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Ultranarrow dual-transmission-band filter based on an all-fiber two-cavity structure composed of three fiber Bragg gratings
Optics Communications 313 (2014) 337–344
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Ultranarrow dual-transmission-band filter based on an all-fiber two-cavity structure composed of three fiber Bragg gratings Ou Xu n School of Information Engineering, Guangdong University of Technology, No. 100 Waihuan Xi Road, Guangzhou Higher Education Mega Center, Panyu District, Guangzhou 510006, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 July 2013 Received in revised form 4 October 2013 Accepted 4 October 2013 Available online 17 October 2013
An all-fiber flexible two-cavity structure composed of three fiber Bragg gratings (FBGs) is proposed to realize a kind of ultranarrow dual transmission band filter. General condition for transmission spectra with symmetric two resonant peaks is presented and analyzed. The influences of various structure parameters on transmission responses are simulated and discussed, including index modulation depths and lengths of three FBGs and cavity lengths. By properly designing the structure parameters, two identical and symmetric transmission peaks can be produced in the stop band. The 3-dB bandwidth of each peak can be as narrow as 1 pm or even less than 1 pm. The spacing between the two transmission peaks can be flexibly designed. The fabrication error tolerances of the structures, including inaccurate cavity lengths and index modulation depths of FBGs, are investigated. Comparing with the conventional single-cavity structure, the proposed two-cavity structures provide better control on the suppression of sidebands. In applications of fiber lasers, the proposed two-cavity structure with the ultranarrow dual transmission-band can help in building the dual-wavelength single-longitudinal-mode operation at room temperature. & 2013 Elsevier B.V. All rights reserved.
Keywords: Fiber grating Ultranarrow Dual-wavelength Fiber filter
1. Introduction Ultranarrow dual-transmission-band filters based on fiber gratings are all-fiber devices helpful for realizing singlelongitudinal-mode dual-wavelength fiber lasers and their potential applications in microwave generation, high resolution spectroscopy, and high sensitive sensing, etc. To achieve dual transmission-band responses of narrow 3-dB bandwidth, various structures based on fiber gratings had been proposed. By cascading two independent πphase-shifted FBGs, an ultranarrow dual-channel filter can be constructed. But it is difficult for the two independent FBGs to keep the same reflectivity and the same bandwidth. Two narrow transmission bands can be realized using a pair of FBGs also [1] with a very short cavity, usually in micrometer range. This traditional single-cavity Fabry–Perot (FP) structure has well-known limitations in resolution, bandwidth and free spectral range. By introducing multiple π-phaseshifted regions into a uniform FBG and properly choosing their locations and magnitudes, the transmission spectrum of the FBG can be tailored, and multiple narrow transmission bands inside the stop band can be opened up [2,3]. But the 3-dB bandwidth of each peak is as large as 0.1 nm in some cases. Phase-shifted FBGs can usually be fabricated by introducing a true phase shift using a piezoelectric
n
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transducer or a phase-shifted phase mask. However, the use of a phase-shifted phase mask is inflexible with high cost, and using a piezoelectric transducer requires nanometer precision [4]. Thus, an equivalent phase-shift technique which controls the sampling period has been proposed to obtain two ultranarrow transmission bands [5,6]. This technique has a significant advantage: it requires only micrometer precision instead of nanometer precision. By introducing one π-phase-shift at the center of a sampled FBG, or multiple π phase shifts in a sampled FBG with a symmetric structure, novel ultranarrow dual-channel filters have been proposed [7,8]. The π phaseshifted sampled FBGs can yield any desired spacing of the neighboring channels on flexibly changing the sampled period. In the fabrication of this kind of sampled FBG, the precision of the translation stage is 10 nm to realize good results. In this paper, we have proposed an all-fiber flexible two-cavity structure composed of three FBGs to realize a new kind of ultranarrow dual transmission band filter. In the former studies [9], this kind of two-cavity structures had been researched as a single wavelength filter with a narrow bandwidth. In this study, our purpose is dual wavelength filters with ultranarrow bandwidth, and the design principles are different from that in [9]. By adjusting the structure parameters, two identical and symmetric transmission peaks can be produced in the stop band. The 3-dB bandwidth of each peak is less than 1 pm. The spacing between the two transmission peaks can be flexibly designed. Comparing with the conventional single-cavity structure, the proposed
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two-cavity structures provide better control on the suppression of sidebands. This paper is organized as follows. In Section 2, the principle of the proposed two-cavity structure is described by an explicit expression of the intensity transmission coefficient. The general condition to produce two symmetric transmission peaks is presented in Section 3. Then, the characteristics of the transmission spectrum are calculated and analyzed with different structure parameters, including index modulation depths, lengths of FBGs and cavity lengths. In Section 4, we investigate the tolerance of various fabrication errors of presented structures, including inaccurate cavity lengths and grating index modulations. And some results are discussed in Section 5.
2. Principles Fig. 1 shows a schematic diagram of the two-cavity structure composed of three FBGs. Three FBGs can be designed to have different lengths li (i¼ 1,2,3) and index modulation depths to meet the design needs. Two cavities can also have different lengths Li (i¼ 1,2). Following the calculations for a two-cavity structure [9], the amplitude transmission can be written in the form t¼
1 ¼ T 11
1þ
t1 t2 t3 P1 P2 t1 n t2 n t1t2 2 r r P þ r r 3 P 22 þ n n r n1 r 3 P 21 P 22 2 1 t n1 1 t n2 2 t1t2
ð1aÞ
P 1 ¼ expðik0 nef f L1 Þ; P 2 ¼ expðik0 nef f L2 Þ; k0 ¼ 2π =λ
ð1bÞ
where r 1;2;3 and t 1;2;3 are the amplitude reflection and transmission coefficients, respectively for the corresponding gratings; nef f is the effective refractive index; and L1;2 are the lengths of the grating-free regions between FBGs, called as cavity lengths in the following part. For a symmetric structure with r 1 ¼ r 3 a r 2 , L1 ¼ L2 ¼ L, the intensity transmission can be derived as T ¼ tt n ¼
ð1 jr 1 j2 Þ2 ð1 jr 2 j2 Þ ð1 jr 1 j2 Þ2 ð1 jr 2 j2 Þ þ ½jr 2 jð1 þ jr 1 j2 Þ þ2jr 1 j cos ðϕr1 þ ϕr2 þ2k0 nef f L π Þ
ð2Þ where ϕr1;2 are the phase of the amplitude reflection coefficients and r 1;2 ¼ r 1;2 expðiϕr1;2 Þ. Setting T ¼ 1 in (2), one finds the general condition for unity transmission jr 2 jð1 þ jr 1 j2 Þ þ 2jr 1 j cos ðϕr1 þ ϕr2 þ 2k0 nef f L π Þ ¼ 0
ð3Þ
From (3), the following analytical results can be achieved: 1. For a given two-cavity structure in which all parameters are fixed, including grating lengths, periods, effective refractive index, refractive index modulation depth and the cavity lengths, Eq. (3) is related only to the wavelength λ. At the wavelengths which satisfy (3), the unity transmission will occur. 2. Differing from a conventional FP based on mirrors, the reflection of fiber gratings functions only in a limited wavelength range around the Bragg wavelength, and reflectivity is wavelength related. If the Bragg wavelengths of three FBGs in the two-cavity structure are the same, Eq. (3) at the Bragg
wavelength λB becomes jr 2 jð1 þ jr 1 j2 Þ þ2jr 1 j cos ð2k0 nef f LÞ ¼ 0
Because ϕr1 ¼ ϕr2 ¼ π =2 at λB . Then, if cos ð2k0 nef f LÞ ¼ 1, we have
L1
l2
L2
l3
Fig. 1. Schematic diagram of the proposed two-cavity structure composed of three FBGs.
2jr 1 j 1 þ jr 1 j2
ð5aÞ
ð2m þ 1ÞλB 4nef f
ð5bÞ
jr 2 j ¼ L¼
Where m is an arbitrary integer. The above results are the same as the derivations in Ref. [9] (Eqs. (9) and (10)). 3. If the cosine factor in (3) is symmetric about λB , the transmission spectrum of the two-cavity structure will also be symmetric about λB . In fact, the symmetric transmission spectrum can be presented when cos ðϕr1 þ ϕr2 þ 2k0 nef f L π Þ equals 1 at λB , from which it can be derived that the cavity length satisfies (5b). This will be proved in the following simulations. Based on the above theoretical model, the transmission spectra of symmetric two-cavity structures formed by three FBGs are calculated and discussed below. The influences of the structure parameters on the transmission spectrum are investigated. Through optimizing the parameters of the structure, the design principles of transmission spectrum with two ultranarrow transmission peaks symmetric about λB are concluded. 3. Simulations and analysis In the simulations below, the effective refractive index of the fiber is nef f ¼1.45. After writing FBGs into the fiber, the average index of the fiber at the locations of FBGs takes the value nef f þ δnef f , in which δnef f is the averaged dc index change. The reflectivity of every forming FBG is determined by its length and index modulation depth Δn ¼ υ δnef f , and υ is the fringe visibility of the index change and always selected as 1. In order to obtain the ideal results, which means two peaks appearing symmetrically about the central wavelength and unity transmission, all three FBGs in two-cavity FP structures have the same center wavelength, here fixed as 1550 nm. For gratings with certain lengths and index modulation depths, central wavelength of 1550 nm can be reached by adjusting the periods. 3.1. General condition for transmission spectra with symmetric two resonant Peaks According to (3), resonant peaks in transmission spectra of two-cavity structures are produced at the cross-points of the following function curves: y1 ¼ cos ðϕr1 þ ϕr2 þ 2k0 nef f L π Þ y2 ¼
jr 2 jð1 þ jr 1 j2 Þ 2jr 1 j
ð6aÞ ð6bÞ
For fixed structure parameters, y1 and y2 are both wavelengthrelated functions. Since y1 equals 1 at λB , y1 4 1 for other wavelengths around λB . So we have y2 4 1 to produce two symmetric resonant peaks about λB . The general condition can be written as jr 2 jo
l1
ð4Þ
2jr 1 j ð1 þ jr 1 j2 Þ
ð7Þ
Fig. 2 shows the transmission spectrum of a two-cavity structure as a function of wavelength λ with parameters satisfying (5b) and (7). The three gratings have the same length of 10 mm.
O. Xu / Optics Communications 313 (2014) 337–344
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1.0
16X10-5
0.6
12X10-5
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20X10-5
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-0.8
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4X10-5 0.6
0.2
T y1 y2
-0.6
8X10-5
159.945
1550.05
100.055
Wavelength (nm) Fig. 2. The transmission spectrum of a two-cavity structure and the curves of y1 and y2. (For interpretation of the reference to color in this figure, the reader is referred to the web version of this article).
150.055
Wavelength (nm) Fig. 3. The transmission spectra as a function of wavelength for various values of Δn2 .
2.2
3.2. Influences of FBGs on transmission spectra We have seen in Section 3.1 that when (5b) and (7) are satisfied, double symmetrical peaks appear in the transmission spectrum. In order to investigate the characteristics of the dual-wavelength spectra, including bandwidth and spacement of two peaks, the influences of parameters about three FBGs in two-cavity structures on transmission curves are simulated and discussed below. In expression (7), the smaller-than symbol means that the values of r2 which can produce two symmetric peaks of unity transmission have a range, if r1 has been fixed. As discussed above, the amplitude reflection coefficient r 1;2;3 are determined by their lengths l1;2;3 and index modulation depths Δn1;2;3 . If condition (7) satisfied, the influences of different grating lengths or index modulation depths on the transmission spectra can be simulated and discussed. Firstly, the influences of different grating index modulation depths on the transmission spectra are calculated and investigated. The transmission spectra around the central wavelength are plotted in Fig. 3 for various values of Δn2 , and Δn1 ¼ Δn3 ¼ 10 10 5 . The three gratings have the same length of 10 mm, and cavity lengths are L1 ¼ L2 ¼ 1:336474 mm. From Fig. 3, when Δn2 ¼ 20 10 5 ((5a) holds), a single peak is produced at the center wavelength. With decreasing the value of Δn2 ((7) holds), two symmetric peaks occur around the center wavelength. The separation between double peaks becomes bigger with smaller Δn2 . Since the separation and bandwidth (BW) of double peaks are important parameters for the structure, these two parameters are plotted against Δn2 in Fig. 4. Here, the bandwidth is defined as the 3dB-bandwidth for one of the double peaks. The separation is the wavelength spacing between the points of the biggest
45 40
2.0
BW (pm)
35 1.8
30 25
1.6
20 15
1.4
Separation (pm)
For symmetric structures, the first and third gratings are the same, and here their index modulations are set asΔn1 ¼ Δn3 ¼ 5 10 5 . The index modulation of middle grating is chosen as Δn2 ¼ 2 10 5 to satisfy (7). The cavity lengths are L1 ¼ L2 ¼ 1:336474 mm, satisfying (5b). In the figure, the curves of y1 and y2 are also shown as red dashed and blue dotted lines. In Fig. 2, the symmetric double peaks are produced about the central Bragg wavelength, as predicted above. And the unity transmission occurs at the cross points of y1 and y2, which are pointed out by arrows in the figure. In fact, this condition can be called an over-coupling state. If r2 increases to the limiting case in which expression (5a) holds, we can find unity transmission at the Bragg wavelength, already investigated thoroughly in Ref. [9].
10 5
1.2
0 0
2
4
6
8
10
12
14
16
Δn2 /10-5 Fig. 4. The influence ofΔn2 on the bandwidth and the separation between double peaks.
transmission coefficient in the two peaks and other parameters in the structure are the same as in Fig. 3. From the calculation results in Fig. 4, it is shown that the bandwidth decreases and the separation increases with decreasing Δn2 . When Δn2 is decreased to 1 10 5, the bandwidth of one of the double peaks is about 1.25 pm. Further decreasing Δn2 can make the bandwidth less than 1 pm easily. In other words, for a smallerΔn2 , double peaks of narrower bandwidth and bigger separation can be achieved. Moreover, for three gratings of the same index modulation depths, double peaks also can be produced by adjusting the length l2 of the middle grating. The transmission spectra around the central wavelength are plotted in Fig. 5 for various values of l2, and l1 ¼ l3 ¼10 mm. The three gratings have the same index modulation depths Δn1;2;3 ¼ 10 10 5 , andL1 ¼ L2 ¼ 1:336474 mm. From Fig. 5, when l2 ¼20 mm ((5a) holds), a single peak is produced at the center wavelength. With decreasing the value of l2 ((7) holds), two symmetric peaks occur around the center wavelength. The separation between double peaks becomes bigger with smaller l2. In Fig. 6, the separation and bandwidth (BW) of double peaks are plotted against l2. With decreasing l2, the spacing between two peaks becomes bigger, and the bandwidth decreases first, but increases when l2 is smaller than 10 mm. From simulations above, whenever index modulation depth or length of the middle grating decreases, the two peaks seem to separate. However, these two kinds of changes affect the whole spectra differently. Since the transmission spectra are symmetric,
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20mm 16mm 12mm 8mm 4mm
1.0
4
0.8
0.6
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Transmission
0.8
8
12
1.0
0.4 0.2
0.6 0.4
16 0.2
20
0.0 0.0
159.945
100.055
150.055 1549.90
Wavelength (nm)
1549.92
1549.94
1549.96
Wavelength (nm)
Fig. 5. The transmission spectra as a function of wavelength for various values of l2. 4.0 80
3.6
20mm
8mm
40
2.4 2.0
20
Separation (pm)
60
2.8
1.6
0.8
Transmission
3.2
BW (pm)
12mm 16mm
1.0
0
2
4
6
8
10
12
14
4mm
0.4 0.2
0
1.2
0.6
0.0
16
l2 (mm)
1549.90
1549.92
1549.96
Transmission
Fig. 7. Transmission spectra in shorter wavelength range with different (a) index modulation depths and (b) lengths of the middle grating. The values of index modulation depths (Δn2 105) or lengths are labeled near the corresponding curves.
1.0 0.8 0.6 0.4 0.2 0.0 1549.90
Transmission
only the range of shorter wavelengths is shown in Fig. 7(a) and (b), respectively for the transmission spectra with different Δn2 or l2 . The other parameters of the structures in Fig. 7(a) (Fig. 7(b)) are the same as in Figs. 3 and 5. There are side-peaks in the shortwavelength range. In Fig. 7(a), the side-peaks tend to move towards the central wavelength with decreasing Δn2 . But in Fig. 7(b), they tend to move away from the central wavelength with decreasing l2 . This phenomenon can be explained by the change of effective length for the middle FBG [10]. Since decreasing the index modulation depth of the grating results in the increase in the effective length of the grating, which increases the cavity length, it is seen that the spacing between side-peaks and central-peaks decreases in Fig. 7(a). On the contrary , decreasing the length of the grating causes decrease in the effective length of the grating. Thus, the spacing grows in Fig. 7(b). Therefore, choosing the middle grating in the structures with large index modulation depth and short length is beneficial for enlarging the spacing between central double-peaks and side-peaks. Although the wavelength spacing between double peaks increases with decrease of Δn2 , as shown in Fig. 3, further decreasing Δn2 will also result in side-peaks moving towards central wavelength. Since decreasing the length of the middle grating can make side-peaks to move away from central wavelength, we can simultaneously decrease the values of index modulation depth and length for middle grating to prevent side-peaks getting closer. Fig. 8(a) shows the transmission spectrum for two-cavity structure with Δn2 ¼ 2 10 5 and l2 ¼ 10 mm, and the other parameters are the same as in simulations of Fig. 3. Two side-peaks are produced besides the
1549.94
Wavelength (nm)
Fig. 6. The influence of l2 on the bandwidth and the separation between double peaks.
1549.95
1550.00
1550.05
1550.10
1550.00
1550.05
1550.10
1.0 0.8 0.6 0.4 0.2 0.0 1549.90
1549.95
Wavelength (nm) Fig. 8. Transmission spectra of two-cavity structures with Δn2 ¼ 2 10 5 . (a)l2 ¼ 10 mm, and (b) l2 ¼ 2 mm.
central double peaks. In Fig. 8(b), the middle grating has Δn2 ¼ 2 10 5 and l2 ¼ 2 mm and the other parameters are unchanged. It can be found that the spacing between central double peaks is further
O. Xu / Optics Communications 313 (2014) 337–344
341
1.0
1.0
Transmission
0.6 0.4 0.2
0.8 0.6 0.4 0.2 0.0
L=5.345094mm L=2.672681mm L=1.336474mm
0.0 -0.2 1549.90
1549.95
1550.00
1550.05
1.0
1550.10
Wavelength (nm) Fig. 9. Transmission spectra with different cavity lengths.
enlarged, and at the same time the two two side-peaks are effectively suppressed.
Transmission
Transmission
0.8
0.8 0.6 0.4 0.2 0.0
3.3. Influences of cavity lengths on transmission spectra
4. Fabrication imperfection analysis In the theoretical analysis section above, all the parameters of the structures are strictly chosento obtain the ideal results, which
Transmission
1.0 0.8 0.6 0.4 0.2 0.0 1549.90
1549.95
1550.00
1550.05
1550.10
Wavelength (nm) Fig. 10. Transmission spectra with different values of cavity lengths: (a) L ¼ 1.34 mm; (b) L ¼1.33 mm;and (c) L ¼1.336474 mm.
1.0 0.8
y (1.34) 1
0.6
y (1.33) 1
0.4
y1 (y2 )
Besides gratings in the two-cavity structures, cavity length is another important parameter. Fig. 9 shows transmission spectra for different cavity lengths L, with the three gratings unchanged. The values of cavity lengths satisfy (5b), and the accuracy is set as 1 nm in order to obtain the ideal results. From the figure, with decreasing L, side-peaks move away from the central wavelength. And at the same time the spacing between two central peaks becomes slightly bigger, which results from the increasing period of the cosine function in (6a). As shown in Fig. 2, decreasing the value of L can increase the period of y1 to increase, with y2 unchanged, so that the spacing between their cross points is enlarged. In simulations above the values of cavity lengths are strictly chosen to satisfy (5b) within an accuracy of 1 nm, in order to obtain the ideal symmetric double peaks around 1550 nm. However, it is practically difficult to precisely control the cavity length with accuracy of nm range. In Fig. 10(a) and (b), the transmission spectra for cavity lengths of 1.34 mm and 1.33 mm are simulated, in contrast to the spectrum for cavity length of 1.336474 mm in Fig. 10(c). From the Fig. 10(a) and (b), we can find that the two resonant peaks do not remain symmetric about 1550 nm, shifting to the longer or shorter wavelengths for cavity length of 1.34 mm or 1.33 mm. Meanwhile the transmission of the two peaks remains about 100%. Thus, the inaccuracy of the cavity length affects the wavelength positions mainly where two resonant peaks appear. Further detailed investigations about the influence of non-ideal cavity lengths on transmission spectra are shown in the next section. In fact, the positions of two peaks are determined by the cross points of (6a) and (6b), seen in Section 3.1. The functions y1 and y2 corresponding to Fig. 10(a) and (b) are shown in Fig. 11. And the cross points of y1 and y2 pointed by red or black arrows are the positions where two peaks appear in Fig. 10(a) or (b). As discussed in Section 2, since the wavelength at which y1 in (6a) equals 1 is not the central wavelength 1550 nm when cavity length is 1.33 mm or 1.34 mm, the transmission spectra are not symmetric, as shown in Fig. 11 In other words, the double peaks with different wavelengths can be achieved by slightly adjusting cavity lengths in the structure.
0.2 0.0 -0.2
y2
-0.4 -0.6 -0.8 -1.0
1549.95
1550.00
1550.05
Wavelength (nm) Fig. 11. The functions y1 and y2 corresponding to Fig. 10(a) and (b).(For interpretation of the reference to color in this figure, the reader is referred to the web version of this article).
are the two resonant peaks appearing symmetrically about the center wavelength 1550 nm, and at the same time their transmission reaching nearly 100% (higher than 0.99 at least) . For example, the length of cavity is with an accuracy of 1 nm, and FBG1 is
O. Xu / Optics Communications 313 (2014) 337–344
completely the same as FBG3, etc. However, these ideal conditions can hardly be satisfied in experimental realizations. Among all the structure parameters, the cavity lengths and grating index modulation are more difficult to control accurately. In the following part, we focus on the cavity length and grating index modulation, and investigate the effects of their imperfections on the two-cavity structure performance, mainly in terms of the shifts of two resonant peaks, FWHM and transmission of the peaks. First, the influence of cavity length was investigated. Assuming that FBGs are written by the phase mask technique [11], the fiber is irradiated through a uniform phase mask with an ultraviolet laser beam. The cavity between two adjacent FBGs is formed by shifting the fiber along the fiber axis between the two exposures using a translation stage. For the cavity length, we suppose that the accuracy can reach 0.1 μm order. Thus, a commercial translation stage containing ESP300 motion controller/driver of Newport Corporation can meet our need. Choosing the same parameters as in Fig. 10(c) with cavity length of 0.1 μm accuracy, the transmission spectrum is simulated and shown in Fig. 12. The parameters concerned are listed in Table 1, including the wavelength shifts of two resonant peaks from the ideal condition in which the cavity length is with an accuracy of 1 nm, FWHM and transmission of two peaks. In the table, Δλ (Δλ) presents the wavelength shift of the short-wavelength peak (long-wavelength peak) from the ideal value, FWHM1 (FWHM2) presents full width at half maximum of the short-wavelength peak (long-wavelength peak), and T1 (T2) presents the transmission of the short-wavelength peak (longwavelength peak). There is one–one correspondence between the curves in Fig. 12 and the data in Table 1. The solid spectrum curve presents an ideal reference, and the dashed and dotted curves present two cases of inaccurate cavity lengths. From Fig. 12, it can be found that if the cavity lengths reach an accuracy of only 0.1 μm, the resonant peaks move away form the ideal condition. When L1¼ L2¼ 1.3365 mm (dashed red line), the two resonant peaks move towards long
wavelengths by about a few picometers, the peak of the longer wavelength gets wider FWHM than the ideal condition of about 0.3 pm, and the other peak of the shorter wavelength gets narrower FWHM than the ideal condition of about 0.3 pm. When L1¼ 1.3364 mm and L2¼1.3365 mm (dotted blue line), two resonant peaks move towards short wavelengths by about a few picometers, the peak of the shorter wavelength gets wider than the ideal condition of about 0.4 pm, and the other peak of the longer wavelength gets narrower than the ideal condition of about 0.3 pm. Meanwhile, the peak transmissions of these two imperfect structures remain very high, although the values are not exactly the same as in ideal condition. Hence, the inaccuracy of the cavity length affects the wavelength positions mainly where two resonant peaks appear, and slightly affects the FWHM and the maximum transmission of the peaks.
1.0 0.8
Transmission
342
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Wavelength(nm)
1.0
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0.6
1549.95
1550.00
1550.05
1550.10
1549.90
1549.95
Wavelength(nm)
1550.00
Wavelength(nm)
Fig. 12. Transmission spectra of the two-cavity structures with inaccurate cavity lengths. (For interpretation of the reference to color in this figure, the reader is referred to the web version of this article).
Fig. 13. Transmission spectra of the two-cavity structures with inaccurate grating index modulations. (For interpretation of the reference to color in this figure, the reader is referred to the web version of this article).
Table 1 Influence of the cavity length on transmission spectral parameters.
Solid line (Ideal) Dashed line Dotted line
L1, L2 /mm
Δλ1, Δλ2 /pm
FWHM1, FWHM2 /pm
T1, T2
1.336474, 1.336474 1.3365, 1.3365 1.3364, 1.3365
0, 0 6.7, 6.6 6.7, 5.7
2.3, 2.3 2.0, 2.6 2.7, 2.0
0.99836, 0.99821 0.99811, 0.99927 0.99278, 0.9921
O. Xu / Optics Communications 313 (2014) 337–344
Table 2 Influence of the grating index modulations on transmission spectral parameters. Δn1 ð10 5 Þ, Δn2 ð10 5 Þ Ideal 10, 10 (a) 11, 10 9, 10 (b) 11, 9 11, 9 (L1 ¼L2¼ 1.3365 mm)
Δλ1,Δλ2 /pm
FWHM1, FWHM2 /pm
T1, T2
0, 0 1.7, 2.8 2.0, 3.4 0.3, 0.7 6.5, 5.8
2.3, 2.3 2.0, 1.7 2.6, 2.7 2.3, 1.7 2.0, 1.8
0.99836,0.99821 0.99481, 0.90846 0.98978, 0.89904 0.97045, 0.68614 0.94981, 0.63939
Transmission
1.0 0.8 0.6 0.4 0.2 0.0
Transmission
1.0 0.8 0.6 0.4 0.2 0.0
1.0
Transmission
Second, the effect of the inaccurate grating index modulation is investigated. For the ideal two-cavity structure, the two end FBGs should be the same, and the reflectivity of the middle FBG r2 needs to satisfy (7) to produce two symmetric resonant peaks. The expression (7) means that r2 has a range for fixed r1. And, as discussed in Section 3, for fixed r1, different values of r2 can produce two symmetric resonant peaks with different spacements. Thus, here we focus on the influences of different grating index modulations for FBG1 and FBG3 on the two resonant peaks. Because of non-ideal fabrication process including variations in the laser beam power or pointing direction, errors in the phase mask, wobbling of the translation stages and non-uniformities of the fiber properties, writing two FBGs with the equal strength is not very easy. Based on the ideal parameters of Fig. 10(c), the two plots labeled (a) and (b) in Fig. 13 show the simulation results of different kinds of grating index modulation deviations, corresponding to data in Table 2. There is one–one correspondence between the curve in each plot and the data in the table. In each plot, the black solid spectrum curve presents an ideal reference. In Fig. 13(a), the transmission spectra are calculated when only the index modulation of FBG1 varies and other parameters are ideal. From the data labeled (a) in Table 2, the effects of increasing (decreasing) the index modulation of FBG1 include shifting of the two resonant peaks toward long (short) wavelength, decreasing (increasing) of the FWHM of two peaks, and a slight reduction of the maximum transmission (the peak transmission is reduced by about 10% in the worst case of our simulations). When only the index modulation of FBG3 varies and other parameters are ideal, simulations show the same results as those of inaccurate index modulation of FBG1, since the structure is symmetric. With further deterioration, the index modulation depths of FBG1 and FBG3 both deviate from the ideal condition in different ways, and calculated spectra are shown in Fig. 13(b). The results show that the transmissions of two peaks are reduced much more, especially the peak of long wavelength (see the red dashed curve in Fig. 13(b)). Combining the inaccurate cavity lengths, two resonant peaks shift to the long wavelength and transmission of two peaks is further decreased, as shown in blue dotted curve of Fig. 13(b). Thus, in order to obtain two peaks with nearly equal transmission of high value (high than 0.9), at least one of the two end FBGs should be carefully fabricated to have accurate index modulation depth, or close to the ideal value. From the analysis presented above, we can conclude that, the main effect of cavity length inaccuracy is shifting the wavelengths of two resonant peaks by about a few picometers, while the FWHM and transmission of two peaks change a little, provided that the accuracy of the cavity length can reach 0.1 μm order; when the two end FBGs do not have the same index modulation depth and deviate from the ideal values, the main influences are shifting of two peaks to long or short wavelengths by about a few picometers and reduction of the peak transmission, while the FWHM changes of two peaks are very small (less than 0.6 pm in the simulation results of Table 2). It is worth noting that the
343
0.8 0.6 0.4 0.2 0.0 1549.95
1550.00
1550.05
Wavelength (nm) Fig. 14. Transmission spectra for (a) single-cavity FP of two FBGs and (b, c) twocavity structure of three FBGs.
FWHM of two peaks remain at picometer order in the simulation results inspite of various imperfections of fabrication. Therefore, the imperfect fabrications have little influence on the ultranarrow bandwidth characteristic of two resonant peaks.
5. Discussion Traditionally, Fabry–Perot structures are based on two equal FBGs, which have a single cavity. For example, the transmission spectrum of a single-cavity FP is plotted in Fig. 14(a), showing four transmission peaks distributed with almost equal spacing in the stop-band. The composing FBGs have the same parameters: grating length 10 mm, index modulation 1 10 4, and central wavelength 1550 nm. The cavity length is about 20 mm. For this kind of FP, changing parameters, including cavity length, grating lengths and index modulations, may result in shifting the transmission peaks on the whole towards longer or shorter wavelengths, or inducing bigger or smaller spacing between two neighboring peaks. This means the changing trends of all the transmission peaks and the spacing between them are almost the same. Hence, if the spacing between two central peaks around 1550 nm becomes smaller, the distribution of all peaks becomes denser and the number of peaks will increase in the stop-band.
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However, the two-cavity structures can overcome this problem. Based on the single-cavity FP in Fig. 14(a), the third FBG is inserted in the middle position of the cavity with grating length 10 mm, index modulation 5 10 5 and Bragg wavelength 1550 nm. As shown in Fig. 14(b), only two peaks remain around central wavelength in the stop-band, and the spacing between them becomes smaller. Increasing the index modulation 1 10 4, the two peaks gets closer in Fig. 14(c). Therefore, the two-cavity structures are helpful to produce two symmetric peaks around central wavelength, at the same time excluding the influences of other side-peaks. From the calculation results of Fig. 14(b), the 3 dB bandwidths of two transmission peaks are both smaller than 0.9 pm. In fact, increasing the index modulation depths of the two end gratings and decreasing the index modulation depth of the central grating can further reduce the value of the 3-dB bandwidth for transmission peaks. For example, the transmission spectrum is simulated for the two-cavity structure in which two end gratings have index modulation depth 2 10 4 and central grating has 4 10 5, and the other parameters are the same as in the calculation for Fig. 14(b). The calculation results show that the 3-dB bandwidths of two transmission peaks are smaller than 0.02 pm. The ultra-narrow transmission-band character is beneficial for many applications. As a direct application, a fiber linear or ring laser incorporating the twocavity structure with ultra-narrow dual transmission-band can help the dual-wavelength single-longitudinal-mode operation at room temperature [4,6]. In [4], a dual-transmission-band FBG filter implemented using the EPS technique has been demonstrated, with an estimated 3-dB bandwidths of 0.2 pm. However, in calculating the transmission spectrum, the maximum index modulation is 1 10 3. From simulations above, the proposed two-cavity structures can realize dual transmission band of 3 dB bandwidth as narrow as 0.02 pm just under the index modulation depth of 2 10 4. Therefore, from the point of view of fabrication, the proposed structure provides an approach the low demand on fiber gratings to realize ultra-narrow transmission bands. From the physical point of view, the filter proposed in this paper is analogous to a multimirror FP interferometer where the mirrors are substituted by equally spaced in-fiber gratings. By allowing the individual grating sections to have different lengths
and index modulation depths, a resonant structure consisting of elements with different individual reflectivities can be obtained. Through designing the parameters of gratings and cavities, two symmetrical transmission bands can be obtained with required spacing and bandwidth.
6. Conclusion Based on a two-cavity structure composed of three FBGs, we have proposed novel ultranarrow dual transmission-band all-fiber structures. The proposed structure can produce symmetrical spectra. The two transmission bands have the same bandwidth and even strength, and the ultranarrow bandwidth can be less than 1 pm. The spacing between the two passbands can be flexibly adjusted by changing the structure parameters, including index modulation depths and lengths of FBGs and the cavity lengths. Detailed spectral characteristics are investigated by numerical simulations. The influences of the structure parameters on two transmission bands are discussed. And we investigated the influence of imperfections of cavity lengths and index modulation depths of two end FBGs on the device performance. The twocavity structure is shown to be tolerant to a variety of potential fabrication errors while maintaining the two-peak operation with ultranarrow bandwidth. This new kind of dual passband filter of ultranarrow bandwidth can help in realizing the dual-wavelength single-longitudinal-mode fiber lasers.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
D. Chen, H. Fu, W. Liu, Y. Wei, S. He, Electronics Letters 7 (2008) 459. G.P. Agrawal, S. Radic, IEEE Photon. Technol. Lett. 6 (1994) 995. R. Zengerle, O. Leminger, J. Lightwave Technol. 13 (1995) 2354. X. Chen, Opt. Lett. 16 (2005) 2068. Y. Dai, X. Chen, D. Jiang, S. Xie, C. Fan, IEEE Photon. Technol. Lett 10 (2004) 2284. Y. Dai, X.F. Chen, J. Sun, Y. Yao, S.Z. Xie, IEEE Photon. Technol. Lett 18 (2006) 1964. X.M. Liu, A. Lin, G. Sun, D.S. Moon, D. Hwang, Y. Chung, Appl. Opt. 30 (2008) 5637. X.M. Liu, J. Lightwave Technol. 13 (2008) 1885. S. Xu, S. Lu, S. Feng, J. Opt. Soc. Am. A 3 (2009) 639. Y.O. Barmenkov, D. Zalvidea, S. Torres-Peiro, J.L. Cruz, M.V. Andres, Opt. Express 14 (2006) 6394. [11] K.O. Hill, Gerald Meltz, J. Lightwave Technol. 15 (1997) 1263.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Ultranarrow dual-transmission-band filter based on an all-fiber two-cavity structure composed of three fiber Bragg gratings Ou Xu n School of Information Engineering, Guangdong University of Technology, No. 100 Waihuan Xi Road, Guangzhou Higher Education Mega Center, Panyu District, Guangzhou 510006, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 July 2013 Received in revised form 4 October 2013 Accepted 4 October 2013 Available online 17 October 2013
An all-fiber flexible two-cavity structure composed of three fiber Bragg gratings (FBGs) is proposed to realize a kind of ultranarrow dual transmission band filter. General condition for transmission spectra with symmetric two resonant peaks is presented and analyzed. The influences of various structure parameters on transmission responses are simulated and discussed, including index modulation depths and lengths of three FBGs and cavity lengths. By properly designing the structure parameters, two identical and symmetric transmission peaks can be produced in the stop band. The 3-dB bandwidth of each peak can be as narrow as 1 pm or even less than 1 pm. The spacing between the two transmission peaks can be flexibly designed. The fabrication error tolerances of the structures, including inaccurate cavity lengths and index modulation depths of FBGs, are investigated. Comparing with the conventional single-cavity structure, the proposed two-cavity structures provide better control on the suppression of sidebands. In applications of fiber lasers, the proposed two-cavity structure with the ultranarrow dual transmission-band can help in building the dual-wavelength single-longitudinal-mode operation at room temperature. & 2013 Elsevier B.V. All rights reserved.
Keywords: Fiber grating Ultranarrow Dual-wavelength Fiber filter
1. Introduction Ultranarrow dual-transmission-band filters based on fiber gratings are all-fiber devices helpful for realizing singlelongitudinal-mode dual-wavelength fiber lasers and their potential applications in microwave generation, high resolution spectroscopy, and high sensitive sensing, etc. To achieve dual transmission-band responses of narrow 3-dB bandwidth, various structures based on fiber gratings had been proposed. By cascading two independent πphase-shifted FBGs, an ultranarrow dual-channel filter can be constructed. But it is difficult for the two independent FBGs to keep the same reflectivity and the same bandwidth. Two narrow transmission bands can be realized using a pair of FBGs also [1] with a very short cavity, usually in micrometer range. This traditional single-cavity Fabry–Perot (FP) structure has well-known limitations in resolution, bandwidth and free spectral range. By introducing multiple π-phaseshifted regions into a uniform FBG and properly choosing their locations and magnitudes, the transmission spectrum of the FBG can be tailored, and multiple narrow transmission bands inside the stop band can be opened up [2,3]. But the 3-dB bandwidth of each peak is as large as 0.1 nm in some cases. Phase-shifted FBGs can usually be fabricated by introducing a true phase shift using a piezoelectric
n
Tel.: þ 86 2039322475. E-mail address: [email protected]
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.10.016
transducer or a phase-shifted phase mask. However, the use of a phase-shifted phase mask is inflexible with high cost, and using a piezoelectric transducer requires nanometer precision [4]. Thus, an equivalent phase-shift technique which controls the sampling period has been proposed to obtain two ultranarrow transmission bands [5,6]. This technique has a significant advantage: it requires only micrometer precision instead of nanometer precision. By introducing one π-phase-shift at the center of a sampled FBG, or multiple π phase shifts in a sampled FBG with a symmetric structure, novel ultranarrow dual-channel filters have been proposed [7,8]. The π phaseshifted sampled FBGs can yield any desired spacing of the neighboring channels on flexibly changing the sampled period. In the fabrication of this kind of sampled FBG, the precision of the translation stage is 10 nm to realize good results. In this paper, we have proposed an all-fiber flexible two-cavity structure composed of three FBGs to realize a new kind of ultranarrow dual transmission band filter. In the former studies [9], this kind of two-cavity structures had been researched as a single wavelength filter with a narrow bandwidth. In this study, our purpose is dual wavelength filters with ultranarrow bandwidth, and the design principles are different from that in [9]. By adjusting the structure parameters, two identical and symmetric transmission peaks can be produced in the stop band. The 3-dB bandwidth of each peak is less than 1 pm. The spacing between the two transmission peaks can be flexibly designed. Comparing with the conventional single-cavity structure, the proposed
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two-cavity structures provide better control on the suppression of sidebands. This paper is organized as follows. In Section 2, the principle of the proposed two-cavity structure is described by an explicit expression of the intensity transmission coefficient. The general condition to produce two symmetric transmission peaks is presented in Section 3. Then, the characteristics of the transmission spectrum are calculated and analyzed with different structure parameters, including index modulation depths, lengths of FBGs and cavity lengths. In Section 4, we investigate the tolerance of various fabrication errors of presented structures, including inaccurate cavity lengths and grating index modulations. And some results are discussed in Section 5.
2. Principles Fig. 1 shows a schematic diagram of the two-cavity structure composed of three FBGs. Three FBGs can be designed to have different lengths li (i¼ 1,2,3) and index modulation depths to meet the design needs. Two cavities can also have different lengths Li (i¼ 1,2). Following the calculations for a two-cavity structure [9], the amplitude transmission can be written in the form t¼
1 ¼ T 11
1þ
t1 t2 t3 P1 P2 t1 n t2 n t1t2 2 r r P þ r r 3 P 22 þ n n r n1 r 3 P 21 P 22 2 1 t n1 1 t n2 2 t1t2
ð1aÞ
P 1 ¼ expðik0 nef f L1 Þ; P 2 ¼ expðik0 nef f L2 Þ; k0 ¼ 2π =λ
ð1bÞ
where r 1;2;3 and t 1;2;3 are the amplitude reflection and transmission coefficients, respectively for the corresponding gratings; nef f is the effective refractive index; and L1;2 are the lengths of the grating-free regions between FBGs, called as cavity lengths in the following part. For a symmetric structure with r 1 ¼ r 3 a r 2 , L1 ¼ L2 ¼ L, the intensity transmission can be derived as T ¼ tt n ¼
ð1 jr 1 j2 Þ2 ð1 jr 2 j2 Þ ð1 jr 1 j2 Þ2 ð1 jr 2 j2 Þ þ ½jr 2 jð1 þ jr 1 j2 Þ þ2jr 1 j cos ðϕr1 þ ϕr2 þ2k0 nef f L π Þ
ð2Þ where ϕr1;2 are the phase of the amplitude reflection coefficients and r 1;2 ¼ r 1;2 expðiϕr1;2 Þ. Setting T ¼ 1 in (2), one finds the general condition for unity transmission jr 2 jð1 þ jr 1 j2 Þ þ 2jr 1 j cos ðϕr1 þ ϕr2 þ 2k0 nef f L π Þ ¼ 0
ð3Þ
From (3), the following analytical results can be achieved: 1. For a given two-cavity structure in which all parameters are fixed, including grating lengths, periods, effective refractive index, refractive index modulation depth and the cavity lengths, Eq. (3) is related only to the wavelength λ. At the wavelengths which satisfy (3), the unity transmission will occur. 2. Differing from a conventional FP based on mirrors, the reflection of fiber gratings functions only in a limited wavelength range around the Bragg wavelength, and reflectivity is wavelength related. If the Bragg wavelengths of three FBGs in the two-cavity structure are the same, Eq. (3) at the Bragg
wavelength λB becomes jr 2 jð1 þ jr 1 j2 Þ þ2jr 1 j cos ð2k0 nef f LÞ ¼ 0
Because ϕr1 ¼ ϕr2 ¼ π =2 at λB . Then, if cos ð2k0 nef f LÞ ¼ 1, we have
L1
l2
L2
l3
Fig. 1. Schematic diagram of the proposed two-cavity structure composed of three FBGs.
2jr 1 j 1 þ jr 1 j2
ð5aÞ
ð2m þ 1ÞλB 4nef f
ð5bÞ
jr 2 j ¼ L¼
Where m is an arbitrary integer. The above results are the same as the derivations in Ref. [9] (Eqs. (9) and (10)). 3. If the cosine factor in (3) is symmetric about λB , the transmission spectrum of the two-cavity structure will also be symmetric about λB . In fact, the symmetric transmission spectrum can be presented when cos ðϕr1 þ ϕr2 þ 2k0 nef f L π Þ equals 1 at λB , from which it can be derived that the cavity length satisfies (5b). This will be proved in the following simulations. Based on the above theoretical model, the transmission spectra of symmetric two-cavity structures formed by three FBGs are calculated and discussed below. The influences of the structure parameters on the transmission spectrum are investigated. Through optimizing the parameters of the structure, the design principles of transmission spectrum with two ultranarrow transmission peaks symmetric about λB are concluded. 3. Simulations and analysis In the simulations below, the effective refractive index of the fiber is nef f ¼1.45. After writing FBGs into the fiber, the average index of the fiber at the locations of FBGs takes the value nef f þ δnef f , in which δnef f is the averaged dc index change. The reflectivity of every forming FBG is determined by its length and index modulation depth Δn ¼ υ δnef f , and υ is the fringe visibility of the index change and always selected as 1. In order to obtain the ideal results, which means two peaks appearing symmetrically about the central wavelength and unity transmission, all three FBGs in two-cavity FP structures have the same center wavelength, here fixed as 1550 nm. For gratings with certain lengths and index modulation depths, central wavelength of 1550 nm can be reached by adjusting the periods. 3.1. General condition for transmission spectra with symmetric two resonant Peaks According to (3), resonant peaks in transmission spectra of two-cavity structures are produced at the cross-points of the following function curves: y1 ¼ cos ðϕr1 þ ϕr2 þ 2k0 nef f L π Þ y2 ¼
jr 2 jð1 þ jr 1 j2 Þ 2jr 1 j
ð6aÞ ð6bÞ
For fixed structure parameters, y1 and y2 are both wavelengthrelated functions. Since y1 equals 1 at λB , y1 4 1 for other wavelengths around λB . So we have y2 4 1 to produce two symmetric resonant peaks about λB . The general condition can be written as jr 2 jo
l1
ð4Þ
2jr 1 j ð1 þ jr 1 j2 Þ
ð7Þ
Fig. 2 shows the transmission spectrum of a two-cavity structure as a function of wavelength λ with parameters satisfying (5b) and (7). The three gratings have the same length of 10 mm.
O. Xu / Optics Communications 313 (2014) 337–344
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1.0
16X10-5
0.6
12X10-5
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Transmission
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0.2 0.0 -0.2 -0.4
-0.8
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4X10-5 0.6
0.2
T y1 y2
-0.6
8X10-5
159.945
1550.05
100.055
Wavelength (nm) Fig. 2. The transmission spectrum of a two-cavity structure and the curves of y1 and y2. (For interpretation of the reference to color in this figure, the reader is referred to the web version of this article).
150.055
Wavelength (nm) Fig. 3. The transmission spectra as a function of wavelength for various values of Δn2 .
2.2
3.2. Influences of FBGs on transmission spectra We have seen in Section 3.1 that when (5b) and (7) are satisfied, double symmetrical peaks appear in the transmission spectrum. In order to investigate the characteristics of the dual-wavelength spectra, including bandwidth and spacement of two peaks, the influences of parameters about three FBGs in two-cavity structures on transmission curves are simulated and discussed below. In expression (7), the smaller-than symbol means that the values of r2 which can produce two symmetric peaks of unity transmission have a range, if r1 has been fixed. As discussed above, the amplitude reflection coefficient r 1;2;3 are determined by their lengths l1;2;3 and index modulation depths Δn1;2;3 . If condition (7) satisfied, the influences of different grating lengths or index modulation depths on the transmission spectra can be simulated and discussed. Firstly, the influences of different grating index modulation depths on the transmission spectra are calculated and investigated. The transmission spectra around the central wavelength are plotted in Fig. 3 for various values of Δn2 , and Δn1 ¼ Δn3 ¼ 10 10 5 . The three gratings have the same length of 10 mm, and cavity lengths are L1 ¼ L2 ¼ 1:336474 mm. From Fig. 3, when Δn2 ¼ 20 10 5 ((5a) holds), a single peak is produced at the center wavelength. With decreasing the value of Δn2 ((7) holds), two symmetric peaks occur around the center wavelength. The separation between double peaks becomes bigger with smaller Δn2 . Since the separation and bandwidth (BW) of double peaks are important parameters for the structure, these two parameters are plotted against Δn2 in Fig. 4. Here, the bandwidth is defined as the 3dB-bandwidth for one of the double peaks. The separation is the wavelength spacing between the points of the biggest
45 40
2.0
BW (pm)
35 1.8
30 25
1.6
20 15
1.4
Separation (pm)
For symmetric structures, the first and third gratings are the same, and here their index modulations are set asΔn1 ¼ Δn3 ¼ 5 10 5 . The index modulation of middle grating is chosen as Δn2 ¼ 2 10 5 to satisfy (7). The cavity lengths are L1 ¼ L2 ¼ 1:336474 mm, satisfying (5b). In the figure, the curves of y1 and y2 are also shown as red dashed and blue dotted lines. In Fig. 2, the symmetric double peaks are produced about the central Bragg wavelength, as predicted above. And the unity transmission occurs at the cross points of y1 and y2, which are pointed out by arrows in the figure. In fact, this condition can be called an over-coupling state. If r2 increases to the limiting case in which expression (5a) holds, we can find unity transmission at the Bragg wavelength, already investigated thoroughly in Ref. [9].
10 5
1.2
0 0
2
4
6
8
10
12
14
16
Δn2 /10-5 Fig. 4. The influence ofΔn2 on the bandwidth and the separation between double peaks.
transmission coefficient in the two peaks and other parameters in the structure are the same as in Fig. 3. From the calculation results in Fig. 4, it is shown that the bandwidth decreases and the separation increases with decreasing Δn2 . When Δn2 is decreased to 1 10 5, the bandwidth of one of the double peaks is about 1.25 pm. Further decreasing Δn2 can make the bandwidth less than 1 pm easily. In other words, for a smallerΔn2 , double peaks of narrower bandwidth and bigger separation can be achieved. Moreover, for three gratings of the same index modulation depths, double peaks also can be produced by adjusting the length l2 of the middle grating. The transmission spectra around the central wavelength are plotted in Fig. 5 for various values of l2, and l1 ¼ l3 ¼10 mm. The three gratings have the same index modulation depths Δn1;2;3 ¼ 10 10 5 , andL1 ¼ L2 ¼ 1:336474 mm. From Fig. 5, when l2 ¼20 mm ((5a) holds), a single peak is produced at the center wavelength. With decreasing the value of l2 ((7) holds), two symmetric peaks occur around the center wavelength. The separation between double peaks becomes bigger with smaller l2. In Fig. 6, the separation and bandwidth (BW) of double peaks are plotted against l2. With decreasing l2, the spacing between two peaks becomes bigger, and the bandwidth decreases first, but increases when l2 is smaller than 10 mm. From simulations above, whenever index modulation depth or length of the middle grating decreases, the two peaks seem to separate. However, these two kinds of changes affect the whole spectra differently. Since the transmission spectra are symmetric,
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20mm 16mm 12mm 8mm 4mm
1.0
4
0.8
0.6
Transmission
Transmission
0.8
8
12
1.0
0.4 0.2
0.6 0.4
16 0.2
20
0.0 0.0
159.945
100.055
150.055 1549.90
Wavelength (nm)
1549.92
1549.94
1549.96
Wavelength (nm)
Fig. 5. The transmission spectra as a function of wavelength for various values of l2. 4.0 80
3.6
20mm
8mm
40
2.4 2.0
20
Separation (pm)
60
2.8
1.6
0.8
Transmission
3.2
BW (pm)
12mm 16mm
1.0
0
2
4
6
8
10
12
14
4mm
0.4 0.2
0
1.2
0.6
0.0
16
l2 (mm)
1549.90
1549.92
1549.96
Transmission
Fig. 7. Transmission spectra in shorter wavelength range with different (a) index modulation depths and (b) lengths of the middle grating. The values of index modulation depths (Δn2 105) or lengths are labeled near the corresponding curves.
1.0 0.8 0.6 0.4 0.2 0.0 1549.90
Transmission
only the range of shorter wavelengths is shown in Fig. 7(a) and (b), respectively for the transmission spectra with different Δn2 or l2 . The other parameters of the structures in Fig. 7(a) (Fig. 7(b)) are the same as in Figs. 3 and 5. There are side-peaks in the shortwavelength range. In Fig. 7(a), the side-peaks tend to move towards the central wavelength with decreasing Δn2 . But in Fig. 7(b), they tend to move away from the central wavelength with decreasing l2 . This phenomenon can be explained by the change of effective length for the middle FBG [10]. Since decreasing the index modulation depth of the grating results in the increase in the effective length of the grating, which increases the cavity length, it is seen that the spacing between side-peaks and central-peaks decreases in Fig. 7(a). On the contrary , decreasing the length of the grating causes decrease in the effective length of the grating. Thus, the spacing grows in Fig. 7(b). Therefore, choosing the middle grating in the structures with large index modulation depth and short length is beneficial for enlarging the spacing between central double-peaks and side-peaks. Although the wavelength spacing between double peaks increases with decrease of Δn2 , as shown in Fig. 3, further decreasing Δn2 will also result in side-peaks moving towards central wavelength. Since decreasing the length of the middle grating can make side-peaks to move away from central wavelength, we can simultaneously decrease the values of index modulation depth and length for middle grating to prevent side-peaks getting closer. Fig. 8(a) shows the transmission spectrum for two-cavity structure with Δn2 ¼ 2 10 5 and l2 ¼ 10 mm, and the other parameters are the same as in simulations of Fig. 3. Two side-peaks are produced besides the
1549.94
Wavelength (nm)
Fig. 6. The influence of l2 on the bandwidth and the separation between double peaks.
1549.95
1550.00
1550.05
1550.10
1550.00
1550.05
1550.10
1.0 0.8 0.6 0.4 0.2 0.0 1549.90
1549.95
Wavelength (nm) Fig. 8. Transmission spectra of two-cavity structures with Δn2 ¼ 2 10 5 . (a)l2 ¼ 10 mm, and (b) l2 ¼ 2 mm.
central double peaks. In Fig. 8(b), the middle grating has Δn2 ¼ 2 10 5 and l2 ¼ 2 mm and the other parameters are unchanged. It can be found that the spacing between central double peaks is further
O. Xu / Optics Communications 313 (2014) 337–344
341
1.0
1.0
Transmission
0.6 0.4 0.2
0.8 0.6 0.4 0.2 0.0
L=5.345094mm L=2.672681mm L=1.336474mm
0.0 -0.2 1549.90
1549.95
1550.00
1550.05
1.0
1550.10
Wavelength (nm) Fig. 9. Transmission spectra with different cavity lengths.
enlarged, and at the same time the two two side-peaks are effectively suppressed.
Transmission
Transmission
0.8
0.8 0.6 0.4 0.2 0.0
3.3. Influences of cavity lengths on transmission spectra
4. Fabrication imperfection analysis In the theoretical analysis section above, all the parameters of the structures are strictly chosento obtain the ideal results, which
Transmission
1.0 0.8 0.6 0.4 0.2 0.0 1549.90
1549.95
1550.00
1550.05
1550.10
Wavelength (nm) Fig. 10. Transmission spectra with different values of cavity lengths: (a) L ¼ 1.34 mm; (b) L ¼1.33 mm;and (c) L ¼1.336474 mm.
1.0 0.8
y (1.34) 1
0.6
y (1.33) 1
0.4
y1 (y2 )
Besides gratings in the two-cavity structures, cavity length is another important parameter. Fig. 9 shows transmission spectra for different cavity lengths L, with the three gratings unchanged. The values of cavity lengths satisfy (5b), and the accuracy is set as 1 nm in order to obtain the ideal results. From the figure, with decreasing L, side-peaks move away from the central wavelength. And at the same time the spacing between two central peaks becomes slightly bigger, which results from the increasing period of the cosine function in (6a). As shown in Fig. 2, decreasing the value of L can increase the period of y1 to increase, with y2 unchanged, so that the spacing between their cross points is enlarged. In simulations above the values of cavity lengths are strictly chosen to satisfy (5b) within an accuracy of 1 nm, in order to obtain the ideal symmetric double peaks around 1550 nm. However, it is practically difficult to precisely control the cavity length with accuracy of nm range. In Fig. 10(a) and (b), the transmission spectra for cavity lengths of 1.34 mm and 1.33 mm are simulated, in contrast to the spectrum for cavity length of 1.336474 mm in Fig. 10(c). From the Fig. 10(a) and (b), we can find that the two resonant peaks do not remain symmetric about 1550 nm, shifting to the longer or shorter wavelengths for cavity length of 1.34 mm or 1.33 mm. Meanwhile the transmission of the two peaks remains about 100%. Thus, the inaccuracy of the cavity length affects the wavelength positions mainly where two resonant peaks appear. Further detailed investigations about the influence of non-ideal cavity lengths on transmission spectra are shown in the next section. In fact, the positions of two peaks are determined by the cross points of (6a) and (6b), seen in Section 3.1. The functions y1 and y2 corresponding to Fig. 10(a) and (b) are shown in Fig. 11. And the cross points of y1 and y2 pointed by red or black arrows are the positions where two peaks appear in Fig. 10(a) or (b). As discussed in Section 2, since the wavelength at which y1 in (6a) equals 1 is not the central wavelength 1550 nm when cavity length is 1.33 mm or 1.34 mm, the transmission spectra are not symmetric, as shown in Fig. 11 In other words, the double peaks with different wavelengths can be achieved by slightly adjusting cavity lengths in the structure.
0.2 0.0 -0.2
y2
-0.4 -0.6 -0.8 -1.0
1549.95
1550.00
1550.05
Wavelength (nm) Fig. 11. The functions y1 and y2 corresponding to Fig. 10(a) and (b).(For interpretation of the reference to color in this figure, the reader is referred to the web version of this article).
are the two resonant peaks appearing symmetrically about the center wavelength 1550 nm, and at the same time their transmission reaching nearly 100% (higher than 0.99 at least) . For example, the length of cavity is with an accuracy of 1 nm, and FBG1 is
O. Xu / Optics Communications 313 (2014) 337–344
completely the same as FBG3, etc. However, these ideal conditions can hardly be satisfied in experimental realizations. Among all the structure parameters, the cavity lengths and grating index modulation are more difficult to control accurately. In the following part, we focus on the cavity length and grating index modulation, and investigate the effects of their imperfections on the two-cavity structure performance, mainly in terms of the shifts of two resonant peaks, FWHM and transmission of the peaks. First, the influence of cavity length was investigated. Assuming that FBGs are written by the phase mask technique [11], the fiber is irradiated through a uniform phase mask with an ultraviolet laser beam. The cavity between two adjacent FBGs is formed by shifting the fiber along the fiber axis between the two exposures using a translation stage. For the cavity length, we suppose that the accuracy can reach 0.1 μm order. Thus, a commercial translation stage containing ESP300 motion controller/driver of Newport Corporation can meet our need. Choosing the same parameters as in Fig. 10(c) with cavity length of 0.1 μm accuracy, the transmission spectrum is simulated and shown in Fig. 12. The parameters concerned are listed in Table 1, including the wavelength shifts of two resonant peaks from the ideal condition in which the cavity length is with an accuracy of 1 nm, FWHM and transmission of two peaks. In the table, Δλ (Δλ) presents the wavelength shift of the short-wavelength peak (long-wavelength peak) from the ideal value, FWHM1 (FWHM2) presents full width at half maximum of the short-wavelength peak (long-wavelength peak), and T1 (T2) presents the transmission of the short-wavelength peak (longwavelength peak). There is one–one correspondence between the curves in Fig. 12 and the data in Table 1. The solid spectrum curve presents an ideal reference, and the dashed and dotted curves present two cases of inaccurate cavity lengths. From Fig. 12, it can be found that if the cavity lengths reach an accuracy of only 0.1 μm, the resonant peaks move away form the ideal condition. When L1¼ L2¼ 1.3365 mm (dashed red line), the two resonant peaks move towards long
wavelengths by about a few picometers, the peak of the longer wavelength gets wider FWHM than the ideal condition of about 0.3 pm, and the other peak of the shorter wavelength gets narrower FWHM than the ideal condition of about 0.3 pm. When L1¼ 1.3364 mm and L2¼1.3365 mm (dotted blue line), two resonant peaks move towards short wavelengths by about a few picometers, the peak of the shorter wavelength gets wider than the ideal condition of about 0.4 pm, and the other peak of the longer wavelength gets narrower than the ideal condition of about 0.3 pm. Meanwhile, the peak transmissions of these two imperfect structures remain very high, although the values are not exactly the same as in ideal condition. Hence, the inaccuracy of the cavity length affects the wavelength positions mainly where two resonant peaks appear, and slightly affects the FWHM and the maximum transmission of the peaks.
1.0 0.8
Transmission
342
0.4 0.2 0.0 1549.90
1550.00
1550.05
1550.10
1550.05
1550.10
1.0
0.8
Transmission
0.8
0.6
0.4
0.6 0.4
0.2
0.2
0.0
0.0
1549.90
1549.95
Wavelength(nm)
1.0
Transmission
0.6
1549.95
1550.00
1550.05
1550.10
1549.90
1549.95
Wavelength(nm)
1550.00
Wavelength(nm)
Fig. 12. Transmission spectra of the two-cavity structures with inaccurate cavity lengths. (For interpretation of the reference to color in this figure, the reader is referred to the web version of this article).
Fig. 13. Transmission spectra of the two-cavity structures with inaccurate grating index modulations. (For interpretation of the reference to color in this figure, the reader is referred to the web version of this article).
Table 1 Influence of the cavity length on transmission spectral parameters.
Solid line (Ideal) Dashed line Dotted line
L1, L2 /mm
Δλ1, Δλ2 /pm
FWHM1, FWHM2 /pm
T1, T2
1.336474, 1.336474 1.3365, 1.3365 1.3364, 1.3365
0, 0 6.7, 6.6 6.7, 5.7
2.3, 2.3 2.0, 2.6 2.7, 2.0
0.99836, 0.99821 0.99811, 0.99927 0.99278, 0.9921
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Table 2 Influence of the grating index modulations on transmission spectral parameters. Δn1 ð10 5 Þ, Δn2 ð10 5 Þ Ideal 10, 10 (a) 11, 10 9, 10 (b) 11, 9 11, 9 (L1 ¼L2¼ 1.3365 mm)
Δλ1,Δλ2 /pm
FWHM1, FWHM2 /pm
T1, T2
0, 0 1.7, 2.8 2.0, 3.4 0.3, 0.7 6.5, 5.8
2.3, 2.3 2.0, 1.7 2.6, 2.7 2.3, 1.7 2.0, 1.8
0.99836,0.99821 0.99481, 0.90846 0.98978, 0.89904 0.97045, 0.68614 0.94981, 0.63939
Transmission
1.0 0.8 0.6 0.4 0.2 0.0
Transmission
1.0 0.8 0.6 0.4 0.2 0.0
1.0
Transmission
Second, the effect of the inaccurate grating index modulation is investigated. For the ideal two-cavity structure, the two end FBGs should be the same, and the reflectivity of the middle FBG r2 needs to satisfy (7) to produce two symmetric resonant peaks. The expression (7) means that r2 has a range for fixed r1. And, as discussed in Section 3, for fixed r1, different values of r2 can produce two symmetric resonant peaks with different spacements. Thus, here we focus on the influences of different grating index modulations for FBG1 and FBG3 on the two resonant peaks. Because of non-ideal fabrication process including variations in the laser beam power or pointing direction, errors in the phase mask, wobbling of the translation stages and non-uniformities of the fiber properties, writing two FBGs with the equal strength is not very easy. Based on the ideal parameters of Fig. 10(c), the two plots labeled (a) and (b) in Fig. 13 show the simulation results of different kinds of grating index modulation deviations, corresponding to data in Table 2. There is one–one correspondence between the curve in each plot and the data in the table. In each plot, the black solid spectrum curve presents an ideal reference. In Fig. 13(a), the transmission spectra are calculated when only the index modulation of FBG1 varies and other parameters are ideal. From the data labeled (a) in Table 2, the effects of increasing (decreasing) the index modulation of FBG1 include shifting of the two resonant peaks toward long (short) wavelength, decreasing (increasing) of the FWHM of two peaks, and a slight reduction of the maximum transmission (the peak transmission is reduced by about 10% in the worst case of our simulations). When only the index modulation of FBG3 varies and other parameters are ideal, simulations show the same results as those of inaccurate index modulation of FBG1, since the structure is symmetric. With further deterioration, the index modulation depths of FBG1 and FBG3 both deviate from the ideal condition in different ways, and calculated spectra are shown in Fig. 13(b). The results show that the transmissions of two peaks are reduced much more, especially the peak of long wavelength (see the red dashed curve in Fig. 13(b)). Combining the inaccurate cavity lengths, two resonant peaks shift to the long wavelength and transmission of two peaks is further decreased, as shown in blue dotted curve of Fig. 13(b). Thus, in order to obtain two peaks with nearly equal transmission of high value (high than 0.9), at least one of the two end FBGs should be carefully fabricated to have accurate index modulation depth, or close to the ideal value. From the analysis presented above, we can conclude that, the main effect of cavity length inaccuracy is shifting the wavelengths of two resonant peaks by about a few picometers, while the FWHM and transmission of two peaks change a little, provided that the accuracy of the cavity length can reach 0.1 μm order; when the two end FBGs do not have the same index modulation depth and deviate from the ideal values, the main influences are shifting of two peaks to long or short wavelengths by about a few picometers and reduction of the peak transmission, while the FWHM changes of two peaks are very small (less than 0.6 pm in the simulation results of Table 2). It is worth noting that the
343
0.8 0.6 0.4 0.2 0.0 1549.95
1550.00
1550.05
Wavelength (nm) Fig. 14. Transmission spectra for (a) single-cavity FP of two FBGs and (b, c) twocavity structure of three FBGs.
FWHM of two peaks remain at picometer order in the simulation results inspite of various imperfections of fabrication. Therefore, the imperfect fabrications have little influence on the ultranarrow bandwidth characteristic of two resonant peaks.
5. Discussion Traditionally, Fabry–Perot structures are based on two equal FBGs, which have a single cavity. For example, the transmission spectrum of a single-cavity FP is plotted in Fig. 14(a), showing four transmission peaks distributed with almost equal spacing in the stop-band. The composing FBGs have the same parameters: grating length 10 mm, index modulation 1 10 4, and central wavelength 1550 nm. The cavity length is about 20 mm. For this kind of FP, changing parameters, including cavity length, grating lengths and index modulations, may result in shifting the transmission peaks on the whole towards longer or shorter wavelengths, or inducing bigger or smaller spacing between two neighboring peaks. This means the changing trends of all the transmission peaks and the spacing between them are almost the same. Hence, if the spacing between two central peaks around 1550 nm becomes smaller, the distribution of all peaks becomes denser and the number of peaks will increase in the stop-band.
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However, the two-cavity structures can overcome this problem. Based on the single-cavity FP in Fig. 14(a), the third FBG is inserted in the middle position of the cavity with grating length 10 mm, index modulation 5 10 5 and Bragg wavelength 1550 nm. As shown in Fig. 14(b), only two peaks remain around central wavelength in the stop-band, and the spacing between them becomes smaller. Increasing the index modulation 1 10 4, the two peaks gets closer in Fig. 14(c). Therefore, the two-cavity structures are helpful to produce two symmetric peaks around central wavelength, at the same time excluding the influences of other side-peaks. From the calculation results of Fig. 14(b), the 3 dB bandwidths of two transmission peaks are both smaller than 0.9 pm. In fact, increasing the index modulation depths of the two end gratings and decreasing the index modulation depth of the central grating can further reduce the value of the 3-dB bandwidth for transmission peaks. For example, the transmission spectrum is simulated for the two-cavity structure in which two end gratings have index modulation depth 2 10 4 and central grating has 4 10 5, and the other parameters are the same as in the calculation for Fig. 14(b). The calculation results show that the 3-dB bandwidths of two transmission peaks are smaller than 0.02 pm. The ultra-narrow transmission-band character is beneficial for many applications. As a direct application, a fiber linear or ring laser incorporating the twocavity structure with ultra-narrow dual transmission-band can help the dual-wavelength single-longitudinal-mode operation at room temperature [4,6]. In [4], a dual-transmission-band FBG filter implemented using the EPS technique has been demonstrated, with an estimated 3-dB bandwidths of 0.2 pm. However, in calculating the transmission spectrum, the maximum index modulation is 1 10 3. From simulations above, the proposed two-cavity structures can realize dual transmission band of 3 dB bandwidth as narrow as 0.02 pm just under the index modulation depth of 2 10 4. Therefore, from the point of view of fabrication, the proposed structure provides an approach the low demand on fiber gratings to realize ultra-narrow transmission bands. From the physical point of view, the filter proposed in this paper is analogous to a multimirror FP interferometer where the mirrors are substituted by equally spaced in-fiber gratings. By allowing the individual grating sections to have different lengths
and index modulation depths, a resonant structure consisting of elements with different individual reflectivities can be obtained. Through designing the parameters of gratings and cavities, two symmetrical transmission bands can be obtained with required spacing and bandwidth.
6. Conclusion Based on a two-cavity structure composed of three FBGs, we have proposed novel ultranarrow dual transmission-band all-fiber structures. The proposed structure can produce symmetrical spectra. The two transmission bands have the same bandwidth and even strength, and the ultranarrow bandwidth can be less than 1 pm. The spacing between the two passbands can be flexibly adjusted by changing the structure parameters, including index modulation depths and lengths of FBGs and the cavity lengths. Detailed spectral characteristics are investigated by numerical simulations. The influences of the structure parameters on two transmission bands are discussed. And we investigated the influence of imperfections of cavity lengths and index modulation depths of two end FBGs on the device performance. The twocavity structure is shown to be tolerant to a variety of potential fabrication errors while maintaining the two-peak operation with ultranarrow bandwidth. This new kind of dual passband filter of ultranarrow bandwidth can help in realizing the dual-wavelength single-longitudinal-mode fiber lasers.
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