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Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings
Optics Communications 281 (2008) 4099–4102
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings Xingchun Chu *, Shanghong Zhao, Lei Shi, Shengbao Zhan, Jie Xu, Zhuoliang Wu Department of Network Engineering, Telecommunication Engineering Institute, AFEU, Xi’an, Shanxi 710077, China
a r t i c l e
i n f o
Article history: Received 25 November 2007 Received in revised form 10 April 2008 Accepted 15 April 2008
Keywords: Laser technology Spectral beam combining Expansion of channel number Cascaded volume gratings
a b s t r a c t Aiming at the problem that the channel number in spectral beam combining (SBC) of fiber lasers array is limited by the narrow spectral selectivity of grating, a new scheme based on cascaded volume Bragg gratings (VBG) is proposed to expand the channel number. A system cascaded by two VBGs is presented and analyzed. A formula used to accurately determine the angle of incidence of all the channels is derived using rigorous coupled-wave analysis (RCWA). The channel number can be multiplied with increasing the number of VBGs. Numerical results show that the system based on cascaded VBGs is a good SBC combiner that has the ability to combine a large number of channels. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Recent advances in fiber lasers have shown that fiber lasers are one of the best solutions for high power laser sources that have the capability to generate kW output [1]. However, to scale up the output power of a single-fiber laser to higher power level faces significant challenges as nonlinear effects, thermal loading, fiber damage as well as pump power and brightness of pump LDs will limit the maximum output power [2]. Beam combining of laser arrays has been considered a promising approach for power and brightness scaling with high efficiency and good beam quality [3]. There are two approaches for beam combining which are under intensive study last years [4]. The first one is called ‘‘coherent beam combining” which is based on splitting of radiation of a master oscillator to a number of beams, power amplification of all beams, and a combining of amplified coherent beams in the interferometer type device by phase equalizing [5,6]. This approach requires extremely high precision and stability of phase retarding elements. The second approach termed ‘‘spectral beam combining (SBC)” is based on a diffractive element, such as a diffraction grating which is used to merge the outputs from multiple lasers with distinct central wavelengths into a single diffraction limited beam. This incoherent approach is inherently scalable as the individual laser elements are un-coupled, requires only modest spectral rather than precise phase control, allows the beams to be overlapped in the near and far fields without spatial interference [7,8]. The number of elements can vary
* Corresponding author. E-mail address: [email protected] (X. Chu). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.04.033
without changing the near- and far-field beam patterns, and the onaxis intensity will scale with the number of elements without any optical system reconfiguration [9]. This makes SBC a promising approach because of significantly simpler design and lower requirements for precision and stability of components [10,11]. The typical SBC implementation, invented at MIT Lincoln Laboratory [12,13], used to demonstrate near ideal beam combining on large laser arrays is shown in Fig. 1. This implementation uses an external cavity containing a grating to simultaneously control each element to emit a different wavelength and to overlap the individual beams in the near and far fields. Here, the output coupler provides optical feedback at different wavelengths to each element of the laser array, which controls the spectrum of each laser. The transform lens overlaps the beams in the near field at the grating, and the beams must be perpendicular to the output coupler and, consequently, the far fields overlap. Obtaining multi-kW power in a combined beam requires combination of a large number of channels. However, typically laser gain bandwidth and application requirements limit available bandwidth in Fig. 1. Assuming that the dispersion of the grating is db/dk, the focal length of the transform lens is f and the total wavelength spread of the optical output is Dk. The dimensional extent of the gain element array is [14] L ¼ f ðdb=dkÞDk
ð1Þ
Eq. (1) shows clearly that the number of element is mainly limited by the finite dispersion power of the grating and the aperture of the transform lens. One way to expand the number of channels is to decrease the spectral spacing between channels. For example, if the bandwidth of the grating is 50 nm and the power in each channel
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X. Chu et al. / Optics Communications 281 (2008) 4099–4102
ƒ
ƒ
λ2 λ1 λ0
Laser Array
λ1 λ2
Grating
Input Plane
Transform Lens ( λ 0,λ1,λ2)
Output Coupler
Fig. 1. Schemetic geometry of SBC invented at MIT Lincoln laboratory.
is about 400 W, it need 250-channel with 0.2 nm channel spacing being combined to generate 100 kW level output power [15]. Although it is possible to combine so many elements with a single diffraction grating in theory, it is a practical challenge to arrange them in finite space and a rather strict requirement to operate them at such slightly different wavelengths. To overcome the above problems, a new scheme of SBC is put forward in this paper. The number of channels can be easily expanded by using cascaded volume Bragg gratings (VBG) while the spectral spacing between channels need not be decreased. In Section 2, a SBC system based on two cascaded VBGs is presented and its basic principle is described. In Section 3, a formula used to determine the incident angle of every channel is derived based on rigorous coupled-wave analysis (RCWA). In Section 4, an example system is designed and numerically analyzed. And a conclusion is given in the end. 2. SBC system based on cascaded VBGs VBG has extremely narrow wavelength and/or angular selectivity. This feature of VBG enables it being a perfect cascaded beam combiner of a large number of channels. SBC by means of VBGs is based on the grating property that diffraction efficiency is high at or near Bragg condition and very low or null at multiple spectral bands corresponding to particular wavelength and/or angular offsets from Bragg condition [15]. Fig. 2 shows the scheme of SBC based on two cascaded VBGs. All the channels with wavelengths of (k1, k2, . . ., km, km+1, . . ., kn) are firstly divided into two groups of (k1, . . ., kc1, . . ., km) and (km+1, . . ., kc2, . . ., kn). The Bragg wavelengths of the two VBGs are designed to be kc1 and kc2, respectively. The first wave band (k1, . . ., kc1, . . ., km), which totally or approximately satisfy the Bragg condition of the VBG1, will be diffracted at high efficiency and go on reach the VBG2 at the Bragg angle hB1. The VBG2 can not only be a efficient combiner for the second wave band (km+1, . . ., kc2, . . ., kn) but also behaves like a transparent plate for the first wave band because they strongly violate the Bragg condition of the VBG2. The two wave bands can then be efficiently power added by the two cascaded
gratings and propagate like a single beam along a common direction determined by the Bragg angle hB1. This scheme can be easily expanded to the case that includes three or multi-gratings. Firstly, all the channels are divided into several groups in term of their wavelengths. Then, several gratings are used to combine them. By designing the VBGs carefully, the diffraction efficiency of a given VBG can be high enough for the specified group of wavelengths and low enough for the other ones. If the wavelength selectivity of these gratings are not overlapped each other, they can be cascaded together as a compound grating with broader spectral bandwidth than that of a single grating. So, the number of channels can be multiplied with the number of gratings increased. 3. Determination of the angle of incidence for every channel In Fig. 2, all the channels are divided into two groups and the central wavelengths of them are kc1 and kc2, which also are the Bragg wavelengths of the two VBGs, respectively. In order to make all the beams diffracted by the corresponding VBG propagate along a common direction, the incident angle of them must be determined accurately. Take the first group of channels for example, the wavevector diagram used to determine the incident angles of the first wave band is shown in Fig. 3. Here, the plane of incidence is assumed to be the x–z plane and the grating vector K1 of the VBG1, which is oriented with respect to the z-axis at angle /1, is in the same plane. The k1, . . ., kc1, . . ., km and h1, . . ., hc1, . . ., hm are the incident wavevectors and incident angles of the first wave band, respectively. kc1 and hB1 are the Bragg wavelength and angle of the VBG1, respectively. The k1d, . . ., kc1d, . . ., kmd are the wavevectors of the 1 diffracted orders of the first wave band. According to RCWA theory [16,17], the first Bragg condition of the VBG1 is cosð/1 hB1 Þ ¼
kc1 2K1 n0
ð2Þ
where K1 is the fringe period of the VBG1. And n0 is the average refractive index. In grating region, the x-component of the space-harmonic fields is determined by the Floquet condition [18] kx1 ¼ k0 A
ð3Þ
with A ¼ n0 sinðhÞ þ B
ð4Þ
B ¼ ðk=K1 Þ sinð/1 Þ
ð5Þ
x
VBG1 km λm
kc1
θB1
θ1
k1
λc1 λ1
θ2
Transform Lens λn λc2 λm+1
θm θc1 K1 θ1
θB1
φ1 z
θB1
kmd kc1d k1d
VBG2 Output (λ1,…,λn) Coupler
Transform Lens
Fig. 2. Schematic geometry of dual-band SBC system based on two cascaded VBGs.
Fig. 3. Wavevector diagram used to determine the incident angles of the first wave band.
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ð6Þ
In transmittance region, the corresponding z-component is h i1=2 kz1 ¼ k0 n2II ðkxi =k0 Þ2
ð7Þ
where nI and nII are the refractive index of the incidence and transmittance regions, respectively. Combining the Eqs. (3) and (7), the angle of diffraction is [19] tanðhd Þ ¼
kx1 A ¼ kz1 fn2II A2 g1=2
ð8Þ
From Eqs. (6) and (8), we can obtain that the angle of incidence is ( ) nII tanðhd Þ B½tan2 ðhd Þ þ 11=2 1 hin ¼ sin ð9Þ nI ðtan2 ðhd Þ þ 1Þ1=2 According to Eq. (2), the Bragg angle of the VBG1 is kc1 hB1 ¼ cos1 þ /1 2K1 n0
ð10Þ
Because all the 1 diffracted order of the first wave band are assumed to propagate in a common direction, there is hd = hB1 and we can obtain ( ) nII tanðhB1 Þ B½tan2 ðhB1 Þ þ 11=2 1 hin ¼ sin ð11Þ nI ½tan2 ðhB1 Þ þ 11=2 Using Eq. (11), we can exactly determine the angles of incidence for the first wave band when the Bragg wavelength and angle of the VBG1 are designed to be kc1 and hB1. Eq. (11) can also be used to determine the angles of incidence for the second wave band by replacing kc1, K1 and /1 with kc2, K2 and /2, respectively.
4. Design of the cascaded VBGs and numerical analysis The design criterion for the cascaded VBGs are that: (1) the two wave bands must be diffracted at high efficiency by the VBG1 and VBG2, respectively; (2) the Bragg angle of the VBG1 should equal to that of the VBG2; (3) the beams that combined by the VBG1 incident on the VBG2 at angle of hB1 will transmit it with little diffractive loss. Two hundred and fifty-channels with 0.5 nm channel spacing are considered in the combining system. They are divided into two groups whose central wavelengths are 1519 nm and 1581.5 nm, respectively. Each group contains 125 channels. For simplicity, the case of lossless dielectric VBGs with sinusoidal permittivity is treated. The incident plan wave polarization is perpendicular to the plane of incidence (TE wave). According to coupled wave theory, the 3-dB wavelength selectivity of a volume grating is given by [20] DkSel
Kkc cos hB ¼ d
ð12Þ
where K and d are the period and thickness of a VBG, respectively. hB is the Bragg angle. Eq. (12) shows that the spectral selectivity of a VBG is inversely proportional to d. Thus, an appropriate thickness of every VBG should be selected to meet the total spectral width of the corresponding wave band. Because the VBG1 is only used to diffract the first wave band, its design is relatively simple. The RCWA theory is used to calculate the parameters of the VBGs and the diffraction efficiency of all
1
Diffraction Efficiency
nI sinðhin Þ ¼ n0 sinðhÞ
the beams. The parameters of the VBG1 we chosen are /1 = 90°, K1 = 2.996 lm, d1 = 73.03 lm, hB1 = 10°, nI = nII = n0 = 1.46 and the amplitude of the modulated index is Dn = 0.03. The incident angle of every channel is determined by Eq. (11). Fig. 4 shows the diffraction efficiencies of every channel of the first wave band by the VBG1. And Fig. 5 shows the parallelism of the propagating direction of the first wave band. We can see that the efficiencies of 1 diffracted order of all the first wave band can reach at least 90% if they incident on the VBG1 at the angles determined by Eq. (11). All the 1 diffracted orders are propagate along a common direction determined by hB1. The parallelism of the propagating direction of all the diffracted beams are less then 1015 deg. But the divergence angle of the combined beam is determined by the maximum one of all the collimated laser beams because the VBG1 can not expand or compress it. Design of the VBG2 is a little complex because it should have the ability to diffract the second wave band at high efficiency but transmit the first wave band with little diffraction loss. In order to achieve this, a preferred method is to slant the fringe of the VBG2. Since VBG has extremely narrow angular selectivity, the angle /2, which is the angle between the grating vector of the VBG2 and the normal of the medium, is deliberately slanted so that the beams combined by the VBG1 incident on the VBG2 at the angle of hB1 will strongly violate the angular selectivity of the VBG2. At the same time, the Bragg angle of the VBG2 can be set equal to that of the VBG1 by selecting a appreciate fringe period according to Eq. (2). All the beams can then be steered to propagate along the common direction determined by hB2. The parameters of the VBG2 are chosen to be /2 = 80°, K2 = 1.583 lm, d2 = 76.04 lm, hB2 = 10°. The average refractive index and the amplitude of the modulated index are the same as that of the VBG1. Fig. 6 shows the angular selectivity of the VBG2. It is clear that only the beams
0. 8
-1 0. 6 0. 4
0 0. 2 1.49
1.5
1.51
1.52
1.53
1.54
1.55
Wavelength / μm Fig. 4. The diffraction efficiency of the first wave band by the VBG1.
-
× 10 15
Deviation / º
where h is the angle of refraction of the incident beam from incident region. k is the free space wavelength, which may be any one wavelength of the first wave band. And k0 = 2p/k is the free space wave number. Thus h is related to the angle of incidence hin on incident region through
5 0 -5
1. 49
1. 5
1. 51
1. 52
1. 53
1. 54
1. 55
Wavelength / μm Fig. 5. The deviation of the propagating direction all the diffracted beams.
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using three or more VBGs, the total channel numbers can be multiplied if the channels contained in every wave band remain unchanged. On the other hand, the channel numbers contained in each wave band can be also decreased to a reasonable range by using more VBGs if the total channel numbers are the same. This will help in practice to overcome the limitation of finite space on the arrangement of fiber arrays while the wavelength space can still be maintained in a reasonable size.
Diffraction Efficiency
1
-1
0.8
+1
0.6 0.4 0.2
5. Conclusion
0 -80 -60
-40
-20
0
20
40
60
80
Angle of incident / º Fig. 6. The angular selectivity of the VBG2.
Diffraction Efficiency
1 0.8
-1
0 0.6 0.4
1st Wave band
2ed Wave band 0
+1
0.2 1.5
We have proposed a new beam combining system based on cascaded VBGs to expand the channel numbers in SBC. A formula used to determine the angle of incidence of every channel is derived using RCWA theory. By carefully designing the parameters of the cascaded VBGs, the channel number can be multiplied with increasing the number of VBGs. Numerical results show that the system is a good SBC combiner and the diffraction efficiency of every channel can be higher than 90%.
1.52
1.54
1.56
1.58
1.6
Wavelength / μm Fig. 7. The diffractive efficiency of the two wave band diffracted by the VBG2.
that incident on the VBG2 at or near the angle of 20° can be efficiently diffracted. Fig. 7 shows the diffraction efficiencies of the two wave bands by the VBG2. We can see that the beams combined by the VBG1 transmit the VBG2 with little power diffracted into +1 order because they incident on it at angle of 10°. The second wave band, however, which incident on the VBG2 at the angles determined by Eq. (11), are efficiently diffracted into 1 order. The diffractive efficiencies of all the two band beams can be higher than 90%. And the angular divergences of the combined beams are still lees than 1015 deg as shown in Fig. 5. From the above numerical results, we can draw a conclusion that the system based on cascaded VBGs is a good SBC combiner that has the ability to combine a large number of channels. By
Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60678018) and the National Science Foundation for Post-doctoral Scientists of China (No. 20070420220). References [1] V. Reichel, K. Morl, S. Unger, et al., Proc. SPIE 5777 (2005) 404. [2] A.P. Liu, M. Roy, V. Tracy, et al., Proc. SPIE 5335 (2004) 81. [3] X. Chu, S. Zhan, S. Zhao et al., Opt. Laser. Technol., 2007, doi:10.1016/ j.optlastec.2007.08.004. [4] A. Sevian, O. Andrusyak, I. Ciapurin, et al., Proc. SPIE 62160V (2006) 1. [5] J. Anderegg, S. Brosnan, E. Cheung, et al., Proc. SPIE 61020U (2006) 1. [6] T.M. Shay, V. Benham, L.J. Spring, Proc. SPIE 61020V (2006) 1. [7] H.L. Thomas, A.P. Liu, P.R. Hoffman, et al., Opt. Lett. 32 (4) (2007) 349. [8] S. Klingebie, F. Röser, B. Ortacß, et al., J. Opt. Soc. Am. B 24 (8) (2007) 1716. [9] T.Y. Fan, A. Sanchez, Proc. SPIE 5709 (2005) 157. [10] V. Ciapurin, L.B. Glebov, V.I. Smirnov, Proc. SPIE 5335 (2004) 116. [11] L.B. Glebov, Proc. SPIE 6216 (2006) 1. [12] C.C. Cook, T.Y. Fan, Trend Opt. Photon. Ser. OSA 26 (1999) 163. [13] S.J. Augst, A.K. Goyal, R.L. Aggarwal, Opt. Lett. 28 (5) (2003) 331. [14] E.J. Bochove, IEEE J. Quantum Electron. 38 (5) (2002) 432. [15] O. Andrusyak, I. Ciapurin, V. Smirnov, Proc. SPIE 64531L (2007) 1. [16] M.G. Moharam, T.K. Gaylord, J. Opt. Soc. Am. 71 (7) (1981) 811. [17] T.K. Gayloed, M.G. Moharam, Appl. Phys. B28 (1982) 1. [18] M.G. Moharam, D.A. Pommet, E.B. Grann, J. Opt. Soc. Am. 12 (5) (1995) 1077. [19] M.G. Moharam, T.K. Gaylord, J. Opt. Soc. Am. 73 (9) (1983) 1105. [20] An Jun Won, Do Duc Dung, Kim Nam, et al., IEEE Photon. Technol. Lett. 18 (6) (2006) 788.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Expansion of the channel number in spectral beam combining of fiber lasers array based on cascaded gratings Xingchun Chu *, Shanghong Zhao, Lei Shi, Shengbao Zhan, Jie Xu, Zhuoliang Wu Department of Network Engineering, Telecommunication Engineering Institute, AFEU, Xi’an, Shanxi 710077, China
a r t i c l e
i n f o
Article history: Received 25 November 2007 Received in revised form 10 April 2008 Accepted 15 April 2008
Keywords: Laser technology Spectral beam combining Expansion of channel number Cascaded volume gratings
a b s t r a c t Aiming at the problem that the channel number in spectral beam combining (SBC) of fiber lasers array is limited by the narrow spectral selectivity of grating, a new scheme based on cascaded volume Bragg gratings (VBG) is proposed to expand the channel number. A system cascaded by two VBGs is presented and analyzed. A formula used to accurately determine the angle of incidence of all the channels is derived using rigorous coupled-wave analysis (RCWA). The channel number can be multiplied with increasing the number of VBGs. Numerical results show that the system based on cascaded VBGs is a good SBC combiner that has the ability to combine a large number of channels. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Recent advances in fiber lasers have shown that fiber lasers are one of the best solutions for high power laser sources that have the capability to generate kW output [1]. However, to scale up the output power of a single-fiber laser to higher power level faces significant challenges as nonlinear effects, thermal loading, fiber damage as well as pump power and brightness of pump LDs will limit the maximum output power [2]. Beam combining of laser arrays has been considered a promising approach for power and brightness scaling with high efficiency and good beam quality [3]. There are two approaches for beam combining which are under intensive study last years [4]. The first one is called ‘‘coherent beam combining” which is based on splitting of radiation of a master oscillator to a number of beams, power amplification of all beams, and a combining of amplified coherent beams in the interferometer type device by phase equalizing [5,6]. This approach requires extremely high precision and stability of phase retarding elements. The second approach termed ‘‘spectral beam combining (SBC)” is based on a diffractive element, such as a diffraction grating which is used to merge the outputs from multiple lasers with distinct central wavelengths into a single diffraction limited beam. This incoherent approach is inherently scalable as the individual laser elements are un-coupled, requires only modest spectral rather than precise phase control, allows the beams to be overlapped in the near and far fields without spatial interference [7,8]. The number of elements can vary
* Corresponding author. E-mail address: [email protected] (X. Chu). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.04.033
without changing the near- and far-field beam patterns, and the onaxis intensity will scale with the number of elements without any optical system reconfiguration [9]. This makes SBC a promising approach because of significantly simpler design and lower requirements for precision and stability of components [10,11]. The typical SBC implementation, invented at MIT Lincoln Laboratory [12,13], used to demonstrate near ideal beam combining on large laser arrays is shown in Fig. 1. This implementation uses an external cavity containing a grating to simultaneously control each element to emit a different wavelength and to overlap the individual beams in the near and far fields. Here, the output coupler provides optical feedback at different wavelengths to each element of the laser array, which controls the spectrum of each laser. The transform lens overlaps the beams in the near field at the grating, and the beams must be perpendicular to the output coupler and, consequently, the far fields overlap. Obtaining multi-kW power in a combined beam requires combination of a large number of channels. However, typically laser gain bandwidth and application requirements limit available bandwidth in Fig. 1. Assuming that the dispersion of the grating is db/dk, the focal length of the transform lens is f and the total wavelength spread of the optical output is Dk. The dimensional extent of the gain element array is [14] L ¼ f ðdb=dkÞDk
ð1Þ
Eq. (1) shows clearly that the number of element is mainly limited by the finite dispersion power of the grating and the aperture of the transform lens. One way to expand the number of channels is to decrease the spectral spacing between channels. For example, if the bandwidth of the grating is 50 nm and the power in each channel
4100
X. Chu et al. / Optics Communications 281 (2008) 4099–4102
ƒ
ƒ
λ2 λ1 λ0
Laser Array
λ1 λ2
Grating
Input Plane
Transform Lens ( λ 0,λ1,λ2)
Output Coupler
Fig. 1. Schemetic geometry of SBC invented at MIT Lincoln laboratory.
is about 400 W, it need 250-channel with 0.2 nm channel spacing being combined to generate 100 kW level output power [15]. Although it is possible to combine so many elements with a single diffraction grating in theory, it is a practical challenge to arrange them in finite space and a rather strict requirement to operate them at such slightly different wavelengths. To overcome the above problems, a new scheme of SBC is put forward in this paper. The number of channels can be easily expanded by using cascaded volume Bragg gratings (VBG) while the spectral spacing between channels need not be decreased. In Section 2, a SBC system based on two cascaded VBGs is presented and its basic principle is described. In Section 3, a formula used to determine the incident angle of every channel is derived based on rigorous coupled-wave analysis (RCWA). In Section 4, an example system is designed and numerically analyzed. And a conclusion is given in the end. 2. SBC system based on cascaded VBGs VBG has extremely narrow wavelength and/or angular selectivity. This feature of VBG enables it being a perfect cascaded beam combiner of a large number of channels. SBC by means of VBGs is based on the grating property that diffraction efficiency is high at or near Bragg condition and very low or null at multiple spectral bands corresponding to particular wavelength and/or angular offsets from Bragg condition [15]. Fig. 2 shows the scheme of SBC based on two cascaded VBGs. All the channels with wavelengths of (k1, k2, . . ., km, km+1, . . ., kn) are firstly divided into two groups of (k1, . . ., kc1, . . ., km) and (km+1, . . ., kc2, . . ., kn). The Bragg wavelengths of the two VBGs are designed to be kc1 and kc2, respectively. The first wave band (k1, . . ., kc1, . . ., km), which totally or approximately satisfy the Bragg condition of the VBG1, will be diffracted at high efficiency and go on reach the VBG2 at the Bragg angle hB1. The VBG2 can not only be a efficient combiner for the second wave band (km+1, . . ., kc2, . . ., kn) but also behaves like a transparent plate for the first wave band because they strongly violate the Bragg condition of the VBG2. The two wave bands can then be efficiently power added by the two cascaded
gratings and propagate like a single beam along a common direction determined by the Bragg angle hB1. This scheme can be easily expanded to the case that includes three or multi-gratings. Firstly, all the channels are divided into several groups in term of their wavelengths. Then, several gratings are used to combine them. By designing the VBGs carefully, the diffraction efficiency of a given VBG can be high enough for the specified group of wavelengths and low enough for the other ones. If the wavelength selectivity of these gratings are not overlapped each other, they can be cascaded together as a compound grating with broader spectral bandwidth than that of a single grating. So, the number of channels can be multiplied with the number of gratings increased. 3. Determination of the angle of incidence for every channel In Fig. 2, all the channels are divided into two groups and the central wavelengths of them are kc1 and kc2, which also are the Bragg wavelengths of the two VBGs, respectively. In order to make all the beams diffracted by the corresponding VBG propagate along a common direction, the incident angle of them must be determined accurately. Take the first group of channels for example, the wavevector diagram used to determine the incident angles of the first wave band is shown in Fig. 3. Here, the plane of incidence is assumed to be the x–z plane and the grating vector K1 of the VBG1, which is oriented with respect to the z-axis at angle /1, is in the same plane. The k1, . . ., kc1, . . ., km and h1, . . ., hc1, . . ., hm are the incident wavevectors and incident angles of the first wave band, respectively. kc1 and hB1 are the Bragg wavelength and angle of the VBG1, respectively. The k1d, . . ., kc1d, . . ., kmd are the wavevectors of the 1 diffracted orders of the first wave band. According to RCWA theory [16,17], the first Bragg condition of the VBG1 is cosð/1 hB1 Þ ¼
kc1 2K1 n0
ð2Þ
where K1 is the fringe period of the VBG1. And n0 is the average refractive index. In grating region, the x-component of the space-harmonic fields is determined by the Floquet condition [18] kx1 ¼ k0 A
ð3Þ
with A ¼ n0 sinðhÞ þ B
ð4Þ
B ¼ ðk=K1 Þ sinð/1 Þ
ð5Þ
x
VBG1 km λm
kc1
θB1
θ1
k1
λc1 λ1
θ2
Transform Lens λn λc2 λm+1
θm θc1 K1 θ1
θB1
φ1 z
θB1
kmd kc1d k1d
VBG2 Output (λ1,…,λn) Coupler
Transform Lens
Fig. 2. Schematic geometry of dual-band SBC system based on two cascaded VBGs.
Fig. 3. Wavevector diagram used to determine the incident angles of the first wave band.
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X. Chu et al. / Optics Communications 281 (2008) 4099–4102
ð6Þ
In transmittance region, the corresponding z-component is h i1=2 kz1 ¼ k0 n2II ðkxi =k0 Þ2
ð7Þ
where nI and nII are the refractive index of the incidence and transmittance regions, respectively. Combining the Eqs. (3) and (7), the angle of diffraction is [19] tanðhd Þ ¼
kx1 A ¼ kz1 fn2II A2 g1=2
ð8Þ
From Eqs. (6) and (8), we can obtain that the angle of incidence is ( ) nII tanðhd Þ B½tan2 ðhd Þ þ 11=2 1 hin ¼ sin ð9Þ nI ðtan2 ðhd Þ þ 1Þ1=2 According to Eq. (2), the Bragg angle of the VBG1 is kc1 hB1 ¼ cos1 þ /1 2K1 n0
ð10Þ
Because all the 1 diffracted order of the first wave band are assumed to propagate in a common direction, there is hd = hB1 and we can obtain ( ) nII tanðhB1 Þ B½tan2 ðhB1 Þ þ 11=2 1 hin ¼ sin ð11Þ nI ½tan2 ðhB1 Þ þ 11=2 Using Eq. (11), we can exactly determine the angles of incidence for the first wave band when the Bragg wavelength and angle of the VBG1 are designed to be kc1 and hB1. Eq. (11) can also be used to determine the angles of incidence for the second wave band by replacing kc1, K1 and /1 with kc2, K2 and /2, respectively.
4. Design of the cascaded VBGs and numerical analysis The design criterion for the cascaded VBGs are that: (1) the two wave bands must be diffracted at high efficiency by the VBG1 and VBG2, respectively; (2) the Bragg angle of the VBG1 should equal to that of the VBG2; (3) the beams that combined by the VBG1 incident on the VBG2 at angle of hB1 will transmit it with little diffractive loss. Two hundred and fifty-channels with 0.5 nm channel spacing are considered in the combining system. They are divided into two groups whose central wavelengths are 1519 nm and 1581.5 nm, respectively. Each group contains 125 channels. For simplicity, the case of lossless dielectric VBGs with sinusoidal permittivity is treated. The incident plan wave polarization is perpendicular to the plane of incidence (TE wave). According to coupled wave theory, the 3-dB wavelength selectivity of a volume grating is given by [20] DkSel
Kkc cos hB ¼ d
ð12Þ
where K and d are the period and thickness of a VBG, respectively. hB is the Bragg angle. Eq. (12) shows that the spectral selectivity of a VBG is inversely proportional to d. Thus, an appropriate thickness of every VBG should be selected to meet the total spectral width of the corresponding wave band. Because the VBG1 is only used to diffract the first wave band, its design is relatively simple. The RCWA theory is used to calculate the parameters of the VBGs and the diffraction efficiency of all
1
Diffraction Efficiency
nI sinðhin Þ ¼ n0 sinðhÞ
the beams. The parameters of the VBG1 we chosen are /1 = 90°, K1 = 2.996 lm, d1 = 73.03 lm, hB1 = 10°, nI = nII = n0 = 1.46 and the amplitude of the modulated index is Dn = 0.03. The incident angle of every channel is determined by Eq. (11). Fig. 4 shows the diffraction efficiencies of every channel of the first wave band by the VBG1. And Fig. 5 shows the parallelism of the propagating direction of the first wave band. We can see that the efficiencies of 1 diffracted order of all the first wave band can reach at least 90% if they incident on the VBG1 at the angles determined by Eq. (11). All the 1 diffracted orders are propagate along a common direction determined by hB1. The parallelism of the propagating direction of all the diffracted beams are less then 1015 deg. But the divergence angle of the combined beam is determined by the maximum one of all the collimated laser beams because the VBG1 can not expand or compress it. Design of the VBG2 is a little complex because it should have the ability to diffract the second wave band at high efficiency but transmit the first wave band with little diffraction loss. In order to achieve this, a preferred method is to slant the fringe of the VBG2. Since VBG has extremely narrow angular selectivity, the angle /2, which is the angle between the grating vector of the VBG2 and the normal of the medium, is deliberately slanted so that the beams combined by the VBG1 incident on the VBG2 at the angle of hB1 will strongly violate the angular selectivity of the VBG2. At the same time, the Bragg angle of the VBG2 can be set equal to that of the VBG1 by selecting a appreciate fringe period according to Eq. (2). All the beams can then be steered to propagate along the common direction determined by hB2. The parameters of the VBG2 are chosen to be /2 = 80°, K2 = 1.583 lm, d2 = 76.04 lm, hB2 = 10°. The average refractive index and the amplitude of the modulated index are the same as that of the VBG1. Fig. 6 shows the angular selectivity of the VBG2. It is clear that only the beams
0. 8
-1 0. 6 0. 4
0 0. 2 1.49
1.5
1.51
1.52
1.53
1.54
1.55
Wavelength / μm Fig. 4. The diffraction efficiency of the first wave band by the VBG1.
-
× 10 15
Deviation / º
where h is the angle of refraction of the incident beam from incident region. k is the free space wavelength, which may be any one wavelength of the first wave band. And k0 = 2p/k is the free space wave number. Thus h is related to the angle of incidence hin on incident region through
5 0 -5
1. 49
1. 5
1. 51
1. 52
1. 53
1. 54
1. 55
Wavelength / μm Fig. 5. The deviation of the propagating direction all the diffracted beams.
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using three or more VBGs, the total channel numbers can be multiplied if the channels contained in every wave band remain unchanged. On the other hand, the channel numbers contained in each wave band can be also decreased to a reasonable range by using more VBGs if the total channel numbers are the same. This will help in practice to overcome the limitation of finite space on the arrangement of fiber arrays while the wavelength space can still be maintained in a reasonable size.
Diffraction Efficiency
1
-1
0.8
+1
0.6 0.4 0.2
5. Conclusion
0 -80 -60
-40
-20
0
20
40
60
80
Angle of incident / º Fig. 6. The angular selectivity of the VBG2.
Diffraction Efficiency
1 0.8
-1
0 0.6 0.4
1st Wave band
2ed Wave band 0
+1
0.2 1.5
We have proposed a new beam combining system based on cascaded VBGs to expand the channel numbers in SBC. A formula used to determine the angle of incidence of every channel is derived using RCWA theory. By carefully designing the parameters of the cascaded VBGs, the channel number can be multiplied with increasing the number of VBGs. Numerical results show that the system is a good SBC combiner and the diffraction efficiency of every channel can be higher than 90%.
1.52
1.54
1.56
1.58
1.6
Wavelength / μm Fig. 7. The diffractive efficiency of the two wave band diffracted by the VBG2.
that incident on the VBG2 at or near the angle of 20° can be efficiently diffracted. Fig. 7 shows the diffraction efficiencies of the two wave bands by the VBG2. We can see that the beams combined by the VBG1 transmit the VBG2 with little power diffracted into +1 order because they incident on it at angle of 10°. The second wave band, however, which incident on the VBG2 at the angles determined by Eq. (11), are efficiently diffracted into 1 order. The diffractive efficiencies of all the two band beams can be higher than 90%. And the angular divergences of the combined beams are still lees than 1015 deg as shown in Fig. 5. From the above numerical results, we can draw a conclusion that the system based on cascaded VBGs is a good SBC combiner that has the ability to combine a large number of channels. By
Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60678018) and the National Science Foundation for Post-doctoral Scientists of China (No. 20070420220). References [1] V. Reichel, K. Morl, S. Unger, et al., Proc. SPIE 5777 (2005) 404. [2] A.P. Liu, M. Roy, V. Tracy, et al., Proc. SPIE 5335 (2004) 81. [3] X. Chu, S. Zhan, S. Zhao et al., Opt. Laser. Technol., 2007, doi:10.1016/ j.optlastec.2007.08.004. [4] A. Sevian, O. Andrusyak, I. Ciapurin, et al., Proc. SPIE 62160V (2006) 1. [5] J. Anderegg, S. Brosnan, E. Cheung, et al., Proc. SPIE 61020U (2006) 1. [6] T.M. Shay, V. Benham, L.J. Spring, Proc. SPIE 61020V (2006) 1. [7] H.L. Thomas, A.P. Liu, P.R. Hoffman, et al., Opt. Lett. 32 (4) (2007) 349. [8] S. Klingebie, F. Röser, B. Ortacß, et al., J. Opt. Soc. Am. B 24 (8) (2007) 1716. [9] T.Y. Fan, A. Sanchez, Proc. SPIE 5709 (2005) 157. [10] V. Ciapurin, L.B. Glebov, V.I. Smirnov, Proc. SPIE 5335 (2004) 116. [11] L.B. Glebov, Proc. SPIE 6216 (2006) 1. [12] C.C. Cook, T.Y. Fan, Trend Opt. Photon. Ser. OSA 26 (1999) 163. [13] S.J. Augst, A.K. Goyal, R.L. Aggarwal, Opt. Lett. 28 (5) (2003) 331. [14] E.J. Bochove, IEEE J. Quantum Electron. 38 (5) (2002) 432. [15] O. Andrusyak, I. Ciapurin, V. Smirnov, Proc. SPIE 64531L (2007) 1. [16] M.G. Moharam, T.K. Gaylord, J. Opt. Soc. Am. 71 (7) (1981) 811. [17] T.K. Gayloed, M.G. Moharam, Appl. Phys. B28 (1982) 1. [18] M.G. Moharam, D.A. Pommet, E.B. Grann, J. Opt. Soc. Am. 12 (5) (1995) 1077. [19] M.G. Moharam, T.K. Gaylord, J. Opt. Soc. Am. 73 (9) (1983) 1105. [20] An Jun Won, Do Duc Dung, Kim Nam, et al., IEEE Photon. Technol. Lett. 18 (6) (2006) 788.