Self-similar pulse compression by defective core photonic crystal fiber with cubic–quintic nonlinearities

Optik - International Journal for Light and Electron Optics 178 (2019) 591–601

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Original research article

Self-similar pulse compression by defective core photonic crystal fiber with cubic–quintic nonlinearities

T

⁎

A. Sharafalia, K. Nithyanandanb,c, , K. Porseziana a

Department of Physics, Pondicherry University, Puducherry 605014, India Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS, Univ. Bourgogne Franche-Comté, 9 Av. A. Savary, B.P. 47870, 21078 Dijon Cedex, France c CNRS/Universite Joseph Fourier, Laboratoire Interdisciplinaire de Physique (LIPHY), 38402 Saint-Martin-d’Héres, Grenoble, France b

ABS TRA CT

We propose a class of photonic crystal fiber called as defective core photonic crystal fiber which can fulfill the self-similar condition for pulse compression in a relatively simple means, without the necessity for a complex approach of changing the size of air holes in the cladding. The proposed model obeys exponential decrease (increase) in dispersion (nonlinearity) along the fiber length and hence compresses the pulse under selfsimilar technique. We also study the effect of system parameters such as initial dispersion, nonlinearity and chirping in the self-similar compression and compared with adiabatic compression. Using extended nonlinear Schrödinger equation, we have taken into account the effect of quintic nonlinearity on the pulse compression. In the proposed configuration, the quintic nonlinearity shows only a weaker effect in the compression as compared to the cubic nonlinearity. In the present system, we show that the self-similar technique is superior with high degree of compression than the adiabatic route of compressing the pulse. The effect of fiber loss is also taken into account and the critical fiber loss coefficient for compression is estimated.

1. Introduction Photonic crystal fibers (PCF) are a class of optical waveguides with numerous advantages and applications in science and engineering [1–3]. PCF shows exceptional properties such as endlessly single mode, high nonlinearity, birefringence and high flexibility in chromatic dispersion. All these parameters can be further enhanced by maneuvering the geometrical parameters and the choice of waveguide materials. This can be achieved by changing the geometry and size of the cladding air holes [4–11,13,12]. Instead of changing the air hole size of the cladding, it is also possible to control the properties of the PCF by introducing a defect in the core, such fibers are called as defective core photonic crystal fibers (DCPCF). Recently, Kim et al. reported a elliptical DCPCF with high birefringence and negative flattend dispersion with high birefringence [14]. Due to these design flexibilities in manufacturing, the DCPCF can be a potential contender for various nonlinear and sensing applications. Among the various nonlinear applications, pulse compression (PC) remained as the central process in nonlinear fiber optics for the generation of high quality ultrashort pulses [15]. Due to the relative simplicity of DCPCF over the conventional PCF, PC in DCPCF is a thrust area of research in fiber optics. Although there are different mechanisms existing for PC, three of them are very popular, namely higher order soliton PC, adiabatic PC and self-similar PC [16–22]. Higher order soliton compression uses soliton effect in order to compress a soliton pulse by a factor depends on soliton order given by N = t0 γP , where β2 is the second order dispersion parameter, γ is the nonlinear coefβ2

ficient, t0 is the initial pulse width and P is the input peak power. Since the soliton order is directly proportional to the input pulse power, better compression can be achieved with an input pulse of high power. It has been observed that, for a compression factor of

⁎

Corresponding author. E-mail address: [email protected] (K. Nithyanandan).

https://doi.org/10.1016/j.ijleo.2018.09.183 Received 29 August 2018; Accepted 30 September 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 178 (2019) 591–601

A. Sharafali et al.

60, 80% of pulse energy is accumulated in the pedestal of the pulse [23]. This accumulation of more energy in the pedestals of the pulse significantly reduces the quality of the pulse. In the case of adiabatic PC technique, researchers typically use a dispersion map of monotonically decreasing dispersion along the fiber length [24]. In a slowly varying dispersion system the soliton self-adjusts in order to maintain the balance between nonlinearity and dispersion. This kind of compression has been experimentally demonstrated with different dispersion decreasing schemes [25]. In all these cases the adiabatic condition required to be satisfied. Besides, if the dispersion varies exponentially exact chirped soliton solution is possible for the nonlinear Schrödinger equation (NLSE) and we can have a better compression without following any adiabatic condition [26]. The self-similar PC technique has been utilized to study the linearly chirped pulses in optical fibers [27–29]. The self-similar compression with longitudinally varying fiber parameters shows that, the interplay between nonlinearity, dispersion and gain can form a linearly chirped pulse which can sustain the catastrophic effects such as wave breaking. In self-similar PC technique with longitudinally varying dispersion and nonlinearity, one can achieve highly compressed pulse with low pedestals [30]. Thus the self-similar pulse compression with longitudinally varying fiber parameters is of wide interest in modern fiber optics research [31,32]. In what follows, we study the pulse compression in the proposed DCPCF taking into account higher order nonlinear effects up to second order using an extended NLSE with two power law nonlinearity. Using appropriate analytical model, we formulate the explicit expression for the pulse envelope and numerically evaluate the compression. For a comprehensive understanding, we compare the compression produced by the self-similar technique with adiabatic compression process. The organization of the paper is as follows: After a detailed introduction, Section 2 describe the fiber modeling and the parameters of choice. Section 3 is the theoretical modeling of the self-similar analysis. Discussion of the result is given in Section 4; Section 5 concludes the paper with summary of results. 2. Design of DCPCF Exponentially varying dispersion and nonlinearity in conventional optical fibers is a challenging task with practical limitations. The invention of PCF has made it possible to tailor the dispersion and nonlinearity by changing the air hole size and pitch [1–3,5,33,34,42,43]. A variety of PCF structures have been proposed to achieve ultrashort pulse generation through different PC techniques. One of the major difficulties in such approach is the design of PCF to support the self-similar condition, which remains as a major challenging with practical limitations. In conventional PCF design, one of the possible ways to control the design parameters is by changing the size of the cladding air hole. In this paper, we propose a class of PCF known as DCPCF, which enable one to change the PCF physical parameters by merely changing the size of the defect in the core, instead of the complex approach of changing the size of the cladding air holes. In this approach, we notice that it is possible to achieve the self-similar condition in a relatively simple means than the conventional PCF modeling. This enable one to control the PC mechanism either by simply varying the radius of the central defect, or alternatively by infiltering suitable material in the defective core. Fig. 1 shows the longitudinal and cross-sectional representation of the PCF designed for the compression. Its worth noting from Fig. 1(a), that there is no defect at the front facet of the fiber, and the size of the defect is increased radially along the fiber to establish the self-similar condition. For the design of DCPCF, we have chosen the fiber parameters d = 1 μm and Λ = 1.2 μm, where d and Λ are the air hole diameter and pitch, respectively. The radius of the defective air hole (r) is increased exponentially up to a maximum value of 0.1 μm along the fiber length. The effective index corresponding to each wavelength has been found by finite element method through Comsol 5.2a. To remove the back scattering at the boundaries of simulation area, a circular perfectly matched layer (PML) is constructed. Unlike other absorbing conditions PML strongly absorb outgoing waves from the computational region without any reflection [35]. The linear refractive index of glass for different wavelengths are evaluated by using the formula [36],

Fig. 1. (a) Longitudinal and (b) cross sectional schematic diagram of the proposed DCPCF. 592

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Fig. 2. Excitaion in (a) non-defective and (b) defective PCF at the fiber end.

n2 − 1 =

λ2

0.6961663λ2 0.4079426λ2 0.8974794λ2 + 2 + 2 . 2 2 − 0. 0684043 λ − 0. 1162414 λ − 9. 861612

(1)

To illustrate the role of the defect in the core, we design PCF without and with defect and the mode excitation are shown in Fig. 2(a) and (b), respectively. The variation of the effective index and area as a function of the wavelength are portrayed in Fig. 3(a) and (b), respectively. As expected the variation of effective index/area of both types of fiber exhibit a similar behavior except the fact that the core with defect show reduced effective index/area than the conventional counterpart. To calculate the nonlinearity coefficient of the fiber, the nonlinear refractive index of silica has been taken as n2 = 3.5 × 10−20. Fig. 4(a) and (b) shows the calculated dispersion λ d2n

(λ )

2πn

and nonlinearity coefficient of the fiber for different wavelengths by using equations, D (λ ) = − c eff2 and γ = λA 2 , respectively. As dλ eff the dispersion is set to vary proportionally with the effective index of the fiber, hence it is possible to decrease the dispersion exponentially by increasing the defective core. As the effective area exhibit inverse relation with the size of the defective core, the nonlinearity increases exponentially along the length of the fiber. Therefore, the increase in the defective air hole size exponentially along the fiber length leads to exponential decrease (increase) of dispersion (nonlinearity), thus inherently satisfying the self-similar condition paving the way for self-similar compression in a relatively simple means. The calculated dispersion and the nonlinear coefficient at the input and the output ends of the fiber corresponding to the operating wavelength 850 nm is given in Table 1. In the conventional method of fiber drawing for dispersion variation, the air hole size in the cladding or pitch of the fiber has to be varied in the desired prearranged form. Varying the entire air hole size in the cladding is a tedious task and the possibility of imperfection is large. Whereas in the proposed model, to get the desired variation in dispersion and nonlinearity coefficients, we only have to change the size of the central defect air hole along the fiber. By this way the possibility of imperfections and the manufacturing effort can be significantly minimized. We have confirmed this by modeling a PCF with conventional method by varying the cladding air hole size and pitch along the fiber. The parameters of the fiber design is given in Table 1.

Fig. 3. Variation of (a) effective index and (b) effective area as a function of wavelength for PCF with defect and without defect. 593

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Fig. 4. Variation of (a) nonlinearity coefficient and (b) dispersion for the defective and non-defective faces of the fiber. Table 1 Fiber parameters of conventional and defective core PCFs. Conventional PCF

Defective core PCF

3-4 Parameters

Value −4

−5.472×10 −1.924×10−4 0.1551 0.2046

β20 (ps^2/m) β2f (ps2/m) γ0 (W−1 m−1) γf (W−1 m−1)

Parameters

Values

β20 (ps /m) β2f (ps2/m) γ0 (W−1 m−1) γf (W−1 m−1)

−0.06669 −0.0243 0.275 0.315

2

3. Self-similar analysis The NLSE with distributed coefficients in a medium exhibiting two power-law nonlinearity can be given as [37]

i

β (z ) ∂2q ∂q + γ1 (z )|q|P q − γ2 (z )|q|2P q = 0 − 2 ∂z 2 ∂t 2

(2)

where q(z, t) is the amplitude, z is the longitudinal coordinate and t is the retarded time. β2(z) is the dispersion coefficient, γ1(z) and γ2(z) are the cubic and quintic nonlinearity coefficients of the fiber related to the terms |q(z, t)|Pq(z, t) and |q(z, t)|2Pq(z, t), respectively. As the dispersion/nonlinear coefficient critically depends on the effective index/area of the core; the proposed exponential variation of the defect core makes the dispersion and nonlinearity to vary exponentially of the form [38] given below

β2 (z ) = β20 exp(−σz )

(3a)

σ = α20 β20

(3b)

γ (z ) = γ0 exp(ρz )

(3c)

where ρ represents the growth rate and α20 is the chirp value. Thus the DCPC enable one to achieve self-similar condition in a relatively simple means than the conventional PCFs. We assume a self-similar solution of Eq. (2) of the form given by [37,39]

q (z , t ) =

1 η (z )

t − tc ⎤ α (z ) × R⎡ exp ⎡iα1 (z ) + i 2 (t − tc )2⎤ ⎢ 2 ⎦ ⎣ ⎣ η (z ) ⎥ ⎦

(4)

with,

α1 (z ) = α10 −

α2 (z ) =

λ1 2

∫0

z

β2 (z ′) dz′ 1 − α20 D (z ′)2

(5a)

α20 1 − α20 D (z )

(5b)

η (z ) = 1 − α20 D (z )

(5c) 594

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D (z ) =

∫0

z

β2 (z ′)dz′

(5d)

where tc represents center of the pulse, D(z) is the accumulated dispersion function, α20 and λ1 are constants. By substituting Eq. (4) in Eq. (2) and equating after separating into real and imaginary part, one can obtain the modified form of NLSE

2γ η2 − P /2 P + 1 2γ η2 − P 2P + 1 2η2 dα1 d 2R R− 1 R R + + 2 =0 2 dθ β2 dz β2 β2

(6) t − tc . η (z )

This equation is in terms of R(θ) which depends only on the scaling variable θ with θ = In general, coefficients of Eq. (6) are functions of variable z, and to find a nontrivial solutions of R(θ) the following coefficients must be constant, i.e.,

λ1 = −

2η (z )2 dα1 β2 (z ) dz

(7a)

λ2 = −

η (z )2 − P /2γ1 (z ) β2 (z )

(7b)

λ3 = −

η (z )2 − P γ2 (z ) β2 (z )

with λ1 =

d 2R dθ 2

2 dα1 | , β20 dz z = 0

(7c) γ10

λ2 = − β

20

γ20

and λ3 = − β

20

by η(0) = 1. For nontrivial case, one can write Eq. (6) as

− λ1 R − 2λ2 RP + 1 − 2λ3 R2P + 1 = 0

(8)

t − tc . 1 − α20 D (z )

with scaling variable, θ = Eq. (8) is the generalized equation for finding amplitude of self-similar pulses with two power law nonlinearity. With P = 2 for cubic–quintic nonlinearity system, it can be reduced as,

d 2R − λ1 R − 2λ2 R3 − 2λ3 R5 = 0. dθ 2

(9)

By taking amplitude part of Eq. (4), we can determine the bright soliton solution by integrating Eq. (9) given by 1/2

⎡ ⎢ 1 q (z , t ) = ⎢ t0 [1 − α20 D (z )] ⎢ 1+ ⎢ ⎣ where, ρ1 =

β2 (z ) γ1 (z )

and ρ2 =

β2 (z ) . γ2 (z )

⎤ ⎥ 1 ⎥ t−t cosh 2 ⎡ t [1 − α cD (z )] ⎤ ± 1 ⎥ 20 ⎥ ⎣0 ⎦ ⎦ β (z )

∓ 2 γ2 (z ) 8ρ12 3ρ2 t02 [1 − α20 D (z )]2

(10)

In order to calculate the second order nonlinear refractive index of silica by using the formula,

(5) n4 = 15χRe /16n 0 , we have used the value of fifth order nonlinear susceptibility as −5.1 × 10−41 W4/V4. The value of n4 for silica is obtained as −3.3 × 10−36m4/W2 [40]. We have calculated the second order nonlinearity coefficients value corresponding to each 2 wavelength by using the formula γ2 = 2πn4 / λAeff and at 850 nm γ2 =−3.54 × 10−5. To understand the effect of quintic nonlinearity in the compression, we have analyzed the self-similar compression produced with cubic nonlinearity term only. Hence γ2 and λ3 terms are assumed to be zero. With this, Eq. (9) reduces to,

d 2R − λ1 R − 2λ2 R3 = 0. dθ 2

(11)

Assuming γ2 = 0 in Eq. (10), the amplitude of bright soliton without quintic nonlinearity can be given as, β (z )

1/2

⎤ ⎡ ∓ 2 γ2 (z ) 1 1 ⎥ ⎢ q (z , t ) = t t − c t0 [1 − α20 D (z )] ⎢ cosh 2 ⎡ ⎤ ± 1⎥ t [1 α D ( z )] − 20 ⎣0 ⎦ ⎦ ⎣

(12)

It can be further reduced to

q (z , t ) =

β2 (z ) 1 t − tc ⎤ × sech ⎡ ⎢ − γ1 (z ) t0 [1 − α20 D (z )] t [1 α20 D (z )] ⎥ 0 ⎣ ⎦

(13)

In the presence of loss Eq. (2) can be modified as,

i

β (z ) ∂2q g ∂q + γ1 (z )|q|P q − γ2 (z )|q|2P + i q = 0 − 2 ∂z 2 ∂t 2 2

(14)

where g represents loss coefficient. By applying the transformation q(z, t) = Γ(z)Q(z, t), Eq. (14) becomes 595

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i where By

β (z )Γ(z ) ∂2Q g ∂ (Γ(z ) Q) − 2 + γ1 (z )Γ(z ) P + 1|Q|P Q − γ2 (z )Γ(z )2P + 1|Q|2P Q + i Γ(z ) Q = 0 2 2 ∂z ∂t 2

(

i

(15)

)

g Γ(z ) = exp − 2 dz . dΓ(z ) using the relation dz

∫

g

= − 2 Γ(z ) , the above equation can be reduced as,

β (z ) ∂2Q (z , t ) ∂Q − 2 + G1 (z )|Q|P Q − G2 (z )|Q (z , t )|2P Q (z , t ) = 0 ∂z 2 ∂t 2

(16)

with G1(z) = γ1(z)Γ(z) and G2(z) = γ2(z)Γ(z) . As q(z, t) = Γ(z)Q(z, t), by using Eq. (9), amplitude part of the solitory wave solution for NLSE in a lossy medium with two power nonlinearity can be written as, P

2P

1/2

q (z , t ) =

1 exp ⎛− t0 [1 − α20 D (z )] ⎝

β2 (z ) G1 (z )

∫

⎡ ⎢ g dz⎞ × ⎢ 2 ⎠ ⎢ 1+ ⎢ ⎣

⎤ ⎥ 1 ⎥ t−t cosh 2 ⎡ t [1 − α cD (z )] ⎤ ± 1 ⎥ ⎥ 20 ⎣0 ⎦ ⎦ β (z )

∓ 2 G2 (z ) 8ρ1′ 2 3ρ2′ t02 [1 − c0 D (z )]2

(17)

β2 (z ) . G 2 (z )

where ρ1′ = and ρ2′ = It can be easily observed that, as compared to the effect of cubic nonlinearity on PC, the effect of quintic nonlinearity is very small. To calculate the critical loss coefficient for solitons with cubic nonlinearity Eq. (17) can be reduced as,

q (z , t ) =

β2 (z ) t − tc 1 ⎤ exp ⎛− × sech ⎡ ⎢ γ1 (z )exp(− ∫ g dz) t0 [1 − α20 D (z )] ⎝ ⎦ ⎣ t0 [1 − α20 D (z )] ⎥

g

∫ 2 dz⎞⎠

(18)

From the above solution it is worth mentioning that the self-similar PC in a lossy medium can only be achieved if the nonlinearity coefficient varies as,

γ1 (z ) = γ0 exp(−

∫ g dz).

(19)

If the nonlinearity coefficient is not varying in accordance with the loss as Eq. (19), then the soliton energy will not be conserved and the pulse becomes dissipative while propagating through the fiber. At critical fiber loss with normalized intensity and unbalanced nonlinearity coefficient with loss, Eq. (18) can be written as

|q (z , t )|2 exp(

∫ gc dz) =

2 β2 (z ) ⎡ t − tc 1 ⎤ = χ (z , t ) ⎤ × sech2 ⎡ × ⎥ ⎢ ⎥ ⎢ γ1 (z ) ⎣ t0 [1 − α20 D (z )] ⎦ ⎣ t0 [1 − α20 D (z )] ⎦

(20)

where gc is the critical loss coefficient. The maximum amplitude of the pulse can be found at t = tc, hence χ(z, t) becomes χ (z , t )|t = tc . By using this identity and Eq. (20), the expression for critical loss coefficient can be deduced as,

gc =

d [Log(χ (z , t )|t = tc )]. dz

(21)

4. Results and discussion By using Eq. (13), we have analyzed the possibility of self-similar PC in the proposed DCPCF. The fiber parameters are given in Table 2 and the variation of nonlinearity and dispersion are depicted in Fig. 5(a) and (b), respectively. Fig. 6 shows the temporal evolution of the self-similar compressed pulse propagating at 850 nm inside DCPCF. We have chosen the initial chirp α20 =−0.75 THz2 and the length of the DCPCF L = 20 m. It is quite evident from Fig. 6, that the proposed model compresses the input pulse more than the self-similar compression using conventional PCF, whose parameters are given in Table 2. An input pulse of width 800 fs has been compressed down to 307.2 fs with an estimated compression factor Cf = 2.6. The compression factor is calculated by using the formula Cf = τ0/τl, where τ0 and τl are the initial and final pulse width respectively. For further insight, we have analyzed the effect of system parameters on PC. The initial dispersion and the dispersion decreasing factor plays a crucial role in compression, which can be conveniently controlled in DCPCF by the changing the size of the defect core. An input pulse of width 800 fs has been compressed to 171 fs with β20 =−0.1 ps2/m. A compression factor of 1.63, 2.6 and 4.67 has Table 2 Compression factor produced by adiabatic and self-similar pulse compression technique. Compression

Compression factor (Cf)

Technique

L = 10 m

L = 20 m

Adiabatic Self-similar

1.041 1.62

1.052 2.9

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Fig. 5. Variation of (a) nonlinearity coefficient and (b) dispersion for the DCPCF along the length of the fiber.

Fig. 6. PC in DCPCF for β20 =−0.06669 ps2/m, other fiber parameters are same as given in Table 2.

been observed for PCF with initial dispersion of −0.0333 ps2/m, −0.06669 ps2/m and −0.1 ps2/m, respectively. Thus, its quite evident that the initial dispersion strongly affects the compression, such that Cf increases with increase in |β20| as shown in Fig. 7, thus enabling the control over the compression by appropriately designing the PCF. Fig. 8 shows the self-similar compression produced by the DCPCF with constant initial dispersion with different final dispersion values. The final dispersion β2f has been varied by changing the value of the initial chirp. Fig. 8 depicts the compression for some representative values of initial chirp −1 THz2, −0.75 THz2 and −0.5 THz2 and the calculated Cf is equal to 4.02, 2.58 and 1.76, respectively. It is evident from Fig. 8, the effective compression produced by the DCPCF is increasing with increase in the initial chirp, as chirp is related to the rate of dispersion. We have used Eq. (10) for analyzing the effect of quintic term along with cubic nonlinearity in PC. It has been noticed that, the value of second order nonlinearity coefficient and its effect on pulse are very less as compared to the cubic nonlinearity. The quintic nonlinearity has an effects only on the intensity of the compressed pulse as I′ = I − 0.51 × 10−3 for β20 =−0.06669 ps2/m with chirp, α20 =−0.75 THz2. Where I′ represents the intensity of the pulse with cubic–quintic nonlinearity and I is the intensity with effect of cubic nonlinearity. Thus, it is quite evident that the quintic nonlinearity only exhibit a feeble effect while comparing with the contribution of cubic nonlinearity. We have also compared the self-similar pulse compression produced by the proposed DCPCF with the well known adiabatic compression. It has been observed that, the self-similar compression can give a high compression factor within a short length of fiber as compared to adiabatic pulse compression. Also, for adiabatic compression the dispersion should vary adiabatically with some prearranged law, and hence, very long fiber is needed for effective compression. Fig. 9 shows the adiabatic pulse compression produced by DCPCF and the comparison of compression factors extracted from adiabatic and self-similar pulse compression techniques is shown in Fig. refAdiabatic_selfsimilar (Fig. 10). Its quite obvious that the compression produced by the self-similar technique is almost thrice the Cf achieved through adiabatic compression. The extracted values of Cf for self-similar and adiabatic 597

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Fig. 7. PC produced by DCPCF with (a) β20 =−0.1 ps2/m, (b) −0.06669 ps2/m and (c) −0.0333 ps2/m.

compression from simulation is tabulated in Table 2. These observations again confirms the superiority of self-similar pulse compression over adiabatic pulse compression as reported in [30].

4.1. Critical fiber loss coefficient In principle, microstructure fibers like PCF, DCPCF the losses cannot be disregarded due to the complex manufacturing process, and therefore the study on the effect of losses becomes important. Therefore, in what follow, we study the effect of loss in the efficiency of compression. In general, soliton pulse propagating in a medium having loss causes continuous depletion of the pulse intensity. This will destroy the important soliton property of conservation of pulse energy while propagating through the fiber. In case of soliton compression, the pulse intensity increases with increase in the compression while propagating through the fiber. If the loss of the medium exceeds a particular limit, pulse will not show this property but shows a less intense compressed pulse at the output. The critical loss coefficient corresponds to the case of adiabatic pulse compression with different dispersion profiles is studied by Vinoj et al. [41]. Following that, we calculate the critical loss coefficient corresponding to the self-similar compressed pulse for the first time to the best of our knowledge. The critical loss coefficient of a fiber is the loss coefficient at which the similariton pulse energy conservation property is completely lost. When the loss coefficient exceeds this value, the intensity of the pulse propagating through the fiber decreases continuously and completely disappears after traveling some distance. At the critical loss point, the pulse amplitude and hence intensity remains constant while propagating through the fiber with efficient compression. The critical loss coefficient of the proposed model is calculated by using Eq. (17) for some representative values of initial dispersion/chirp and tabulated in Table 3 It has been observed that the critical loss coefficient value is increasing with increase in dispersion and chirping, which is due to the increase in the compression factor with initial dispersion/chirp. Since the compression factor produced by self-similar technique is significantly higher than that of adiabatic compression, it is obvious that the critical loss will be higher for the pulse compressed by self-similar technique.

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Fig. 8. PC produced by DCPCF with chirp (a) α20 =−1 THz2, (b) α20 =−0.75 THz2 and (c) α20 =−0.5 THz2.

Fig. 9. Adiabatic PC produced by DCPCF.

5. Conclusion A new class of photonic crystal fiber with central defect has been proposed and its optical characteristics are studied. The proposed design enable one to achieve the self-similar condition of exponential decrease (increase) of dispersion (nonlinearity) coefficient in a relatively simple means than the other existing fiber systems. Due to relative simplicity in design and less manufacturing difficulties the DCPCF could be a potential contender for PC. Through detailed analytical study, we show that the pulse compression through self-similar technique is superior in terms of compression factor, as well as the quality of the compressed pulse. We have also studied the effect of system parameters such as dispersion, nonlinearity and chirp on the compression factor. We observed that the compression factor increases with increase in the initial dispersion and chirp. Using the extended NLSE with two power law nonlinearity, we have taken into account both the cubic and quintic nonlinearity. We observe that the effect of quintic nonlinearity is feeble in the compression process. As losses in the microstructure fibers cannot be disregarded, we studied the role of fiber losses and reported the critical loss coefficient for PC. For a comprehensive understanding on different compression technique, we have presented a comparative study between the adiabatic and self-similar compression technique. We noticed that the self599

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Fig. 10. Comparison of compression factors by adiabatic and self-similar pulse compression produced by DCPCF. Table 3 Critical loss variation for different initial dispersion and chirp. β20 =−0.06669 (ps2/m)

α20 =−0.75 (THz2)

3-4 α20 (THz2)

gc

β20 (ps2/m)

gc

−0.5 −0.75 −1

0.026745 0.043417 0.06009

−0.0333 −0.0666 −0.1

0.018375 0.043417 0.0684

similar technique compress the pulse almost thrice than the adiabatic route of compression. Owing to the relative simplicity in design, we believe that the proposed DCPCF could be an obvious choice for the future generation of ultrashort pulse generation using selfsimilar PC technique. As there are no experimental results on the PC using DCPCF, the aforementioned theoretical results can stimulate new experiments in the context of ultrashort pulse generation using PC. Acknowledgements The work is supported by the major projects DST-SERB, CSIR NBHM funded by the Government of India, for the financial support through major projects. KN acknowledge CNRS for post doctoral fellowship at Universite de Bourgogne, Dijon, France. He also thanks Agence Nationale de la Recherche (ANR) for the research fellowship at Universite Grenoble-Alpes, Grenoble, France. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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