As described in Section 1.4, the multipole method is capable of accurately calculating the confinement losses of the modes of a holey fibre. The confinement losses for the first higher- order mode present in the multi-mode fibres included in this study have been calculated by Vittoria Finazzi at the ORC and are listed in Table 5.2 for 6 rings of holes [54]. It has been shown that in holey fibres with a six-fold symmetric cladding, the fibre modes can be classified in an analogous way to the symmetry based classification used for the modes in a step-index fibre [148]. As such, the first higher-order multiplet in a holey fibre is a linear combination of modes that are equivalent to the TE01, the TM01 and the two degenerate HE21 modes of a step-index fibre. In the large core, low NA regime, these four modes are near-degenerate and possess similar levels of confinement losses. Here the confinement losses are calculated for 6 rings of holes for the first higher-order mode with the highest effective index, which represents the higher-order mode with the strongest level of confinement. The modal intensity profiles for the first higher-order mode guided in each fibre are shown in Fig. 5.13 (a), (b), (c) and (d). Transverse cross sections of these intensity profiles in thex
direction are shown in (e), (f), (g) and (h), and transverse cross sections of the intensity profile in thexdirection are shown in (i), (j), (k) and (l) respectively. Note that the higher- order mode for the multi-mode fibre with the smallest holes (HFF, d/Λ = 0.44), is broad, extending well into the cladding region. As d/Λ increases, the modal field of the HOM becomes smaller and more confined to the core. This change in mode size is reflected in the confinement losses of the HOM, which reduce dramatically with increasing values of d/Λ. For the fibres shown here, the confinement losses of the HOM decreases from 3.4 dB/m to less than 1×10−8 dB/m as the relative hole size is increased from 0.44 to 0.63 (both values of confinement loss quoted for 6 rings of holes). The high value of confinement loss for the higher-order mode in fibre HFF (C
HOM
loss = 3.4 dB/m for 6 rings of holes), means that it is unlikely that this mode would be observed in lengths of fibre greater than a few 10’s of cm. As as result, we define this fibre to be effectively single-mode and thus extend the range of d/Λ that results in single-mode guidance in practice. The fact that fibres with values
of d/Λ > (d/Λ)c can be effectively single-mode is confirmed by the experimental results
presented in Section 5.4.
Table 5.2: Calculated modal properties for a range of holey fibre structures. Note that SM stands for single-mode, MM stands for multi-mode, FM refers to a property of the fundamental mode, and HOM refers to a property of the first higher-order mode.
Fibre Λ d d/Λ AFMeff RFMc Modes CHOMloss for 6 rings RHOMc [µm] [µm] [µm2] [cm] of holes [dB/m] [cm] HFF 12.56 5.53 0.44 189 3.7 MM 3.40 >14.5 HFG 12.70 5.73 0.45 188 3.6 MM 0.46 >14.5 HFH 13.40 6.70 0.50 190 3.0 MM 1.75×10−5 14.5 -17.1
HFI 15.00 9.46 0.63 189 1.8 MM negligible 2.7 - 2.8
For fibres HFG, HFH, HFI listed in Table 5.2, the confinement losses of the higher- order modes are low enough to ensure that these modes would be observed in lengths of straight fibre of a few metres at least. However, it may still be possible to selectively guide only the fundamental mode by exploiting the fact that the fundamental mode is the least sensitive to bend induced loss and that the bend loss is exponentially dependent on the radius of curvature. If the higher-order modes suffer a sufficiently greater bend loss than the fundamental mode, it may be possible to selectively remove these modes by bending the fibre. The technique of removing unwanted higher-order modes by bending the fibre is well known in conventional fibres [149, 150] and can be evaluated by calculating the bending losses of the first higher-order mode, since all other higher-order modes present in a fibre will be less bend resistant. Whether this is also possible in holey fibres can be determined by using the numerical methods outlined in Chapter 3, which are capable of calculating the modal properties of higher-order modes. In the scalar version of the orthogonal function method outlined in Section 3.2, the second and third solutions of the eigenvalue equation approximate to the linearly polarised LPx
11and LPy11modes, examples of which can be seen in Figs 5.14 (b) and (c) for fibre HFH.
Unfortunately, whilst the method of calculating bend loss used here is applicable to higher-order modes, it cannot be used for fibres HFF and HFG due to the fact that they are so close to cut-off. As mentioned in Section 3.6, the modedness of a holey fibre is most accurately determined by the multipole approach. However, the multipole approach
Figure 5.13: Contour plots of intensity for the first higher-order mode in fibres (a) HFF, (b) HFG, (c)
HFHand (d) HFI. The contour lines are separated by 2 dB. Transverse cross sections of the intensity profile
in thexdirection are shown in (e), (f), (g) and (h), and transverse cross sections of the intensity profile in thexdirection are shown in (i), (j), (k) and (l) respectively. The fibre parameters are listed in Table 5.1.
is incapable of considering the graded index profile that is required to generate the modes of a bent fibre. Here we use an orthogonal function method to calculate the modes of the bent fibre, which systematically produces modal effective indices that are slightly lower than those calculated by the multipole method (also discussed in Section 3.6). As a result, the orthogonal function method (incorrectly) predicts that fibres HFF and HFG are single- mode at 1064 nm. The bend loss for the higher-order modes cannot be calculated in this instance due to the fact that the point at which all radiation is lost from the fibre, defined as xr =Ro(βFSMβb −1) in Eq. 3.12, is a negative value for βb < βFSM, whereβb and βFSM
are the propagation constant of the bent fibre mode and the fundamental cladding mode respectively. However, since the higher-order modes for fibres HFF and HFG, as calculated by the multipole approach and shown in Fig. 5.13, have higher confinement losses and extend further into the cladding region than those in fibre HFH, it is reasonable to assume that the bending losses of the higher-order modes in fibres HFF and HFG will be greater than those in HFH. Consequently, by calculating the bend losses for fibres HFH, a lower bound can be placed on the critical bend radius for the higher-order modes of fibres HFF and HFG.
The LPx
11 and LPy11 modes, calculated for fibre HFH using the orthogonal function method, are shown in Figs 5.14 (b) and (c) respectively together with the fundamental mode for the same fibre, shown in Fig. 5.14 (a). The effective indices of the modes in Figs 5.14 (a), (b) and (c) are 1.449167 1.448540 and 1.448538 respectively. The same modes for the bent fibre are shown in Figs 5.15 (a), (b) and (c) for Ro = 15 cm. The
bending losses associated with the LPx11 and LPy11 modes for the four multi-mode holey fibres listed in Table 5.1 are evaluated below.
(a) (b) (c)
Figure 5.14: Modal intensity profile for (a) the fundamental mode, (b) the LPx
11 and (c) LPy11 for holey
fibre HFH, Λ = 13.4µm,d/Λ = 0.5 at 1064 nm. Contours are separated by 2 dB.
The bend loss for each of the modes present in fibre HFH is shown in Fig. 5.16 (a) as a function of bend radius. The solid line represents the bend loss of the fundamental mode and the dotted and dashed lines represent the bend loss of the LPx
11and LPy11respectively. The critical bend radius for the LPx
11 and LPy11 modes is extracted as 14.5 and 17.1 cm respectively. The smallest of these is still ≈ 5 times larger than the critical bend radius for the fundamental mode (3.0 cm), demonstrating that the LP11 modes can be stripped without perturbing the fundamental mode. In Fig. 5.16 (b), the sum of the bend loss from
(a) (b) (c) Figure 5.15: Modal intensity profile for (a) the fundamental mode, (b) the LPx
11mode and (c) the LPy11
mode, forRo= 15.0 cm for holey fibre HFH, Λ = 13.4µm,d/Λ = 0.5 at 1064 nm. Contours are separated
by 2 dB. 0.1 0.2 0.3 0 1 2 3 4 0.1 0.2 0.3 (a) (b)
Figure 5.16: (a) Pure bend loss as a function of bend radius for the fundamental mode (solid line) and the LPx
11 and LPy11 modes in fibre HFH (dotted and dashed lines respectively) (Λ = 13.4µm, d/Λ = 0.5).
(b) The sum of the loss from the fundamental and the two higher-order modes shown in (a), assuming a ratio of 3:1:1 respectively.
the fundamental mode and the LPx
11and LPy11modes is plotted as a function of bend radius assuming a ratio of 3:1:1 respectively. This ratio is purely chosen to produce a graph of total bend loss that best matches with some experimental results presented in the next section. As such, the first rise in loss in Fig. 5.16 (b) represents loss from the LPx
11and LPy11modes and the second, sharper rise in loss corresponds to the loss from the fundamental mode.
Note that the loss from the LPx
11 and LPy11 modes appears as a single curve in Fig. 5.16 (b) due to the fact that the bending losses of the two modes are similar compared with the fundamental mode. This plot of total bend loss for fibre HFHdemonstrates that the LP11 modes can be stripped from the fibre using a few turns with a bend radius of≈10−15 cm or a single loop of ≈8 cm, without suffering loss from the fundamental mode. Indeed, as can be seen from Fig. 5.15 (a), the fundamental mode is not perturbed to any significant degree forRo= 15 cm. However, approximately 2 dB of power is lost from the fibre in this
process, which is less than ideal from a power handling perspective.
The numerical results presented above demonstrate that it is possible to remove higher- order modes by introducing a bend in a holey fibres for d/Λ = 0.5. In the following, a similar evaluation is performed for fibre HFI, in which d/Λ = 0.63, and hence possesses lower overall bend loss, in order to determine the upper limit in which this approach can be successfully used to eliminate higher-order modes. The LPx
11 and LPy11 modes calculated for fibre HFI using the orthogonal function method are shown in Figs 5.17 (b) and (c) respectively together with the fundamental mode for the same fibre, shown in Fig. 5.17 (a). The same modes for the bent fibre are shown in Figs 5.18 (a), (b) and (c) forRo = 3.0 cm.
The bending losses associated with the LPx
11and LPy11modes for the fibre HFIare evaluated below.
(a) (b) (c)
Figure 5.17: Modal intensity profile for (a) the fundamental mode, (b) the LPx
11 and (c) LPy11 for holey
fibre HFI, Λ = 15.0µm,d/Λ = 0.63 at 1064 nm. Contours are separated by 2 dB.
The calculated bending losses for the first three modes present in fibre HFIare plotted as a function of bend radius in Fig. 5.19 (a). The solid line represents the bend loss of the fundamental mode and the dotted and dashed lines represent the bend loss of the LPx
11 and LPy respectively. In contrast to the loss curve of the fundamental mode, in which
(a) (b) (c) Figure 5.18: Modal intensity profile for (a) the fundamental mode, (b) the LPx
11mode and (c) the LPy11
mode, forRo = 3.0 cm for holey fibre HFI, Λ = 15.0µm,d/Λ = 0.63 at 1064 nm. Contours are separated
by 2 dB. 0.02 0.03 0.04 0 1 2 3 4 0.02 0.03 0.04 (a) (b)
Figure 5.19: (a) Bend loss as a function of bend radius for the fundamental mode (solid line) and the LPx
11 and LPy11modes in fibre HFI(dotted and dashed lines respectively) (Λ = 15.0µm,d/Λ = 0.63). (b)
The sum of the loss from the fundamental and the two higher-order modes shown in (a), assuming a ratio of 3:1:1 respectively.
the loss begins sharply at Ro ≈ R
FM
c , the bend loss for the higher-order modes increases more gradually as function of bend radius, reflecting the fact that the modes are not so well confined to the core. The fact that the higher-order modes become more distorted than the fundamental mode for the same bend radius can be seen in Fig. 5.18 and Fig. 5.19. The critical bend radius for the LPx
Once again, the sum of the bend loss from the fundamental mode and the LPx
11 and LPy11 modes is plotted as a function of bend radius assuming a ratio of 3:1:1 respectively in Fig. 5.19 (b). The critical bend radii for the LP11 modes are ≈1.5 larger than the critical bend radius of the fundamental mode, which is equal to 1.8 cm. From Fig. 5.19 (b) it can be seen that the higher-order modes can be stripped from the fibre using several turns of fibre with a bend radius of ≈2.5−3.0 cm or a single loop of≈2 cm. However, although the bend loss for the fundamental mode is not significant for Ro = 2−3 cm, the modal
distortion does affect the mode area of the fundamental mode in this range. As shown above in Fig. 5.10 (n) for Ro = 2.0 cm and in Fig. 5.18 (a) for 3.0 cm, the fundamental
mode is somewhat squashed. For these radii, Abenteff of fibre HFI reduces to ≈ 96% and 85% of that of the straight fibre, reducing the power handling capabilities of the fibre. As such, these results suggest that for the mode area of≈190µm2 considered here, fibre HFI with d/Λ = 0.63, probably represents the upper limit in d/Λ in which the higher-order modes can be successfully stripped by bending the fibre, without significantly perturbing the fundamental mode.