1. Introduction

The light dispersion relation is commonly linear or close to linear in complex refractive media and therefore the frequency mode spacing is uniform or close to uniform. However, a nonlinear dispersion can be important in some cases such as for obtaining photon Bose-Einstein condensation (BEC) in one-dimensional (1D) cavities [1,2]. The unusual mode separation changes the frequency dependence of the density of mode states to meet the requirement for BEC in 1D. Photon-BEC occurs when the population in the modes above a cutoff mode are filled and from that point any additional pumping populates the lowest frequency mode near the cutoff. This scenario of the exhaustion of the high modes depends on the density of states that needs to be sublinear in the photon 1D case. The conditions for atomic boson BEC with a quadratic dispersion (kinetic energy) and a trapping potential is described in [3].

In this work we study chirped gratings based Fabry-Perot with analytical calculations and an experimental demonstration of sublinear dispersion, nonuniform comb structure and a density of modes states (DOS) with a nonzero exponent as required for BEC in 1D. There were a few former works [4–8] on nonlinear mode combs by chirped gratings, especially in [4], but as far as we know, they gave only numerical simulations with no analytical results, nor experimental demonstration, except for important applications, such as for sensing [5,6,8] and filters for CW lasers [9]. We recently used such FP to observe photon BEC in a fiber cavity.

2. Analysis of chirped fiber Bragg gratings (CFBGs) Fabry-Perot (FP) that gives a nonlinear light mode dispersion and a nonuniform mode comb

We first give an analytical study of a Fabry-Perot (FP) with chirped fiber Bragg gratings CFBGs and give analytical expressions for the mode spacing showing a nonuniform comb, the DOS and the mode dispersion. We start the calculation with a FP that consists of two similar gratings with opposite chirp directions. In each chirped grating the Bragg wavelength changes along the grating according to: ${z_1} = {C^{ - 1}}{[({\lambda _0} - \lambda )/n]^\alpha }$. C that has a dimension of ${(meter)^{\alpha - 1}}$ is a positive or negative chirp strength in the fiber, $\lambda$- the wavelength in free space, and n - the fiber refractive index. The chirp nonlinearity is given by the exponent $\alpha $. We fabricated and used linearly chirped gratings with $\alpha = 1$ that has a dimensionless chirp strength C. In our experiment $C = 2 \times {10^{ - 6}}$. ${\lambda _0}$ is the wavelength where the chirp starts at ${z_1} = 0$. The distance between the two CFBGs ${l_0}$ was taken to be small, a few mm in our experiments, but the calculations are done for an arbitrary ${l_0}$. Therefore, for similar gratings with an opposite chirp direction and adjacent long period sides (shown in Fig. 1) the effective FP length is wavelength dependent:

 

Fig. 1. Schematic of the chirped fiber Bragg gratings Fabry-Perot (CFBG FP).

Download Full Size | PDF

Then the FP mode spacing varies as [4]:

For the linear chirp case ($\alpha = 1$) and an arbitrary ${l_0}$ we have:

It gives, as it should, $({\lambda _0} - \lambda ) = 0$ for $m = 0$. For ${l_0} \approx 0$ ($\alpha = 1$):

The general $\alpha$ case can be mapped to a nonlinear dispersion $(\omega - {\omega _0}) = \,\,a\,{(k - {k_0})^\eta }$ that gives for the DOS of the FP modes, $\rho (\varepsilon ) \propto \,\,1/(d\omega /dk) \propto {\varepsilon ^{(1/\eta ) - 1}}$, where $\eta = 1/(\alpha + 1)$ or $\alpha = (1/\eta ) - 1$. The linear case ($\alpha = 1$) corresponds to a square-root dispersion: $(\omega - {\omega _0}) = a\,\,{(k - {k_0})^{1/2}}$, where $\eta \equiv 1/2$.

We used such CFBG FP in a EYDF cavity to have a sublinear (square-root) dispersion for observing photon BEC at the cutoff wavelength of $1560.5\,nm$ [1], while we formerly showed BEC with a regular linear light dispersion that was possible only in a finite system size [1].

Similar calculations can be made for different chirped gratings (different C and ${\lambda _0}$ values) and chirp directions. For example, flipping the chirp direction of both FP gratings makes${\lambda _0}$ be the lowest wavelength. Then $({\lambda _0} - \lambda )$ and $(\omega - {\omega _0})$ in the former expressions will be changed to $(\lambda - {\lambda _0})$ and $({\omega _0} - \omega )$, where now $\lambda \ge {\lambda _0}$ and $\omega \le {\omega _0}$. Thus, for example, we have here an increasing mode spacing when $\omega$ increases towards ${\omega _0}$. Flipping only one grating will provide a wavelength independent spacing (a uniform mode comb) in the chirp wavelength range [7].

3. Experimental demonstration of sublinear (square-root) mode dispersion and nonuniform mode spacing with a linear CFBG FP (${\bf{\alpha = 1}}$)

We now show experimental results using a Fabry-Perot with linearly chirped gratings ($\alpha = 1$), schematically shown in Fig. 1. We fabricated the FP with the two chirped gratings in a single-mode fiber on the same fiber by a UV light illumination with a 244 nm wavelength using the phase mask method and we increased its sensitivity by hydrogenation [10]. The nominal chirp strength the mask provided was $C = 2\,nm/1\,mm = 2 \times {10^{ - 6}}$ in the fiber. The mask length was 15mm, but we used in this experiment only ${\approx} 10.5\,mm$ of it. The fiber was made sensitive to UV by hydrogen loading at a high pressure during a weak. The UV illumination through the mask evolves into an index grating. The mask and therefore the two FP gratings had approximately a linear chirp and the separation between them was ${l_0} \approx 2.8\,\,mm$.

Figure 2 shows the transmission spectrum measurement of the CFBG FP with $\alpha = 1$ that gives a nonuniform spacing and therefore a nonuniform mode-comb. Figure 3 shows a zoomed transmission spectrum of the FP in Fig. 2 in a small wavelength band of ${\approx} 8.9\,nm$ near ${\lambda _0}$. We can see there the FP modes with the nonuniform spacing. For the measurements we used a spectrum analyzer APEX AP2041B with a resolution of 1pm.

 

Fig. 2. Broad spectrum in the range of the wave-shaper of the FP transmission with the nonuniform spacing with a linear CFBG FP ($\alpha = 1$). The spacing between modes increases towards ${\lambda _0}$.

Download Full Size | PDF

 

Fig. 3. Zoomed spectrum near ${\lambda _0}$ in a ${\approx} 8.9\,nm$ range of Fig. 2. We can see the FP modes with the nonuniform spacing.

Download Full Size | PDF

Figures 4 and 5 show the experimental mode dispersion of the CFBG FP, obtained from the transmission spectra in Fig. 2, by the modes’ peak wavelength values with respect to ${\lambda _0}$ as a function of the FP mode number m for the linear chirp ($\alpha \approx 1$) case. The fits, red and green curves (hardly seen in the figures since they fall on the experimental points) are given by:

 

Fig. 4. Experimental mode dispersion given by the modes’ wavelength (peaks) as a function of the FP mode number m with a linear chirp ($\alpha \approx 1$) for all CFBG FP modes. The fits that are hardly seen are given in red and green curves with slightly different dispersion exponent of 0.481 and 0.5, respectively, follow the theoretical expression given by Eq. (11). Here the best fit is with the red curve that gives a dispersion exponent of 0.481 that corresponds to . However, the green fit with the 0.5 exponent is quite accurate.

Download Full Size | PDF

 

Fig. 5. Zoom at the first 52 CFBG FP modes mode dispersion in Fig. 4 showing the experimental modes’ wavelengths (peaks) as a function of the FP mode number m with a linear chirp ($\alpha \approx 1$). The fits shown in red and green curves with slightly different dispersion exponent of 0.483 and 0.5, respectively, follow the theoretical expression given by Eq. (11). Here the fit is excellent with an exponent of 0.5 ($\alpha \approx 1$).

Download Full Size | PDF

The first fit (red line) is accurate for all modes, and the second fit (green line) with the 0.5 exponent follows accurately the first few hundred modes (Fig. 3 shows only 52 modes) but is quite accurate for all modes. Both curves are not far from the theoretical values of Eq. (11) with ${l_0} = 2.8\,mm$ and $C = 2 \times {10^{ - 6}}$ that for the exponent of 0.5 gives:

As mentioned above since in this experiment ${l_0} = 2.8\,mm$ is small, the first modes spacing is relatively large but not infinite, as it would be for ${l_0} = 0$, and therefore the slope of the curves in Figs. 4 and 5 at $\omega = {\omega _0}$ are not infinite and the DOS $\rho (\omega = {\omega _0})$ is small but not zero. However, as we said above, the overall fitting is excellent. The experimental dispersion exponent 0.481 slightly deviates from a square-root (0.5) dependence, reflecting a slight deviation from a linear chirp, having $\alpha = (1/0.481) - 1 = 1.079$.

We had in the experiment $m \sim 530$ FP modes in a 31.7 nm range. The theoretical number of modes can be derived from Eq. (9) that with ${l_0} \approx 2.8\,mm$ gives $m \sim 490$ FP modes.

4. Conclusion

We have shown analytically and experimentally a nonlinear mode-comb and a nonlinear mode dispersion using CFBGs FP. We demonstrated with linear chirped gratings an effective close to a square-root dispersion dependence on mode number that is important for various uses such as for obtaining photon-BEC in 1D systems, like fiber cavities [1,5,6,8].