polarized (along /out). The angle /0 is the bisector of the angle between /in and /out and is given by /0 ¼
/in þ /out . 2
ð8Þ
A method to extract a and b knowing /0 has been proposed in [16]. In this paper a new method is presented. It simply consists to launch in the fiber a linear polarization oriented at 45 with respect to /0 and to analyse the Stokes parameters of the corresponding output state of polarization. In the fiber basis (X, Y), this input state, oriented at 45 of the basis axes, writes 1 1 Ein ¼ pffiffiffi . ð9Þ 2 1 The corresponding output state is calculated as follows: Eout ¼ MEin ;
ð10Þ
where M is given by Eqs. (2)–(5). Using Appendix A to calculate the Stokes parameters of Eout, we find:
S 0 ¼ 1; a S 1 ¼ sin L; S 2 ¼ cos L; b S 3 ¼ sin L.
ð11Þ ð12Þ ð13Þ ð14Þ
We see that simple relations link Stokes parameters to fiber parameters. From Eqs. (12)–(14), we easily find: 1 ¼ ½arccos S 2 2kp; L S1 a ¼ 2 ; S 1 þ S 23 S3 b ¼ 2 . S 1 þ S 23
ð15Þ ð16Þ ð17Þ
3.3. Experimental set-up We present in Fig. 4 the experimental set-up to measure the Jones matrix of the fiber as a function of the wavelength in the range 1080– 1094 nm. The tunable laser source is a linearly polarized ytterbium-doped double-clad fiber laser similar to that described in [17] but delivering here only few mW. The quarter-wave plate at 1080 nm transforms the linear polarization into a circular one. The polarizer allows to inject the appropriate linear polarization in the fiber. This association of a quarter-wave plate and a polarizer is convenient to control the input state of polarization since it is wavelength independent. The output state of polarization is analyzed with a polarization analyzer (Thorlabs polarimeter). We performed measurements in
Fig. 4. Experimental set-up.
T. Chartier et al. / Optics Communications 253 (2005) 352–361 100
φin (deg.)
50
φout (deg.)
(a)
0 -50 -100
70
(b)
60 50 40 30 20 60
φ0 (deg.)
the range 1080–1094 nm, with a step of 0.25 nm. The procedure is the following. First, by rotating the polarizer, we find the input linear polarization that exits the fiber linearly polarized. This gives both angles /in and /out and we calculate /0 according to (8). Then, we inject in the fiber the linear polarization oriented at 45 with respect to /0 and measure the Stokes parameters s1, s2 and s3 of the output polarization with the polarimeter in the laboratory basis (x, y). A simple transformation of s1, s2 and s3 will give S1, S2 and S3 in the fiber basis. In order to use these results to explain experimental tunability of the fiber laser, the very important point here was not to move the fiber between both experiments on the laser and on the fiber.
357
(c)
40 20 0 -20 -40 0.5
(d)
3.4. Results S1
0.0
-0.5
0.5
(e)
S2
0.0 -0.5 -1.0 1.0
(f)
0.5
S3
Fig. 5 represent evolution of the six parameters /in, /out, /0, S1, S2 and S3 as a function of the wavelength. First of all, let us point out the periodicity of 3 nm for each parameter. Secondly, it seems that each parameter obeys a simple law. Dots represent experimental data while lines are fitting functions. Fig. 5(a) represents the measured values of /in together with a fit by linear functions of slope equal to 60/nm. Fig. 5(b) gives the measured values of /out and the corresponding fit by the sine curve of equation: /out = 45 + 22.5 sin (2pk/3 + p/3). Wavelength k is expressed in nm. Fig. 5(c) is deduced from data of both Figs. 5(a) and (b) according to Eq. (8). Figs. 5(d)–(f) represent the normalized Stokes parameters S1, S2 and S3 of the output polarization state in the basis (X, Y) of the fiber. We used Appendix B with the experimental values of /0 to change the basis of the Stokes parameters. In Fig. 5(d), S1 is fitted by S1 = 0.3 + 0.6 sin (2pk/ 3 + 5p/6) while in Fig. 5(e) S2 is fitted by S2 = 0.4 + 0.6 sin (2pk/3 + p/6). Fig. 5(f) represents S3 with its fitting func-tion deduced from S1 and S2 according to: S 20 ¼ S 21 þ S 22 þ S 23 with S0 = 1. Using experimental data of Fig. 5, we can extract parameters a, b and according to Eqs. (15)–(17). Corresponding fitting function for each parameter can also be calculated from Eqs. (15)–(17) using the fitting functions given
0.0 -0.5 -1.0 1080
1082
1084
1086
1088
1090
1092
1094
Wavelength (nm)
Fig. 5. Evolution of measured parameters versus wavelength.
above. Fig. 6 represents the extracted values of aL, bL and L. In Eqs. (15)–(17), the signs + or have been chosen in order to avoid discontinuities in the evolution of parameters. We set the value of k = 0, this leads to oscillations of L around p. In reality L, which corresponds to the total phase shift, is greater. For example, with a birefringence of 105 and a fiber length of L = 20, L should be of the order of 185 · 2p rad. However, the value of k has no real importance for our study since appears in Eqs. (2)–(5) as
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T. Chartier et al. / Optics Communications 253 (2005) 352–361 M1
αL, βL, εL (rad.)
3π/2
MLT(λ)
M2
PT(θ)
y
π π/2
A
θ
z
0 −π/2
x
αL βL εL
−π
ML (λ)
−3π/2 1080
P(θ)
Fig. 7. Schematic representation of the cavity. 1082
1084
1086
1088
1090
1092
1094
Wavelength (nm)
Fig. 6. Extracted values of aL, bL and L and their corresponding fitting function.
the argument of trigonometric functions or through ratios a/ or b/ which are independent of k (see Eqs. (16) and (17)). Figs. 6 and 5(c) show that both ellipticity and azimuth of fiber birefringence evolves versus wavelength. The fiber birefringence is generally elliptical but for some particular wavelengths (1080, 1083, 1086 nm, etc.) the fiber birefringence is linear (a = 0) and the value of the phase shift L is such that the fiber acts like a quarter-wave plate (L = p/2 or 3p/2). At this stage no explanation is given to describe wavelength evolution of the fiber birefringence. This point should be the subject of further work. 4. Theory 4.1. The round-trip operator The round-trip operator method describes, in term of Jones matrices, the resonance condition stipulating that a state of polarization must remains unchanged after one round-trip in the laser cavity. This method has already been successfully used to describe polarization properties of lowpower CW fiber lasers [18,19]. The laser cavity of Fig. 1 can be resumed to the schematic representation of Fig. 7 if we are only interested in polarization effects. We have assumed that only two anisotropic elements are present in the cavity: the fiber and the polarizer. Let ML
and P(h) be, respectively, the Jones matrices of the fiber and the polarizer in the laboratory basis (x, y) for light propagating from left to right (forward direction). Matrix ML is given by Eq. (6) and P(h) by cos2 h sin h cos h PðhÞ ¼ . ð18Þ sin h cos h cos2 h The Jones matrices of these components when light propagates in the backward direction are the transpose matrices MTL and PT ðhÞ [18]. We immediately note that PT(h) = P(h). The roundtrip operator C at point A is the Jones matrix experienced by a polarization starting from point A and propagating through the cavity C ¼ PðhÞML MTL PðhÞ.
ð19Þ
Using Eq. (6) for ML, we find C ¼ PðhÞRð/0 ÞMMT Rð/0 ÞPðhÞ;
ð20Þ
where M is the Jones matrix of the fiber in its own basis in the forward direction and is given by Eqs. (2)–(5). Eigenvectors and eigenvalues of C represent, respectively, the eigenpolarizations of the laser at point A and the transmission of the eigenvectors through the cavity. Using experimental results of Section 3, we can, at this stage, calculate C for each angle h and each wavelength in the range 1080–1094 nm. Moreover, using the fitting functions found for a, b, and /0 we can simulate a continuous evolution of C versus h and k.
T. Chartier et al. / Optics Communications 253 (2005) 352–361
4.2. Simulations It is obvious that one eigenvalue of C is always null, corresponding to the eigenvector oriented at 90 of the transmission axis of the polarizer. Let c be the non-zero eigenvalue corresponding to the polarization oriented along the polarizer axis. Modulus |c| corresponds to the transmission of this polarization after one round-trip through the cavity. Fig. 8 represents the wavelengthdependence of |c| for three particular values of h. We note a strong modulation of |c|. Longitudinal modes for which the transmission coefficient |c| is below a certain value cannot satisfy the oscillation condition: gain = loss. Consequently, the lasing wavelengths observed in the experimental spectra correspond to the maximums of |c|. Note that the round-trip operator analysis without the intra-cavity polarizer does not give any wavelength selection but a constant value equal to unity for both eigenvalues of C. We have verified that wavelength-selection always occurs for any values of h. We have also verified that, for a simply linear birefringence model for the fiber, no tuning occurs. Tunability of the spectrum is also visible in Fig. 8. In order to compare both experimental
1.0
0.8
1092 1090 1088 1086 1084 1082 1080 0
30
60
90
120
150
180
Angle θ (deg.)
Fig. 9. Theoretical (lines) and experimental (crosses) evolution of the laser spectrum peak-wavelengths versus h.
and theoretical tunability, we performed simulation of |c| from h=0 to 180 by step of 1. Then we reported wavelengths corresponding to the maximums of |c| as a function of h. Results are shown in Fig. 9 together with the experimental data of Fig. 3. We first note the good adequacy between theoretical and experimental data. In particular, splitting of the peak wavelengths is clearly visible. However, some discrepancies occur. In particular, a red-shift of theoretical prediction and the presence of additional wavelength around h = 0, h = 90 and h = 180. The first one may be due to a drift of fiber parameters between both experiment on the laser and on the fiber, due for example to temperature changes. The second one may be attributed to the mismatch between experimental data of Fig. 5 and corresponding fitting functions. However, to the authors knowledge, Fig. 9 represents the first direct comparison between experimental and theoretical tunability of a multiwavelength fiber laser with an intracavity polarizer.
|γ |
0.6
1094
Peak wavelength (nm)
Note that the round-trip operator method is not valid for high-power lasers or ultrashort pulsed lasers because of the intensity dependance of birefringence.
359
0.4
0.0 1080
5. Conclusion
20˚ 40˚ 60˚
0.2
1082
1084
1086
1088
1090
1092
1094
Wavelength (nm)
Fig. 8. Wavelength-dependence of |c| for three particular values of h.
We have presented multiwavelength operation of a Nd-doped fiber laser with an intracavity polarizer. We have shown tunability and splitting of the lasing peaks. In order to explain the experimental observations, we have presented a novel
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T. Chartier et al. / Optics Communications 253 (2005) 352–361
approach based on the Jones matrix analysis of fiber birefringence and on the derivation of the round-trip operator of the cavity. Excellent agreement between theory and experiment validates the principle of the method. Let us point out that this approach is not limited by the spectral range of the laser and can be applied to other types of multiwavelength fiber lasers based on polarization selection. Possible extension of this work concerns the comprehension of the wavelength evolution of fiber birefringence.
Acknowledgements The authors thank Pr. Pierre Pellat-Finet (LAUBS, Lorient, France) for the idea to inject a polarization oriented at 45, Dr. Se´bastien Coe¨tmellec (CORIA, Saint-Etienne du Rouvray, France) for helpfull discussions and Pr. Pascal Besnard (UMR FOTON, Lannion, France) for his comments on the manuscript.
Appendix A. Relations between Jones vectors, Stokes parameters and ellipticity and azimuth In the most general case, the Jones vector V representing an elliptical state of polarization S, has two complex coordinates and can be written as ! Ax expðiux Þ ax þ ibx V¼ . ðA:1Þ ¼ Ay expðiuy Þ ay þ iby The Stokes parameters of S write: s0 ¼ A2x þ A2y ¼ a2x þ b2x þ a2y þ b2y ; s1 ¼
A2x
A2y
¼
a2x
þ
b2x
a2y
b2y ;
ðA:2Þ ðA:3Þ
s2 ¼ 2Ax Ay cosðuy ux Þ ¼ 2ðax ay þ bx by Þ;
ðA:4Þ
s3 ¼ 2Ax Ay sinðuy ux Þ ¼ 2ðax by ay bx Þ.
ðA:5Þ
The ellipticity v and the azimuth / of S can be expressed as following: s3 sin 2v ¼ ; ðA:6Þ s0 s2 ðA:7Þ tan 2/ ¼ . s1
Appendix B. Change of basis for the Stokes parameters Let s0, s1, s2 and s3 be the Stokes parameters of a polarization state S in a basis (x, y) of linear polarizations. As a function of ellipticity v and azimut / of S, they write: s0 ¼ 1; s1 ¼ s0 cos 2v cos 2/;
ðB:1Þ ðB:2Þ
s2 ¼ s0 cos 2v sin 2/;
ðB:3Þ
s3 ¼ s0 sin 2v.
ðB:4Þ
Let us now consider a new basis (X, Y) of linear polarizations that make an angle /0 with respect to the basis (x, y). In this new basis the Stokes parameters of S write: S 0 ¼ 1;
ðB:5Þ
S 1 ¼ S 0 cos 2v cos 2ð/ /0 Þ ¼ s1 cos 2/0 þ s2 sin 2/0 ;
ðB:6Þ
S 2 ¼ S 0 cos 2v sin 2ð/ /0 Þ ¼ s2 cos 2/0 s1 sin 2/0 ;
ðB:7Þ
S 3 ¼ S 0 sin 2v ¼ s3 .
ðB:8Þ
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