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Recently, the generation of coherent, octave-spanning, and recompressible supercontinuum (SC) light has been demonstrated in optical fibers with all-normal group velocity dispersion (GVD) behavior by femtosecond pumping. In the normal dispersion regime, soliton dynamics are suppressed and the SC generation process is mainly due to self-phase modulation and optical wave breaking. This makes such white light sources suitable for time-resolved applications. The broadest spectra can be obtained when the pump wavelength equals the wavelength of maximum all-normal GVD. Therefore each available pump wavelength requires a specifically designed optical fiber with suitable GVD to unfold its full power. We investigate the possibilities to shift the all-normal maximum dispersion wavelength in microstructured optical fibers from the near infra red (NIR) to the ultra violet (UV). In general, a submicron guiding fiber core surrounded by a holey region is required to overcome the material dispersion of silica. Photonic crystal fibers (PCFs) with a hexagonal array of holes as well as suspended core fibers are simulated for this purpose over a wide field of parameters. The PCFs are varied concerning their air hole diameter and pitch and the suspended core fibers are varied concerning the number of supporting walls and the wall width. We show that these two fiber types complement each other well in their possible wavelength regions for all-normal GVD. While the PCFs are suitable for obtaining a maximum all-normal GVD in the NIR, suspended core fibers are well applicable in the visible wavelength range.
©2011 Optical Society of America
1. Introduction
The invention of photonic crystal fibers (PCF) [1] has opened a new field of supercontinuum (SC) generation supported by both a versatile freedom in designing the waveguide dispersion and minor demands on pulse energy available by standard femtosecond lasers. It took short time to discover that in the ultrashort pulse regime octave-spanning spectra could be obtained when pump laser and group velocity dispersion (GVD) are matched in the way that the pump wavelength is close to the zero dispersion wavelength (ZDW) at the anomalous dispersion side [2].
Single ZDW setups have been examined in detail in the past due to the inherent GVD material behavior of silica with a ZDW around 1300 nm. The advantages concerning spectral width of the generated SC when ZDW and pump wavelength are matched appropriately led to extensive numerical investigations on how to push the ZDW below the material ZDW to a wide range of wavelengths in the visible and near-infrared regime [3]. According to these investigations a strand of silica in air constitutes the low wavelength limit when the air filling fraction of the PCF reaches unity.
The extraordinary broadening relies on soliton dynamics only possible in the anomalous dispersion regime (ADR). The drawback of utilizing soliton dynamics for SC generation is the occurrence of soliton fission that splits up the input pulse into a series of subsequent soliton pulses, each exhibiting unique temporal and spectral distributions that are very susceptible to pump pulse shot noise. Therefore these SC can hardly be used for time critical applications like optical coherence tomography, pump-probe spectroscopy, metrology or non-linear microscopy. In addition compression to few or even single cycle pulses is impossible. A calculated spectrogram of such a typical SC pulse is illustrated in Fig. 1a ).
Fig. 1 Resulting spectrograms for typical SCG in PCFs with a) a single ZDW ([2], Λ = 1.6 µm, d hole = 1.4 µm, 100 fs, 10 kW peak power, 800 nm) and c) SCG in all-normal GVD optical fibers ([5], Λ = 1.44 µm, d hole = 0.56 µm, 50 fs, 8 nJ pulse energy, 1050 nm). b) Illustration of corresponding GVD behavior. Arrows mark recommended pump wavelengths.
After extensive studies about single ZDW SC generation, the focus changed to PCFs that possess two closely spaced ZDWs and which were pumped in between them. Due to the minimized influence of the ADR, spectra with stabilized temporal and spectral properties and decreased noise behavior could be generated that are beneficial for time critical applications [4]. However, the generated SC show two major spectral peaks separated by a depleted spectral region a few hundred nanometers in width corresponding to the ADR. Thus applications are limited to the wavelength ranges of the major spectral peaks of the generated SC.
Recently further improvement of spectral homogeneity and width were achieved in all-normal dispersion PCFs without any ZDW [5], (Fig. 1b). In the normal dispersion regime (NDR) all noise sensitive effects are suppressed, resulting in excellent pulse-to-pulse coherence. Furthermore, the breakup of the injected pulse into multiple pulses is suppressed and a single recompressible pulse remains in the time domain. Combining all characteristics, these SC are promising candidates for numerous time critical applications as listed above. Figure 1c) illustrates the extraordinary properties by a calculated projected axes spectrogram.
The all-normal GVD is essential for the recently demonstrated coherent and broadband single pulse SCG in optical fibers. It has been shown that the broadest spectra are generated when the fiber is pumped at the wavelength with its dispersion being closest to zero [6]. Since this dispersion is the maximum of the dispersion curve this wavelength is called maximum dispersion wavelength (MDW). For a GVD curve exhibiting both normal and anomalous dispersion the MDW is only connected with the maximum of the dispersion curve and the corresponding dispersion is not the one closest to zero. We discuss the various possibilities of silica air microstructured optical fibers for tailoring the waveguide dispersion to shift the normal MDW across the interesting wavelengths ranges in the visible and near infrared.
2. All-normal dispersion optical fibers
A high refractive index contrast like silica to air in combination with a small core diameter in the order of one wavelength is required for a large waveguide dispersion to counteract material dispersion. This can nowadays be realized in microstructured optical fibers by multiple air hole inclusions running along the optical fiber. In this paper two kinds of fibers are numerically investigated in detail. In suspended core fibers (SCF), a single ring of large air holes runs around the small core that is supported by thin silica walls as shown in Fig. 2a . Design parameters are the number N and the width w of the supporting walls. In PCFs these air hole inclusions are arranged on a hexagonal lattice and a single lattice defect represents the guiding core (Fig. 2b). The hole diameter d hole and the inter hole spacing Λ are the two degrees of freedom that can be independently set to a wide range of values to tailor the waveguide dispersion. From the investigation of how to push the ZDW into the visible in PCFs [3] it is known that an optical nanofiber is the ultimate limit if the air filling fraction d hole /Λ approaches unity. We will see that the SCF approaches the nanofiber as well if N approaches infinity and w zero.
Fig. 2 Scanning electron microscope images of an all-normal dispersion a) SCF and b) PCF. Both fibers were stacked and drawn at the IPHT Jena. The scale bar is a) 1 µm and b) 100 nm.
The calculated GVD of an optical nanofiber surrounded by air as a function of the fiber diameter is shown in Fig. 3a . In this paper all wavelength and geometry dependent propagation constants required for GVD calculations are simulated via a fully vectorial finite elements method (FEM). The refractive index of air was taken as unity and material dispersion of silica has been taken into account.
All GVD curves of the calculated fiber diameters shown in Fig. 3a exhibit a MDW. At a fiber diameter of 500 nm the GVD maximum is located in the ADR and decreases with decreasing fiber diameter. The maximum is close to zero at a diameter of 470 nm and enters the NDR for smaller diameters. Thus optical nanofibers below this diameter are suitable for coherent single pulse SCG. The reduction of the diameter has negligible influence on the short wavelength side of the dispersion curve that stays nearly fixed but shows a huge impact on the long wavelength side. The evolution of the MDW is illustrated in Fig. 3a as well (dark green). It exhibits a blue shift with decreasing fiber diameter. The largest MDW situated in the NDR is around 490 nm thus pumping at this wavelength is recommended. The MDW has a remarkable linear dependence on the fiber diameter as shown in Fig. 3b. We found no simple physical origin for this circumstance. In first-order approximation the GVD maximum value can be seen as a linear function of the fiber diameter as well but slight deviations are obvious in Fig. 3b.
2.1 Suspended Core Fibers (SCF)
The handling of bare and delicate nanofibers can be simplified by suspending them by thin radial running walls of silica in the center of a conventionally sized optical fiber of 125 µm. In addition to the core diameter d as examined in the case of the nanofiber, the waveguide geometry and thereby the overall dispersion now is influenced by the number N and width w of supporting walls. Different core geometries are possible and some of them are illustrated in Fig. 4 . Three and four supporting walls can be manufactured quite easily just by stacking three or four capillaries with thin walls together and draw them to the desired size. The individual walls will melt and a core will be formed at the central region that is enclosed by the capillaries. A SCF supported by six or twelve walls potentially could by manufactured by blowing up the first or second ring of hexagonally arranged capillaries as common for PCFs. The idealized numerical geometry is defined by the number N and width w of the walls and by the incircle core diameter d. Assuming the core boundaries are formed by circular arcs, the SCF geometry is clearly defined by these parameters.
Fig. 4 Various SCF geometries discussed in the text. The upper row presences ideal polygonal SCFs (left to right: N = 3, 4, 6, 12) with vanishing wall width w. Dark blue represents silica and light blue represents air. In the lower row polygonal SCFs with a wall width of w = 50 nm are illustrated. The incircle core diameter is 700 nm in all cases.
We first examine the impact of the core geometry at vanishing wall width w = 0 due to the closest agreement with the unsuspended nanofiber. In Fig. 5a the GVD for the trigonal (N = 3) geometry for a diameter range that exhibits both anomalous and normal dispersion at the MDW is shown. In agreement with the nanofiber and typical for all other SCF geometries the dispersion maximum and the MDW decreases with decreasing fiber diameter. The short wavelength side of the GVD curve is nearly unaffected by the fiber diameter and a strong impact can be seen at the long wavelength side. The largest MDW situated in the NDR is at about 615 nm and at a fiber diameter of around 530 nm. Both values are significant larger than the corresponding nanofiber values.
Fig. 5 a) GVD of a SCF with N = 3 and w = 0. GVD maximum enters NDR at 615 nm wavelength. b) Maximum dispersion for various SCFs with w = 0 as a function of core diameter. With increased number of walls N the SCF MDW approaches the MDW of the nanofiber. The SCF core sizes evaluated in b) are identical to those presented in a).
A comparison of the MDW as a function of core diameter for all mentioned polygonal SCF geometries with w = 0 is shown in Fig. 5b. The trigonal SCF has the largest MDW in the NDR at around 615 nm. An increased number of walls leads to a blue shift of this MDW (N = 4: 568 nm; N = 6; 534 nm; N = 12: 506 nm) which successively approaches the nanofiber value of 490 nm. This approach in GVD can be explained by the increased circularity of the core shape with increasing N, as obviously seen in Fig. 4.
The influence of the wall width is exemplarily illustrated in Fig. 6a for N = 6 and for two different core diameters (d = 700 nm and d = 500 nm). With increasing wall width a decrease of the dispersion maximum occurs at nearly constant MDW. This is typical for all simulated combinations of N and d. The interplay of core diameter and wall width is shown in Fig. 6b. Thus if a GVD with desired MDW is found, the maximum dispersion value can be pushed into the NDR by increasing the wall width w. For fixed N the smallest MDW in the NDR can be reached at vanishing w. Larger MDWs in the NDR close to zero GVD can be obtained by increasing both wall width and core diameter. The influence of the wall width increases dramatically with the number of walls as illustrated in Fig. 6c. While the MDW of the trigonal SCF shows nearly no dependence concerning the wall width, the MDW of the dodecagonal SCF strongly depends on it. This impact can in an extreme case even turn an initially lower MDW due to a more circular core shape at a small wall width to a larger MDW at a larger wall width. This increased influence is explained by the larger summarized amount of silica in the cladding due to a higher N at equal wall width w as explained in the following example. The summarized circular arc length of pure silica for a wall width w = 100 nm is around N*w = 300 nm for the trigonal SCF and around 1200 nm for the dodecagonal SCF. For an incircle diameter of d = 300 nm the core circumference is π*d = 942 nm. This value is well above the summarized arc length of the trigonal SCF thus the wall width has minor influence on the core boundaries and on the GVD of the trigonal SCF. On the contrary, the summarized arc length of the dodecagonal SCF is larger than the core circumference. The original core boundaries do no longer define core shape and GVD resulting in a significant modification of the latter.
Fig. 6 a) Influence of wall width on GVD. The maximum GVD decreases with increasing wall width at nearly constant MDW. b) Interplay of core and wall size. Smallest MDW in the NDR is reached at vanishing wall width. Higher MDW in the NDR is possible when both wall and core size is increased. c) Impact of wall width on GVD increases dramatically with wall number N.
2.2 Photonic Crystal Fibers (PCF)
First experimental results concerning broadband and coherent single pulse SC generation were demonstrated in all-normal dispersion PCFs with a MDW in the near infrared around 1020 nm and in the visible around 650 nm [5]. The different employed MDWs already point to the flexibility in designing the waveguide dispersion by the hole diameter d hole and the inter hole spacing Λ. The full versatility is shown in Fig. 7 where the MDW in the NDR is shifted from around 550 nm up to 1300 nm. A small pitch and a high air filling fraction near unity is required for small MDWs in the visible and a large pitch and a low air filling fraction for large MDWs located in the near infrared. Within the displayed geometry range the nonlinear parameter γ and mode field diameter dMF at the MDW are varying from around γ = 1.35 (Wm)−1 and dMF = 520 nm at the short wavelength side (Λ = 500 nm) to around γ = 0.0046 (Wm)−1 and dMF = 7.6 µm at the long wavelength side (Λ = 2300 nm). They were calculated as denoted in [7].
Fig. 7 Various GVD curves illustrating the richness in designing the dispersion of PCFs by pitch Λ and air filling fraction d hole/Λ. Within the NDR the lower MDW limit approaches the nanofiber value and the upper MDW limit around 1300 nm is set by the disappearance of any GVD maximum for larger Λ.
The lower limit for the GVD and MDW approaches the nanofiber values and upper limit around 1300 nm is set by the characteristic GVD behavior. An increase of the MDW in the NDR is not possible due to flattening of the GVD with increasing pitch at the long wavelengths side until disappearance of any GVD maximum as illustrated in Fig. 8a . When approaching this limit a very low GVD around −10 ps/nmkm can be achieved over several hundreds of nanometers of wavelength promising large spectral broadening by self phase modulation (Fig. 8a), dotted).
Fig. 8 a) Upper limit for an all-normal GVD maximum is set by disappearance of any GVD maximum around Λ = 2500 nm. b) Influence of air filling fraction at constant pitch on GVD.
Due to the low air filling fraction accompanied by a large mode field diameter a large number of rings (> 15) have to be simulated and hence fabricated for MDWs around 1300 nm. In contrast only a small ring number (approx. 5) is sufficient for mode confinement in PCFs with high air filling fraction d/Λ as is the case for MDWs in the visible.
The influence of the air filling fraction d hole/Λ on the GVD at constant pitch Λ is illustrated in Fig. 8b. The pitch Λ basically defines the position of the MDW while the hole diameter d hole serves for reduction of the GVD into the NDR accompanied by a slight decrease of the MDW. This behavior is similar to SCFs where the core radius basically defines the MDW position and decreasing of the GVD into the NDR can be reached by increasing the wall width.
For the hexagonal SCF a PCF with identical core shape can be constructed provided the following equations are valid: N = 6, Λ = d-w and dhole = d-2w. Identical core shape and index step raises the question if the GVD of the two fibers are identical as well. In Fig. 9 the GVD of a hexagonal SCF and a corresponding PCF with identically shaped core are shown. The obvious differences in the GVD for the considered waveguides can be explained by accounting for an effective cladding index and not for the identical index step. The effective cladding index of the PCF can be accurately calculated by the fundamental space filling mode (FSFM) of the infinite and defect free photonic crystal cladding. Since there is no infinite but periodic extension of the SCF cladding, the effective SCF cladding index can only be specified by rough estimations. Due to the differences in the calculated GVD the effective SCF cladding index has to be significant different from the FSFM index. This difference increases with wavelength since longer wavelengths extend more into the cladding than shorter ones.
Fig. 9 Comparison of GVD for a SCF and PCF with identically shaped core and indexed step. The PCF parameters are Λ = 550 nm and dhole = 500 nm.
From SCF to PCF a downshift of the GVD maximum is visible as is the fact when increasing the wall width at fixed core size. Thus at identical core shape and index step the PCF behaves like a SCF with larger wall width due to the increased amount of silica in the photonic crystal cladding.
3. Conclusion
We examined in detail the possibilities in designing the GVD of SCF and PCF to all-normal behavior. SCFs are well suited for constructing all-normal GVD optical fibers in the visible wavelength range. PCFs can be utilized in the near infra red as well. In both cases the low wavelength limit for an all-normal GVD is set by a strand of silica in air. The long wavelength limit for PCFs is induced by the disappearance of any GVD maximum. Therefore a MDW located in the NDR can by tuned from around 500 nm up to 1300 nm. Within this range any wavelength can be addressed by appropriate fiber geometry.
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