Second harmonic generation in thin optical fibers via cladding modes

Since silica goes under the category of amorphous materials, it is difficult to investigate important processes such as second harmonic generation (SHG) in silica-based fibers. In this paper, we proposed a method for SHG relaying on cladding modes as pump modes. Cladding modes are introduced in optical fibers through tilted long period grating (TLPG), where power of core mode is transferred into cladding modes. By functionalizing T-LPG with nonlinear coating, the interaction occurs between cladding modes and the coating material, consequently second harmonic signal (SHS) is generated with efficiency up to 0.14%. ©2016 Optical Society of America OCIS codes: (070.7345) Wave propagation; (130.2790) Guided waves; (190.4370) Nonlinear optics, fibers; (310.6860) Thin films optical properties; (050.1950) Diffraction gratings. References and links 1. H. Ishio, J. Minowa, and K. Nosu, “Review and status of wavelength-division multiplexing technology and its application,” J. 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Introduction
Silica-based fibers are the base of optical technology.Optical fibers are used in optical communication systems, sensing and biomedical applications [1,3].However, their major limitation is the lack of nonlinear effects [4][5][6].Silica possess third order nonlinearity such as four-wave-mixing, but lakes second order nonlinearity.The forbiddance of second order nonlinearity in silica is due to considering silica as amorphous materials [4][5][6].Consequently, it is difficult to investigate important processes such as second harmonic generation (SHG) in optical fibers [6].
SHG is the phenomenon that an input wave generates another wave with twice the optical frequency.This phenomenon has many applications in optical communication systems and biomedical applications [7,8].Hence, the demand has increased to develop new methods that can functionalize the surface of a silica fiber with nonlinear materials to generate second order nonlinearity.Many studies in this area use core containing different mole fractions of GeO 2 [9], fibers with germania, phosphorous and fluorine doping both in the core and the cladding regions [10], and slot micro/nano-fiber (SMNF) [11].
All mentioned studies use optical fibers with major changes to generate a second harmonic signal (SHS).Y. Xu et al. have theoretically proposed a general approach that can produce thermally stable second order nonlinearity on the traditional silica fiber surface [3].C. Daengngam et al. developed another technique to generate Second Harmonic Signal (SHS) based on a coated uniform fiber taper with nonlinear thin film.This coated fiber taper can produce SHS with theoretical efficiency of order 10 -3 % [12,13].Recently, C. Daengngam et al. [14] implemented Quasi Phase Matching (QPM) in nonlinear fiber tapers that enhance the SHG efficiency to reach the order of 10%.
Unfortunately, fabricating a nonlinear fiber taper is a difficult process for many reasons.First, the intermodal phase matching condition between fundamental core mode and its corresponding SHS is satisfied in case of fiber taper with diameter in the range of submicron.This range of tapering is very difficult to achieve during fabrication process, as the fiber can be easily broken.Furthermore, the efficiency of SHG is quite low due to the phase mismatching between the fundamental core mode and generated SHS, and the small area of interaction between fundamental core mode and the nonlinear material.QPM can solve the phase mismatching issue but still concerns about the small diameter and the efficiency itself [14].
In this paper, we propose a methodology for SHG in optical fiber.Cladding modes of tilted long period grating (T-LPG) are exploited as fundamental modes to overcome the problems of using fiber taper.First, cladding modes allow the intermodal phase matched condition to be satisfied at the cladding region of fiber with a proper diameter, and consequently, the fiber will not be fragile.Second, the proposed method promotes the fundamental cladding mode to interact with the nonlinear material at a larger area.As a result, the efficiency should increase.The cladding modes are introduced in the optical fiber via writing a tilted grating in the fiber core.T-LPGs possess a periodic refractive index modulation along the fiber axis, with a certain tilt angle between the grating plane and fiber cross section [15,16].Owing to the existence of a tilt angle, coupling light power between the core modes and cladding modes becomes possible [15][16][17].T-LPGs facilitate the coupling of the propagating core mode to cladding modes propagating in forward direction too.The grating period, Λ range is typically from less than 10 μm to 700 μm and the tilt angle, θ is preferred to be greater than 80° to couple the core mode with cladding modes with high azimuthal orders (l > 1) as inferred in Refs [17,18].LPG whose grating planes are tilted by 0° angle is a special case of T-LPG.LPGs promote coupling between the propagating core mode and the co-propagating cladding modes with l = 1 only [17][18][19][20][21][22][23].
Our proposed model is separated into two main parts.Firstly, the coupling between core mode and cladding mode, which is the (pump mode), is analyzed.Identifications of the coupling details are introduced through dealing with coupled mode theory (CMT).In the other part, the conversion is supposed to occur from cladding pump mode of frequency ω and wavelength λ into cladding mode of frequency 2ω and wavelength λ/2 which is the SHS mode.Based on these identifications, the efficiency is calculated.Then, the results and discussions are stated with all computational calculations of solving intermodal phase matching condition, coupled mode equations (CMEs), power of both pump and SHS modes as a function of wavelength and longitudinal distance, and finally the power efficiency.The paper is ended with the concluded vision of the introduced model with clarification of pros and cons with respect to other models in the literature review.

Characterization and analysis
In this section, we introduce an illustrative summary of the propagation constant and fields of cladding modes, and the coupling through T-LPG using CMT.Moreover, we derive a formula of SHS power and conversion efficiency.
The introduced method is based on generating SHS through using cladding modes as pump modes.The T-LPG transfers the energy of fundamental core mode,  Optical fibers support different kind of modes such as guided, radiation, and leaky modes.Full descriptions of guided modes are represented in [17][18][19].Cladding and core-guided modes are confined in cladding, and core regions, respectively.The schematic presented in Fig. 1 shows T-LPG coated with nonlinear material of thickness δ.The core and cladding refractive indices are n 1 and n 2 , and their radii are a 1 and a 2 , respectively.The surrounding medium is assumed to be infinite with a refractive index n 3 .
The dispersion equation of cladding modes is derived in [25] by applying the boundary conditions at the interfaces between core and cladding regions, and between cladding and outside region.[20,25].
The T-LPG permits the coupling of the propagating core mode to forward propagating cladding modes.The CMEs of T-LPG are defined as [16-18, 20,21] ( ) ( ) ( ) A ω , that interacts with the nonlinear layer to generate SHS.Our target is to find the amplitude pump mode, , cl pump A ω , as a function of wavelength, λ, and longitudinal distance, Z.This can be achieved by solving the CMEs given by Eq. ( 1) via the standard Runga-Kutta numerical algorithm.In order to solve the CMEs, it is assumed that the amplitudes of field at the input side of the grating are initially equal to 1 for the core mode ( ( ) , and 0 for the cladding modes ( , (0) 0 [18,20].
The second order susceptibility, χ (2) , in optical fibers is introduced by coating the fiber surface with organic nonlinear molecules of poly (allylamine hydrochloride)/procion brown (PAH/PB) through hybrid covalent/ionic self-assembled multilayer (HCISAM) technique [26] where the nonlinear coefficient is dominated by radial component ( ) χ which is ~21 pm/V [25].The CMT with nonlinear vision explains the relation between the field of the second harmonic mode and the cladding mode (pump mode) as follows [5,6,12,13,27] where ν g (2ω) is the group velocity of the propagating wave at frequency 2ω, and Δβ represents the intermodal phase mismatching between pump and SHS modes and is given by The effective nonlinear coefficient, ( ) where δ is the thickness of the coating layer.The cylindrical symmetry allows further decomposition of the electric field components [6], to be ( ) ( ) . By substitution in Eq. ( 4), one gets If l SHS = 2l pump , then ( ) 2 eff χ will have nonzero value.This condition is known as the selection rule [6,26].For example, when the pump mode with azimuthal order, l pump = 1, the possible SHS must have azimuthal order, l SHS = 2.
The SHS field amplitude, ( ) 2), is integrated with respect to Z and is rewritten as [6] ( ) ( ) The actual power of each propagating mode is calculated by [6] 2 o where A is the field amplitude and P o is the initial power that makes each mode satisfying the orthonormal condition [6].Here, we apply the orthonormality condition such that P o equals 1 W.
The total conversion efficiency of SHG through T-LPG coated with nonlinear organic material (PHA/PB) can be calculated from where L is the grating length,

Results and discussion
For coating the fiber with nonlinear film, the ionic self -assembled multilayer (ISAM) technique is used.This technique is fully described in [12][13][14].The one bilayer is composed of positively charged polyelectrolyte which is poly (allylamine hydrochloride) (PAH) followed by negatively charged polyelectrolyte which is the nonlinear nanomaterial (Procion Brown (PB)).Many bilayers of PAH/PB could be achieved by repeating the ISAM film process.It is important to consider the effect of absorption of the nonlinear organic material used in this work (PB) and the limitation of available electronic equipments.Hence, a pump signal needs to be within the wavelength window 1100-1300 nm, to generate SHS within the wavelength range 550-650 nm [4,12,13].Such consideration can be achieved by controlling the grating parameters [21][22][23][24].

SHG through T-LPG and LPG
SHG could be achieved with high efficiency if the intermodal phase matching condition is satisfied.The condition given by Eq. ( 3) states that the mode to be converted (pump cladding mode) and generated one (mode of SHS) should have the same refractive indices, i.e.
( ) ( ) given in [19,24].Consider a single mode fiber (SMF) with the following parameters: n 1 = 1.4681, n 2 = 1.4628, n 3 = 1, and a 1 = 4.15 μm.The effective refractive indices of pump cladding modes with azimuthal order l pump = 2 at wavelength 1230 nm and of the SHS with azimuthal order l SHS = 4 at wavelength 615 nm are plotted versus fiber cladding radius in Fig. 2. The intersections between the red and blue curves mean that the nonlinear intermodal phase matching condition is satisfied.Since the thickness of the nonlinear coating is much lower than a 2 (50 nm in our work), the effect of nonlinear material on effective refractive indices is not considered in the dispersion relation.HE ω mode at wavelength 615 nm intersect at cladding radius equals 14 μm.The choice of this particular point is because at this point the cladding radius is the smallest one, hence, the field amplitude is increased, and as a result the efficiency of SHG is enhanced.When the surrounding refractive index (SRI) increases, the required cladding radius to satisfy Eq. ( 3) decreases, due to the dependence of dispersion equation on the SRI.This is shown in Fig. 3.After satisfying the intermodal phase matching condition, we have to analyze the conversion from core mode to cladding pump mode by solving CMEs given by Eq. ( 1) and consequently finding the field amplitude of pump mode.Then, the power of the cladding pump mode is calculated from Eq. ( 7).The analysis of T-LPG of a thin cladding radius verifies that the energy of core mode is transferred to a single cladding pump mode, 2,1 , cl HE ω , over the entire possible spectrum (i.e.1100 nm to 1300 nm).
Since the cladding mode with azimuthal order l pump = 2 is used as a pump mode, a grating with tilt angle is needed.As tilt angle becomes greater than 80°, the coupling coefficient increases and the coupling conversion between core and cladding modes, with azmuthal order greater than 1, takes place.Practically, the largest possible tilt angle written in optical fibers equals 81° [19] Amplitude of SHS, given by Eq. ( 6), is maximized by transferring the power of core mode totally to the pump cladding mode that propagate at a wavelength equals 1230 nm.This can be achieved by controlling the grating period and the index modulation depth.Table 1 contains the needed parameters of T-LPG to fulfill the demands discussed before at different SRI.In all cases, the grating length is 5 cm.The chosen values of modulation refractive index amplitude are in the practical range [3].The power of pump mode versus wavelength, λ, and longitudinal distance, Z, is calculated and displayed in Fig. 4. One can observe that, the pump power at the end of the grating equals 1 W, which means that the core mode power completely coupled to the cladding pump mode to grantee maximum pump mode amplitude.We keep this criterion in all studied cases irrespective of the values of SRI and cladding radius.By knowing the suitable cladding radius that satisfy the phase matching condition and finding the field amplitude of cladding pump mode as function of both wavelength and longitudinal distance, we are ready to find the field amplitude of SHS to be able to calculate the total efficiency.Figure 5 shows the power of the generated SHS versus wavelength and longitudinal distance at different SRI.The percentage of total efficiencies are calculated numerically from Eq. ( 8) at different values of SRI, and are obtained as 4.8 × 10 -6 %, 1.27 × 10 -3 % and 1.4 × 10 -1 % for SRI equals 1, 1.33, and 1.4, respectively.Hence, it is worthy to say that, as SRI increases the total efficiency increases too due to the incensement of effective nonlinear coefficient, ( ) χ , that confirms the enhancement of electric field on the surface.
The effective nonlinear coefficient, Eq. ( 5) is the key parameter in SHS field amplitude calculation.It depends on the cladding pump mode field, SHS field, and the area of interaction with the nonlinear material.As the fields on the surface are magnified, the effective nonlinear coefficient increases, hence, the total efficiency of SHG is raised.As an evidence, the electric fields of pump and SHS are plotted in Fig. 6 as a function of the radial component in the case when SRI, n 3 , = 1, 1.33 and 1.4.The same procedure is applied for the case of LPG to generate SHS.The LPG provides coupling with azimuthal order l = 1.Assume SMF-28 with the following parameters: n 1 = 1.4681, n 2 = 1.4628, n 3 = 1, and a 1 = 2.625 μm.By considering the selection rule, intermodal phase matching condition is satisfied for cladding pump modes with azimuthal order l pump = 1 at wavelength 1230 nm, and for the SHS with azimuthal order l SHS = 2 at wavelength 615 nm, as shown in Fig. 7.We focus on the first point of intersection, which illustrate the conversion of HE ω mode at wavelength 615 nm.The following results of the SHS are obtained for a LPG having a grating tilt angle, θ = 0°, and a grating length, L = 5 cm, and the rest of grating parameters are summarized in Table 2.As in T-LPG case, the power of core mode is fully delivered to cladding pump mode by optimizing the modulation index of the grating.Passing the needed calculations from Eq. ( 8), the power of generated SHS at different values of SRI are summarized in Table 3 for both T-LPG and LPG.It is concluded that, efficiencies of SHG through T-LPG are higher than those efficiencies of obtained in LPG case, due to the increase in the fields on the surface.As the field of higher azimuthal mode order is extending extensively to the surface [23].
When the fiber grating is surrounded by air, the calculated overall efficiency is still good compared with previous work.In order to enhance the overall efficiency by order 10 4 in case of LPG, or 10 5 in case of T-LPG, the fiber is to be immersed in a medium with a refractive index equals 1.4.This incensement in efficiency occurs as result of extending more electric fields on the surface.The cladding modes with higher azimuthal order have higher fields on the surface, so, the efficiency is higher in case of T-LPG.The highest obtained efficiency is about 1.4 × 10 -1 %.
Experimentally, cladding modes could be recoupled from the cladding region to the core region using different methods of tapering [28,29].

Related work and comparison
In previous studies [12,13], researchers used the core mode as a pump mode.Since the SMF has a single core mode, their choices were limited.They were able to satisfy the intermodal phase matching condition between the core mode and the second harmonic mode by tapering the core radius to be in the range of submicron.Theoretically, they achieved conversion efficiency ~10 -3 % at pump power 5 W. Experimentally, the conversion efficiencies are quite poor ~10 -5 % due to the invalidity to meet the intermodal phase matching condition.
Recently [14], the QPM technique was applied in nonlinear fiber tapers in order to solve the problem of mismatching between pump and second harmonic modes [12,13].Under 400 W of pump power, the SHG efficiency is enhanced theoretically to ~10%, and experimentally to ~10 -4 %.Slot micro/nano-fiber (SMNF) was used to enhance the SHG [11].The efficiency reached 2.7 × 10 -2 %, however, the used input power is relatively high.
Alternative methods were introduced to generate SHS through modified optical fiber [10,11].SHS was generated efficiently in doped SMF and multi-mode fiber (MMF) [10].The efficiency through SMF under pump power 3 W is higher than MMF under pump power of 400 W.
Table 4 is introduced to compare our results with other reports considering the main parameters such as input power, fiber dimensions, and interaction length.Although fiber taper model [12][13][14] introduces the idea of using traditional optical fibers to produce SHS with acceptable efficiency, the model has many practical problems such as the assumption of higher input pump mode and difficulties to reach fiber taper with such dimensions.We figure out that, the maximum total efficiency of our model is around 10 2fold enhancement compared with that calculated from nonlinear fiber taper [12,13] without the need to apply the QPM technique.In our proposed model, we are dealing with cladding modes as a pump mode, so, we have a variety of options that verify the intermodal phase matching condition at a thinner cladding radius, which could be practically achieved in many sensor applications [2,23].We achieved promising efficiency under a pump power equals 1 W. Finally, we introduce a simple system that practically can be implemented.

Conclusion
In this study, the T-LPG is used to transfer energy from the core mode into the cladding modes with azimuthal order, l > 1.Here, we use a T-LPG that promotes a core mode propagating into forward direction to couple with cladding mode propagating in forward direction too.Practically, the maximum grating tilt angle which can be fabricated is about 81°.The simplest type of T-LPG is named LPG at which grating tilt angle equals zero and hence, coupling occurs between core mode and cladding modes having azimuthal order, l = 1.
By adjusting the grating tilt angle and period, we control the cladding mode to propagate at frequency ω.A SHG is introduced due to the interaction between cladding mode field and nonlinear coating on the surface of the T-LPG.The selection rule of modes (l SHS = 2l pump ), and phase matching condition are required.
Our theoretical model is based on using cladding modes as pump modes.Through this approach, promising results are obtained which are better than that obtained by previous studies in this field.Using a thin cladding layer instead of tapering the fiber increases the interaction between the cladding field and the nonlinear material coating the surface.The cladding modes with azimuthal order greater than 1 provide an enhancement in the SHG efficiency.A maximum total efficiency of order 10 -1 % is achieved without any sophisticated fabrication process.The obtained efficiencies are based on using a pump power equals 1 W. If the pump power increases, the efficiency would increase in a quadratic way and a higher efficiency could be achieved.
the cladding mode interacts with the nonlinear layer on

Fig. 1 .
Fig. 1.Schematic representation of coupling through T-LPG coated with organic nonlinear molecules and the generated SHS.
The solution of the dispersion equation gives the effective refractive indices, , cl eff lm n of cladding modes ordered by l, m, where cl eff n depends on the refractive indices of the three regions, radii of core and cladding regions, and the wavelength.The dispersion equation has been reported in

A
the azimuthal order, m is the radial number, r is the radial distance measured from the fiber axis, θ is the tilt angle (θ = 0 for LPG).1,1 , between cladding modes with each other.Further details about calculating coupling coefficients are explained in[16-18, 20,21].The amplitudes of both fundamental core mode and the amplitudes of cladding modes subscribed by l, m in forward direction, are 1, respectively.The amplitude of cladding mode will be considered as the pump mode, , cl pump and amplitude of the core mode at the beginning of grating region.
the selection rule.If a fundamental pump mode and its second harmonic mode propagate in the traditional silica glass fiber, they would always fail to compensate phase matching condition, since refractive index of a waveguide material is wavelength dependent and is, in the same time, sensitive to the optical fiber geometry, the difference between effective refractive indices of pump and second harmonic modes can be compensated through decreasing the cladding radius.The effective refractive indices for cladding mode, cl eff n is determined by solving the dispersion relation

Fig. 2 . 2 (
Fig. 2. Possible cladding radii for which the intermodal phase matching condition is satisfied, at the intersections (green crosses) between cl eff n for modes with azimuthal order l pump = 2

Fig. 3 .
Fig. 3. Possible cladding radii for which the intermodal phase matching condition is satisfied, at the intersections (green crosses) between blue curves and red curves.SRI takes values of 1, 1.33 and 1.4.

Fig. 4 .
Fig. 4. Power of pump mode as a function of: (a) wavelength, λ and (b) longitudinal distance, Z.

Fig. 7 .
Fig. 7. Possible cladding radii for which the intermodal phase matching condition is satisfied, at the intersections (green crosses) between modes with azimuthal order l pump = 1 (Blue curves) and modes with azimuthal order l SHS = 2 (Red curves).