All-silica, large mode area, single mode photonic bandgap fibre with Fabry–Perot resonant structures

Optical Fiber Technology 28 (2016) 1–6

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All-silica, large mode area, single mode photonic bandgap fibre with Fabry–Perot resonant structures Zoltán Várallyay a,b,⇑, Péter Kovács c a

FETI Ltd., Késmárk utca 28/A, H-1158 Budapest, Hungary ELI-HU Nonprofit Ltd., Dugonics tér 13, H-6720 Szeged, Hungary c Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary b

a r t i c l e

i n f o

Article history: Received 1 July 2015 Revised 11 October 2015

Keywords: Photonic crystal fibre Photonic bandgap fibre Dispersion management Nonlinear propagation

a b s t r a c t All-silica, photonic crystal fibres consisting of a low index, silica core surrounded by higher index inclusions embedded in a silica matrix to form a photonic bandgap cladding were numerically analysed. The aim of the investigations was to modify the guiding properties of the fibre by introducing resonant structural entities. These structural modifications are realised by altering the refractive index of certain high index inclusions in the photonic crystal cladding resulting in mode coupling between the core mode and the mode propagated in the modified index region. This results in an increased effective core area of the fundamental core mode and consequently decreased nonlinearity as well as modified effective index compared to the effective index of the unmodified structure and resonant dispersion profile that can be used for pulse compression or optical delay purposes. Ó 2016 Elsevier Inc. All rights reserved.

1. Introduction Normal optical fibres, consisting of a high refractive index core and low refractive index cladding, propagate light by means of total internal reflection. Photonic bandgap (PBG) fibres, however, consist of a core with a lower refractive index than the average cladding index and waveguidence only happens if the cladding is able to open up a bandgap in a plane perpendicular to the propagation. In this case, the core modes remain confined in the low-index region and can propagate along the fibre. The photonic bandgap effect can be achieved by periodic low and high index structural entities surrounding the core region [1]. PBG fibres with high and low index cladding patterns show the typic dispersion functions for PBG structures. This is true for one (dielectric mirrors) or more dimensional forms of the bandgap structures. The dispersion within the bandgap increases monotonically from the shorter to the longer wavelengths, resembling a third order function having steeper courses close to the edge of the bandgap and a positive dispersion slope in the entire wavelength range where the fibre has low loss, namely in the bandgap [2]. Recent ultra-short fibre laser and amplifier developments lack effective, monolithic dispersion management for higher orders ⇑ Corresponding author at: FETI Ltd., Késmárk utca 28/A, H-1158 Budapest, Hungary. E-mail address: [email protected] (Z. Várallyay). http://dx.doi.org/10.1016/j.yofte.2015.12.003 1068-5200/Ó 2016 Elsevier Inc. All rights reserved.

and the common practice is to use chirped-mirrors to compress high power, ultra-short laser pulses [3]. The higher order dispersion contribution to the pulse evaluation from the optical components (optical fibres, isolators, couplers, splitters etc.) introduces a net third order dispersion (TOD) to the propagating pulse. This may also introduce some temporal distortions on the pulse if the pulse is short enough but TOD can be compensated by an optical element having negative dispersion slope. This requires dispersion modification of existing PBG fibres and broad resonances in the dispersion profile may result in negative dispersion slope at the long-wavelength edge of the resonance which can compensate the positive TOD contribution of other optical elements [4,5]. In telecommunication applications, specialty fibres may provide new possibilities in all-optical switching and ultra-fast wavelength division multiplexing (WDM) systems. This field requires fibres having resonance peaks in the dispersion function with relatively low losses in order to introduce some delay between the different channels. PBG fibres have different dispersion values at different frequencies but these differences are too small to achieve significant delay between channels using sufficient lengths of fibres. Dispersion tailored PBG fibres with narrow resonances and high peak values could be effectively used in time-gated filters [6], photonic time-stretched analog to digital converters [7] and in systems applying phase ripple correction [8]. The attempt to produce resonances in the bandgap and modify the dispersion for the fundamental optical mode in a waveguide is

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Z. Várallyay, P. Kovács / Optical Fiber Technology 28 (2016) 1–6

not a novel approach and was first demonstrated and calculated for Bragg fibres applying a dielectric confinement region around the core [4,9]. Negative dispersion slope was also achieved by modifying the hole size and shape of the first and second periods in a hollow-core PBG fibre with honey-comb cladding [10]. It turned out that the modification of the first period only in the same type of hollow-core fibre may yield resonant dispersion behaviour with more than 100 nm bandwidth [5]. We note however that the stored energy of the light in the resonant cladding structures cause mode field distortion which may lower the focusability of the output or the coupling efficiency to an other optical fibre [11]. Resonance structures in the cladding region of a PBG waveguide also can suppress higher order modes (HOM) in the air-core in a narrow wavelength range [12]. Increasing the core size is essential for high power pulse transmission and amplification in any kind of optical fibres to be appropriate for the recent demand of ultrashort and high power fibre lasers [13–17]. Increased core area can reduce the problems caused by the nonlinearity, mostly self-phase modulation (SPM) but in the same time it can lead to HOM propagation which lower the beam quality and it makes impossible such applications like reliable attosecond generation [18]. The usage of endlessly single mode photonic crystal fibre is reasonable to avoid HOM propagation [19]. Index guiding microstructured fibres have an embedded central core in a two-dimensional photonic crystal lattice with a hexagonal array of air holes. These type of fibres can be single mode for any wavelength and any core size [19]. In this case, the higher-order modes escapes between the holes with the fundamental mode confined in the core. The same single-mode operation can be achieved with photonic bandgap fibres with the same structure but replacing the air holes with high index inclusions [20]. The attempt to introduce resonant structures in the cladding in such microstructured and PBG fibre is presented by changing the hole size of the first period and the hole-to-hole spacing (pitch) between the first and second ring of high index inclusions [2]. These complicate structural modifications may be difficult to realise with the traditional stack and draw manufacturing process of these fibres. In this paper, finite element calculations on such PCF structures are presented in which the refractive index of some certain inclusions are modified whilst the fibre geometry, uniform diameter and distance of the inclusions across the fibre, remains constant. These modified refractive index regions may introduce a wavelength dependent mode coupling with the fundamental core mode. This may result in resonant guiding properties of the fibre which can be utilised where special dispersion properties are required. We finally demonstrate by numerical analysis solving the nonlinear propagation equation for short pulses that the designed fibres can be applied effectively for pulse compression purposes.

2. Results We solve the Helmholtz eigenvalue equation on the fibre crosssection using the Finite Element Method (FEM) [21] and a perfectly matched layer (PML) around the geometry to obtain the complex effective index (eigenvalue) which is used to calculate the dispersion and loss properties of the fibre. We use the obtained mode profile (eigenvector) to calculate the effective core area and the stored energy in the fibre structure. Fig. 1 shows a PCF structure of a SiO2 glass fibre with the higher index inclusions arranged in a hexagonal lattice with large pitch values in order to obtain large mode field area. The high index inclusions in similar fibres are typically GeO2 doped glass and we choose a refractive index 0.015 larger in absolute value than the low index silica at any calculated wavelength which is rheologically adequate. The silica refractive

Λ

d

nL

nH

nM

Fig. 1. All-silica PCF with low index silica matrix (nL ), high index inclusions (nH ) in the cladding and modified index inclusions (nM ) in the second period of the photonic crystal cladding. The pitch of the fibre is K while the diameter of the inclusions are uniform with a diameter of d.

index is calculated from the Sellmeier equation [22] and the refractive index of the GeO2 doped glass is adjusted to that value:

nH ðkÞ ¼ nL ðkÞ þ Dn

ð1Þ

where k denotes the wavelength of the propagating light, nH and nL are referring to the high index regions and low index regions in the fibre geometry, respectively and Dn is the difference between the high and low refractive indices what we choose now as 0.015. The fibre cross-section in Fig. 1 differentiates the modified refractive indices of certain inclusions which are now the ones in the second period around the core. We will refer to the modified index period around the core by nM1 if the modified period is the first period of inclusions around the core, by nM2 if the modified period is the second period around the core and so on. 2.1. The introduction of the resonances In the first calculation series, we present the dispersion and loss properties of the fibre if all inclusions are uniform and also in that case if we lower the refractive index of the second period of inclusions around the core using the parameter set given in Table 1. One can see that the second period of inclusions has just a 0:005 higher refractive index value than the low index silica matrix which can be easily prepared by the usual stack-and-draw technique using doped glass rods in the fibre preform with lower GeO2 concentration in the second period around the core. Fig. 2 shows the dispersion and the confinement loss of the propagating, fundamental core mode for three different nM2 . For nM2 ¼ nL þ 0:005, there is a resonance in the dispersion profile and a negative dispersion slope between 1332 nm (1.332 lm) and 1452 nm (1.452 lm) which is a 120 nm wide bandwidth can be achieved that way that the first order dispersion is anomalous in the whole wavelength range. Decreasing nM2 to nL þ 0:004 results in narrowing and blue shifting the resonance. There is a greater confinement loss of the fibre compared to the unresonant fibre structure (nM2 ¼ nH ) but it remains below 1 dB/m over the Table 1 Parameter list of the fibre cross-section given in Fig. 1. Refractive index values are given at 1 lm. Parameter

Value

d K nL nH nM2

2.9 lm 8.7 lm 1.4504 1.4654 1.4554

Z. Várallyay, P. Kovács / Optical Fiber Technology 28 (2016) 1–6

103

150

Dispersion [ps/(nm·km)]

1

10

0

50

10

10−1 0

10−2

−50

disp: nM2 = nH loss: nM2 = nH disp: nM2 = nL+0.004 loss: nM2 = nL+0.004 disp: nM2 = nL+0.005 loss: nM2 = nL+0.005

−100 −150 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

10−3 10−4 −5

Confinement loss [dB/m] (log)

2

10 100

10

10−6 1.8

Wavelength [μm] Fig. 2. Dispersion and confinement loss of the guided, fundamental core mode with the regular PC cladding structure and different, resonant nM2 values.

0.992

0.99

2

Effective core area [μm ]

700 600 0.988

500 nM2 = nL+0.005 nM2 = nL+0.005 nM2 = nH nM2 = nH

400

0.986

300 0.984 200 100 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Energy − group delay ratio [J/s]

800

0.982 1.7

Wavelength [μm] Fig. 3. Effective core area of the propagating core mode and the related U=s ratio with the regular structure and with nM2 ¼ nL þ 0:005.

whole calculated wavelength range which is an acceptable value for usages which only require a few meters of fibre, like high power pulse shaping. The effective core area (ECA) of the fibre and its variation as a function of wavelength is an important parameter as at high power levels, the fibre nonlinearities will become significant if the ECA is small [22] and the larger the ECA, the higher power pulse can be transmitted through the fibre without significant spectral and temporal distortions. An other important parameter investigated is the

3

stored energy in the fibre structure divided by the group delay of the pulse. This ratio is related to the amount of mode field distortion and the fibre dispersion [11] and should approach unity. Fig. 3 shows the calculated effective core area and the ratio of the stored energy and group delay for the regular structure (nM2 ¼ nH ) and for a fibre cross-section with one modified period of inclusions (nM2 ¼ nL þ 0:005). Fig. 4 also shows mode profiles present. As the mode coupling appears between the core mode and around the second period of inclusions the mode field area starts to increase as the mode expands into the cladding region. The ECA is more than 500 lm2 at the long wavelength edge of the 1400 nm resonance curve in Fig. 2. This mode field expansion is advantageous, from the point of nonlinearity, as the obtained mode profile (Fig. 4(b) and (c)) is similar to a higher order mode profile and has a significantly larger area than the non-resonant mode. This higher order mode-like propagation is a very similar way of light transfer using higher order mode (HOM) indexguiding fibres [23] where the primary aim of the HOM injection into the fibre is the reduction of the nonlinear coefficient by the increased effective core area [22].The same effect is achieved here in a photonic bandgap fibre introducing a resonant layer of inclusions around the core in the deeper regions of the cladding which structure this way supports this resonant mode propagation. Fig. 3 shows that this condition is satisfied and this E=s value is greater than 0.98 for the whole bandgap of the fibre and decreases at those regions where the fibre loss is elevated compared to the low loss wavelength ranges. This shows that the fibre loss is not significant [11]. 2.2. Dispersion profile tailoring The position of the resonance peak can be adjusted by selecting different pitch size and different diameter for the inclusions. Instead of calculating similar structures with different pitch K and inclusion diameter d, the refractive index change of other periods of inclusions on a previously calculated resonant dispersion profile was calculated. Fibre structure having nM2 ¼ nL þ 0:005 were chosen and the refractive index of other periods of inclusions in the PCF cladding such as the first and third ones around the core were varied. Fig. 5(a) and (b) shows the effect of setting different refractive indices for the first and third periods of inclusions on the dispersion function and confinement loss profile while the corresponding ECAs and U s ratios are plotted in Fig. 6(a) and (b). Modifying the refractive index of the first period of inclusions along with the second period ones results in stronger resonances in the dispersion function than having only the resonant second period of inclusions around the core. This is the case when the refractive index of the inclusions in the first period are increased compared to the original high index inclusions (See Fig. 5(a)

Fig. 4. Mode profiles for the smaller but broader resonance nM2 ¼ nL þ 0:005 at (a) 1275 nm, (b) 1325 nm and (c) 1375 nm.

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Z. Várallyay, P. Kovács / Optical Fiber Technology 28 (2016) 1–6

100

500 nM1 = nL+0.009 nM1 = nL+0.011 nM1 = nL+0.013 nM1 = nL+0.015 nM1 = nL+0.017 nM1 = nL+0.019 nM1 = nL+0.021

400 300 200

10

−2

10

−3

10

−4

10

−5

10−6 10

100

−7

10−8

0

10

−100 1.1

1.15

1.2

1.25

1.3

1.35

1.4

−9

nM3 = nL+0.009 nM3 = nL+0.011 nM3 = nL+0.013 nM3 = nL+0.015 nM3 = nL+0.017 nM3 = nL+0.019 nM3 = nL+0.021

80 60 40

4

10

2

10

20

0

10

0 −2

−20

10

−40 −4

10

−60

1.45

Confinement loss [dB/m] (log)

Dispersion [ps/(nm·km)]

600

100

−1

Dispersion [ps/(nm·km)]

10

Confinement loss [dB/m] (log)

700

1.1

1.15

1.2

1.25

Wavelength [μm]

1.3

1.35

1.4

1.45

Wavelength [μm]

(a)

(b)

Fig. 5. Dispersion and confinement loss of PCF with different nM1 (a) and nM3 (b) values keeping nM2 ¼ nL þ 0:005. (a) The first ring of inclusions are varied nM1 ¼ nL þ 0:009; . . . ; nL þ 0:021 and (b) the third ring of inclusions are modified: nM3 ¼ nL þ 0:009; . . . ; nL þ 0:021.

400

0.988 0.987 0.986 0.985

200 0 1.1

0.984 1.15

1.2

1.25

1.3

1.35

1.4

0.983 1.45

0.99

1000

0.989 800 600 400

nM3 = nL+0.009 nM3 = nL+0.011 nM3 = nL+0.013 nM3 = nL+0.015 nM3 = nL+0.017 nM3 = nL+0.019 nM3 = nL+0.021

0.988 0.987 0.986 0.985

200 0 1.1

0.984 1.15

1.2

1.25

1.3

1.35

Wavelength [μm]

Wavelength [μm]

(a)

(b)

1.4

Energy−group delay ratio [J/s]

600

nM1 = nL+0.009 nM1 = nL+0.011 nM1 = nL+0.013 nM1 = nL+0.015 nM1 = nL+0.017 nM1 = nL+0.019 nM1 = nL+0.021

Effective core area [μm ]

0.989 800

0.991

2

0.99

1000

1200

Energy−group delay ratio [J/s]

0.991

2

Effective core area [μm ]

1200

0.983 1.45

Fig. 6. The effective core area and the ratio of energy and group delay are shown for different refractive indices of the first and third periods of inclusions. Fig. 6(a) and (b) corresponds to Fig. 5(a) and (b), respectively.

Table 2 Properties of the resonant peaks at different refractive indices for the first and third periods of inclusions. Refractive indices are represented by the difference value with that the actual period has larger index than the low index region: DnMi ¼ nMi  nL ; i ¼ 1 . . .. In case of nM1 ¼ nL þ 0:009 and 0.011, no any resonance can be found.

DnM1

DnM2

DnM3

FWHM (nm)

DL (nm)

DD (ps/nm/km)

0.009 0.011 0.013 0.015 0.017 0.019 0.021

0.005 0.005 0.005 0.005 0.005 0.005 0.005

0.0015 0.0015 0.015 0.015 0.015 0.015 0.015

81.8 77.4 49.0 15.8 7.6

114.6 120.1 93.2 53.7 35.2

4.809 23.709 73.947 287.822 624.205

0.015 0.015 0.015 0.015 0.015 0.015 0.015

0.005 0.005 0.005 0.005 0.005 0.005 0.005

0.009 0.011 0.013 0.015 0.017 0.019 0.021

61.0 66.7 72.3 77.4 79.6 82.6 82.1

108.6 113.6 117.6 120.1 121.9 123.5 122.7

55.084 40.06 30.395 23.709 18.803 15.053 12.107

nM1 ¼ nL þ 0:021 compared to nM1 ¼ nL þ 0:015 ¼ nH ). If the refractive index of the first period of inclusions are decreased, weaker but wider resonances on the dispersion function can been seen. Fig. 5(a) nM1 ¼ nL þ 0:011 is a good example of this behaviour. As the first period of inclusions has large contribution to the effective index of the propagating core mode, the drastic changes are also seen in the ECA where values of 700 lm2 ECA (Fig. 6(a) nM1 ¼ nL þ 0:017) are obtained. Subtle changes in the dispersion curves can be observed by keeping the first period of inclusions the same as in the original structure (nM1 ¼ nH ) and the second period of inclusion the same found in the former section nM2 ¼ nL þ 0:005 and varying the refractive index of the third period of inclusions(Fig. 5(b)). This modification still enable the tuning of the strength of the resonance and subsequently the slope of the dispersion. Table 2 summaries the resonances calculated using parameters in Fig. 7. Parameter DD is the height of the resonance peak expressed with the difference between the peak value and the zero slope value of it at the long wavelength edge, FWHM is the full width at half maximum as shown in Fig. 7 and DL is the

5

Z. Várallyay, P. Kovács / Optical Fiber Technology 28 (2016) 1–6

Fig. 7. Resonance peak in the dispersion and the parameters that we characterise the peak with.

Table 3 Dispersion parameters of the original fibre (nM2 ¼ nH ) and the one with resonant second period (nM2 ¼ nL þ 0:005) using Eq. (2) and k0 ¼ 1370 nm. Loss and effective core area are also mentioned at 1370 nm. Parameter

Original fibre

Resonant struc.

Units

D S T

26:8034 0:2307

53:1318 0:6225

ps/(nm km) ps/(nm2 km) ps/(nm3 km)

5:4994  104

F

3:5046  10

Loss (a) Effective core area (Aeff )

6

2:8766  103 6:8019  104

2:42  103 128:4

17:51  103 405:3

ps/(nm4 km) dB/m

lm2

wavelength region where the dispersion slope is negative. This table clearly indicates that the 120 nm reversed dispersion slope can be achieved and these broad resonances have small peaks, consequently shallow broadband slopes and in some cases nearly flat dispersion profile can be seen. For instance, the third period of inclusions are modified, nM3 ¼ nL þ 0:021 where the dispersion changes are only 12:1 ps=ðnm kmÞ over 122 nm wavelength range which could support sub-15 fs pulse compression in this wavelength range.

3. Nonlinear compression in PBG fibres

450

Peak power (kW)

400 350 300 250

T F DðkÞ ¼ D0 þ Sðk  k0 Þ þ ðk  k0 Þ2 þ ðk  k0 Þ3 2 6

The nonlinear refractive index of the fibre is 2:6  1020 m2 =W [22] and all other parameters required for the calculations can be seen in Table 3. The propagation of the input pulse through the two different fibres are shown in Fig. 8. Fig. 8(a) shows the pulse shapes corresponding to 8.1 cm of propagation in both type of fibre structures and also after 6.3 cm propagation in the case of the original, unmodified fibre. The propagation lengths for the output pulses are selected based on the peak power evaluation in Fig. 8(b). The highest peak power is associated with the largest compression 500

input original 8.1 cm original 6.3 cm resonant 8.1 cm

1.4 1.2 1 0.8

400

0.4 0.2 0 1250 1300 1350 1400 1450 1500 1550

Wavelength (nm)

28.5 fs

150

350 300 250 200

8.1 cm

150

100

100

50

50

0 −200

original resonant

450

0.6

200

ð2Þ

where k is the wavelength, k0 is the reference wavelength, D0 is the linear dispersion, S is the dispersion slope, T is the third order dispersion and F is the fourth order dispersion. The fitted dispersion parameters of the two selected fibres for k0 ¼ 1370 nm is collected in Table 3 along with the loss and effective core area parameters. We consider pulse parameters which are easily obtainable from today’s popular master oscillator power amplifier (MOPA) arrangement [24] or from chirped pulse amplification systems [25]. We assume 12.5 nJ, high energy output pulses corresponding to a system with 1 W average power at 80 MHz repetition rate. We calculate with a Gaussian spectrum having 60 nm bandwidth and also second order and third order input chirps as much as 5000 fs2 and 30,000 fs3, respectively [26]. These are contributions from the optical components in the oscillator and amplifier systems.

Peak power (kW)

500

Spectral intensity (arb. units)

We apply a similar concept to Ref. [5] to demonstrate the pulse compression capability of the designed fibres. Namely, we use the

dispersion function of the above designed fibres to calculate the pulse propagation of an input pulse originating from a fibre oscillator or amplifier which produces chirped output. We use a second order split-step Fourier method to calculate the propagation of the slowly varying, complex envelope function of the input pulse through the investigated fibre [22]. We compare then the output pulse shapes and spectra from the two different PBG fibres. We note here that the designed fibres has a reversed dispersion profile in such a wavelength range where rear-earth doped fibres are rarely used but simply scaling the geometry of the designed fibre will shift the bandgap accordingly. This way the designed fibre can be used in any nearby wavelength region by scaling the cross-section of it. We calculate with the dispersion function of the resonant model where nM2 ¼ nL þ 0:005 and also the unmodified fibre when nM2 ¼ nH (See Fig. 2). We fit the dispersion curve in the following Taylor form

6.3 cm

0 −150 −100

−50

0

50

100

150

200

0

2

4

6

8

10

Duration (fs)

Length (cm)

(a)

(b)

12

14

16

Fig. 8. (a) Pulse shape of the input pulse and the output ones in three different cases. ‘‘Original” refers to the unmodified fibre cross-section while ‘‘resonant” refers to the fibre with resonant structure in the second period of insets (nM2 ¼ nL þ 0:005). Inset in the graph shows the input and output spectra in the same cases as temporal shapes (same line type belongs to the same fibre and propagation length) and (b) evolution of the peak power of the pulse along the two different fibres.

6

Z. Várallyay, P. Kovács / Optical Fiber Technology 28 (2016) 1–6

ratio and consequently the shortest pulse [26]. One can see that the propagation in the two fibres results in the maximum peak power after 8.1 cm of propagation. While the pulse shape is a nice, compressed pulse in the case of the modified fibre, it is highly distorted in the case of the original cross-section (Fig. 8(a)). The modified, resonant cross-section results in a pulse shape after 8.1 cm of propagation in which the main peak contains the 96:8% of the total pulse energy and the full-width at the half maximum (FWHM) is shorter than 30 fs. This pulse duration is possible due to the nonlinear spectral broadening which results in a 131 nm spectral FWHM at around 1370 nm at the fibre output. The spectral shape shows some sign of self-phase modulation (SPM) but remains almost completely symmetric which means that the higher than second order dispersion did not contribute to the spectral formation significantly, namely the fibre can compensate the chirp on the pulse effectively. The pulse shape at the output of the original, unmodified fibre cross-section is highly distorted (Fig. 8(a)). Only 55:7% pulse energy remains in the main peak after 8.1 cm of propagation. The spectral shape of it is also significantly broadened due to the relatively high nonlinearity (small core area) and it is also highly distorted due to the uncompensated higher order dispersion contributions. Therefore we looked for a distance in the original fibre where the pulse break-up did not occur. We found that this was at 6.3 cm length of propagation. At 6.4 cm, 20% of the total pulse energy was outside of the main peak. At 6.3 cm, 100% of the total pulse energy is located in a highly asymmetric pulse which is hardly compressed due to the remaining, uncompensated higher order chirps. We showed here that in the used power range, the designed fibre with resonant dispersion profile can be used effectively to compress chirped pulses on the femtosecond time scale since it can successfully eliminate the third order chirp contribution originating from other optical fibres and components.

[3]

[4]

[5]

[6]

[7] [8] [9]

[10] [11]

[12] [13]

[14]

[15]

[16]

[17]

[18]

4. Conclusions It has been shown, by numerical calculations, that introducing resonant structural entities in a photonic bandgap cladding may result in advantageous physical properties of the waveguide. One of these beneficial characteristic is the obtainable resonant dispersion function which may be used for pulse compression or the purposes of dispersive guidance. The other improved characteristic of the resonant waveguide is the increased mode field area or effective core area which can result in reduced nonlinearity and consequently elevated power of beam delivery. This is possible without irreversible nonlinear distortions. Changing the refractive indices of inclusions in combination with other periods of inclusions may result in a huge variety of shapes in the dispersion function of the propagating fundamental mode. References [1] A. Bjarklev, J. Broeng, A.S. Bjarklev, Photonic Crystal Fibres, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003. [2] Zoltán Várallyay, Kunimasa Saitoh, Photonic Crystal Fibre for Dispersion Controll, Frontiers in Guided Wave Optics and Optoelectronics, Bishnu Pal

[19] [20]

[21]

[22] [23]

[24]

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