Abstract Segmented cladding fiber (SCF) is capable of single mode operation over anextended range of wavelengths while maintaining large mode area. In this paper we reportthe design of an SCF with mode area as large as 1,825μm2, suitable for delivery of highpeak power femtosecond laser pulses at 1550 and 1064 nm wavelengths. An SCF with sucha large-mode area is a few-moded fiber and its design requires careful choice of designparameters to have robustness against mode-coupling effects and bend loss. In this paper weaddress these issues and report a design of an SCF showing near distortion-free propagationof 100-fs, 53-kW peak power pulses at 1550-nm wavelength with 1,825-μm2 mode areathrough fundamental mode. The same fiber can also deliver 250-fs, 15-kW peak powerpulses at 1064-nm wavelength with 1,793-μm2 mode area. The fiber has been analyzed byusing the radial effective-index method in conjunction with transfer matrix method and thepulse propagation has been studied by solving the nonlinear Schroedinger equation by split-step Fourier method. Such a fiber would find applications in multiphoton microscopy and inbiomedical engineering.
Keywords Ultra-short pulse (USP) · Large mode area · Mode stability · Pulse propagation ·Segmented cladding fiber
1 Introduction
Ultra-short pulse (USP) lasers find applications in micro-machining (Liu et al. 1997), laserablation of solids (Liu et al. 1997; Shirk and Molian 1998), femtochemistry (Zewail 1988),multiphoton fluorescence microscopy (Xu et al. 1996), terahertz generation and detection
B. Hooda (B) · V. RastogiDepartment of Physics, Indian Institute of Technology Roorkee,Roorkee 247667, Indiae-mail: [email protected]
(Takayanagi et al. 2008), and frequency combs (Washburn et al. 2005). Advancements inhigh-power fiber amplifiers and lasers, and specialty optical fibers have opened new doorsfor generation of USPs (Limpert et al. 2002; Malinowsky et al. 2004). Pulse energy andpower from the amplified fiber laser system now compete in performance with conventionalbulk-optic based USP systems. The inherently flexible nature of such lasers makes themattractive to fulfill the present laser source requirements, and to scale in performance, formfactor, and cost in order to keep pace with industrial growth. Apart from generation of USPsthrough fiber-based lasers, transport of high energy USPs through optical fibers has alsofound importance in various applications. Fiber delivery of USPs gives much flexibilityfor laser integration and allows easy access to otherwise inaccessible regions. However,propagation of USP through small core standard silica fiber leads to significant nonlineareffects and results in distortion of the pulses. To circumvent this problem researchers haveused specialty fibers such as photonic crystal fibers (PCFs) or photonic band gap fibers(PBGs) (Tempea et al. 2009; Konorov et al. 2003; Saitoh et al. 2006; Peng et al. 2011;Luan et al. 2004; Mosley et al. 2010; Göbel et al. 2004), OmniGuide fibers (Johnson etal. 2001), multi-mode high dispersion fibers (Ramachandran et al. 2005), rigid glass rodfibers (Zaouter et al. 2008; Limpert et al. 2006) and higher-order mode fibers (Nicholsonet al. 2006; Ramachandran et al. 2006). Glass rod has drawn much attention because oflarge mode area with low susceptibility to mode coupling. However, these structures arenot suitable for long length devices because of difficulty in compact packaging. PCFs canbe designed with a large differential leakage loss between fundamental mode (LP01) andLP11 mode, and therefore, can give higher modal purity at the output. However, modalcoupling remains an issue with the PCFs (Ramachandran et al. 2006). The largest modearea reported in PCFs is approximately 1,400 μm2. Hollow core photonic bandgap fiberscan be the promising choice because optical nonlinearities are reduced by a factor of 1000compared with silica-core fibers. However, the guidance of the laser beam through the hollowcore negates the possibility of constructing a distributed amplifier by incorporating rareearth dopants. Recently, the higher-order-mode (HOM) fibers with the very large mode areaand low susceptibility to mode coupling emerged for USP delivery, however, they requireselective higher-order-mode excitation using a long-period grating (Nicholson et al. 2006;Ramachandran et al. 2006). In this paper we carry out a design of segmented cladding fiber(SCF) for delivery of high energy fs-pulses through the fundamental mode of the fiber. TheSCF has been originally proposed for high data rate transmission as it shows extended single-mode operation with large core (Rastogi and Chiang 2004; Chiang and Rastog 2002; Yeunget al. 2004). The fiber has been fabricated in polymer (Yeung et al. 2009; Duan et al. 2009)and silver halide glass (Millo et al. 2006).
In this paper we propose the use of large-mode-area SCF for delivery of high powerfs-pulses. The SCF being capable of single moded over extended wavelength range andwith large mode area seems a favorable choice. However, it requires a careful choice ofdesign parameters to address the issues of mode-coupling effects and bend loss whilemaintaining large mode area. We have optimized design of an SCF for high peak powerultra-short laser pulse delivery through LP01 mode at 1550 and 1064 nm wavelength. Thedesign has low susceptibility to mode coupling, leaky higher-order modes except LP11. Fordelivery of ultra-short pulses we have considered 15 cm as the minimum bending radiusat which loss should be as low as possible or at least lower than 0.1 dB/m. We numeri-cally demonstrate the distortion-free propagation of 100-fs, 53-kW peak power laser pulsesover 4-m length of the fiber with mode area of about 1,825μm2 at 1550-nm wavelengthand 250-fs, 15-kW peak power laser pulses at 1064-nm wavelength with a mode area of1,793 μm2.
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Design of segmented cladding fiber 399
2 Theory
We have used radial effective index method to analyze the fiber in conjunction with transfermatrix method (Rastogi and Chiang 2004; Thyagarajan et al. 1991). The fiber consists of auniform core (0 < r < a) of refractive index n1 and a segmented cladding (a < r < b)having alternate high index medium (n1) of angular width (2θ1) and low index medium (n2)of angular width (2θ2). The relative index difference between the two regions defined as
� = (n21−n2
2)
(2n21)
. The period and the duty cycle of segmentation are given by � = θ1 + θ2 and
γ = θ2/� respectively. The fiber is uniform in the axial direction which is the directionof wave propagation. The relative index difference between the high and low index regionsis small therefore the transverse component of the electric field satisfies the scalar waveequation which can be expressed in the cylindrical coordinate system as:
∂2φ
∂r2 + 1
r
∂φ
∂r+ 1
r2
∂2φ
∂θ2 + k2[n2(r, θ) − n2
eff
]φ = 0 (1)
where φ(r, θ) is the field, k = 2 π /λ is the free-space wave number, λ is the wavelength,n(r, θ) is the refractive-index distribution, and neff is the mode index.
φ(r, θ) can be written as:
φ(r, θ) = φr (r)φθ (r, θ) (2)
This is assumed that variation of φθ (r, θ) with r is small in comparison to variation of φr (r)
with r . By considering this assumption azimuthally uniform effective index profile neffr(r)can be obtained from the equation given below by applying suitable boundary conditions:
d2φθ (ri , θ)
dθ2 + k2[n2(ri , θ) − n2
e f f (ri )]
r2i φθ (ri , θ) = 0 (3)
Mode index and leakage loss then can be calculated from the following equation with theknowledge of φr (r) and φθ (r, θ).
d2φ
dr2 + 1
r
dφ
dr+ k2
[n2
effr(r) − n2eff
]φr = 0 (4)
Light transmission through the single mode fiber undergoes material and waveguide disper-sion. Dispersion is also an important property of the fiber design as higher order dispersiveeffects can distort ultra-short optical pulses in both the linear and nonlinear regime. Totalinduced dispersion (D) in ps/km/nm is defined as (Agrawal 2001):
D = −λ
c
d2neff
dλ2 (5)
where c is the velocity of light in free space and λ is the free space wavelength. Total dispersionconsists of material dispersion as well as waveguide dispersion. Material dispersion of puresilica, fluorine dope silica and germanium doped silica has been taken into account by usingfollowing Sellmier relations:
n2 − 1 = A1λ2
λ2 − λ21
+ A2λ2
λ2 − λ22
+ A3λ2
λ2 − λ23
(6)
Where A1 = 0.6961663, A2 = 0.4079426, A3 = 0.897479 λ1 = 0.0684043 μm, λ2 =0.1162414 μm, and λ3 = 9.896161μm for pure silica. For 1 % Fluorine doped silica valuesof constants are A1 = 0.69325, A2 = 0.3972, A3 = 0.86008, λ1 = 0.06723987 μm,
Another optical property of the fiber which describes the power density within the fiberis mode effective area (Aeff). Nonlinear effects in the fiber strongly depend on Aeff. We havecalculated Aeff of the fiber by using the following relation.
Aeff = 2π[∫ ∞
0 |φr (r)|2 rdr]2
∫ ∞0 |φr (r)|4 rdr
(7)
To study pulse propagation through the fiber we have used non-linear Schrondinger equa-tions given by [28]:
∂ A(t, z)
∂z+ α
2A + iβ2
2
∂2 A
∂T 2 − β3
6
∂3 A
∂T 3 = i�
(|A|2 A + i
ω0
∂(|A|2 A)
∂T− TR A
∂ |A|2∂T
)(8)
A frame of reference moving with the pulse at the group velocity υg is used by makingtransformation given below:
T = t − z
υg(9)
where α is the fiber loss, � is the fiber nonlinearity, β2 is the fiber’s second-order dispersionand β3 is the fiber third order dispersion. ω0 is the central frequency of the pulse. TR is Ramantime constant and A(t, z) represents the amplitude of the pulse envelope in time t at the spatialposition z. β2 and β3 can be calculated by using the following relations (Agrawal 2001):
β2 = −Dλ2
2πcand β3 = ∂ D
∂λ(10)
Nonlinearity parameter � is defined as:
�(ω0) = N2(ω0)ω0
cAeff(11)
where N2 is the nonlinear refractive index of the fiber material.Split-step Fourier method has been used to solve Eq. (8). In Eq. (8), dispersion and non-
linearity act together along the length of the fiber. The solution of the equation is obtained byassuming that in propagating the optical field over a small distance h the dispersive and non-linear effects are assumed to act independently. Propagation from z to z+h is carried out in twosteps, in the first step there exists only non-linearity and dispersion is zero, in the second stepthere acts dispersion only and nonlinearity is zero. Distortion-free USP propagation througha fiber requires a balance between pulse broadening due to dispersion and pulse distortiondue to self phase modulation (SPM). These two effects are characterized by the dispersionlength L D and nonlinear length L N L given by the following equations (Agrawal 2001):
L D = −2πc
λ20
τ 2
Dand L N L = Aeffλ
2π N2 P0(12)
where c is the speed of light in vacuum, τ is the pulse duration, λ0 is the center wavelength,N2 is the nonlinear refractive index of the fiber material, and P0 is the peak power. Aef f isthe effective area of the mode of the fiber which describes the power density within the fiber.Nonlinear effects in the fiber strongly depend on the effective area.
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Design of segmented cladding fiber 401
3 Fiber design
In order that nonlinear effects do not dominate and are balanced by dispersion one requiresa very large value of Aeff . Typically to propagate 100 fs pulse of peak power 53 kW througha fiber having LD of 50 cm one requires Aeff approximately equal to 2,500μm2. Scalingup the core dimensions of the fiber for high energy USP propagation requires lowering thenumerical aperture (NA) of the fiber. This makes the fiber vulnerable to mode instabilitydue to mode-coupling and high bend loss. Mode stability in different large-core-fibers hasbeen examined by Ramachandran et al. and it has been shown that mode spacing betweenthe adjacent modes, δneff = neff (LP0m) − neff (LP1m), should be sufficiently large to avoidmode-coupling (Ramachandran et al. 2006). In conventional large mode area fibers the modearea due to mode-coupling is limited to 800 μm2 (Ramachandran et al. 2006). The bend loss ofthe fiber should also be sufficiently low for delivery of pulses over few meter long distances.Therefore, it is very difficult to design a fiber having large Aeff , with significantly smallbend loss and sufficient mode spacing to avoid intermodal coupling. To address the issuesstated above we have implemented the design of large mode area SCF. The transverse crosssection of SCF with field profile of LP01 mode is shown in Fig. 1. The value of relative indexdifference (Δ) in the proposed design is 0.9 %. The outer jacket of the fiber having refractiveindex n3 is of pure silica whereas n1 and n2 correspond to Ge-doped and F-doped silicarespectively. We have considered standard cladding diameter of 125 μm. The parametersof the fiber are so chosen as to have mode area 1,825 μm2 and mode spacing greater than1 × 10−4. To achieve these values of mode area and mode spacing the optimum value of ais 29 μm. We have considered standard cladding diameter of 125 μm.
The modal properties of the proposed fiber have been analyzed by solving Eq. (4) usingwell established transfer matrix method (TMM) (Thyagarajan et al. 1991). Aeff and D ofthe fiber have been calculated using Eqs. (5) and (8) respectively. Aeff , mode stability andδneff of the fiber can be controlled by the duty cycle of segmentation as shown in Fig. 2.We have considered δneff for LP01 and LP11 mode. We can see that with duty cycle theδneff increases whereas Aeff decreases and there is a tradeoff between Aeff and δneff whiledesigning the fiber. We have chosen 60 % duty cycle to achieve sufficiently high value ofAeff (1,825μm2) and δ neff (1.68 × 10−4) at 1550-nm wavelength which is higher than thelimits observed in the 800 μm2-mode area conventional large mode area fiber and in the PCF
Fig. 1 Transverse cross-sectionof SCF with field profile of LP01mode at 1550 nm wavelength
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402 B. Hooda, V. Rastogi
Fig. 2 Variation of Aeff andδ neff with duty cycle
Fig. 3 Variation of Aeff anddispersion coefficient D withwavelength
(Ramachandran et al. 2006). The mode stability at 1064-nm wavelength, however, reduces toabout 1 × 10−4 with mode area 1,793 μm2. Aeff and D of the fiber at different wavelengthsfor 60 % duty cycle are plotted in Fig. 3. We can see that the Aeff does not vary significantlyover the entire wavelength range because of dispersive cladding of the fiber. Such a featureof the SCF makes it attractive for USP delivery at several wavelengths. Aeff , D and β3 ofthe fiber is 1,825μm2, 21.76 ps/nm/km and 1.495 × 10−4 ps3/m respectively at 1550-nmwavelength, where β3 is a third order dispersion coefficient. The first higher-order mode(LP11) of the fiber is guided, however, due to large modal spacing and its small overlapwith LP01 mode, this will not affect the pulse propagation. The LP02 mode has 18-dB/mleakage loss at 1550 nm wavelength and is not sustained over the length considered here.Mode spacing which plays a crucial role for mode stability is a function of wavelength anddecreases with wavelength. While designing the fiber for delivery of USPs the value of δ neff
larger than 1 × 10−4 is preferred (Ramachandran et al. 2006). The bend loss of the fiberhas been calculated at different bending radii by using the approach of Sakai and Kimura
Fig. 4 Variation in bending losswith bending radii
123
Design of segmented cladding fiber 403
(1978) and is shown in Fig. 4. The fiber shows bend loss of 0.023 dB/m at 15-cm bendradius. At 1064-nm wavelength Aeff , D and β3 of the fiber are 1,793μm2, 27.4 ps/nm/kmand 5.91 × 10−4 ps3/m respectively. The leakage loss of LP02 mode is 0.002 dB/m and themode spacing is close to 1 × 10−4. Hence, the same fiber can be used for delivery of pulsesat 1550 and 1064-nm wavelengths.
4 Femtosecond pulse propagation
Propagation of pulses through the fiber can be characterized by ratio L D/L N L , where L D
and L N L are defined by Eq. (11). When L D/Lit N L < 1, pulse evolution along the fiberis initially dominated by dispersion and pulse will get broaden in the absence of any chirpotherwise it stretches or compresses in time, depending on the sign of initial chirp. However,if L D/L N L > 1 then the nonlinearity dominates during the evolution of the pulse and pulsessuffer spectral narrowing or broadening and do not retain the original shape. In order to avoidnonlinearities, one needs to limit the peak power to make L N L larger than L D . It makes surethat linear stretching happens in shorter time so that the nonlinear interactions do not buildup. However, if L D/L N L ∼ 1, which implies that D is balanced by the nonlinear effectof self-phase modulation (SPM), then fundamental soliton propagation can occur. A typicalsoliton pulse is defined as a sech pulse given by the following equation (Agrawal 2001).
A(z = 0, t) = P1/20 × sech(t/τ)1+iC (13)
Where, P0 is the peak power, C is chirp parameter and τ is 1/e half width. The peak powerrequired for fundamental soliton propagation is given by (Agrawal 2001):
P0 = λ30 D Aeff
4π2cN2τ 2 (14)
Apart from the sech pulse we have also studied the propagation of a Gaussian pulse definedby the following equation through the fiber
The propagation dynamics of the pulse has been studied by solving Eq. (8) using split stepFourier method (Agrawal 2001). In the simulation of pulse propagation through the fiber,we have considered second order Group velocity dispersion (GVD), third order GVD, SPM,and transmission loss. Since the pulses considered here are ultra-short, we have also takeninto account self-steepening and Raman scattering.
4.1 Pulse propagation at 1550-nm wavelength
To study the propagation dynamics of sech pulse at 1550-nm center wavelength we haveconsidered the pulse with C = 0. The peak power for soliton propagation of 100-fs pulseduration comes out to be 53 kW, where we have used N2 = 2.36 × 10−20 m2/W. Thetransmission loss coefficient (α) of the fiber at 1550-nm wavelength has been consideredas 0.2 dB/km. The value of Raman time constant (τR = TR/τ ) for silica has been takenas 3 fs. Evolution of pulse, corresponding contour plot and input-output profiles after 4-mpropagation through the fiber are shown in Fig. 5a–c respectively. We can see distortion-freepropagation of the pulse.
For studying the propagation dynamics of 100-fs Gaussian pulse through the fiber andthe peak power has been adjusted to 71 kW so as to obtain near distortion-free propagation.
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404 B. Hooda, V. Rastogi
Fig. 5 a Propagation of a sech pulse in the SCF at 1550 nm wavelength. b Corresponding contour plot. cInput and output temporal profiles of the pulse
Fig. 6 a Propagation of a Gaussian pulse at 1550 nm wavelength. b Corresponding contour plot. c Input andoutput temporal profiles of the pulse
The pulse evolution along with the contour plot and input-output temporal profiles is shownin Fig. 6a–c, respectively. Initially we can see compression of the pulse due to nonlinearitybut as the pulse evolves nonlinearity and dispersion balance each other and we get neardistortion-free propagation of the pulse through the fiber. The pulse shape is retained withslight broadening. The near distortion-free propagation could be achieved for a 2-m longfiber.
4.2 Pulse propagation at 1064-nm wavelength
1064 nm wavelength lies in normal dispersion regime D < 0 and at this wavelength dis-persion induced chirping and nonlinearity induced chirping both are positive. This leads tothe chirping of output pulse and distortion-free propagation of high power pulses becomesdifficult. Segmented cladding fiber maintains a large mode area with small dispersion, there-fore, the total chirping of the pulse is not significant and can be balanced by a slight negativechirping in the input pulse. We have studied propagation of 250-fs chirped sech and Gaussianpulses at 1064-nm wavelength with a peak power of 15 kW through the same fiber. We haveconsidered an input pulse with C = −1.1, which is sufficient to counterbalance the totalchirping induced due to dispersion and nonlinearity. The transmission loss coefficient (α) ofthe fiber at 1064-nm wavelength has been taken as 0.76 dB/km and the Raman time constant
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Design of segmented cladding fiber 405
Fig. 7 a Propagation of a sech pulse in the SCF at 1064 nm wavelength. b Corresponding contour plot. cInput and output temporal profiles of the chirped and unchirped pulses (c)
Fig. 8 a Propagation of a Gaussian pulse at 1064 nm wavelength. b Corresponding contour plot. c Input andoutput temporal profiles
as 5 fs. The evolution of the sech pulse through the fiber is shown in Fig. 7a with corre-sponding contour plot in Fig. 7b. The input and output temporal profiles for a 1-m long fiberfor C = 0 and C = −1.1 are shown in Fig. 7c. Evolution of the pulse shows that the pulseis initially compressed due to nonlinearity and then broadens. Figure 7c shows that in caseof chirped pulse dispersion and nonlinearity partially balance each other whereas in caseof un-chirped pulse dispersion dominates and the pulse is broadened. The propagation ofa Gaussian pulse is shown in Fig. 8a along with its contour plot in Fig. 8b and input andoutput pulse profiles in Fig. 8c. We can again see near distortion-free propagation for thecase C = −1.1 and broadening in case of un-chirped pulse.
5 Comparision of SCF with conventional fiber
The pulse propagation in SCF has also been compared with that in a conventional fiber. TheAeff of SCF is 1,825μm2 while that of conventional fiber is 109 μm2 at 1550 nm wavelength.L D/LNL ratio for SCF is approximately equal to 1 whereas for conventional fiber its valueis 25. Hence nonlinearity dominates in case of conventional fiber during the evolution of thepulse and pulses suffer spectral narrowing. Evolution of sech pulse and its contour plot at1550 nm wavelength is shown in Fig. 9a, b respectively which clearly shows that just after 2
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406 B. Hooda, V. Rastogi
Fig. 9 a Propagation of sech pulse at 1550 nm wavelength through conventional fiber. b Correspondingcontour plot
cm propagation through the fiber the pulse is compressed and is not able to retain its shape.Aeff of the conventional fiber will be even smaller at 1064 nm wavelength and dispersion willbe negative. Therefore distortion free propagation through conventional fiber will be evenmore difficult at 1064 nm wavelength than that of propagation at 1550 nm wavelength.
6 Conclusions
We have designed a large-mode-area SCF for high-peak power, fs-duration pulse delivery.The design has been optimized for delivery of ultra short laser pulses at 1064 and 1550 nmwavelength. We have reported that fiber has mode areas of 1,825 μm2 at 1550-nm wavelengthand 1,793μm2 at 1064-nm wavelength. Proposed fiber design exhibits very high large modearea with very low bending loss and low susceptibility to mode coupling. We have shownthat the fiber can deliver 100-fs sech and Gaussian pulses at 1550-nm wavelength and 250-fspulses at 1064-nm wavelength without significant distortion. The proposed design shouldbe useful in the area of multiphoton microscopy and biomedical applications for delivery ofultra-short pulses.
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