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Cladding index modulated fiber grating
Optics Communications 259 (2006) 587–591 www.elsevier.com/locate/optcom
Cladding index modulated fiber grating Na Chen 1, Binfeng Yun, Yiping Cui
*
Advanced Photonics Center, Southeast University, Nanjing 210096, China Received 8 May 2005; received in revised form 29 August 2005; accepted 8 September 2005
Abstract A new type of fiber grating with only cladding index modulation is presented. Characteristics of both cladding index modulated shortperiod fiber grating (FBG) and long-period fiber grating (LPFG) are analyzed. The calculation of the modes involved in this paper is based on a model of three-layer step-index fiber geometry. Transmission of a mode guided by the core through a cladding index modulated grating when evanescent field coupling occurs is analyzed with couple-mode theory. Evanescent field coupling causes a power flowing from the core to the cladding, so the attenuation of the new grating is analyzed as well. Lower attenuation, flexible spectral characteristics are demonstrated in comparison with traditional fiber core index modulated grating. Ó 2005 Elsevier B.V. All rights reserved. PACS: 42.81.Wg; 42.79.Dj; 42.81.Dp Keywords: Fiber gratings; Cladding index modulation; Couple-mode theory
1. Introduction Since Hill et al. [1] discovered the photosensitivity of germanium-doped-core optical fiber twenty years ago, photoimprinting grating in the core of optical fiber has been a primary method of fiber grating fabrication [2,3]. And fiber gratings as important fiber optics components have been widely used in fiber communication and sensor system [4–6]. Recently, various improvements have been made to fabricate more complex fiber grating with photosensitivity fiber [7–9]. However, this kind of doped-core photosensitive optical fiber always has inevitable great attenuation due to its heavy doped core. For example, the attenuation at 1550 nm of Single-Mode Photosensitive Fiber and Radiation Mode Suppression Photosensitive Fiber of INO is 63.6 dB/km and 53.2 dB/km, respectively, while the atten-
*
Corresponding author. Tel.: +86 025 83601769x18; fax: +86 025 83601769x17. E-mail addresses: [email protected] (N. Chen), [email protected] (Y. Cui). 1 Tel.: +86 025 83601769x13; fax: +86 025 83601769x17. 0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.09.017
uation of an ordinary Single-mode Fiber that is used in communication at the same wavelength is only 0.20 dB/km. In many applications, such as FBG as a basic reflector used as the end mirror of fiber lasers, such loss affects the performance seriously. In order to mitigate the attenuation of a fiber grating, we are proposing a new type of fiber grating in this paper. The core of the new grating, the area that contains the dominating energy distribution of the core mode in a fiber, is unperturbed, while the index modulation is only in fiber cladding, where the core mode is evanescent field. The structure of the new grating is depicted in Section 2. Furthermore, we can modulate the cladding index in certain ways such as through exposing the fiber to periodic ultraviolet directly if the cladding is photosensitive. The fabrication of core index modulated FBG and LPFG are mainly done using the well-known phase mask technique and amplitude mask technique, respectively. Provided fiber with only photosensitive cladding is available, cladding index modulated FBG and LPFG can be realized with the same techniques easily. Furthermore, the state of the art of photosensitive fiber fabrication assures such fiber is not difficult to be realized. This new modulation scheme
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could minimize the attenuation of the grating greatly, which will be proved in Section 3. Although the field of fundamental core mode carries much less energy in the cladding region than the fiber core, the periodic index change still could affect the fundamental core mode through its evanescent field and modes coupling occurs when proper phase matching condition is met. Both cladding index modulated FBG and LPFG are analyzed. The analysis procedure is detailed in Section 4. Section 5 gives the calculated spectra of the new class grating and some discussion compared with traditional core index modulated gratings. Conclusion is in Section 6. 2. Grating structure and index modulation The structure of this new grating is analyzed under the model of three-layer step-index fiber geometry, which is shown in Fig. 1. The three layers are the core layer, the cladding layer and the surrounding layer respectively. The core of the grating is with the diameter of 2a1. The index is uniform in the core with the value of n1. The cladding surrounds the core with the diameter of 2a2. The index of the cladding is not uniform but modulated in a periodical way. The period of the modulation is in distance. Therefore, the index in the cladding is n2(z). Both the core and cladding are immersed in the surrounding, which has the index of n3. Usually the surrounding is air and n3 = 1. In our analysis we assume that, when a phase grating is induced in the fiber, it exits only in the fiber cladding, changing the cladding index to n2 (z), but leave the core and surrounding index unchanged, as follows 8 r 6 a1 ; > < n1 ; nðr; zÞ ¼ n2 ðzÞ ¼ n2 þ dnðzÞ½1 þ m cosð2pz=KÞ; a1 < r 6 a2 ; > : n3 ; r > a2 . ð1Þ
Here, n1 is the fiber core index, K is the period of the grating, m is the induced index fringe modulation, where 0 6 m 6 1, dnðzÞ is the average induced index change. 3. Attenuation of cladding index modulated fiber grating The absorbing fiber can be treated as the perturbation of the nonabsorbent fiber when the absorption is slight. The absorption can be described by the addition of imaginary constants to the core and cladding index. Thus, the correction to the perturbed propagation constant is imaginary. If
Fig. 1. Diagram of cladding index modulated fiber grating, showing the refractive index, grating period and the radius of the core (a1) and of the cladding (a2).
we denote the correction by bi, then bi ¼ k gdnico þ ð1 gÞdnicl g, where g is the fraction of the modeÕs power flowing within the core [10]. Power absorption coefficients for the core and cladding is defined as aco ¼ 2kdnico and acl ¼ 2kdnicl . The power attenuation coefficient c is then expressible as c ¼ gaco þ ð1 gÞacl , and the modeÕs power is reduced to exp (cz) of its original value after distance z. The shape of the intensity distribution on the absorbing fiber does not change as the mode propagation, but its amplitude is reduced to exp(cz) of its original value after distance z. If the power absorption coefficient is uniform over the whole cross-section, then c = a and power lost by attenuation at any position is exactly equal to the power absorbed there. The two classes fiber gratings considered here correspond to the case aco 5 acl, when power flows across the interface to maintain the shape of the intensity distribution. In the case of core index modulated grating ðaco cl ¼ 0Þ, power flows to the core, while for the cladding index modulated grating ðaclco ¼ 0Þ, power flows to the cladding. For the convenience of comparison, we assume that the index modulated fractions of both gratings have the same power absorption coefficients aclcl ¼ aco co ¼ a and the unmodulated fractions are nonabsorbent aclco ¼ aco cl ¼ 0. Thus, the power attenuation coefficient c of the both classes grating are ccl ¼ gaclco þ ð1 gÞaclcl ¼ ð1 gÞa and cco ¼ gaco co þ cl co ð1 gÞaco cl ¼ ga, respectively. Therefore, we have c =c ¼ ð1 gÞa=ga ¼ ð1 gÞ=g, that means the power attenuation coefficient ratio of the two grating with different index modulated area is determined only by the fraction of the modeÕs power flowing within the core g. Because the power transmitted in fiber is mainly confined in the core, the cladding index modulated gratingÕs power attenuation coefficient will be much smaller than fiber core index modulated gratingÕs. The typical parameter of single-mode fiber used for communications purposes is fiber core radius a1 = 4.5 lm, cladding radius a2 = 62.5 lm. The fraction of the modeÕs power flowing within the core can be expressed as [10] u2 w2 K 20 ðwÞ g¼ 2 2þ 2 . ð2Þ v u K 1 ðwÞ Then we can obtain g ¼ 0:91796. Thus, the ratio of power attenuation coefficient of the two class fiber gratings is cco =ccl ¼ ð1 gÞ=g ¼ 11:19. So the insertion loss of the fiber grating is cL, where the L is the whole length of the photosensitive fiber inserting in the system. Because there is only a small fraction of EM field intensity in the fiber cladding, in order to have the same transmission/reflection characteristics, the claddingindex-modulated fiber grating should be longer or have stronger index modulation compared to the core index modulated one. The detailed analysis will be given in Section 5. However, the whole length inserting in the system of both class gratings can be regarded as the same. Therefore, the attenuation of fiber core index modulated
N. Chen et al. / Optics Communications 259 (2006) 587–591
grating is 11.19 times over cladding index modulated grating fabricated with the same type fiber. 4. Coupling coefficient The field expressions of fundamental core mode and cladding modes can be obtained by solving MaxwellÕs equations with proper boundary conditions. These modes transmit independently in the ordinary fiber. While in the grating periodical index is induced, this perturbation would cause the evanescent field coupling among the forward fundamental core mode, backward fundamental core mode and the cladding modes. Coupled-mode theory can be employed to analyze such interactions. According to the solving procedure of the coupled-mode theory, all the necessary propagation constants and coupling coefficients need to be calculated first. Propagation constants of both fundamental core mode and cladding modes can be obtained through solving dispersion relations in core and cladding, respectively [11,12]. Thus, the resonant wavelength can be determined by the phase matching condition, as well. Since fundamental core mode is so familiar and the exact field expressions of the cladding modes have been detailed in [11,12] we do not list the field components here. The following analysis follows that of [11], while the difference is the interaction between modes considered in this paper occurs in the cladding rather than in the fiber core. The transverse coupling coefficient between two modes m and l is K tml ¼
x 4
Zþ1Z
dx dy De Etm Etl .
ð3Þ
1
To calculate coupling coefficient, permittivity change De caused by the induced cladding index change should be obtained first. Because the grating considered is weak grating, dnðzÞ is in the order of 103–104, we can make the approximation De = e0D(n2) ffi 2e0nD(n). Combine with (1), we obtain 8 r 6 a1 ; > < 0; Deðr; zÞ 2e0 n2 dnðzÞ½1 þ m cosð2pz=KÞ; a1 < r 6 a2 ; > : 0; r > a2 .
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Using the expression for fundamental core mode in the cladding region, and performing the overlap integrals of (6), we obtain the coupling constant for core-mode-coremode coupling 2p 2u2 1 1 2 2 a ½K ðwa2 =a1 Þ k V 2 K 21 ðwÞ a21 2 0 K 21 ðwa2 =a1 Þ a21 ½K 20 ðwÞ K 21 ðwÞ .
coco ðzÞ ¼ dnðzÞ j0101
ð8Þ
However, substituting the field expressions of both fundamental core mode and cladding mode in the cladding into (7), the analytical solution for the overlap integrals is not available. Coupling constant for core-mode-cladding-mode coupling must be evaluated numerically. Provided that all the necessary propagation constants and coupling coefficients have been obtained, simple twomode coupled-mode theory that involves constant-coefficient differential equations with slowly varying amplitudes can be employed. For a uniform grating considered in this paper, analytical solutions for the transmission in both FBG and LPFG are available. 5. Results and discussion In this section, we calculated the reflection spectra of cladding index modulated FBG and the transmission spectra of cladding index modulated LPFG based on the theory described above. 5.1. Cladding index modulated FBG Fig. 2 shows the theory value of the reflection spectrum of the cladding index modulation fiber Bragg grating (solid line). The parameters of the fiber grating under calculation are as follows: fiber core radius a1 = 2.625 lm, cladding radius a2 = 62.5 lm, fiber core index n1 = 1.458, cladding index n2 = 1.45, surrounding index n3 = 1, grating period K = 0.557846 lm, grating length L = 8 mm, average index
ð4Þ Then, the coupling coefficient can be written as K tml ðzÞ ¼ jml ðzÞ½1 þ m cosð2pz=KÞ.
ð5Þ
Then, we can deduce the coupling constant for core-modecore-mode coupling and core-mode-cladding-mode coupling by (3)–(5) as follows: Z a2 Z co 2 co 2 xe0 n2 dnðzÞ 2p E þ E Þ; ð6Þ jcoco ðzÞ ¼ d/ r drð 0101 r / 2 a1 0 Z a2 Z 2p xe0 n2 dnðzÞ cl co jclco d/ r drðEclr Eco r þ E / E/ Þ. 1m01 ðzÞ ¼ 2 a1 0 ð7Þ
Fig. 2. Theoretically calculated reflection spectra (in decibels) through a cladding index modulated FBG (solid line) and a core index modulated FBG (dash line).
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change dn ¼ 5 104 , and the induced index fringe modulation m = 0.6. A comparison spectrum of the fiber core index modulated grating is shown in the same figure (dash line). It shows that cladding index modulated FBG has similar characters to core index modulated FBG. The main reflection peak is due to the evanescent field coupling between the forward guided mode and the backward one in the present of the periodic cladding index modulation. 5.2. Cladding index modulated LPFG We calculated the transmission spectra of the cladding index modulated LPFG to demonstrate the cladding-mode coupling is present and the result shows transmission spectrum of the new grating demonstrates the band stop filter characteristic. Based on the phase-matched condition described in forward sections the relationship between grating period and wavelength can be obtained by the calculation of the propagation constants of the guided and the various cladding modes of a fiber at each specific wavelength. The result is shown in Fig. 3. The wavelength at which mode-coupling will be enabled by a particular grating period can now be predicted. The vertical dash line in Fig. 3 shows the position of the period involved in our calculation. Fig. 4 shows the theory value of the transmission spectrum of the cladding index modulation fiber grating. The parameters of the fiber grating under calculation are as follows: fiber core radius a1 = 2.625 lm, cladding radius a2 = 62.5 lm, fiber core index n1 = 1.458, cladding index n2 = 1.45, surrounding index n3 = 1, grating period K = 576 lm, grating length L = 11.72 mm, average index change dn ¼ 5 104 , and the induced index fringe modulation m = 0.6. In order to compare the new grating with an ordinary grating, the transmission spectrum of a typical uniform fiber core index modulated grating with the same parameters is given in Fig. 5. The transmission spectra of both grating were presented simultaneously for the convenience of com-
Fig. 3. Theoretical determination of the relationship between grating period and wavelengths.
Fig. 4. Theoretically calculated transmission spectra (in decibels) through a uniform cladding index modulated LPFG.
prehension. We can conclude from Fig. 5 that under the same circumstances, the coupling in the cladding index modulated grating is much weaker than that in the core index modulated grating. Same conclusion can be drawn from Fig. 2. It is reasonable because the part of the fundamental core mode in the cladding carries much less energy than the one in the fiber core. The interactions due to the induced perturbation between the evanescent field of the fundamental core mode and the field of cladding modes in the cladding is much weaker than the field in the fiber core. Furthermore, there is also a difference of the resonance wavelength between the fiber core index modulated grating and the fiber cladding index modulated grating, which is because the center wavelength is determined by two factors, one is the phase matching condition, the other is coupling coefficient. Although the phase matching condition is the same under the assumed condition of same grating period, the coupling coefficients in both fiber gratings have a little difference due to the different interactions occurring in both gratings. Therefore, the center wavelength is a bit
Fig. 5. Comparison of transmission spectra (in decibels) through a uniform cladding index modulation fiber grating (dash line) and a uniform fiber core index modulated grating with the same parameters (solid line).
N. Chen et al. / Optics Communications 259 (2006) 587–591
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than the losses of the core index modulated fiber grating under the circumstance of similar transmission amplitudes. 6. Conclusion
Fig. 6. Growth of transmission spectrum with the grating length.
different, too. From another point of view, the indexes of cladding and the core are different. Although the grating period is the same, the two types of grating are not the same for a certain wavelength. However, this not means that the cladding index modulation LPFG cannot gain strong resonant coupling. It is well known that the spectral characteristics of the fiber grating are quite flexible, since practically one can vary numerous parameters to meet the various requirements. Both the induced index change and the grating length affect the coupling strength. In this paper, we lengthen the cladding index modulated grating temperately and get a better coupling. Fig. 6 shows the growth of the transmission spectrum with the increase of the grating length. When the grating length increases to 17 mm, the maximum transmissivity dip reached 60 dB. Additionally, we can compare the grating length of both class fiber gratings under the same coupling strength from Figs. 5 and 6. In Fig. 5, when the grating length of the core index modulated fiber gating is 11.72 mm, the minimum transmissivity of the main dip is 29.47 dB. Then, from Fig. 6 we can see that in order to obtain the same transmissivity, the grating length of the cladding index modulated grating is around 15 mm, which is only 1.28 times of core index modulated fiber gating. However, from the calculation of Section 3 the ratio of power attenuation coefficient of the two class fiber gratings is 11.19. Therefore, we can draw the conclusion that the losses of the cladding index modulated fiber grating are smaller
In this paper, we present a new type of fiber grating with only cladding index modulated. The new grating is analyzed under a three-layer model with the coupling-mode theory and demonstrates its reliable spectrum filter characteristic. Meanwhile, we have theoretically shown that the new cladding index modulated fiber gratings have lower attenuation compared with traditional fiber core index modulated grating. Furthermore, the configuration of the structure of the new grating is flexible. We can reconfigure the grating to meet various requirements. Acknowledgments This work is supported by the National Science Fund for Distinguished Young Scholars under Project 60125513 and Jiangsu Province Natural Science Foundation of China under Project BK2004207. References [1] K.O. Hill, Y. Fujii, D.C. Johnson, B.S. Kawasaki, Appl. Phys. Lett. 32 (1978) 647. [2] K.O. Hill, B. Malo, F. Bilodeau, D.C. Johnson, J. Albert, Appl. Phys. Lett. 62 (10) (1993) 1035. [3] J. Zhang, P. Shum, N.Q. Ngo, X.P. Cheng, J.H. Ng, IEEE Photon. Technol. Lett. 15 (11) (2003) 1558. [4] Jean-Luc Archambault, S.G. Grubb, J. Lightwave Technol. 15 (8) (1997) 1378. [5] C.R. Giles, J. Lightwave Technol. 15 (8) (1997) 1391. [6] A.D. Kersey, M.A. Davis, H.J. Patrick, M. LeBlanc, K.P. Koo, C.G. Askins, M.A. Putnam, E. Joseph Friebele, J. Lightwave Technol. 15 (8) (1997) 1442. [7] Kai-Ping Chuang, Yinchieh Lai, Lih-Gen Sheu, OFC 2004 1 (2004) 74. [8] Kai-Ping Chuang, Yai-Ping Lai, Lih-Gen Sheu, Opt. Lett. 29 (4) (2004) 340. [9] Kai-Ping Chuang, Yinchieh Lai, Lih-Gen Sheu, IEEE Photon. Technol. Lett. 16 (3) (2004) 834. [10] A.W. Snyder, J.D. Love, Optical Waveguide Theory, Chapman and Hall, London, 1991 (Secs. 11-8 and 18-8). [11] Turan Erdogan, JOSA A 14 (1997) 1760. [12] Turan Erdogan, JOSA A 17 (2000) 2113.
Cladding index modulated fiber grating Na Chen 1, Binfeng Yun, Yiping Cui
*
Advanced Photonics Center, Southeast University, Nanjing 210096, China Received 8 May 2005; received in revised form 29 August 2005; accepted 8 September 2005
Abstract A new type of fiber grating with only cladding index modulation is presented. Characteristics of both cladding index modulated shortperiod fiber grating (FBG) and long-period fiber grating (LPFG) are analyzed. The calculation of the modes involved in this paper is based on a model of three-layer step-index fiber geometry. Transmission of a mode guided by the core through a cladding index modulated grating when evanescent field coupling occurs is analyzed with couple-mode theory. Evanescent field coupling causes a power flowing from the core to the cladding, so the attenuation of the new grating is analyzed as well. Lower attenuation, flexible spectral characteristics are demonstrated in comparison with traditional fiber core index modulated grating. Ó 2005 Elsevier B.V. All rights reserved. PACS: 42.81.Wg; 42.79.Dj; 42.81.Dp Keywords: Fiber gratings; Cladding index modulation; Couple-mode theory
1. Introduction Since Hill et al. [1] discovered the photosensitivity of germanium-doped-core optical fiber twenty years ago, photoimprinting grating in the core of optical fiber has been a primary method of fiber grating fabrication [2,3]. And fiber gratings as important fiber optics components have been widely used in fiber communication and sensor system [4–6]. Recently, various improvements have been made to fabricate more complex fiber grating with photosensitivity fiber [7–9]. However, this kind of doped-core photosensitive optical fiber always has inevitable great attenuation due to its heavy doped core. For example, the attenuation at 1550 nm of Single-Mode Photosensitive Fiber and Radiation Mode Suppression Photosensitive Fiber of INO is 63.6 dB/km and 53.2 dB/km, respectively, while the atten-
*
Corresponding author. Tel.: +86 025 83601769x18; fax: +86 025 83601769x17. E-mail addresses: [email protected] (N. Chen), [email protected] (Y. Cui). 1 Tel.: +86 025 83601769x13; fax: +86 025 83601769x17. 0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.09.017
uation of an ordinary Single-mode Fiber that is used in communication at the same wavelength is only 0.20 dB/km. In many applications, such as FBG as a basic reflector used as the end mirror of fiber lasers, such loss affects the performance seriously. In order to mitigate the attenuation of a fiber grating, we are proposing a new type of fiber grating in this paper. The core of the new grating, the area that contains the dominating energy distribution of the core mode in a fiber, is unperturbed, while the index modulation is only in fiber cladding, where the core mode is evanescent field. The structure of the new grating is depicted in Section 2. Furthermore, we can modulate the cladding index in certain ways such as through exposing the fiber to periodic ultraviolet directly if the cladding is photosensitive. The fabrication of core index modulated FBG and LPFG are mainly done using the well-known phase mask technique and amplitude mask technique, respectively. Provided fiber with only photosensitive cladding is available, cladding index modulated FBG and LPFG can be realized with the same techniques easily. Furthermore, the state of the art of photosensitive fiber fabrication assures such fiber is not difficult to be realized. This new modulation scheme
588
N. Chen et al. / Optics Communications 259 (2006) 587–591
could minimize the attenuation of the grating greatly, which will be proved in Section 3. Although the field of fundamental core mode carries much less energy in the cladding region than the fiber core, the periodic index change still could affect the fundamental core mode through its evanescent field and modes coupling occurs when proper phase matching condition is met. Both cladding index modulated FBG and LPFG are analyzed. The analysis procedure is detailed in Section 4. Section 5 gives the calculated spectra of the new class grating and some discussion compared with traditional core index modulated gratings. Conclusion is in Section 6. 2. Grating structure and index modulation The structure of this new grating is analyzed under the model of three-layer step-index fiber geometry, which is shown in Fig. 1. The three layers are the core layer, the cladding layer and the surrounding layer respectively. The core of the grating is with the diameter of 2a1. The index is uniform in the core with the value of n1. The cladding surrounds the core with the diameter of 2a2. The index of the cladding is not uniform but modulated in a periodical way. The period of the modulation is in distance. Therefore, the index in the cladding is n2(z). Both the core and cladding are immersed in the surrounding, which has the index of n3. Usually the surrounding is air and n3 = 1. In our analysis we assume that, when a phase grating is induced in the fiber, it exits only in the fiber cladding, changing the cladding index to n2 (z), but leave the core and surrounding index unchanged, as follows 8 r 6 a1 ; > < n1 ; nðr; zÞ ¼ n2 ðzÞ ¼ n2 þ dnðzÞ½1 þ m cosð2pz=KÞ; a1 < r 6 a2 ; > : n3 ; r > a2 . ð1Þ
Here, n1 is the fiber core index, K is the period of the grating, m is the induced index fringe modulation, where 0 6 m 6 1, dnðzÞ is the average induced index change. 3. Attenuation of cladding index modulated fiber grating The absorbing fiber can be treated as the perturbation of the nonabsorbent fiber when the absorption is slight. The absorption can be described by the addition of imaginary constants to the core and cladding index. Thus, the correction to the perturbed propagation constant is imaginary. If
Fig. 1. Diagram of cladding index modulated fiber grating, showing the refractive index, grating period and the radius of the core (a1) and of the cladding (a2).
we denote the correction by bi, then bi ¼ k gdnico þ ð1 gÞdnicl g, where g is the fraction of the modeÕs power flowing within the core [10]. Power absorption coefficients for the core and cladding is defined as aco ¼ 2kdnico and acl ¼ 2kdnicl . The power attenuation coefficient c is then expressible as c ¼ gaco þ ð1 gÞacl , and the modeÕs power is reduced to exp (cz) of its original value after distance z. The shape of the intensity distribution on the absorbing fiber does not change as the mode propagation, but its amplitude is reduced to exp(cz) of its original value after distance z. If the power absorption coefficient is uniform over the whole cross-section, then c = a and power lost by attenuation at any position is exactly equal to the power absorbed there. The two classes fiber gratings considered here correspond to the case aco 5 acl, when power flows across the interface to maintain the shape of the intensity distribution. In the case of core index modulated grating ðaco cl ¼ 0Þ, power flows to the core, while for the cladding index modulated grating ðaclco ¼ 0Þ, power flows to the cladding. For the convenience of comparison, we assume that the index modulated fractions of both gratings have the same power absorption coefficients aclcl ¼ aco co ¼ a and the unmodulated fractions are nonabsorbent aclco ¼ aco cl ¼ 0. Thus, the power attenuation coefficient c of the both classes grating are ccl ¼ gaclco þ ð1 gÞaclcl ¼ ð1 gÞa and cco ¼ gaco co þ cl co ð1 gÞaco cl ¼ ga, respectively. Therefore, we have c =c ¼ ð1 gÞa=ga ¼ ð1 gÞ=g, that means the power attenuation coefficient ratio of the two grating with different index modulated area is determined only by the fraction of the modeÕs power flowing within the core g. Because the power transmitted in fiber is mainly confined in the core, the cladding index modulated gratingÕs power attenuation coefficient will be much smaller than fiber core index modulated gratingÕs. The typical parameter of single-mode fiber used for communications purposes is fiber core radius a1 = 4.5 lm, cladding radius a2 = 62.5 lm. The fraction of the modeÕs power flowing within the core can be expressed as [10] u2 w2 K 20 ðwÞ g¼ 2 2þ 2 . ð2Þ v u K 1 ðwÞ Then we can obtain g ¼ 0:91796. Thus, the ratio of power attenuation coefficient of the two class fiber gratings is cco =ccl ¼ ð1 gÞ=g ¼ 11:19. So the insertion loss of the fiber grating is cL, where the L is the whole length of the photosensitive fiber inserting in the system. Because there is only a small fraction of EM field intensity in the fiber cladding, in order to have the same transmission/reflection characteristics, the claddingindex-modulated fiber grating should be longer or have stronger index modulation compared to the core index modulated one. The detailed analysis will be given in Section 5. However, the whole length inserting in the system of both class gratings can be regarded as the same. Therefore, the attenuation of fiber core index modulated
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grating is 11.19 times over cladding index modulated grating fabricated with the same type fiber. 4. Coupling coefficient The field expressions of fundamental core mode and cladding modes can be obtained by solving MaxwellÕs equations with proper boundary conditions. These modes transmit independently in the ordinary fiber. While in the grating periodical index is induced, this perturbation would cause the evanescent field coupling among the forward fundamental core mode, backward fundamental core mode and the cladding modes. Coupled-mode theory can be employed to analyze such interactions. According to the solving procedure of the coupled-mode theory, all the necessary propagation constants and coupling coefficients need to be calculated first. Propagation constants of both fundamental core mode and cladding modes can be obtained through solving dispersion relations in core and cladding, respectively [11,12]. Thus, the resonant wavelength can be determined by the phase matching condition, as well. Since fundamental core mode is so familiar and the exact field expressions of the cladding modes have been detailed in [11,12] we do not list the field components here. The following analysis follows that of [11], while the difference is the interaction between modes considered in this paper occurs in the cladding rather than in the fiber core. The transverse coupling coefficient between two modes m and l is K tml ¼
x 4
Zþ1Z
dx dy De Etm Etl .
ð3Þ
1
To calculate coupling coefficient, permittivity change De caused by the induced cladding index change should be obtained first. Because the grating considered is weak grating, dnðzÞ is in the order of 103–104, we can make the approximation De = e0D(n2) ffi 2e0nD(n). Combine with (1), we obtain 8 r 6 a1 ; > < 0; Deðr; zÞ 2e0 n2 dnðzÞ½1 þ m cosð2pz=KÞ; a1 < r 6 a2 ; > : 0; r > a2 .
589
Using the expression for fundamental core mode in the cladding region, and performing the overlap integrals of (6), we obtain the coupling constant for core-mode-coremode coupling 2p 2u2 1 1 2 2 a ½K ðwa2 =a1 Þ k V 2 K 21 ðwÞ a21 2 0 K 21 ðwa2 =a1 Þ a21 ½K 20 ðwÞ K 21 ðwÞ .
coco ðzÞ ¼ dnðzÞ j0101
ð8Þ
However, substituting the field expressions of both fundamental core mode and cladding mode in the cladding into (7), the analytical solution for the overlap integrals is not available. Coupling constant for core-mode-cladding-mode coupling must be evaluated numerically. Provided that all the necessary propagation constants and coupling coefficients have been obtained, simple twomode coupled-mode theory that involves constant-coefficient differential equations with slowly varying amplitudes can be employed. For a uniform grating considered in this paper, analytical solutions for the transmission in both FBG and LPFG are available. 5. Results and discussion In this section, we calculated the reflection spectra of cladding index modulated FBG and the transmission spectra of cladding index modulated LPFG based on the theory described above. 5.1. Cladding index modulated FBG Fig. 2 shows the theory value of the reflection spectrum of the cladding index modulation fiber Bragg grating (solid line). The parameters of the fiber grating under calculation are as follows: fiber core radius a1 = 2.625 lm, cladding radius a2 = 62.5 lm, fiber core index n1 = 1.458, cladding index n2 = 1.45, surrounding index n3 = 1, grating period K = 0.557846 lm, grating length L = 8 mm, average index
ð4Þ Then, the coupling coefficient can be written as K tml ðzÞ ¼ jml ðzÞ½1 þ m cosð2pz=KÞ.
ð5Þ
Then, we can deduce the coupling constant for core-modecore-mode coupling and core-mode-cladding-mode coupling by (3)–(5) as follows: Z a2 Z co 2 co 2 xe0 n2 dnðzÞ 2p E þ E Þ; ð6Þ jcoco ðzÞ ¼ d/ r drð 0101 r / 2 a1 0 Z a2 Z 2p xe0 n2 dnðzÞ cl co jclco d/ r drðEclr Eco r þ E / E/ Þ. 1m01 ðzÞ ¼ 2 a1 0 ð7Þ
Fig. 2. Theoretically calculated reflection spectra (in decibels) through a cladding index modulated FBG (solid line) and a core index modulated FBG (dash line).
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N. Chen et al. / Optics Communications 259 (2006) 587–591
change dn ¼ 5 104 , and the induced index fringe modulation m = 0.6. A comparison spectrum of the fiber core index modulated grating is shown in the same figure (dash line). It shows that cladding index modulated FBG has similar characters to core index modulated FBG. The main reflection peak is due to the evanescent field coupling between the forward guided mode and the backward one in the present of the periodic cladding index modulation. 5.2. Cladding index modulated LPFG We calculated the transmission spectra of the cladding index modulated LPFG to demonstrate the cladding-mode coupling is present and the result shows transmission spectrum of the new grating demonstrates the band stop filter characteristic. Based on the phase-matched condition described in forward sections the relationship between grating period and wavelength can be obtained by the calculation of the propagation constants of the guided and the various cladding modes of a fiber at each specific wavelength. The result is shown in Fig. 3. The wavelength at which mode-coupling will be enabled by a particular grating period can now be predicted. The vertical dash line in Fig. 3 shows the position of the period involved in our calculation. Fig. 4 shows the theory value of the transmission spectrum of the cladding index modulation fiber grating. The parameters of the fiber grating under calculation are as follows: fiber core radius a1 = 2.625 lm, cladding radius a2 = 62.5 lm, fiber core index n1 = 1.458, cladding index n2 = 1.45, surrounding index n3 = 1, grating period K = 576 lm, grating length L = 11.72 mm, average index change dn ¼ 5 104 , and the induced index fringe modulation m = 0.6. In order to compare the new grating with an ordinary grating, the transmission spectrum of a typical uniform fiber core index modulated grating with the same parameters is given in Fig. 5. The transmission spectra of both grating were presented simultaneously for the convenience of com-
Fig. 3. Theoretical determination of the relationship between grating period and wavelengths.
Fig. 4. Theoretically calculated transmission spectra (in decibels) through a uniform cladding index modulated LPFG.
prehension. We can conclude from Fig. 5 that under the same circumstances, the coupling in the cladding index modulated grating is much weaker than that in the core index modulated grating. Same conclusion can be drawn from Fig. 2. It is reasonable because the part of the fundamental core mode in the cladding carries much less energy than the one in the fiber core. The interactions due to the induced perturbation between the evanescent field of the fundamental core mode and the field of cladding modes in the cladding is much weaker than the field in the fiber core. Furthermore, there is also a difference of the resonance wavelength between the fiber core index modulated grating and the fiber cladding index modulated grating, which is because the center wavelength is determined by two factors, one is the phase matching condition, the other is coupling coefficient. Although the phase matching condition is the same under the assumed condition of same grating period, the coupling coefficients in both fiber gratings have a little difference due to the different interactions occurring in both gratings. Therefore, the center wavelength is a bit
Fig. 5. Comparison of transmission spectra (in decibels) through a uniform cladding index modulation fiber grating (dash line) and a uniform fiber core index modulated grating with the same parameters (solid line).
N. Chen et al. / Optics Communications 259 (2006) 587–591
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than the losses of the core index modulated fiber grating under the circumstance of similar transmission amplitudes. 6. Conclusion
Fig. 6. Growth of transmission spectrum with the grating length.
different, too. From another point of view, the indexes of cladding and the core are different. Although the grating period is the same, the two types of grating are not the same for a certain wavelength. However, this not means that the cladding index modulation LPFG cannot gain strong resonant coupling. It is well known that the spectral characteristics of the fiber grating are quite flexible, since practically one can vary numerous parameters to meet the various requirements. Both the induced index change and the grating length affect the coupling strength. In this paper, we lengthen the cladding index modulated grating temperately and get a better coupling. Fig. 6 shows the growth of the transmission spectrum with the increase of the grating length. When the grating length increases to 17 mm, the maximum transmissivity dip reached 60 dB. Additionally, we can compare the grating length of both class fiber gratings under the same coupling strength from Figs. 5 and 6. In Fig. 5, when the grating length of the core index modulated fiber gating is 11.72 mm, the minimum transmissivity of the main dip is 29.47 dB. Then, from Fig. 6 we can see that in order to obtain the same transmissivity, the grating length of the cladding index modulated grating is around 15 mm, which is only 1.28 times of core index modulated fiber gating. However, from the calculation of Section 3 the ratio of power attenuation coefficient of the two class fiber gratings is 11.19. Therefore, we can draw the conclusion that the losses of the cladding index modulated fiber grating are smaller
In this paper, we present a new type of fiber grating with only cladding index modulated. The new grating is analyzed under a three-layer model with the coupling-mode theory and demonstrates its reliable spectrum filter characteristic. Meanwhile, we have theoretically shown that the new cladding index modulated fiber gratings have lower attenuation compared with traditional fiber core index modulated grating. Furthermore, the configuration of the structure of the new grating is flexible. We can reconfigure the grating to meet various requirements. Acknowledgments This work is supported by the National Science Fund for Distinguished Young Scholars under Project 60125513 and Jiangsu Province Natural Science Foundation of China under Project BK2004207. References [1] K.O. Hill, Y. Fujii, D.C. Johnson, B.S. Kawasaki, Appl. Phys. Lett. 32 (1978) 647. [2] K.O. Hill, B. Malo, F. Bilodeau, D.C. Johnson, J. Albert, Appl. Phys. Lett. 62 (10) (1993) 1035. [3] J. Zhang, P. Shum, N.Q. Ngo, X.P. Cheng, J.H. Ng, IEEE Photon. Technol. Lett. 15 (11) (2003) 1558. [4] Jean-Luc Archambault, S.G. Grubb, J. Lightwave Technol. 15 (8) (1997) 1378. [5] C.R. Giles, J. Lightwave Technol. 15 (8) (1997) 1391. [6] A.D. Kersey, M.A. Davis, H.J. Patrick, M. LeBlanc, K.P. Koo, C.G. Askins, M.A. Putnam, E. Joseph Friebele, J. Lightwave Technol. 15 (8) (1997) 1442. [7] Kai-Ping Chuang, Yinchieh Lai, Lih-Gen Sheu, OFC 2004 1 (2004) 74. [8] Kai-Ping Chuang, Yai-Ping Lai, Lih-Gen Sheu, Opt. Lett. 29 (4) (2004) 340. [9] Kai-Ping Chuang, Yinchieh Lai, Lih-Gen Sheu, IEEE Photon. Technol. Lett. 16 (3) (2004) 834. [10] A.W. Snyder, J.D. Love, Optical Waveguide Theory, Chapman and Hall, London, 1991 (Secs. 11-8 and 18-8). [11] Turan Erdogan, JOSA A 14 (1997) 1760. [12] Turan Erdogan, JOSA A 17 (2000) 2113.