Supercontinuum Generation in Silica-Based Photonic Crystal Fibers for High-Resolution Ophthalmic Optical Coherence Tomography

Optical coherence tomography (OCT) is a new technology for high-resolution cross-sectional images of biological tissues. In this research, we have designed a photonic crystal fiber (PCF) made of silica with proper dispersion characteristics about the center wavelength of 800 nm to simulate supercontinuum generation (SCG) which is desired for a high-resolution OCT in ophthalmology. Several types of PCFs with different air-hole diameters are designed where squared hyperbolic secant pulses are provided to simulate SCG by solving the generalized nonlinear Schrodinger equation (GNLSE) via split-step Fourier method. To obtain more accurate SCG, dispersion coefficients up to 9th order, Raman scattering and self-steepening are taken into account. We examine the impacts of air-hole diameter, input pulse width and pulse peak power on the SCG bandwidth as well as the OCT resolution through which suitable parameters for maximum axial resolution in ophthalmology are determined.


Introduction
Photonic crystal fibers (PCFs) as a new generation of optical fibers have attracted great attention in recent years [1,2].PCFs have different cladding structures compared with the standard optical fibers where there is an arranged air holes running parallel to the fiber axis [3,4].The size of air-holes is about the micrometer that is comparable with the wavelength of the lightwave, guided in the PCF [5].The advantages of PCFs can be attributed to their high flexibility in design which means that by varying their geometrical parameters such as the cross-sectional area, size or arrangement of the air holes, different optical characteristics can be obtained for various applications [2,[6][7][8][9].
High mode confinement in the PCFs and enhanced nonlinearity make them very good candidates for observation of nonlinear phenomena specially production of new frequency components [10] which has been the subject of extensive research since 1990s [6,11].The process in which the narrow-bandwidth input pulse undergoes nonlinear spectral broadening to spectrally generate a wide-bandwidth output is called supercontinuum generation (SCG).SCG is a complex nonlinear phenomenon that results from the expansion of the main spectrum of a laser pulse propagating in a dispersive nonlinear medium [12].The widest spectrum is obtained when the pump pulse wavelength is set very close to the zero-dispersion wavelength (ZDW) of the PCF [13].By using a PCF as a nonlinear medium with a longer interaction length and a higher nonlinear parameter compared to standard fibers, it is possible to dramatically reduce the input peak power required for SCG observation [14].Light sources based on SCG provide a combination of desirable features which includes: high output power [15], wide and controlled spectrum and high degree of flexibility.These features make SCG sources ideal for many applications such as frequency metrology [16], ultra-short pulse compression [17], spectroscopy of materials and photonic structures [18], and optical coherence tomography (OCT) [19].
OCT is a new technology for obtaining high-resolution cross-sectional imaging.OCT was introduced first in 1991 by David Huang and named by Fujimoto [20,21].The advantage of OCT is that it can act as a non-destructive biopsy and provide information of pathological sample at the exact time and space domains with micrometer resolution [20,22].OCT generally uses broadband light sources in various wavelength regions such as super luminescent diode, Ti: Al 2 O 3 lasers, fem to second Kerr-lens mode locked (KLM) lasers, infrared lasers, etc. [20,23].OCT systems operate based on low-coherence interferometry and require broadband light sources to produce high-resolution images [24].Among the different sources used for OCT, the PCF-based SCG is one of the most promising sources that can provide wide and flat spectrum for highresolution OCT images [25].
As the eye is fundamentally transparent around the wavelength of 800 nm, the light can be easily transmitted with minimal loss to access the retina layers [24,25].
The ophthalmic OCT systems at the center wavelength of 800 nm have been optimum since they fulfill the imaging criteria due to the accessibility of Ti: Sapphire lasers with a very broad bandwidth resulting in a significant resolution ~3 μm.Also, super luminescent diodes (SLDs) were mainly utilized in the commercial ophthalmic OCT systems owing to their good output characteristics and cost-effective price.Therefore, OCT systems operating around 800 nm are very good for the candidates to resolve all essential intra-retinal layers and provide high-resolution images of small morphological changes in these layers [25].
In this research, PCFs are designed with proper dispersion characteristics around the center wavelength of 800 nm which is desired for ophthalmic OCT.Several designs of PCFs with different air-hole diameters are simulated and then short/ultra-short pulses are input into the PCFs to generate supercontinuum by solving generalized nonlinear Schrodinger equation (GNLSE) via split-step Fourier method.Impacts of air-hole diameter, input pulse width and pulse peak power on the SCG bandwidth as well as the OCT resolution are investigated and finally suitable parameters for maximum axial resolution in ophthalmology are determined.It should be noted that our proposed PCF design at 800 nm center wavelength is innovative compared with other PCF designs operating at different wavelengths and/or different materials [26][27][28][29][30][31].

Theory and Simulation Method
In order to design PCFs, we begin with one of the relatively accurate methods which is known as the finitedifference time-domain (FDTD) method, since it solves the Maxwell equations with the least approximation and high accuracy [26,27].In fact, the FDTD method is a common method for electromagnetic numerical simulation in an optical environment.When the PCF is designed and its dispersion curve is plotted for given parameters, then it is used as a dispersive nonlinear medium for pulse propagation and supercontinuum generation.The equation governing the pulse propagation in optical fibers is GNLSE which is [11]: where A(z,T), β m (m = 2 to 9), α, γ and R(t) are pulse envelope, dispersion coefficients of m th order, loss coefficient, nonlinear parameter and Raman response function, respectively.Also, ω 0 is the center frequency of pump pulse and T is the retarded time measured in the reference frame moving with pulse.The terms in the first parentheses on the right-hand side is related to the linear effects (dispersion and loss) while the remaining terms correspond to the nonlinear effects such self-steepening, Raman scattering, self-phase modulation and cross-phase modulation.
Eq. ( 1) is a nonlinear partial differential equation that generally does not have algebraic solutions except in some specific situations, so a numerical solution is often required.Split-step Fourier (SSF) is a method often used in nonlinear dispersive environments to solve the pulse propagation problem [32].This means that in SSF method it is assumed that dispersion and nonlinearity act independently along the length of the fiber.Here, the fiber length is divided into several sections each of them with small distance of h where propagation from z to z + h is carried out in two steps so that in the first step, the nonlinearity acts alone and in the second step, dispersion acts alone.
When the simulation of SCG is attained and the output spectrum is obtained; one should evaluate the output spectral width which is used for OCT axial resolution.The OCT axial (1) resolution l c is determined by the coherent length of the light source (here is the PCF-based SCG) [29] which is given by [20,33]: where λ 0 is the center wavelength of the SCG (here for ophthalmology λ 0 is about 800 nm) and Δλ corresponds to the 10-dB spectral width of output SCG.Therefore, it is obvious that for obtaining higher OCT resolution, a wider and flatter SCG source is needed.

Results and Discussion
In this section, we perform simulations for a 1 m-long PCF and the input squared hyperbolic secant pulse with the envelope ofA(0, t) = √ P 0 sech 2 Here, T 0 is the pulse width and P 0 is the peak power.The central wavelength of the pulse is set at 800 nm which is desired for OCT in ophthalmology [34].
Figure 1 shows the cross section of the designed silica-based PCF used for generation of supercontinuum for (2) Δλ OCT in ophthalmology.The air-hole radii of the first, second, third, and fourth ring are r 1 = 0.26 μm, 0.31 μm, 0.36 μm and 0.41 μm, respectively.Also, the lattice pitch is 0.85 μm.A method which is very common for fabricating PCFs is known as the stack-and-draw technique where glass capillaries with thick walls are assembled to a multicapillary stack with a desired lattice.Then, the stack is put into an oven and heated up to a temperature at which the glass is softened.Finally, the softened stack is drawn to a final fiber where the hole diameters of capillaries are reduced to a much smaller size [4,5].Dispersion curves of the designed PCF for different values of first ring air-hole radii r 1 are shown in Fig. 2. We have used the total dispersion relation as D = − c d 2 n eff d 2 to plot dispersion curves where n eff is the effective refractive index of the fundamental mode [11].As it is seen, by changing r 1 , the dispersion curve is changed where the flattest curve around 0.8 μm is obtained for r 1 = 0.41 μm.It should be noted that both the material and the waveguide dispersions have been included in the total dispersion relation D.We have also plotted in Fig. 3a the cross section of a symmetric PCF with equal air-hole diameter of 0.41 μm as well as the corresponding dispersion curve shown in Fig. 3b.As it is seen, the dispersion curve does not show a flat behavior around the center wavelength of 0.8 μm and therefore the SCG spectrum will not be wide and smooth.
By taking successive derivatives of each dispersion curve in Fig. 2    simulation of GNLSE.Therefore, the linear and nonlinear parameters of the PCF calculated at 0.8 μm for each airhole diameter are listed in Table 1.Using these parameters and solving GNLSE via SSF method, one can simulate the SCG spectra as shown in Fig. 5.The pump peak power is P 0 = 150Wand pulse width is T 0 = 1.5 ps.As it is evident, the spectra depend on the air-hole diameter where the widest spectrum is achieved for r 1 = 0.41 μm.This is in consistent with the result obtained from Fig. 2 where the flattest dispersion curve was obtained for r 1 = 0.41 μm.The 10-dB width is usually used as a measure of the spectral width which is given in Table 2.As it is obvious, the 10-dB width for the PCF with r 1 = 0.41 μm is maximum.According to Table 2, the best bandwidth and resolution are obtained for the PCF with r 1 = 0.41 μm; therefore, we fix the air-hole diameter at r 1 = 0.41 μm for following simulations.In Fig. 6, the SCG spectra are shown when the pump width T 0 is changed.Evidently, as the pulse width decreases, the spectrum is widened so that the widest spectrum is obtained for T 0 = 30 fs.This means that for T 0 = 30 fs, the 10-dB spectrum bandwidth is Δλ =62.0 nm which corresponds to the highest axial resolution of l c = 4.5 μm.
Table 3 lists the 10-dB bandwidths as well as the OCT axial resolutions calculated at each pump width T 0 for the fixed air-hole diameter of r 1 = 0.41 μm.As it is seen the best bandwidth and resolution are obtained for the pulse width of T 0 = 30 fs; therefore, we fix the simulation parameters at T 0 = 30 fs and r 1 = 0.41 μm and then examine the role of pulse peak power in SCG spectrum and its 10-dB bandwidth.This is shown in Fig. 7 where the SCG spectra are plotted when the pump peak power is changed from P 0 = 150 W to P 0 = 750 W.
Table 4 lists the 10-dB bandwidths Δλ as well as the OCT resolutions l c calculated at different pump peak powers P 0 .It can be seen that by increasing the peak power, the SCG bandwidth and the OCT resolution are improved so that the best bandwidth and resolution are obtained for the pump with the peak power of P 0 = 750 W. Therefore, totally the best SCG bandwidth and OCT resolution are achieved when the squared hyperbolic secant pump pulse with P 0 = 750 W and T 0 = 30 fs is input into the PCF with r 1 = 0.41 μm.Obviously, this best bandwidth results in the best OCT resolution of 2.6 μm for ophthalmology.This means that the supercontinuum generated from the designed PCF (see Fig. 1) is a very good candidate as an OCT source for ophthalmology since it can provide high quality images with resolutions as high as 2.6 μm.
Fig. 6 The SCG spectra for different pump widths T 0 and fixed air-hole diameter r 1 = 0.41 μm

Conclusion
In this research we have designed a unique silica-based PCFs which consists of a cladding of 5 rings of air holes where the diameter of the first ring d 1 was changed to obtain suitable dispersion characteristics around 0.8 μm.We have simulated SCG in the designed PCFs to obtain a wide and flat spectrum to be used as a source of OCT for application in ophthalmology.By solving the GNLSE and using the SSF method, the SCG bandwidths as well as the OCT resolutions were calculated for different effective parameters including the air-hole diameter, pulse width and peak powers.The results show that, in general, the best SCG bandwidth of 109.0 nm and equivalent OCT resolution of 2.6 μm can be achieved when a squared hyperbolic secant pump pulse with P0 = 750 W and T0 = 30 fs was provided into the PCF with r 1 = 0.41 μm.To the best of our knowledge, this is the maximum bandwidth and the highest OCT resolution obtained for ophthalmology via the PCF-based SCG.Therefore, the proposed PCF may be considered as a very good design for the SCG to obtain high-resolution images in the ophthalmic OCT systems.

Fig. 3 aFig. 4
Fig. 3 a The cross section of a symmetric PCF with equal air-hole diameters of 0.41 μm; and b corresponding dispersion curve , one can obtain the curves for different orders of the dispersion coefficients ( m = d m d m = 0 ) versus the wavelength for the given d 1 as shown in Fig. 4.However, for sake of brevity, only the curves for r 1 = 0.41 μm are shown in Fig. 4. Since the pump wavelength is chosen at 0.8 μm, the values of dispersion coefficients at this specific wavelength are required for

Fig. 5
Fig.5SCG spectra simulated at different air-hole radius r 1 .The pump peak power is 150 W and pulse width is 1.5 ps

Fig. 7
Fig.7The SCG spectra for different pump peak powers P 0 .The air-hole diameter and the pump width are fixed at r 1 = 0.41 μm and T 0 = 30 fs, respectively

Table 1
Dispersion coefficients and nonlinear parameter of the PCF calculated at 0.8 μm for different air-hole radii r 1

Table 2
The 10-dB bandwidths Δλ of SCG as well as the OCT axial resolutions l c calculated at each air-hole radius r 1

Table 3
The 10-dB bandwidths of SCG as well as the OCT axial resolutions calculated at each pump width T 0 .The airhole diameter is r 1 = 0.41 μm

Table 4
The 10-dB bandwidths of SCG as well as the OCT axial resolutions calculated at different pump peak powers.The air-hole radius and the pump width are fixed at r 1 = 0.41 μm and T0 = 30 fs, respectively