1. Introduction

Photonic crystal fibers (PCFs) have been widely investigated for many years due to its attractive properties [1]. A PCF has a core surrounded by a complicated cladding which consists of an arrangement of inclusions in background materials. Different kinds of PCFs can be obtained by reasonably designing the distribution of inclusions and the refractive indices of the core and the inclusions. The core refractive index of index guiding PCF is the same as that of cladding background material, and air inclusions are employed in the cladding. Most of the mode field can be confined in the core region based on improved total internal reflection because the real part of core mode effective refractive index is higher than that of cladding [2]. If the air filling ratio is high enough in the cladding, core modes can be supported by photonic bandgap effect even in an air core [3]. The photonic bandgap fibers with high-index inclusions in the cladding have also been reported, and the guiding properties of these PCFs are mainly determined by the individual properties of high-index inclusion rather than their distribution [4]. Recently PCFs with air and high-index inclusions in the cladding at the same time is studied, and they exhibit some new features [5, 6]. Although light is mostly confined in the core of these kinds of PCFs, some light can also leak out of the cladding with finite rings of inclusions through the channels between every two of the inclusions. Generally modes in PCF are leaky.

Since modes in PCF are leaky, the coupling between them is usually leaky mode coupling. Mode coupling is a common and useful phenomenon in multiple core fibers and directional couplers. In addition to the usages in couplers, mode coupling has also been used for some special purpose in PCFs [7], OmniGuide fibers [8] and so on. Roberts et al. introduced mode coupling between the fundamental LP₀₁-like mode of the PCF and a mode associated with a ring of relatively small holes to obtain large negative dispersion at the anti-crossing point [9]. An effectively single mode large hollow-core fiber was designed by Kunimasa Saitoh et al. based on the index-matching mechanism of central air-core modes with defected outer core modes [10]. The difference between bound mode coupling and leaky mode coupling was not described in the researches partly because their appearance is similar. In fact these coupling modes are leaky modes. Gilles Renversez et al. exhibited and explained a core mode transition induced by avoided crossing between a core localized leaky mode and a high-index cylinder leaky mode in anti-resonant reflecting optical waveguide microstructured optical fibers [11]. But they did not discuss the effect of the loss in leaky mode coupling.

The loss of leaky modes should be paid enough attention in mode coupling because the imaginary part of propagation constant is also special for a leaky mode like the real part of that. The role of loss in leaky mode coupling has not been discussed clearly in PCF to the best of our knowledge because the loss matching condition was satisfied in most of the reported situations. In this paper, the dependence of leaky mode coupling on loss in photonic crystal fiber with a hybrid cladding which includes low-index and high-index inclusions at the same time is numerically investigated based on a full vector finite element method (FEM) [12]. In PCFs with hybrid cladding described later, the losses of leaky modes can be changed easily by altering the diameters of air holes, and meanwhile the phase matching state is kept. This is convenient to investigate the influence of loss matching on leaky mode coupling in PCFs. At last a theoretic explanation is introduced to analyze the different mode coupling behaviors.

2. PCF with hybrid cladding and numerical method

The PCF with hybrid cladding under study consists of a silica core surrounded by an inner ring of air holes and an outer ring of high-index inclusions in a hexagonal lattice, as shown in Fig. 1(a). The indices of core and cladding background n ₀ are the same and equal to 1.45. The refractive indices of inner air holes n ₁ and outer high-index inclusions n ₂ are 1 and 1.6 respectively. The diameters of outer inclusions are fixed at 1.60µm, and the diameters of inner holes will be changed during the following analysis. The pitch Λ is set to 4.00µm. This PCF can support core modes in the core shown in Fig. 1(b) and cladding defect super modes in the outer high-index inclusions shown in Fig. 1(c). The core modes are similar to modes in holey fiber where the mode behavior can be controlled easily by the cladding air hole distribution. Here the losses of core mode and defect mode will be changed contrarily by altering the diameters of air holes. The defect mode effective index will be affected less than core mode when changing the diameters of air holes unless mode coupling happened because most of the mode field is confined in high-index inclusions. Both kinds of modes in this PCF are leaky as their real part of mode effective index is lower than the refractive index of cladding background. In this paper, the coupling between core fundament leaky mode and a defect super mode will be discussed.

 

Fig. 1 Schematic of PCF with hybrid cladding (a), intensity distribution of the core fundamental mode (b) and of the cladding defect super mode (c).

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FEM with anisotropic perfectly matched layers (PMLs) allows us to evaluate the effective index of a leaky mode including the real part and imaginary part in PCFs. It has been used in various optical waveguides to numerically investigate mode loss [13], group velocity dispersion [14] and so on. FEM is also chosen to evaluate the effective indices of leaky modes in this paper.

3. Numerical results and discussion

The coupling between core fundament leaky mode and a defect super mode will be taken into account in the following discussion. The core mode and cladding defect mode are separated by the inner ring of air holes. When increasing the diameters of air holes, the loss of core decreases and defect mode loss increases. That is the loss difference is larger between the two leaky modes for bigger air holes and smaller for smaller air holes.

3.1 Complete coupling between two leaky modes

At first, a coupling between core fundamental mode and a cladding defect super mode is analyzed to understand the properties of leaky mode coupling when small air holes are used. The diameters of air holes d ₁ are set to 1.36µm, and the other parameters are the same as those given in Section 2.

It can be seen from Fig. 2 (a) that the two modes with phase matching couple each other to splitting into another two new modes. For the lower mode, the mode field resides in central core region mostly at shorter wavelengths, more field leaks to cladding high-index inclusions as wavelength increases, there is nearly the same field intensity in high-index inclusions as in the core near the anti-crossing point, and almost all of the field transfers from core to the high-index inclusions at longer wavelengths. The transition for the upper mode is just contrary to that for lower mode. This course is similar to the coupling of two bound modes. The transition from core mode to defect super mode also happens to the losses of leaky modes which are in proportion to the imaginary part of leaky mode effective index. For the blue curve, the mode loss follows the core mode-like part at shorter wavelengths, and follows the defect mode-like part at longer wavelengths. There is an intersection between the loss curves of the two newly formed leaky modes in Fig. 2 (b). That is the losses of coupled leaky modes are the same at the wavelength corresponding to the anti-crossing point shown in Fig. 2 (a). Apart from the intersection the cladding defect super mode-like parts have larger loss than core-like parts. It seems that phase matching and loss matching are true when leaky mode coupling happens.

 

Fig. 2 Real part (a) and imaginary part (b) of the mode effective index as a function of wavelength when a complete coupling happens. Dashed curves represent the dispersion curves of core fundamental mode and cladding defect super mode respectively when there is no coupling between them at all. The insets are corresponding to intensity distribution of the coupled modes at selected wavelengths. The parameter d ₁ equals to 1.36µm.

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3.2 Incomplete coupling between two leaky modes

It has been known widely that phase matching condition must be satisfied for mode coupling. In order to make out whether loss matching is also a necessary condition for leaky mode coupling as phase matching, the diameters of air holes are increased to 1.76µm to increase the loss difference between the two leaky modes. The results are shown in Fig. 3.

 

Fig. 3 Real part (a) and imaginary part (b) of the mode effective index as a function of wavelength when the coupling does not happen completely. The insets are corresponding to intensity distribution of the coupled modes at selected wavelengths. The parameter d ₁ equals to 1.76µm.

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It is found that the two crossing leaky modes do not split into another two new modes as modes shown in Fig. 2. But abnormal dispersion for core leaky mode appears around the intersection. This can be explained by checking the transformation course of mode field intensity exhibited in the insets of Fig. 3 (a). At shorter wavelengths the two leaky modes are well confined in core and cladding defect regions respectively. A little amount of core mode field starts to leak into cladding high-index inclusions and cladding defect super mode into core region as wavelength increasing. The real part of core mode effective refractive index becomes a little higher near the phase matching point because the refractive index of high-index inclusion is higher than that of the core. The leaked fields from core and cladding defect regions return to their respective regions when the wavelength becomes further longer. There is not a complete transition between the two leaky modes in the whole course. In another word, a complete coupling between the two leaky modes does not happen. This can also be confirmed from the losses of modes shown in Fig. 3 (b). It is clear that there is notable difference between the mode losses. Although the loss curves of the two leaky modes do not intersect, a dip on the upper loss curve and a peak on the lower loss curve appear at the phase matching point due to field “wobbling” of leaky modes. It can be said now that a complete coupling between leaky modes requires phase and loss matching at the same time.

3.3 Leaky mode couplings for different losses

It is comprehensive to look over a serial of effective refractive indices of phase matching leaky modes to make sure the loss influence on leaky mode coupling by changing the air hole diameter d ₁. The results are shown in Fig. 4 where the parameter d ₁ changes from 1.36µm to 1.92µm.

 

Fig. 4 Real part (a) and imaginary part (b) of the mode effective index as a function of wavelength for different air hole diameter d ₁ from 1.36µm to 1.92µm. The coupling becomes weaker and weaker as the loss difference increasing.

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It can be seen in Fig. 4 that the real part of core mode effective index decreases with d ₁ increasing, and the same thing also happens to the cladding defect mode but smaller variation. The phase matching condition is satisfied all the time for different air hole diameters though losses between the two leaky modes are not equal all through. The strength of leaky mode interaction becomes weaker as d ₁ becomes larger, and the complete coupling disappears for large enough d ₁ in Fig. 4 (a). Smaller air hole diameter makes the coupling transition smoother in a wider wavelength range, and lager d ₁ makes the coupling more abrupt while the coupling takes place. When complete coupling does not happen, lager d ₁ induces smaller abnormal dispersion. It is also found that losses of the two leaky modes vary contrarily when changing the diameters of air holes in Fig. 4 (b). So the loss difference becomes larger with d ₁ increasing. Furthermore, the loss curves of coupled modes intersect at the phase matching point, and there is only some fluctuation rather than intersection for uncoupled modes. This is also easy to distinguish coupling or not from the loss curves. It is also noted that almost all intersections of loss curves are corresponding to the same loss value. This may results from the contrary change of mode loss at the same time. Considering both real and imaginary part of those leaky mode effective indices, one can find that it suffers from a stronger coupling with wider field transition wavelength range for the two leaky modes between which the loss difference far away from the intersection is smaller than other groups of leaky modes. The above results exhibit another side to manage the coupling between two leaky modes based on the controlling of their losses.

3.4 Theoretical explanation and discussion

In addition to the above qualitative descriptions for leaky mode coupling, coupled-mode theory can also be used to explain them. For two coupled modes, the coupled-mode equations can be written as:

where E ₁ and E ₂ are the mode fields of two coupling modes respectively, β ₁ and β ₂ are their changed propagation constants due to the waveguide geometry changes, and κ is the coupling strength. Assume that E ₁=Aexp(iβz) and E ₂=Bexp(iβz), then substitute them into Eqs. (1), thus

By using $\det \left(\begin{array}{cc}\beta -{\beta }_1& -\kappa \\ -\kappa & \beta -{\beta }_2\end{array}\right)=0$ , the propagation constant of coupling mode β can be drawn

where β_ave=(β ₁+β ₂)/2, and δ=(β ₁-β ₂)/2. Now the discussion can be divided into two cases corresponding to the bound mode coupling and leaky mode coupling respectively. For bound modes, β ₁ and β ₂ are both real. So δ is also real and $\sqrt{{\delta }^2+{\kappa }^2}$ is larger than zero. β ₊ is not equal to β _- for any wavelength even at anti-crossing point where δ(_λ) goes through zero and the regular anti-crossing happens.

For leaky modes, β ₁ and β ₂ are both complex. So δ may also be complex and written as δ=δ _r+iδ_i. At phase matching point, the real parts of the two leaky mode propagation constants are equal, and in another word δ_r=0. It can be derived that

when δ_i<κ, β ₊ and β _- have different real parts but equal imaginary parts and a complete coupling (regular anti-crossing) between two leaky modes happens. When δ_i>κ, β ₊ and β _- have equal real parts but different imaginary parts and an incomplete coupling appears. That is two leaky modes actually cross. For the fiber structure described above, κ becomes less than δ_i when d ₁ is increased large enough, thus moving the system from anti-crossing regime to the abnormal crossing regime. From the above discussion, there is a conclusion that a regular anti-crossing is obtained if the coupling strength κ is larger than one half the difference between the losses (imaginary part of the propagation constant) of the two leaky modes. If the coupling strength κ is less than this value, an actual crossing takes place because it is possible for the two resulting modes to have equal real parts.

Generally speaking, leaky mode coupling is not easy to happen in optical waveguides because it requires phase matching and loss matching between the two leaky modes come true at the same time. This may be a reason why it is not found or ignored in most of the situations. Another cause is the small wavelength range where the field diverts from one leaky mode to another. The typical value is less than 5nm in our computation as shown in Fig. 4. In fact, most of the related coupling between core modes and cladding defect modes in PCF is leaky mode coupling except for those with doped high-index core region. The cladding defect mode should be also a leaky mode once it couples with a core leaky mode because its real part of effective refractive index is lower than that of cladding due to the phase matching condition. Usually leaky mode coupling is treated in a similar way as bound mode coupling in some literatures owing to their similar appearance. A sufficient coupling takes place in those PCFs where the loss difference is smaller enough at wavelengths without coupling and varies mildly with structure parameters. In PCF with hybrid cladding described in this paper, the loss difference can be changed evidently when changing the air hole diameter.

In addition to the above mentioned PCF, other structures can also be applied to control leaky mode coupling conveniently. For example, an annular high-index region can be used to substitute for high-index inclusions of PCF described in this paper. Although the control of coupling is accomplished by the management of mode loss in our analysis where the phase matching condition is satisfied all the time, the coupling between leaky modes with small enough loss difference can also be manipulated by changing the real part of mode effective refractive index.

4. Conclusion

A special designed PCF with hybrid cladding has been adopted to investigate the coupling between two leaky modes. FEM with PML boundary condition make it possible to take the real and imaginary parts of leaky modes into consideration at the same time. A leaky mode coupling happens or not is decided not only by phase matching condition but also loss matching condition. Abnormal dispersion will appear for uncoupled leaky modes when phase matching condition is satisfied but not for loss matching condition. Furthermore, there is a minimum for the loss difference between the two leaky modes near the phase matching point. According to the theoretic analysis, a regular anti-crossing is obtained if the coupling strength κ is larger than one half the difference between the losses of the two leaky modes; otherwise abnormal crossing takes place. It should be paid enough attention to leaky mode coupling in practice because it can affect both dispersion and loss. In some situation, it can also be used to design PCFs for special purpose, such as dispersion compensating, high-order mode eliminating and distributed filters.