HE11 mode in water-filled hollow core Bragg fiber with a gold layer and dispersive materials

A recent analytical method is used to determine the spectral and amplitude sensitivities for HE11 mode in a hollow core Bragg fiber with or without a gold layer when the dispersion relations for all layers are considered. The fiber without a gold layer is made by a water core, surrounded by periodic reflector layers of polystyrene and polymethyl methacrylate with high and low refractive indices, respectively. The thicknesses of the cladding layers are calculated by using the quarter wave condition for two assumed initial bandgap wavelengths (λ0 = 0.5 μm and λ0 = 0.68 μm). The optical confinement for the HE11 mode in the core, at the minimum-loss wavelength, is increased over four orders (104) of magnitude when a high index material just before the outermost region of a hollow core Bragg fiber is replaced with a gold layer with an optimized thickness. Also, the amplitude sensitivity at the minimum-loss wavelength is increased for an optimized thickness of the gold layer.


Introduction
The transfer matrix method has been used extensively for the analysis of the fiber based plasmonic sensors [1][2][3] and hollow core Bragg fibers [4][5][6][7][8]. Simulations based on the transfer matrix method are validated by the experimental results [6][7][8]. For a water-filled hollow core Bragg fiber, the geometry of the structure is chosen to have its fundamental bandgap located at the wavelength λ 0 =0.68 μm [7].
In the recent papers [1][2][3][4][5], our transfer matrix method is applied to some optical fiber where the radial solutions of the Maxwell equations are written as a combination of the Hankel functions H 1 and H 2 in the gold region. For a hollow core Bragg fiber with or without a gold layer [4,5], a Hankel function of the first kind H 1 is used in the external infinite medium of the fiber. When a high index material just before the outermost region of a hollow core Bragg fiber (r c =r 1 =13.02 μm, n c =n 1 =1, d H =0.086303 μm, d L =0.310 248 μm, n 2 =n 4 =K=n N−3 =4.6, n N−1 =n gold , n 3 =n 5 K=n N =1.6), with large refractive-index contrast in periodic layers of the reflector cladding, is replaced by a gold layer, the optical confinement for the TE 01 mode in the core is increased about ten times [4]. If the gold layer is located between the first and the penultimate layer, the loss for the same TE 01 mode is increased because the parts before and after the reflector gold layer of the fiber are decoupled [4]. The amplitude sensitivity for the leaky core mode HE 11 near the lowest loss point for the hollow-core Bragg fiber without and with a gold layer increases when the number of the layers increases from N=5 to N=11 [5]. We have demonstrated the accuracy of our method for a hollow core Bragg fiber without a gold layer with N=34 layers (32 reflector layers, 16 pairs), r c =1.3278 μm, n c =n 1 =1, d H =0.2133 μm, d L =0.346 μm, n 2 =n 4 =K=n 34 =1.49, n 3 =n 5 K=n 33 =1.17, λ=1 μm, where our effective index b/k=0.891 067 2175+1.422 604 6712×10 −8 i [5] for the TE 01 mode is very close to b/k=0.891 067+1.4226 10 −8 i calculated in [9].
In this paper we extend the research to the HE 11 mode of a hollow core Bragg optical fiber when the dispersion of H 2 O, PS, PMMA and gold layers is considered. The optical confinement and amplitude sensitivity at the minimum-loss wavelength for the HE 11 mode are increased when the layer with high index just before the outermost region of a hollow core Bragg fiber is replaced with a gold layer with an optimized thickness.
2. HE 11 mode in water-filled hollow-core Bragg fiber without and with a gold layer The water-filled hollow-core Bragg fiber with N=11 layers and without a gold layer (figure 1) is made by a water core (refractive index n c =n 1 =n a =n H O 2 and radius r c =25 μm), surrounded by periodic reflector layers of polystyrene (PS) and polymethyl methacrylate (PMMA) with high (n H =n PS ) and low (n L =n PMMA ) refractive indices, respectively. The thicknesses of the cladding layers d H =d PS and d L =d PMMA are calculated by using the quarter wave condition [10]: From the relations (1) and (2) we obtain the dependence of λ 0 on n PS , n PMMA , n H2O , d PS and d PMMA [11]: we must to use d H and d L in nm and the result is in nm RIU −1 . Figure 1 also presents a hollow-core Bragg fiber with N=11 layers and with a gold layer which replaces the layer with high index just before the outermost region. The refractive index of the distilled water [12], PS [13] and PMMA [14] materials are calculated through a Sellmeier-type relation. The refractive index of the gold layer is calculated by the Drude model [15].

Results and discussions
We discuss the results for the effective indices, loss, spectral and amplitude sensitivities of a leaky core mode HE 11 in a hollow core Bragg optical fiber with or without a gold layer, for two assumed initial bandgap wavelengths (λ 0 =0.68 μm and λ 0 =0.5 μm), when the dispersion of all layers is considered. Figure 2 shows the real part of the effective index versus wavelength for the leaky core mode HE 11 near the lowest loss points: λ=0.4937 μm for n a and λ=0.4927 μm for n a +0.001 for a structure without gold layer and λ=0.4932 μm for n a and λ=0.4922 μm for n a +0.001 for an optical fiber with a gold layer when λ 0 =0.5 μm. Figure 3 shows the loss spectra for the same mode near the same lowest loss points for a structure without and with a gold layer. For t g =d H =141.109 65 nm and λ=0.4932 μm, the loss is about 3.27 times smaller in comparison with a structure without gold layer where λ=0.4937 μm. Thus for a non-fundamental bandgap, the gold in the penultimate layer is a better reflector than a PS layer with the same thickness. Figure 4 shows the loss spectra for the leaky core mode HE 11 near the lowest loss point λ=0.6678 μm for n a and a structure without a gold layer when λ 0 =0.68 μm. A similar figure is for n a +0.001 where λ=0.6663 μm. Assuming that 0.1 nm spectral shift in the position of a bandgap center can be detected [10], the variation a D of the losses is 2.562 788 81×10 −10 dB cm −1 when the wavelength λ is changed from 0.6677 to 0.6678 μm. Also, a D =3.763 773 42×10 −11 dB cm −1 when λ is changed from 0.6679 to 0.6678 μm. Figure 5 shows the loss spectra for the leaky core mode HE 11 near the lowest loss point λ=0.6506 μm for n a and a structure with a gold layer when λ 0 =0.68 μm. A similar figure is for n a +0.001 where λ=0.6493 μm. For t g =d H =197.6997 nm and λ=0.6506 μm, the loss is about 11.68 times smaller in comparison with a structure without gold layer where λ=0.6678 μm. Thus for a fundamental bandgap, the gold in the penultimate layer is a better reflector than a PS layer with the same thickness. Similarly, a D =2.138 632 65×10 −11 dB cm −1 when λ is changed from 0.6505 to 0.6506 μm and a D =2.892 473 39×10 −12 dB cm −1 when λ is changed from 0.6507 to 0.6506 μm. Figure 6 shows the amplitude sensitivity for the leaky core mode HE 11 of a hollow core Bragg fiber with N=11 layers versus wavelength near the lowest loss points (λ=0.4937 μm, S A =16.26 RIU −1 for a structure without gold layer and λ=0.4932 μm, S A =3.73 RIU −1 for a structure with a gold layer) when λ 0 =0.5 μm. Figure 7 shows the amplitude sensitivity for the same mode versus wavelength near the lowest loss points Figure 2. The real part of the effective index versus wavelength for the leaky core mode HE 11 near the lowest loss points: λ=0.4937 μm for n a and λ=0.4927 μm for n a +0.001 for a structure without gold layer (a) and λ=0.4932 μm for n a and λ=0.4922 μm for n a +0.001 for a structure with a gold layer (b) when N=11. Figure 3. The loss spectra for the leaky core mode HE 11 near the lowest loss points: λ=0.4937 μm for n a and λ=0.4927 μm for n a +0.001 for a structure without gold layer (a) and λ=0.4932 μm for n a and λ=0.4922 μm for n a +0.001 for a structure with a gold layer (b).
(λ=0.6678 μm, S A =19.86 RIU −1 for a structure without gold layer and λ=0.6506 μm, S A =4.97 RIU −1 for a structure with a gold layer) when λ 0 =0.68 μm. Figure 8 shows the loss versus the thickness t g of the gold layer for the leaky core mode HE 11 at the lowest loss point λ=0.4932 μm when λ 0 =0.5 μm. For t g =28 nm, the loss is about 272 times smaller in comparison with a structure without gold layer. Figure 9 shows the loss versus the thickness t g of the gold layer for the same mode at the lowest loss point λ=0.6506 μm when λ 0 =0.68 μm. Note that the optimized thickness t g of the gold layer corresponds to change from positive (for small t g as in the structure without a gold layer) to negative (for large t g ) in the imaginary part of the effective index. Figure 10 shows the real part of the effective index versus wavelength for the leaky core mode HE 11 near the lowest loss points: λ=0.4942 μm for n a and λ=0.4940 μm for n a +0.001 for a structure with a gold layer when t g =28 nm. The same figure shows the logarithmic in base 10 of the loss a versus the wavelength. For an optimized structure (t g =28 nm and λ=0.4942 μm), the loss is about 18 399 times smaller in comparison with a structure without a gold layer. Figure 11 shows the real part of the effective index versus wavelength for the same mode near the lowest loss points: λ=0.6503 μm for n a and λ=0.6502 μm for n a +0.001 for a structure with a gold layer when t g =41 nm. The same figure shows the logarithmic in base 10 of the loss a versus the wavelength for the same structure. For t g =41 nm and λ=0.6503 μm, the loss is about 33 187.4 times smaller in comparison with a structure without gold layer where λ=0.6678 μm. Thus an optimized thin gold layer is a better reflector than a thick PS layer. The loss for the leaky core mode HE 11 in the fundamental bandgap (λ 0 =0.68 μm, t g =41 nm) is about 1186.2 times smaller than in the non-fundamental bandgap (λ 0 =0.5 μm, t g =28 nm).   Figure 12 shows the amplitude sensitivity for the leaky core mode HE 11 of a hollow core Bragg fiber versus wavelength near the lowest loss points (λ=0.4942 μm, S A =9948.6 RIU −1 ) for a structure with a gold layer when t g =28 nm and λ 0 =0.5 μm. Figure 13 shows the amplitude sensitivity for the same mode versus wavelength near the lowest loss points (λ=0.6503 μm, S A =1023.3 RIU −1 ) for a structure with a gold layer when t g =41 nm and λ 0 =0.68 μm. The maximum value of the amplitude sensitivity (S A m =2106.0 RIU −1 for λ=0.6504 μm) for the leaky core mode HE 11 in the fundamental bandgap (λ 0 =0.68 μm, t g =41 nm) is about 4.7 times smaller than in the non-fundamental bandgap (λ 0 =0.5 μm, t g =28 nm). Tables 1 and 2 show the values of the effective index b/k, loss a and the lowest loss wavelength λ for a hollow-core Bragg fiber with or without a gold layer when λ 0 =0.5 μm and λ 0 =0.68 μm, respectively. The imaginary part of the effective index b/k is very sensitive to the number of the layers and if the structure is with or without a gold layer. The values for the spectral and amplitude sensitivities for an optimized thickness (t g =28 nm for λ 0 =0.5 μm and t g =41 nm for λ 0 =0.68 μm) of the gold layer can be determined from the tables 1 and 2 (modes 3 g, 3 g′ and 3 g″). Tables 3 and 4 show the values of the shift dl res towards lowest loss shorter wavelengths for an increase Dn a of the analyte refractive index by 0.001 RIU, the spectral sensitivity l S , the spectral resolution l SR , the amplitude sensitivity S A at the minimum-loss wavelength and the corresponding resolution SR , A the transmission loss a, the propagation length L and the minimum-loss wavelength λ. Note  that the loss is decreased when the high index material just before the outermost region of a hollow core Bragg fiber is replaced by a gold layer. The calculated spectral sensitivity ( l S =1500 nm RIU −1 ) for a fiber without a gold layer is close to the theoretical value ( l S =1634.67 nm RIU −1 ) for the HE 11 mode when λ 0 =0.68 μm. Also, the calculated spectral sensitivity ( l S =1000 nm RIU −1 ) for a fiber without a gold layer is close to the theoretical value ( l S =1166.98 nm RIU −1 ) for the HE 11 mode when λ 0 =0.5 μm. Note that this sensitivity is smaller than in the fundamental bandgap (λ 0 =0.68 μm).

Conclusions
An analytical method is used to determine the propagation characteristics for HE 11 mode in a hollow core Bragg fiber with or without a gold layer for two assumed initial bandgap wavelengths (λ 0 =0.68 μm and λ 0 =0.5 μm) when the dispersion relations for all layers (water, PS, PMMA and gold) are considered.       , n 2 =n 4 =K=n 10 =n H =n PS , n 3 =n 5 K=n 11 =n L =n PMMA for a fiber without a gold layer and n 1 = n H O 2 , n 2 =n 4 =n 6 == n 8 =n PS , n 10 =n g , n 3 =n 5 =n 7 =n 9 =n 11 =n PMMA for a fiber with a gold layer. , n 2 =n 4 =K=n 10 =n H =n PS , n 3 =n 5 K=n 11 =n L =n PMMA for a fiber without a gold layer and n 1 = n H O 2 , n 2 =n 4 =n 6 = n 8 =n PS , n 10 =n g , n 3 =n 5 =n 7 =n 9 =n 11 =n PMMA for a fiber with a gold layer.
Mode; λ 0 ;n 1 ; t g (nm) b/k α (dB cm −1 ) λ (μm)  Table 3. Values of dl res (nm), l S (nm RIU −1) , l SR (RIU), S A (RIU −1 ), SR A (RIU), a (dB cm −1 ), L (μm) and λ (μm) where r c is the radius of the core, N is the number of layers, n H is the high refractive index, n L is the low refractive index and n a is the refractive index of the analyte.
Mode HE 11 ; λ 0 (μm) (r c ; N; n H ; n L; n a ; n N−1 ) dl res When a high index material just before the outermost region of a hollow core Bragg fiber is replaced by a gold layer with an optimized thickness, the optical confinement at the minimum-loss wavelength for the HE 11 mode in the core is increased about 33 187.4 times for λ 0 =0.68 μm (t g =41 nm) and about 18 399 times for λ 0 =0.5 μm (t g =28 nm) for N=11 layers. Also, the amplitude sensitivity S A at the minimum-loss wavelength is increased from S A =16.3 RIU −1 (λ=0.4937 μm) to S A =9948.6 RIU −1 (λ=0.4942 μm) for λ 0 =0.5 μm and from S A =19.9 RIU −1 (λ=0.6678 μm) to S A =1023.3 RIU −1 (λ=0.6503 μm) for λ 0 =0.68 μm. However, the spectral sensitivity is smaller when a gold layer is used in the place of a PS layer.
The optical confinement for the HE 11 mode in the water-filled core for a fundamental and also for a nonfundamental bandgap wavelength is increased over four orders (10 4 ) of magnitude when a high index material just before the outermost region of a hollow core Bragg fiber is replaced by a gold layer with an optimized thickness. Thus an optimized thin gold layer is a better reflector than a thick PS layer. In such devices, the light of a high power laser can be transmitted with very low loss due to the large confinement in the core of the fiber.

ORCID iDs
Vasile A Popescu https:/ /orcid.org/0000-0002-4928-9007 Table 4. Values of dl res (nm), l S (nm RIU −1 ), l SR (RIU), S A (RIU −1 ), SR A (RIU), a (dB cm −1 ), L (μm) and λ (μm) where r c is the radius of the core, N is the number of layers, n H is the high refractive index, n L is the low refractive index and n a is the refractive index of the analyte.