Mid-infrared sources based on the solitonself-frequency shift

Alaa M. Al-kadry* and Martin Rochette

Department of Electrical and Computer Engineering, McGill University, 3480 University Street, Montreal,Quebec H3A 2A7, Canada

*Corresponding author: alaa.al‐kadry@mail.mcgill.ca

Received December 13, 2011; revised March 21, 2012; accepted March 21, 2012;posted March 22, 2012 (Doc. ID 159747); published May 22, 2012

We present a method to maximize the soliton self-frequency shift (SSFS) in microwires with diameter profilesvarying nonuniformly along the soliton propagation path. The method is divided into two steps. The first stepconsists in selecting the input microwire diameter that leads to the highest rate of frequency shift per unit of pro-pagation length. The second step consists in increasing gradually the microwire diameter along the soliton path tosuppress dispersive wave emission and maintain a large rate of frequency shift per unit of propagation length. Wefirst propose and apply a rule to select the initial diameter using the adiabatic theory. The optimal diameter profileis then achieved by maintaining the redshifting soliton at a fixed spectral separation from the zero-dispersionwavelengths. The optimized profile supports solitons with different input energies that allow a wavelength shiftup to 650 nm from the 2100 nm pump wavelength in a 20 cm microwire length. We compare our results with theSSFS generated in microwires with uniform diameter profile to illustrate the enhancement of wavelength shift inthe designed nonuniform microwire. © 2012 Optical Society of America

OCIS codes: 060.2310, 060.2280, 060.4370.

1. INTRODUCTIONMid-IR laser sources are currently required for applicationssuch as spectroscopy, instrumentation, and biomedicine. Apromising new way of developing mid-IR sources is to exploitnonlinear wavelength shift via the soliton self-frequency shift-ing (SSFS) effect triggered by a pulsed laser operating at awavelength at the transmission limit of silica glass, e.g., a fiberlaser based on Tm or Ho [1,2]. In SSFS, the longer wavelengthpart of the pulse spectrum experiences an amplificationpumped by the shorter wavelength part through intrapulseRaman scattering. As a result, the central wavelength of thepulse undergoes a continuous redshift [2,3]. The SSFS effectdepends mainly on (i) the pump pulse duration, (ii) the wave-guide nonlinearity, and (iii) chromatic dispersion.

Since its observation in single-mode fibers, SSFS was inves-tigated in different fiber profiles with a goal of maximizing thewavelength shift per unit of propagation length. This includes,for example, polarization maintaining fibers [4], highly non-linear fibers [5], tapered [6] and untapered [7,8] microstructuredoptical fibers, bandgap hollow fibers [9], and tapered single-mode fibers [10]. The use of microstructured [11] and taperedoptical fibers has enhanced the waveguide nonlinearity (γ) andthus reduced the amount of energy Eo required to generate in-put solitons through Eo ∝ 1∕γ [12]. In addition, the microstruc-tured and tapered optical fiber profiles have allowed thetailoring of group-velocity dispersion (GVD) over a wide spec-tral range [13]. This has opened the prospect of realizing SSFSat low input powers [7], in short fiber lengths [6,10], and atdifferent pump wavelengths [8,14,15,16]. Furthermore, the re-cent advancement of photonic crystal fiber fabrication [17]and the tapering technique [18] is promising in maximizingthe wavelength shift per unit of propagation length.

In spite of tuning the soliton wavelength over hundreds ofnanometers, the transmission window of silica glass between300 and 2200 nm limits the wavelength conversion toward themid-IR. In contrast with this, chalcogenide glasses (ChGs) aretransparent up to ∼10 μm and thus can serve for the purposeof SSFS in the mid-IR. Moreover, these glasses have an intrin-sic nonlinear refractive index between 2 and 3 orders of mag-nitude higher than that of silica [19], and this nonlinearity canbe even further increased by strong modal confinement in ta-pered fibers with a diameter of submicrometer (microwires)[20]. The use of ChG microwires has also allowed the genera-tion of supercontinuum in the region of anomalous dispersionusing commercially available femtosecond laser sources andlowered the energy requirements for wide spectral broaden-ing. A limitation of microwires is their extreme mechanicalfragility and optical interaction with the surrounding environ-ment. To alleviate this limitation, we have recently fabricateda As2Se3 microwire with polymethyl methacrylate (PMMA)cladding, leading to an improved mechanical strength andoptical insulation from the surrounding environment [21].Very recently, a nonsilica axially nonuniform tapered fiberhas been proposed [22] in which the dispersion and the non-linearity are tailored by varying the fiber diameter. It wasfound that, by shifting the long zero-dispersion wavelength(ZDW2) of the tapered fiber toward shorter wavelengths,the long-wavelength edge of the supercontinuum generationcan be extended further in the mid-IR while using low pumppowers.

In this paper we propose a method to optimize the diameterprofiles of both uniform and nonuniform As2Se3 microwiressurrounded by a protective layer with the purpose of maximiz-ing the extent of SSFS from a 2100 nm laser pump. For the

A. Al-kadry and M. Rochette Vol. 29, No. 6 / June 2012 / J. Opt. Soc. Am. B 1347

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uniform microwire, the optimal diameter profile is the one inwhich an incident soliton acquires a large waveguide nonli-nearity and undergoes wide spectral shifting. The final wave-length shift in these microwires is limited by the emission ofdispersive waves (DWs) in the normal dispersion region. ADW is generated when a soliton redshifts in the proximityof the ZDW2 of the microwire dispersion profile [23]. The uni-form microwire presented here allows a soliton of 29 fs induration, 43 W in peak power (energy of 1.29 pJ), and centralwavelength of 2100 nm to redshift over 600 nm after propaga-tion through a 20 cm microwire. The pulse parameters arechosen based on a commercially available femtosecondmode-locked fiber laser source operating at 2100 nm. Thelaser pulses can be compressed down to 29 fs through asystem generating optical wave breaking followed by a med-ium of anomalous dispersion [24] or through soliton self-compression [25].

A nonuniform microwire design method is proposed to ex-tend the long-wavelength limit of the uniform microwire andreduce power requirements. This is achieved by shifting theZDW2 to longer wavelengths, which is enabled by increasingthe microwire diameter. Based on this concept, a nonuniformmicrowire profile was previously proposed for a specific inputsoliton energy and duration [26]. As the soliton enters this mi-crowire, the diameter increases gradually along the solitonpath such that the soliton experiences the highest possiblewaveguide nonlinearity and an appropriate level of GVD toavoid DWs emission. The diameter profile along the solitonpath was determined by imposing a minimum value rule onthe GVD that a soliton must experience. However, an increasein the input soliton energy, with the purpose of extending thewavelength shift along the microwire profile, leads to an ex-ponential growth of DWs along the propagation path and thusa significant reduction in the soliton energy. Losses experi-enced by the soliton slow down the rate of SSFS per unit ofpropagation length and prevent the soliton from large wave-length shifts. We follow an alternative approach where wedivide the nonuniform microwire into two segments and op-timize the rate of SSFS in each segment. In the first segment,an analytic expression is used to select the diameter corre-sponding to the maximum rate of self-frequency shifting of afundamental soliton. In the second segment, we manage toavoid DW emission and preserve the redshifted soliton energyas input energy increases by imposing a rule of minimum spec-tral separation from the ZDW2 of the microwire (instead of therule of minimum GVD value). We show that the optimizationof the selection of the first segment and the rule of minimumspectral separation enables an improvement on SSFS andenergy conservation of the soliton in nonuniform microwires.

2. THEORY AND NUMERICALSIMULATIONSA. Design of Nonuniform Diameter ProfileAs shown in Fig. 1(a), the nonuniform As2Se3-layer microwireconsists of two segments: a uniform input segment followedby a nonuniform segment. Figure 1(b) schematically showsthe GVD spectrum of the microwire with two points of zerodispersion, ZDW1 and ZDW2. In the uniform segment, the dia-meter remains fixed at constant diameter value, which allowsthe input soliton to spectrally shift until it reaches wave-lengths near the ZDW2. The soliton then enters the nonuni-form segment where the diameter increases gradually to trackthe rate of SSFS per unit of propagation length. Subsec-tions 2.A.1 and 2.A.2 explain the method we use to optimizeSSFS in each segment that leads to the maximum possible wa-velength shift in nonuniform microwires.

1. Uniform Segment and Adiabatic ApproximationTheoryThe starting point in optimizing the design of nonuniformmicrowires consists in selecting the diameter of a uniform seg-ment leading to the highest rate of SSFS per unit of propaga-tion length. To achieve this, we investigate the self-frequencyshifting of a fundamental soliton using the adiabatic theory.

Our theoretical description is based on the generalized-nonlinear Schrödinger equation (GNLSE). In the time domain,the soliton propagation in optical fibers is well described bynumerically solving the GNLSE given in [27]:

∂A∂z

− iXn≥2

inβn�z; δ�∂nτn!

A�z; τ�

� iγ�1� i

δo∂

∂t

��A�z; τ�

Z∞

−∞

R�τ0�jA�z; τ − τ0�j2dτ0�; (1)

where R�τ� � �1 − f R�θ�τ� � f RhR�τ� is the response function,f R � 0.1 represents the fractional contribution of the delayedRaman response, and hR�τ� is the temporal Raman responsefunction. The convolution term of R�τ� with the field intensityjA�z; τ − τ0�j2 provides a full description of the effect of theshape of the Raman gain spectrum gR�δ� in Raman response.The terms βn are the dispersion coefficients associated withthe Taylor expansion of the propagation constant β�ω� aroundthe carrier frequency δo, and γ is the As2Se3-layer waveguidenonlinearity of the fiber fundamental mode. The protectivelayer considered in the model has a refractive index similarto PMMA and is considered transparent in the mid-IR. In prac-tice, PMMA has an attenuation of ∼4.3 dB∕cm at 2100 nm [28]and thus partially absorbs the propagating mode from the

Fig. 1. (Color online) (a) Schematic of the nonuniform As2Se3-layer microwire. The microwire consists of a uniform and a nonuniform segments.(b) Typical GVD profile for a single-mode microwire.

1348 J. Opt. Soc. Am. B / Vol. 29, No. 6 / June 2012 A. Al-kadry and M. Rochette

evanescent wave. However, this loss is a specific characteris-tic of PMMA, and we instead keep the model more general bysimulating results of an ideal case with a negligibly lossy layer.Polymers with optical and mechanical properties compatiblewith the hybrid microwire fabrication are available but withbetter transparency in the mid-IR than PMMA [29]. The linearrefractive indices of As2Se3 and the layer (nAs2Se3 � 2.79,nlayer � 1.46) are calculated at a wavelength of 2100 nm byusing the Cauchy relation n2�λ� � A� B∕λ2 � C∕λ4, whereA, B, and C are the Cauchy coefficients given for As2Se3[30] and the layer as A � 7.56, B � 1.03 μm2, and C �0.12 μm4 and A � 2.149, B � 0.028 μm2, and C � −0.002 μm4,respectively. The large index contrast between the core andlayer allows more than 96% of mode power to propagate in thecore, while only 4% propagate in the layer. Since the attenua-tion of As2Se3 is typically between 0.0015–0.0020 dB∕cm inthe mid-IR [31] and only a small ratio of the evanescent fieldis propagating in the layer, all losses are neglected along theconsidered microwire length.

The adiabatic approximation is based on the assumptionthat, while a fundamental soliton undergoes a frequency shift,its evolution is weakly perturbed by the high-order linear andnonlinear effects. In what follows, we are first going to carryout calculations under this approximation, which will providean analytical expression of the frequency shift per unit of pro-pagation length. The expression is used to determine the rateof SSFS per unit length in uniform segments with different dia-meters. To provide a full description of the SSFS in As2Se3-layer microwire, we are then going to numerically solve theGNLSE in Subsection 2.B for uniform and nonuniform micro-wires and obtain the redshift wavelengths from these spectra.The numerical calculation is carried out with a Runge–Kuttainteraction picture method [32].

To simplify the theoretical description of the soliton dy-namics in optical fiber, we consider the propagation of a fun-damental soliton with a pulse duration of 29 fs. Because of thestability of fundamental solitons far from ZDWs, the higher-order dispersion effects (βn≥3 � 0) and the nonlinear disper-sive effects (self-steepening) can be treated as perturbations.In this case, the soliton propagation is adiabatic (i.e., no radia-tion scattering), and the rate of SSFS per unit of propagationlength can be estimated by analytically solving Eq. 1 with n �2 and neglecting the derivative term.

Gordon [3] was the first to give a theoretical description ofSSFS based on the adiabatic approximation. He predicted therate of SSFS per unit of propagation length to be inversely pro-portional to the fourth power of the pulse width. This is underthe assumption the soliton spectrum is still narrow comparedto the Raman spectrum width. For durations less than the per-iod of molecular oscillation (τR), Herrmann and Nazarkin [33]showed a change in the behavior of soliton shift in which itbecomes inversely proportional to pulse duration. Recently,a rearrangement of the Gordon equation was provided byJudge et al. [26] to include the actual Raman gain spectrumin the analytic expression. The theory is used as it is less timeconsuming than solving the GNLSE and provides an approx-imate estimation of the frequency shift a soliton undergoes perunit length.

Since the priority in this section is to optimize the diameterprofile of the uniform segment, we use the adiabatic theory toprovide only a qualitative estimation of the rate of SSFS per

unit of propagation length. The rate of SSFS can be writtenas [34]

d�δdz

� −jβ2��δ�jτ3s��δ�

f R1 − f R

R�τs��δ��; (2)

where R�τs��δ��, shown in Fig. 2, is the dimensionless spectralresponse function containing the effect of the shape of the Ra-man gain spectrum gR�δ�, given as

R�τs��δ�� �π2τ4s��δ�

6

Z∞

0

dδ2π

gR�δ�δ3sinh2�πτsδ∕2�

: (3)

For the ultrashort pulse duration considered in this study,the investigation shows that the spectral response functionR�τs��δ�� plays a minor role in determining the rate of SSFSper unit of propagation length. For As2Se3-layer microwires,the dependence of this function on the evolution of the pulseduration along a uniform core section, shown in Fig. 2, peaksat τm � 29 fs. In Eq. 3, the Raman gain spectrum is calculatedfrom gR�δ� � 16πn2f R Im�hR�τ��∕3λo [35], where Im�hR�τ�� isthe imaginary part of the Fourier transform of hR�τ�. The non-linear refractive index (n2) of As2Se3 is taken equal to n2 �1.1 × 10−17 m∕W at the 2100 nm pump wavelength [30]. Basedon the Lorentzian approximation of the Raman gain spectrum,the Raman response function hR�τ� is expressed in the timedomain as [27]

hR�τ� �τ21 � τ22τ2τ22

exp�−ττ2

�sin

� ττ1

�: (4)

The parameter values τ1 � 23 fs and τ2 � 195 fs are chosen tobest fit with the experimental measurements [36] of the Ra-man gain spectrum gR�δ� in As2Se3. Using those values, thepeak gain coefficient is located at the frequency shift of6.9 THz in accordance with experimental values.

For a fundamental soliton, with N � 1, the evolution of thepulse width τs��δ� as a function of the soliton center frequency��δ� is written as [37]

τs��δ� � 2jβ2��δ�jγ�δ�E�δ� : (5)

By substituting Eq. 5 in Eq. 2, the rate of SSFS per unit of pro-pagation length is

Fig. 2. Spectral response function of As2Se3-layer microwire calcu-lated for a 0.88 μm core diameter. The function is calculated fromEq. 3, and peaks at a pulse width of 29 fs.

A. Al-kadry and M. Rochette Vol. 29, No. 6 / June 2012 / J. Opt. Soc. Am. B 1349

d�δdz

� −γ3��δ�E3��δ�8jβ2��δ�j2

f R1 − f R

R�τs��δ��: (6)

The equation shows that the rate of SSFS per unit of propaga-tion length depends on the soliton energy along the microwire.This illustrates the importance of DW suppression along thenonuniform segment in enhancing the overall rate of SSFS perunit of propagation length. However, at the pump wavelength,Eq. 6 is rewritten as

d�δdz

� −γ3�δo�E3�δo�8jβ2�δo�j2

f R1 − f R

R�τs�δo��; (7)

where γ�δo�, and β2�δo� are the waveguide nonlinearity and theGVD parameters at the pump wavelength, respectively. For aspecific input soliton width τs�δo�, the R�τs�δo�� parameter isconstant. Therefore, the rate of SSFS per unit of propagationlength increases as the κ � γ3�δo�E3�δo�∕8jβ2�δo�j2 ratio in-creases. By calculating the ratio values (κ) for fundamentalsolitons incident at uniform segments with different dia-meters, we select the diameter of the uniform segment thatleads to the highest rate of SSFS per unit of propagationlength.

Figure 3(a) depicts the normalized ratio (κ) values for fun-damental solitons incident at uniform segments with diametervalues ranging from 0.84 to 0.92 μm. The fundamental solitonscorrespond to same input pulse width τs�δo�, same pump fre-quency δo but different input energies to maintain N � 1. Itshows that the highest rate of fundamental soliton shift isachieved in uniform segments with diameters of 0.88 and0.89 μm. However, solving numerically Eq. 6, as depicted inFig. 3(b), shows that the diameter of 0.88 μm provides higherrate of shift along the segment length. Therefore, based on theadiabatic theory, the nonuniform profile optimal for an inputfundamental soliton consists of a uniform segment of 0.88 μmdiameter. The length of the uniform segment required for thesoliton to shift from the pumpwavelength to wavelengths nearZDW2 is determined numerically using the GNLSE.

2. Nonuniform SegmentIn the proposed nonuniform microwire, the uniform segmentis followed by a nonuniform segment with a diameter that in-creases gradually along the soliton propagation path. The non-uniform segment has to be designed such that the soliton doesnot transfer energy into normal dispersion region as it shiftsin the vicinity of ZDW2. Therefore, the rate of increase of the

diameter along the propagation length must follow the rate ofSSFS per unit of propagation length.

In earlier work [26], a theory was developed to design non-uniform microwires for a specific input soliton energy andduration. As a soliton enters this microwire, the diameter in-creases gradually along the microwire length such that the so-liton experiences the highest possible waveguide nonlinearityand an appropriate level of GVD to avoid DW emission. How-ever, the problem arises when there is a trial to enhance SSFSby increasing the input soliton energy. Because the rate ofthe diameter profile follows the rate of SSFS per unit of pro-pagation length, any change in the input energy leads solitonto overlap the ZDW2 along the spectral shift length and thustransfers its energy to DWs. The growth of DWs slows downSSFS and reduces the output energy radiating at the solitonfrequencies. To avoid this, another diameter profile compati-ble with the new input soliton energy has to be used.

We propose here an approach that maintains a larger frac-tion of soliton energy under substantial change in the inputsoliton energy. This opens the prospect of using a nonuniformmicrowire in tuning the soliton spectrum over a wide wave-length range in the mid-IR. The same concept in avoidingDW emission is followed as in the previous approach but con-sidering instead a fixed spectral separation from the ZDWpoint, considered as a threshold for DW emission. We call thisthreshold condition the constant spectral bandwidth separa-tion (CSB).

Dictated by the phase-matching condition, the transfer ofenergy from the soliton to the DWs occurs at a specific fre-quency. The phase-matching condition arises from the equalityof the soliton and DW propagation constant expressed in Tay-lor expansion. However, in the vicinity of the ZDW, the solitonpropagation is mainly influenced by the third-order dispersion(TOD), and thus the Taylor expansion can be truncated at thethird-order term (n � 3 of the nonlinear Schrödinger equa-tion). In this situation, the spectral separation (ΔΩ) betweenthe soliton and the DWs is approximated as [23]

ΔΩ � �δ − δDW � 3β2��δ�β3��δ�

� Δ�δ� δZDW2− δDW; (8)

where �δ is the soliton instantaneous central frequency, δDW isthe DW resonance frequency, δZDW2

is frequency correspond-ing to the long ZDW, and β2��δ� and β3��δ� are the GVD and TODparameters at �δ, respectively.

The two approaches mentioned above to avoid DW emis-sion can be explained using Eq. 8.

0.84 0.86 0.88 0.9 0.92

0.96

0.98

1

κ

2100 2200 2300 2400 2500100

200

300

400

500

Soliton central wavelength (nm)

δ-1

-1

diameter = 0.88µmdiameter = 0.89µm

Uniform segment diameter (µm)

Fig. 3. (a) Coefficient (κ) calculated as a function of the uniform segment diameter of As2Se3-layer microwire. The value is normalized to themaximum among coefficient values. (b) Rate of SSFS per unit of propagation length given in Eq. 6 calculated for two uniform segments withdiameters of 0.88 and 0.89 μm.

1350 J. Opt. Soc. Am. B / Vol. 29, No. 6 / June 2012 A. Al-kadry and M. Rochette

1. GVD-threshold approach:

jΔΩjmin � 3�jβ2��δ�jmin ≡ GVD�

jβ3��δ�j. (9)

For the approach in [26], setting a threshold on the GVDvalues near ZDW2 �jβ2��δ�jmin� keeps the soliton at a spectralseparation equal to jΔΩjmin. In this case, the soliton acquires asingle value of �jβ2��δ�jmin� along the spectral shift in the non-uniform segment as shown in Fig. 4(a) (blue solid curve).

2. CSB-threshold approach:

jΔΩjmin � �jΔ�δjmin ≡ CSB� � jδZDW2− δDWj. (10)

In this approach, we employ a threshold on the spectralseparation of the center wavelength of the redshifted solitonfrom the ZDW2�CSB � jΔ�δjmin � �δ − δZDW2

�. The value ofjΔ�δjmin is obtained at the soliton center wavelength wherethe redshifted soliton in the uniform segment initiate DWemission. In the two approaches, the incident soliton acts dif-ferently along the nonuniform segment, which leads to differ-ent results at the fiber output. According to Eq. 8, a constantspectral separation from the ZDW2 of microwires with differ-ent diameters corresponds to different values of jβ2��δ�jmin

along the core sections. Following this approach, the disper-sion profile the soliton center wavelength experience inthe nonuniform microwire is depicted in Fig. 4(a) (greendotted curve).

From Eqs. (9 and 10), the magnitude of jΔ�δjmin andjβ2��δ�jmin required to avoid DW emission has to be adjustedwith the input soliton order. This is because an increase inthe input soliton order increases also its spectral bandwidth.In this case, the redshifted soliton has to remain at a largerspectral distance to avoid the overlap with ZDW2 as shownin Fig. 4(b). This is manifested in nonuniform segments asa change in the rate of diameter increase per unit of propaga-tion length.

B. Numerical SimulationsNext, we perform complete simulations for SSFS in As2Se3-layer microwires using the GNLSE solved with an input sechpulse centered at a wavelength of 2100 nm, A�z � 0; τ� �������Po

psech�τ∕τo�, with τo � 29 fs and a varying input peak

power Po. The peak power values correspond to solitons withorders of N varying between N � 1 and 1.4. For N > 1.4, theconversion efficiency from the pump sech pulse into the so-liton pulse declines.

The GNLSE is numerically solved in uniform and nonuni-form As2Se3-layer microwires with the purpose of obtainingthe maximum possible wavelength shift for the consideredsech pulse. From the generated SSFS spectra, we comparethe redshift wavelengths to show the enhancement of the rateof wavelength shift in nonuniform microwires.

1. Uniform MicrowiresTo investigate the potential of the As2Se3-layer microwires ingenerating SSFS at 2100 nm pump wavelength, the solitonshifting is first studied in a uniform microwire with 20 cmlength and a diameter of 0.97 μm. The schematic diagramof the uniform microwire and its dispersion profile are shownin Fig. 5. The dispersion profile contains two ZDW points:ZDW1 and ZDW2 at 1760 and 2940 nm, respectively. The mi-crowire parameters chosen are based on two criteria imposedhere on the diameter of uniform microwire in order to achievea wide soliton shifting. First, the soliton must acquire largewaveguide nonlinearity at the pump wavelength as it entersthe microwire. Second, the two ZDWs, ZDW1 and ZDW2, mustbe widely spaced such that the soliton occurs in the anoma-lous dispersion regime and undergoes a wide spectral shiftingalong the microwire length before reaching ZDW2.

In Fig. 6, we consider the wavelength shifting of solitonswith input peak powers of 22, 34, and 43 W, which correspondto soliton order N of 1, 1.25, and 1.4, respectively. From Fig. 6,the fundamental soliton (i.e., N � 1) undergoes a wavelengthshift of 320 nm. The small relative wavelength shift is due tothe large magnitudes of GVD values the soliton experiencesalong the spectral shifting length. However, at higher powers,more energy shifts to longer wavelengths, and the amount ofwavelength shift increases as shown in Fig. 6. For instance, asoliton with N � 1.4 redshifts to a maximum output wave-length of 2715 nm (615 nm of SSFS). At this wavelengththe soliton overlaps the ZDW2 and sheds energy in the normaldispersion regime. Consequently, pumping at soliton ordersabove N � 1.4 is manifested as higher ratio of soliton energylying in the normal dispersion region but no further wave-length shift.

To avoid DW emission and to extend the limit of outputwavelength shift, a larger anomalous dispersion window is

Fig. 4. (Color online) (a) Dispersion profile along the propagation path of nonuniform As2Se3-layer microwires designed using the thresholdconditions: GVD [jβ2��δ�jmin in Eq. 9; blue solid curve] and CSB (jΔ�δjmin in Eq. 10; green dotted curve). (b) Schematic diagram of the spectralseparation and magnitude of GVD threshold required to avoid DW emission for redshifted solitons generated from soliton of N � 1; 1.25, and 1.4.

A. Al-kadry and M. Rochette Vol. 29, No. 6 / June 2012 / J. Opt. Soc. Am. B 1351

required. For this purpose, two approaches can be followed incontrolling the anomalous dispersion window.

1. The use of uniform microwires with diameters largerthan 0.97 μm. For this approach, the ZDW2 is located at longerwavelengths and thus provides a larger anomalous dispersionwindow as shown in Fig. 5(b). Nonetheless, the soliton in thiscase experiences smaller waveguide nonlinearity, and largermagnitudes of GVD manifest as shorter output wavelengthshift.

2. Gradual increase of the diameter size along the propa-gation length leading to a nonuniform microwire. As the soli-ton enters this microwire, it experiences the highest rate ofSSFS per unit of propagation length in the uniform segmentand maintains in the nonuniform segment a constant spectralseparation bandwidth from the ZDWs to avoid DW emission.

Since our goal is to maximize the rate of SSFS per unit ofpropagation length, nonuniform microwires are desired asthey maintain higher waveguide nonlinearity and avoid DWemission.

2. Nonuniform MicrowiresTo determine the optimum diameter profile for maximumwavelength shift and transfer of energy toward longer wave-lengths, the SSFS is investigated in nonuniform microwireprofiles for soliton orders varying between N � 1 and 1.4.The nonuniform microwires are designed specifically for eachsoliton order. The uniform segment of all nonuniform micro-wires is chosen based on the approach proposed in this paperwith a diameter of 0.88 μm. For the nonuniform segment, theGVD and CSB-threshold conditions are used in order to deter-mine the influence of each condition on the characteristics ofoutput spectra.

In Fig. 7(a), the output wavelength shifts from nonuniformmicrowires of a nonuniform segment designed with GVDconditions are shown for different input soliton orders.Figure 7(b) shows the corresponding ratio of the output en-ergy radiating at the soliton frequencies �Es∕Et�, where Es isthe soliton energy and Et is the total output energy includingthe soliton and DW energy.

Figures 8(a) and 8(b) show the output wavelength shiftsobtained from nonuniform microwires of nonuniform seg-ments designed with the CSB condition and the ratio of theoutput energy radiated from the redshifted solitons, respec-tively. From Figs. 7(a) and 8(a), the input soliton order andthreshold conditions can be extracted for maximum outputwavelength shift and soliton energy. Then from Figs. 7(b)and 8(b), the corresponding ratio of the output soliton energycan be obtained.

Comparing these two sets of graphs shows that soliton be-havior differs when using either the rule of minimum GVD orthe rule of minimum CSB to design nonuniform segments.From Figs. 7(a) and 8(a), it can be concluded that the valuesof input soliton orders and the magnitudes of GVD and CSBconditions that enable a maximum output wavelength shiftare different. In Fig. 7(a), the maximum wavelength shift is2760 nm, obtained for input soliton of orderN � 1.4 propagat-ing in a nonuniform profile of nonuniform segments designedwith a GVD threshold appropriate for N � 1.4. While fromFig. 8(a), it can be seen that the maximum wavelength shiftis obtained when a soliton of order N � 1.4 propagates in anonuniform profile designed with a CSB condition appropri-ate for N � 1.25. The output wavelength-shift value is similarto that obtained in Fig. 7(a).

For the output wavelength-shift values chosen fromFigs. 7(a) and 8(a), Figs. 7(b) and 8(b) indicate that ∼87% and95% of output energy would be emitted at soliton frequencies,respectively. Therefore, the results reveal that the approachfollowed in this paper provide an enhancement of 8% in com-parison to that proposed in [26].

In Fig. 9(a) we present the soliton spectra generated at thenonuniform microwire’s output for the two maximum outputwavelength-shift values chosen in Figs. 7(a) and 8(a). The re-sults are obtained numerically using the input pulse param-eters: a sech pulse shape of N � 1.4, peak power � 22 W(1.21 pJ), and τo � 29 fs. By comparison, it can be seen thatthe two spectra are centered at the same wavelength, i.e.,2760 nm but with a slight DW emission in the profile proposedin this paper. The microwire profiles leading to Fig. 9(a) areshown in Fig. 9(b) as a function of the soliton propagationlength. The diameter of the two optimum microwires consid-ered in this paper increase gradually from 0.88 μm, i.e., the

Fig. 5. (Color online) (a) Schematic diagram of a uniform microwire with 0.97 μm diameter. (b) Dispersion profile of uniform microwire withdiameters of 0.97 μm (black solid curve) and 1.01 μm (red dotted curve).

Fig. 6. (Color online) SSFS after propagation through a uniform mi-crowire for soliton ordersN � 1,N � 1.25, andN � 1.4. The length ofthe As2Se3-layer microwire used is 20 cm with 0.97 μm diameter.

1352 J. Opt. Soc. Am. B / Vol. 29, No. 6 / June 2012 A. Al-kadry and M. Rochette

value chosen from the adiabatic theory, along a 20 cm micro-wire length. The difference between the two microwires is thelength of the uniform segment and the rate of increase of thediameter per unit of propagation length. From a practicalpoint of view, using pulses of duration longer than 29 fs leadsto a reduced DW spectral feature at the output of the two mi-crowires. This is due to the decrease of the rate of SSFS perunit of propagation length with the increase of pulse duration[3]. In contrast, a reduction in the pulse duration causes asignificant emission of DWs due to an increased overlap ofthe soliton spectrum with ZDW2. Figure 9(b) shows alsothe microwire profile designed based on [26] where the fun-damental soliton acquires the maximum possible waveguide

nonlinearity as it enters the microwire, but with no overlapwith the ZDWs. However, the rate of increase of the diameterprofile follows the rate of SSFS of N � 1.4 in order to reducethe emission of DWs. The corresponding output spectra fromthis profile are shown in Fig. 9(a) (blue dashed curve).

To illustrate the enhancement between using a uniform andnonuniform microwire, we show in Fig. 10 the maximum wa-velength shifts received after the propagation of input solitonswith N varying from 1 to 1.4 for constant input pulse durationof 29 fs. We insist here that the energy required to generatesolitons in the nonuniform microwires is much less than thatrequired in uniform microwires. From Fig. 10, the wavelengthshifts change continuously as the soliton order, i.e., input peak

Fig. 7. (Color online) Contour plot of the (a) output wavelength shift and (b) remaining energy obtained using GVD-threshold conditions appro-priate to input soliton orders varying between N � 1 and 1.4.

Fig. 8. (Color online) Contour plot of the (a) output wavelength shift and (b) remaining energy obtained using GVD-threshold conditions appro-priate to input soliton orders varying between N � 1 and 1.4.

0 5 10 15 20

0.8

0.85

0.9

0.95

1

Microwire local position (cm)

µ

Optimum profile based on GVDOptimum profile based on CSBNon-uniform profile by Ref. [26]

1800 2000 2200 2400 2600 2800 3000 3200 34000

0.2

0.4

0.6

0.8

1

Wavelength (nm)

Nor

m. I

nten

sity

(a.

u.)

Input SpectrumOptm. profile, GVDOptm. profile, CSBProfile of Ref. [26]

Fig. 9. (Color online) (a) Maximumwavelength shift obtained for a soliton withN � 1.4 in nonuniformmicrowires designed with a GVD conditionappropriate for N � 1.4 (green dotted–dashed curve) and CSB condition appropriate for N � 1.25 (red solid curve). (b) Optimum nonuniformmicrowire profiles designed based on GVD- (green dotted–dashed curve) and CSB-threshold (red solid curve) conditions with a uniform segmentbased on the approach proposed above, and optimum profiles designed based on [26] (blue dashed curve).

A. Al-kadry and M. Rochette Vol. 29, No. 6 / June 2012 / J. Opt. Soc. Am. B 1353

power, is changed for each soliton order. A dramatic increasein the magnitude of the output wavelength shift occurs whenthe nonuniform microwires are used as a hosting media forsoliton shifting. This is because soliton experiences a higherwaveguide nonlinearity along nonuniform microwires.

3. CONCLUSIONWe investigated SSFS in uniform and nonuniform As2Se3-layermicrowires with the purpose of maximizing the extent of so-liton output wavelength shift. Numerical simulations were car-ried out for solitons propagating in uniform and nonuniformmicrowires with varying input peak power values correspond-ing to soliton orders between N � 1 and 1.4. Using a simpleadiabatic theory, we found that the selection of a transversedimension of the initial segment in nonuniform microwireshas a dramatic impact in the wavelength shift. Despite the in-crease of input peak power values, it was shown that, in anyparticular nonuniform microwire profile optimized for aspecific soliton order, most of the soliton energy could be pre-served at the output end of the microwire when a minimumbandwidth is imposed between the soliton wavelength and theZDW. This has illustrated the usefulness of the proposed ap-proach in suppressing the DW emission of input solitons withN > 1 and thus in tuning the output wavelength shift.

Using contour plots, we demonstrated a nonuniform micro-wire profile that maximizes the output wavelength shift andpreserves the soliton energy along the propagation lengthin comparison to microwires based on previously proposeddesign approaches. The nonuniform microwire profile allowsa 29 fs input soliton with power varying from 21 to 41 W (1.21to 2.37 pJ) to reach a maximum output wavelength shift of650 nm. We showed the ability to tune the output wavelengthshift with more than 95% of the total output energy radiating atthe soliton frequencies. We compared the output wavelengthshifts with the ones from a uniform microwire to illustrate theadvancement of using nonuniform microwires in maximizingthe extent of the soliton wavelength shift in the mid-IR.

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