To suppress the GDR, several apodization profiles have been tested [8]. In this paper, we optimize the following apodization profiles in which we consider the bandwidth to be the effective element as GDR.
Half Tanh- half uniform profile (HTHU):
$$ {\overline{\delta n}}_{eff}\ (z) = \left\{\begin{array}{c}\hfill \frac{ \tan {h}^2\left(2a\frac{z}{L_g}\right)}{tan{h}^2(a)},\ 0\le z < \frac{L_g}{2}\hfill \\ {}\ 1\kern5.65em ,\ \frac{L_g}{2}\le z\le {L}_g\hfill \end{array}\ \right. $$
(3)
Half Exponential-half uniform profile (HEHU):
$$ {\overline{\delta n}}_{eff}\ (z)=\left\{\begin{array}{c}\hfill exp\left(-a\frac{{\left(z-\frac{L_g}{2}\right)}^4}{{L_g}^4}\right), 0\le z < \frac{L_g}{2}\hfill \\ {}1\kern8.3em ,\ \frac{L_g}{2}\le z\le {L}_g\hfill \end{array}\right. $$
(4)
Half Hamming-half uniform profile (HHHU):
$$ {\overline{\delta n}}_{eff}\ (z) = \left\{\begin{array}{c}\hfill \frac{\left(1+ acos\left(\pi \frac{2z-{L}_g}{L_g}\right)\right)}{\left(1+a\right)},\ 0\le z < \frac{L_g}{2}\hfill \\ {}1\kern9.35em ,\ \frac{\ {L}_g}{2}\le z\le {L}_g\hfill \end{array}\right. $$
(5)
Sharpness parameter a is used to control the sharpness of the apodization profile.
Reflectivity and group delay vary with the refractive index change, \( {\overline{\delta n}}_{eff}\ (z) \). Eq. 3, Eq. 4 and Eq. 5, show that the sharpness parameter a is the variable that determines the value of \( {\overline{\delta n}}_{eff}\ (z) \). Therefore, it can also be used to optimize the reflectivity and GDR. Varying parameter α will vary both reflectivity FWHM bandwidth, and GDR. However, minimizing GDR will be accompanied by the loss of reflectivity bandwidth. Figure 2 (a) and (b) respectively show the effect of implementing HTHU profile (Eq. 3) with α = 1 to the reflectivity bandwidth and GDR. It can be noticed that the bandwidth reduces significantly as the targeted GDR is achieved. This trade-off shows that the optimization of the apodization profile is very important to obtain optimum performance.
Selecting the optimum point of α solves a tradeoff between the maximum FWHM bandwidth and minimum GDR. Thus, we rely on a new method to manage this conflict. This method depends on the relationship between the normalized apodized mean GDR and the apodized FWHM bandwidth. Equations (6) and (7) represent the normalized GDR, F
ripple
and normalized FWHM bandwidth, F
bandwidth
, where A
apod
and A
unapod
are the apodized and unapodized peak to peak GDR amplitude respectively. BW
apod
and BW
unapod
are the apodized and unapodized FHWM bandwidth respectively.
$$ {F}_{ripple}=\frac{A_{apod}(a)}{A_{unapod}} $$
(6)
$$ {F}_{bandwidth}=\frac{B{W}_{apod}(a)}{B{W}_{unapod}} $$
(7)
The relationship between F
ripple
and F
bandwidth
for the HEHU apodization profile (Eq. 4) are shown in Fig. 3.
To obtain the optimum value of α, the distance d which is defined by Eq. (8) is evaluated.
$$ d(a)=\sqrt{\left({\left({F}_{ripple}(a)\right)}^2+\right(1-{\left({F}_{bandwidth}(a)\right)}^2} $$
(8)
Distance d is measured from the point F
ripple
= 0 and F
bandwidth
= 1, to any point on the curve, as depicted in Fig. 3.
The optimum point of α can be obtained at the minimum value of d, denoted as d
min
. Figure 4 (a), Fig. 4 (b) and Fig. 4 (c) show the values of d at different value of sharpness parameter for different apodization profile. d
1
, d
2
and d
3
are the shortest distance for HTHU, HEHU and HHHU profiles respectively. These results occur at a = 13 for HTHU profile, a = 47 for HEHU and a = 1 for HHHU (d
1
= 0.1209, d
2
= 0.1935, d
3
= 0.2556).
Figure 5 shows the relationship between F
bandwidth
and F
ripple
for HTHU, HEHU and HHHU apodization profiles during the optimization process. All apodization profiles manage to reach the minimum value of F
ripple
of around 0.12. In contrast, different values of bandwidth are achieved for different apodization profile. The FWHM bandwidth for HTHU profile is the closest to the targeted value (0.8 nm), which is approximately 96.27 %. HEHU profile obtained 85.56 % and 77.71 % is recorded for HHHU profile. Therefore, we decided that HTHU is the best among the 3 optimized apodization profiles, and is the best to be used in the design of the CFBG.
Figure 6 shows the reflectivity response of CFBG and corresponding group delay after applying the HTHU apodization profile with α = 13. In comparison to Fig. 1 (unapodized case), the FWHM bandwidth is reduced by around 0.03 nm. This reduction is acceptable as it may not significantly affect the characteristic of the system. Figure 7 and Fig. 8 show the reflectivity responses and corresponding group delay after applying HEHU and HHHU apodization profiles respectively. The values of α are 47 and 1 for HEHU and HHHU apodization profiles respectively. The percentage bandwidth loss in case of HEHU and HHHU are 14.5 % (0.115 nm) and 22 % (0.18 nm) respectively. On the other hand, with the best optimized apodization profiles, the GDR is reduced to the minimum extent. These results show that our target to design a CFBG that can become the CDC with optimum performance has been achieved.
