Single LP 0 , n mode excitation in multimode ﬁbers

: We analyze the transmission of a Single mode - Multimode - Multimode (SMm) ﬁber structure with the aim of exciting a single radial mode in the second multimode ﬁber. We show that by appropriate choice of the length of the central multimode ﬁber one can obtain > 90% of the total core power in a chosen mode. We also discuss methods of removing undesirable cladding and radiation modes and estimate tolerances for practical applications.


Introduction
Single mode excitation of large core area fibers is attracting a lot of interest with increasing requirement of large power handling capacity of all-fiber lasers and amplifiers [1][2][3][4].Small core single mode fibers (SMFs) can give a single transverse mode but limits the achievable power output due to non linear effects.Various techniques have been proposed for exciting a single mode in a large core area fiber such as using coiled fiber to remove the higher order modes [3], using helical core fibers, which have higher losses for higher order modes [5], designing appropriate dopant distribution to provide high gain to a single mode [6] etc.A versatile and simple way to achieve single mode beams from multimode fibers (MMFs) has also been proposed using Multimode Interference (MMI) effect [7,8].
In this paper, we propose a method to excite a single LP 0,n mode in the core of an MMF using Single mode -Multimode -Multimode (SMm) fiber structure shown in Fig. 1.Since a higher order mode forms a better approximation to a bessel beam, it is advantageous to excite such modes to achieve a better diffraction free beam from fiber lasers as compared to the gaussian like beam of an LP 0,1 mode.Also, bending induced mode coupling between LP 0,n and LP 1,n modes reduces with increasing n, due to relatively large differences between effective refractive indices of these higher order modes [9].In yet another method, a higher order mode excitation has been achieved using a Long Period Fiber Grating (LPFG) in a fiber with specially designed refractive index profile [10].In this method, the LPFG is used to couple the power from a single inner core mode to a higher order outer core mode of a three layered fiber structure.
The SMm structure proposed here achieves a near single mode in the core region of output MMF (MMF2) by exciting it with a specific field profile obtained by interference of propagating modes in another large core area MMF (MMF1), which in turn is excited by an SMF.Our structure is simple to fabricate and provides a novel way to excite a single LP 0,n core mode in the standard MMFs available.Guided mode propagation analysis is employed to understand the transmission properties of this structure, and is presented in section 2. In general, with a large core area MMF1 excitation both core and cladding modes are excited in the small core area MMF2 [11][12][13].In section 3, we show that with appropriate choice of the length of MMF1 a single LP 0,n core mode can be excited in the core of MMF2.A fraction of the total power is also distributed in several cladding modes of MMF2.In section 4, we show the use of an external medium (say, a liquid or a polymer coating) surrounding MMF2 to remove the cladding modes from the structure.Section 5 discusses the use of this structure to generate a single mode beam for all-fiber lasers and amplifiers, and MMF optical sensors.In section 6, practical aspects of fabrication of the structure are presented followed by conclusions.

Guided mode propagation analysis of SMm
For the transmission analysis of our structure, we use core and cladding refractive indices of SMF-28 (n co = 1.4504, n cl = 1.4447) and AFS105/125Y (n co = 1.4446, n cl = 1.4271) for SMF and MMF1, respectively [14].MMF2 is similar to MMF1, except that the core diameter is less.We used LP approximation with an assumption of infinite cladding to calculate the transverse field profiles and propagation constants of the guided core modes in fibers.The wavelength employed is 1.55µm.If we assume an ideal alignment, only radial modes are excited in the structure due to symmetry [14].Fresnel's equation is employed to calculate the average power reflection coefficient at the two junctions as At the first junction (between SMF and MMF1), n i and n j are the effective refractive indices of the guided core modes of SMF and MMF1, respectively.At this junction, N 1 = 1.Similarly, at the second junction (between MMF1 and MMF2), n i and n j are the effective refractive indices of the guided core modes of MMF1 and MMF2, respectively.Equation 1 assumes equal power distribution in all the modes of these fibers.Using eq. 1, we get power reflection coefficients of just 0.0012% and 5.58 × 10 −4 % at the first and the second junctions, respectively.A small reflection at the junction of SMF and MMF has already been reported in the Single mode-Multimode-Single mode (SMS) fiber structure of [14].Reflections at the second junction can be expected to be less because the effective refractive indices of the guided modes in these MMFs are very close.Thus, we considered only the forward propagation in our analysis.Each transverse mode field is normalized to carry unity power as where ψ is the calculated real valued mode field profile, c the velocity of light in vacuum, ε o the permittivity of space and n the effective refractive index of the mode.At the first junction, the SMF field can be written as a linear superposition of the propagating modes of MMF1 as where Φ s and Ψ n are the normalized transverse mode field profiles of the single mode fiber and the n th mode of MMF1, respectively.We will show that the core modes of MMF1 are sufficient to define the input single mode field.c n is the field coupling coefficient of the n th radial mode of MMF1.It can be calculated using the orthogonality property of modes of MMF1 as This gives, where n n and n s are the effective refractive indices of MMF1 and SMF modes.c ′ n is defined as | c n | 2 gives the fraction of input power in the n th radial mode of MMF1.For the above mentioned fibers, we computed field coupling coefficients and obtained ∑ n | c n | 2 = 0.99.Thus, we numerically verified that the core modes of MMF1 are sufficient to describe the input SMF field.As the field propagates, individual modes develop relative phase differences between each other.The transverse field profile at any length of the MMF1 is an interference pattern of all the propagating modes.These patterns result in different power distributions on the transverse plane of the MMF1.Thus, the launch field of MMF2 depends upon the length of MMF1.This would allow the modes of MMF2 to be excited with variable power profile.To calculate the field excitation coefficients of the radial modes of MMF2, we write this launch field as where Ψ ′ m , Ψ ′′ k and Ψ ′′′ p are the transverse field profiles of the core, cladding and the radiation modes of MMF2, respectively.b m , d k and p(β ) are the corresponding excitation coefficients, respectively.Equation 7 is valid when the refractive index of the medium outside MMF2 is less than that of the cladding.In such a case, cladding forms a guided wave structure with the outside medium.If the refractive index of the outside medium is greater than that of the cladding, there would be leaky modes in addition to the core modes and the radiation modes in MMF2.
Using the orthogonality property of modes, the field excitation coefficients for the core modes (b m ) of MMF2 can be obtained as This gives, where α m is the effective refractive index of the m th radial mode of MMF2, and a mn is given by Transmission of the SMm structure for various lengths of MMF1 can be written as If the output fiber is identical to the input fiber (m = 1; α 1 = n s and ψ ′ 1 = φ s ), eq.11 reduces to the earlier result for the transmission of SMS structure [14].Thus, we have a generalised form of the earlier result.Transmission through the SMm structure includes only the power coupled to the core modes of MMF2.This is because the power coupled to the cladding modes would ultimately be lost in the outer jacket or can be made to radiate away by using cladding mode strippers.To verify the above analysis, we used wide angle beam propagation method with pad é order (4,4) [15], using a commercially available software package (BeamProp; Rsoft, Ver.9).We calculated the fraction of power coupled to the radial core modes (| b m | 2 ) of a 50µm core MMF2 for different lengths of MMF1 (core 105µm).Results from both the approaches are in agreement and are shown in Fig. 2 for LP 0,1 , LP 0,2 and LP 0,7 modes of MMF2.
Transmission loss in the structure occurs because the power from MMF1 is not completely T < 0 dB and some power is coupled to the cladding/leaky modes and the radiation modes in MMF2 depending upon the refractive index contrast between cladding and the outside medium.Thus, a transmission of -3 dB would mean a transmission loss of +3 dB.To show the power coupling from MMF1 to the MMF2 cladding modes, we calculated radial mode field profiles of the cladding modes of MMF2 using three layer analysis of the fiber [16].The refractive index of the medium outside MMF2 cladding is chosen to be 1.41.This maintains the validity of LP approximation used in calculating the modes.Field excitation coefficients of the cladding modes of MMF2 are calculated in a manner similar to that of the core modes (see eq. 9 and 10).Fraction of power coupled to the core and the cladding modes of MMF2 is shown in Fig. 3 for various lengths of MMF1.The length of MMF1 at which the fraction of power coupled to the core modes of MMF2 reaches a minimum, we find that the power fraction of cladding modes reaches a maximum, and vice -versa.Figure Hence, the core and cladding modes of MMF2 are sufficient to describe the MMF1 field when the refractive index of the medium surrounding MMF2 is equal to 1.41.
From the above discussion, we can conclude that for a given set of fibers, the fraction of power launched into the core and cladding modes of MMF2 can be changed by changing the length of MMF1.

Single LP 0,n mode excitation in MMF2
Changing the length of MMF1 also distributes variable power amongst different core modes of MMF2.We calculated the power coupling coefficients of the radial core modes of a 50µm core MMF2 for different lengths of MMF1.Core diameter of MMF1 was chosen to be 105µm.The results are shown in Fig. 4. Normalized radial power profile of the excitation field of MMF2 are also shown in the figure.Thus, it is possible to excite a set of core and cladding modes in MMF2 such that a single core mode receives a large fraction of the power coupled to all the core modes.Remaining power is coupled to the cladding modes of MMF2.Power coupled to the cladding modes can be removed by using cladding mode strippers leaving only a core mode to propagate.This technique results in the loss of power to achieve a single core mode propagation in MMF2.
To study the efficiency of this technique and to qualify the excitation of a single core mode, we put a criterion such that the desired single LP 0,n mode in MMF2 receives more than 90% of the total power coupled to all the core modes.This condition can be written as   For various core diameters of MMF2, we calculated the lengths of MMF1 for different core diameters, that satisfies our 90% criterion.Table 1 shows the power coupling coefficients of all the radial modes of a 20µm and a 30µm core MMF2.Core diameters and the lengths of MMF1 required to achieve a single LP 0,1 or LP 0,2 core mode in these output fibers are also shown.Table also gives the fraction of power (γ n ) of the selected n th radial mode of MMF2, and the corresponding transmission loss of the SMm structure.
Excitation of a higher order LP 0,n mode requires larger core diameters of MMF2.Due to the large number of modes, the power coupled to the desired higher order mode (say, LP 0,10 ) of MMF2 would be less as compared to that of a smaller order mode in a small core MMF2.However, the relative power levels of these higher order modes still satisfy the 90% criterion.
Table 2 shows a few design examples of exciting a higher order mode in MMF2.A value of γ as high as 0.99 is also obtained for the LP 0,10 mode in a 64µm MMF2.However, this mode is excited with only 9.7% of the input power.

Removal of unwanted cladding power
Since the core diameter of MMF2 is smaller than that of the MMF1, unwanted power can also propagate in the form of cladding or leaky modes in the cladding of MMF2.Normally, fibers are coated with a polymer layer with refractive index slightly higher than that of the cladding [13].In such a case, core modes, leaky modes and radiation modes would be excited in MMF2.Relative power coupled to the leaky modes and the radiation modes depends upon the refractive index contrast of the cladding and the outside medium.Using BeamProp, we will show that a large fraction of this unwanted power (not coupled to core modes) can be removed from the structure when an outside medium with a refractive index slightly greater than that of the cladding is used.As a result, only a single core mode propagates in MMF2.
To study this effect, we removed the earlier constraint of infinite cladding medium for MMF1 and MMF2.The outside medium is assumed as air.To see the effect of limiting the cladding medium on the designed length of MMF1, we calculated the length of MMF1 to achieve maximum γ n using BeamProp.We performed only radial calculations employing wide angle beam propagation method with pad é order of (4,4) [15].These results are presented in table 3 for all the lower order modes (LP 0,1 -LP 0,3 ).For all these modes, we get approximately the same length (maximum difference is of 6µm) as shown in tables 1 and 2. However, the length of MMF1 to excite LP 0,1 mode in the 20µm MMF2 is quite different in the two cases.This is because the LP 0,1 mode is excited with a large amount of power for various lengths of MMF1.Even at the length of 74866µm of MMF1 (from table 1) nearly same value of γ 1 (= 99.72%) is obtained using BeamProp.Now, we would like to study the effect of an outside medium surrounding MMF2, with a refractive index slightly higher than that of the cladding.For this purpose, we consider an SMm structure with MMF1 and MMF2 core diameters of 99µm and 30µm, respectively.The refractive index of the medium outside MMF2 is chosen to be 1.4272.Boundaries of the simulation region were kept transparent.This enables the radiating field to escape from the simulation region, without any back reflections.Field propagating in the structure is shown in Fig. 5(a).Figure 5(b) shows the power that would couple to the LP 0,2 mode of a 30µm core fiber, when the structure is terminated at various lengths (z).The power coupled to the LP 0,2 mode as a fraction of the input power is shown in solid green line.Similarly, the power coupled to the LP 0,2 mode as a fraction of the local propagating power is shown in dashed blue line.Total power propagating in the structure (as a fraction of input power) is also shown in dotdashed black line.From Fig. 5(b), it can be seen that for z < 10881µm (in MMF1), power that would couple to the LP 0,2 mode of a 30µm core fiber varies with the length of the structure.This happens due to multimode interference in MMF1.Total propagating power in MMF1 remains equal to the input power as there is no loss due to radiation.For z > 10881µm (in MMF2), power that would couple to the LP 0,2 mode of a 30µm core fiber becomes constant.This is because MMF2 itself has a core diameter of 30µm.Thus, this is the power with which LP 0,2 mode is launched in MMF2.Also, after a certain length of propagation in MMF2, total power propagating in the structure saturates nearly equal to that of the LP 0,2 core mode.This happens due to the radiation loss.As a result, power that would couple to the LP 0,2 mode, as a fraction of local propagating power, increases with the length of MMF2.This implies that the power in LP 0,2 mode of MMF2 dominates the power propagating in the structure.This power saturates at 96.78% for this design.Figure 5(b) also shows that at a length of 9131µm of MMF1, a larger fraction of the input power would be coupled to LP 0,2 mode as compared to the length of 10881µm.However, this length corresponds to a lower value of γ 2 (= 69.50%) and is undesirable.Propagation of the LP 0,2 mode in MMF2 can be clearly seen in Fig. 5(a).
Excited higher order modes (LP 0,10 − LP 0,23 ) in the SMm structure are very close to their respective cut-off conditions.The radial power profiles of LP 0,9 and LP 0,10 modes of a 64µm core fiber are shown in Fig. 6.Refractive index of the cladding is 1.4271.Thus, LP 0,10 is nearly cut-off with n e f f = 1.4271043, and carries a large amount of power in the cladding region.In contrast, the LP 0,9 mode, which is just one order less, is well confined with an effective refractive index of 1.4302373.The LP 0,10 mode has a relatively larger field amplitude at the cladding-outer medium boundary.Hence, for a smaller cladding diameter, power would leak away from this mode making it essentially leaky.This can also be seen from the effective refractive index of this mode calculated with the assumption of infinite cladding.This value is less than the refractive index of the outside medium (1.4272).Now, the effect of outside medium on the power propagating in this mode can be reduced by increasing the cladding diameter.With This excites LP 0,2 mode in MMF2.White line shows the refractive index discontinuities in the structure.R is the length in radial direction.(b) Power that would couple to the LP 0,2 mode of a 30µm core fiber, as a fraction of local propagating power (dashed blue line) and input power (solid green line) for various lengths (z) in the structure.For z > 10881µm, this defines the propagating power in LP 0,2 mode of MMF2.Total power propagating in the structure is also shown (dot-dashed black line).
cladding diameters of 250µm and 300µm, we calculated the effective refractive indices of the LP 0,10 core mode employing three layer analysis [16].The values obtained were 1.4271035 and 1.4271039, respectively.This shows that if the diameter of cladding is increased, effective refractive index of the mode approaches the infinite cladding limit.To show this effect, we simulated our structure with different cladding diameters for the excitation of LP 0,10 mode in a 64µm MMF2.We chose the core diameter and the length of MMF1 to be 128µm and 31641µm, respectively.Figure 7(a) shows the propagating field in the structure for the MMF2 cladding diameter of 150µm.We can see that the LP 0,10 mode is excited at the junction of MMF1 and MMF2 but fails to propagate beyond a certain length.Figure 7(b) shows the propagation of LP 0,10 mode in the same structure but with a larger cladding diameter of 250µm.
Propagation of these higher order modes can also be achieved by the use of an outside medium with a refractive index value slightly less than that of the cladding.Although, this technique allows LP 0,10 core mode to propagate without any loss but results in degradation of single mode field in MMF2 core.This happens due to propagation of the cladding modes in MMF2.Propagation of the LP 0,10 mode in MMF2 is shown in Fig. 8(a) for an outer medium having refractive index of 1.4270 surrounding MMF2.In this case, 2 cladding modes are excited in MMF2 with effective refractive indices of 1.4270996 and 1.42701813.The power coupling coefficients of these modes (calculated from eq. 9) are 14.59% and 18.28%, respectively.Fractional core (cladding + outside medium) power of these modes are 43.87%(56.12%) and Fig. 6.Normalized radial power profile of LP 0,9 and LP 0,10 modes of 64µm core diameter fiber.LP 0,10 is near to cut-off and carry a large fraction of power in cladding.Inset shows the decaying fields of LP 0,9 and LP 0,10 modes in the cladding region.15.77% (84.22%), respectively.The fraction of cladding mode power in the core region can be reduced by reducing the refractive index contrast between the cladding and the outside medium until the outer boundary becomes transparent.Cladding modes can be completely removed by matching the refractive index of the outside medium to that of the cladding.In such a case, neither cladding nor leaky modes are excited, and a large fraction of power would be radiated away.Propagation of the field in SMm structure for this configuration is shown in Fig. 8(b).For a small section of length of 5 cm, the medium outside MMF2 is matched with that of the cladding.In practice, this can be achieved using an index matching liquid or a polymer coating.After removal of the radiation modes in a small length of MMF2, the LP 0,10 mode propagates in the core of MMF2 with air as the medium outside cladding region.

Uses of SMm structure
We studied SMm structure only for a specific value of numerical aperture (0.22) of MMF1 and MMF2.Our structure gives a way to excite any desired mode in the output fiber with proper design parameters.In this section, we will discuss the use of SMm structure for exciting a single LP 0,n mode for all-fiber lasers and amplifiers, and MMF optical sensors.

In all-fiber lasers and amplifiers
There are two different ways in which an SMm structure can be employed for all-fiber lasers.One of the techniques would be to use an active MMF1.This is in accordance with the work of Zhu et.al. with reference to the SMS structure [7,8,18].However, with MMF1 as an active medium, different modes would see different gains depending upon their amplitudes and field profiles.This technique can affect the choice of the length of MMF1, degrade the γ factor or even lead to the loss of single mode propagation in MMF2.Thus, it would be better to filter out the mode first and then amplify it using an active medium.
Hence, we propose the use of MMF2 as an active medium for all-fiber lasers and amplifiers.An active fiber can also be spliced to MMF2 after the removal of cladding modes as depicted in Fig. 9.There is no restriction on the length of active fiber as long as mode coupling is avoided due to bending.Moreover, since a higher order mode is propagating in the core, mode coupling due to bending would be less [9].Thus, a single higher order mode beam with very high peak intensity is possible with the structure.To demonstrate the configuration shown in Fig. 9, we excite the LP 0,3 mode in a 50µm core MMF2.The design parameters for this structure are listed in table 3. The refractive index of a small section outside the MMF2 (from 5 cm to 11 cm) is kept equal to that of the cladding.Simulation domain is terminated inside this external region with transparent boundary conditions.This configuration does not allow any reflections from the boundary of the outside medium.Figure 10 shows the results.We can see that before the matching of the cladding refractive index with that of the outside medium, the power is propagating in both the core and the cladding of MMF2.Total propagating power remains equal to the input power in MMF2.However, when the cladding refractive index is matched with that of the outside medium, the power propagating in cladding region radiates away.As a result, total propagating power reduces and saturates near to the excitation power of pure LP 0,3 mode which is now available for amplification in an active fiber.Power propagating in LP 0,3 mode as a fraction of local power saturates to 93.94%.

In optical sensing
Excited higher order modes in SMm structure are very close to their cut-off conditions.In ray optics, these modes corresponds to the rays which are launched near to the critical angle.Now, for a given V parameter, a higher order mode carries a larger amount of power in the cladding

Cladding Mode Stripper
Active Fiber SMm Fiber Structure  9. Core diameter of MMF1 and MMF2 are 176µm and 50µm, respectively with cladding diameter of 200µm.MMF1 length is 31941µm.(a) When the medium outside MMF2 is air, power from MMF1 is coupled to both the core and cladding region of MMF2.Index matching liquid is applied from 5 cm -11 cm, to radiate away this power in the cladding modes.White line shows the refractive index discontinuities in the structure.R is the length in radial direction.(b) Power that would couple to the LP 0,3 mode of a 50µm core termination fiber, as a fraction of the local propagating power (dashed line) and the input power (solid line) for various lengths (z) along the structure.For z > 31941µm, this defines the propagating power in LP 0,3 mode of MMF2.Total power propagating in the structure is also shown (dot-dashed line).region as compared to a lower order mode [19].Increasing power in the cladding region is beneficial for many sensing applications employing MMFs [20,21].For a sensor based on the absorption of evanescent power, absorbance (A) is given by [20] A = P clad P αL 2.303 (13) where P clad is the power in the cladding region, P the total launch power, α the absorption coefficient, and L the length of exposed fiber.P clad /P naturally reduces with a large number of tightly confined propagating modes.However, for a single higher order mode propagation in MMF2, A can reach to its maximum possible value for a given MMF and enhance the sensitivity of the sensor.

Sensitivity to length of MMF1
Since the fraction of power coupled to the core of MMF2 depends on the length of MMF1, a good amount of accuracy would be required while splicing fibers.For all the cases presented in tables 1 and 2, we calculated the fractional power in the selected n th radial mode (γ n ) near the length of its maximum value.These results for the small core MMFs presented in table 1 are plotted in Fig. 11.For large core fibers and higher order modes, presented in table 2, the results are plotted in Figs. 12 and 13.We observe from Figs. 11, 12 and 13 that the γ n for higher order modes (LP 0,10 and higher) is highly immune to changes in MMF1 length.LP 0,1 and LP 0,2 modes of the 30µm core fiber also show high stability to changes in MMF1 length.Fig. 13.Change in γ factor with change in length of MMF1 for the higher order modes (LP 0,10 -LP 0,23 ) presented in table 2.

Effect of temperature
Length of a fiber can also change with temperature of the surroundings.For use in fiber lasers and amplifiers, it is thus necessary to investigate the change in length of MMF1 with temperature.Coefficient of thermal expansion (α l ) of pure silica is 4 × 10 −7 / • C [22].For a small change of δ T in temperature, change in the length of a fiber can be written as δ L = α l Lδ T .For L = 10 cm, this corresponds to a change of 4 nm in the length per 1 • C change in temperature.With reference to Figs. 11, 12 and 13, we can conclude that the γ factor remains stable for a wide range of change in temperature.Also, if MMF2 is made active for use in fiber lasers and amplifiers, the junction of MMF1 and MMF2 can be sealed away in temperature controlled unit to keep the length unchanged.

Conclusions
Single mode -Multimode -Multimode (SMm) fiber structure provides a simple way of achieving a near single LP 0,n core mode in standard MMFs including large core fibers.The chosen single core mode receives more than 90% of the power in the core modes.Fabrication tolerances for various designs to changes in the MMF1 lengths were also studied.The fractional power of a single LP 0,n mode in the output fiber was observed to be highly stable.With the possibility of filtering out a single core mode, SMm forms an ideal structure to provide a seed in all-fiber laser amplifiers for achieving high intensity beams.We also suggest that the excitation of single higher order modes can be used to improve the sensitivity of fiber optic sensors based on evanescent absorption.

1Fig. 3 .
Fig. 3. Fraction of the power coupled to core ( ∑ m | b m | 2 , solid blue line) and cladding modes ( ∑ k | d k | 2 , dotted red line) of MMF2 as a function of the length of MMF1.Total power coupled to MMF2 ( ∑ m | bm | 2 + ∑ k | d k | 2 ,dotted black line) is also shown.

Fig. 4 .
Fig.4.Power coupling coefficients for the radial modes of a 50µm core MMF2 for different lengths of MMF1 (105µm core) with corresponding normalized radial launch field.Dimensions of x-axis in inset is in µm.

Fig. 5 .
Fig.5.(a) Propagation of field in SMm structure with MMF1 core diameter and length of 99µm and 10881µm, respectively.MMF2 core diameter is 30µm.Cladding diameter of MMF1 and MMF2 are 125µm.MMF2 is surrounded by a medium with RI of 1.4272.This excites LP 0,2 mode in MMF2.White line shows the refractive index discontinuities in the structure.R is the length in radial direction.(b) Power that would couple to the LP 0,2 mode of a 30µm core fiber, as a fraction of local propagating power (dashed blue line) and input power (solid green line) for various lengths (z) in the structure.For z > 10881µm, this defines the propagating power in LP 0,2 mode of MMF2.Total power propagating in the structure is also shown (dot-dashed black line).

Fig. 7 .
Fig. 7. Propagation of the field in SMm structure for excitation of LP 0,10 mode in MMF2.Core diameters of MMF1 and MMF2 are 128µm and 64µm.RI of the medium outside MMF1 is 1.0 (air) and MMF2 is 1.4272.Cladding diameter of MMF2 in (a) is 150µm leading to leakage of power into the outer medium and in (b) is 250µm leading to propagation of the mode in the core.

Fig. 8 .
Fig. 8. Propagation of field in SMm structure for excitation of LP 0,10 mode in MMF2.Core diameters of MMF1 and MMF2 are 128µm and 64µm with cladding diameter of 150µm.RI of the medium outside MMF1 is 1.0 (air).(a) shows the propagation of LP 0,10 mode in MMF2 with outer medium RI of 1.4270.(b) shows the propagation of LP 0,10 mode with outer medium RI of 1.0 (air).In this case, cladding modes are removed by exposing 5 cm of length of MMF2 (31641µm − 81641µm) to an outside medium with an RI value equal to that of the cladding.

Fig. 9 .Fig. 10 .
Fig. 9. Scheme to use SMm structure in high power fiber lasers and amplifiers.

1 2 Fig. 11 .
Fig. 11.Change in γ factor with change in length of MMF1 for the cases listed in table 1.

Fig. 12 .−
Fig.12.Change in γ factor with change in length of MMF1 for the lower order modes (LP 0,2 and LP 0,3 ) presented in table 2.

Table 1 .
Fraction of power in each mode of MMF2 for different lengths and core diameters of MMF1.LP 0,1 and LP 0,2 modes are selected in MMF2.

Table 2 .
Selected lower and higher order radial core modes of large core diameter MMF2 for different lengths and core diameters of MMF1.

Table 3 .
Length of MMF1 required to excite near single LP 0,n mode with finite cladding diameter and outside medium as air.