**4. Higher-order-mode fibers**

layer in Figure 16(a). A 3.5m fiber coiled in 53cm diameter was used to demonstrate a slope efficiency of 75% in an amplifier [37]. A single stage gain of 33dB was demonstrated using this

The PM LCF (see Figure 16(b)) had a core diameter of 80μm and had a fluorine-doped pump cladding, providing a pump NA of ~0.28. Low index features with an inner layer d/Λ of 0.8 and an outer layer d/Λ of 0.7 were used. This active PM LCF had a pump guide diameter of ~400μm (flat-to-flat), a fiber outer diameter of ~835μm, and was coated with standard acrylic coating. Pump absorption was estimated to be ~12dB/m. The mode field diameter was measured to be ~62μm. The fiber was used as an amplifier in a single coil 76cm in diameter with a length of straight section at each end, demonstrating a slope efficiency of ~74% and a maximum single-path gain in excess of 30dB [37]. It demonstrated direct amplification of 14.2ps pulses to 190kW peak power with pulse energy of 2.74μJ and negligible SPM spectral

was measured to be below 1.35 for the entire output power range.

**Figure 17.** (a) The ytterbium-doped LCF with an index depression in the core center, (b) measured near field intensity of the guided mode, (c) measured mode intensity distribution, (d) simulated effective mode area versus the index de‐

Recently, a flat-top mode has been demonstrated in an ytterbium-doped LCF with a ~50μm core by introducing an area ~30μm in diameter in the core center with a refractive index of ~2×10-4 lower than that of the background glass (Figure 17(a)) [40]. The flat-top mode (see Figure 17(b) and (c)) increased the effective mode area of the LCF from ~1200μm2 to

limited efficiency (see Figure 17(e)). Lasing wavelength was 1026nm and the pump was at

, a ~50% increase (see Figure 17(d)). The LCF also demonstrated near quantum-

pression, and (d) measured laser output and near field patters at various powers [40].

fiber. It was also used to directly amplify 15ps pulses to a peak power of ~1MW.

238 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

broadening. M2

~1900μm2

976nm.

The concept and demonstration of robust propagation of *higher-order-modes* (HOM) with large effective mode areas were first reported in 2006 [11]. It was argued that the stability of mode propagation in multimode fibers is critically dependent on the effective mode index difference between the propagating mode and its nearest neighbor anti-symmetric mode. This effective mode index difference is actually larger for the higher-order LP04-LP07 modes than for the fundamental mode (see Figure 18). In addition, a *long period grating* (LPG), a fairly matured technology, can be used for broad band mode conversion to and from those higher-order LP0n modes.

**Figure 18.** Effective index difference between nearest neighbor anti-symmetric mode versus Aeff. Top, near-field images of LPG-excited HOMs after >2m propagation with 7cm bend and with Aeff ranging from 2100 to 3200μm2 [11].

Mode coupling first requires phase-matching. Perturbations on the fiber need to provide a vector which equals the difference in the propagation constants of the two modes involved. The larger the mode index difference between the two modes, the larger is the wave vector required for phase matching. A large wave vector implies high spatial frequency and small spatial period. For a mode index difference of 1.5×10-4, the required spatial period of the perturbation at 1.55μm is ~10mm. Due to geometric constraints, the spatial frequency distri‐ bution of perturbations on fibers usually cuts off at a certain upper frequency limit. This would minimize mode coupling between modes with larger effective mode index differences. The second requirement for mode coupling is that the overlap integral among the two modes and the perturbation needs to be non-zero. This requires the perturbation to break up the mode orthogonality. Although it was not explicitly spelled out in [11], it was assumed that the perturbations on fibers are mostly anti-symmetric. In this case, mode coupling dominates between LP0n and its anti-symmetric counterpart LP1n modes in optical fibers.

**Figure 19.** Characteristics of an HOM fiber. The horizontal and vertical scales of the images are identical. (a) Near-field image of a fiber facet, showing the 86μm inner cladding. (b) Refractive-index profile of the HOM fiber, with a core similar to an SMF with an 86μm inner cladding and a down-doped outer trench. (c) Near-field image of the LP07 mode at 1600 nm after 12m propagation with a 4.5cm radius bend. (d) Intensity line scan of (c) and theoretical profile: Aeff=~2100μm2 [11].

The effective mode index difference for LP01, LP04, LP05, LP06 and LP07 modes with their nearest neighbor anti-symmetric modes are shown versus effective mode area in Figure 18 [11]. The near-field images of the measured HOM modes are shown at the top of in Figure 18 and in Figure 19. It is clear that the effective mode index difference is significantly larger for the LP04- LP07 modes than for the LP01 mode with the same effective mode area. The effective mode index difference also increases slightly for higher order LP0n modes with the same effective mode area. For the same fiber, the effective mode area is actually larger for lower-mode-order LP0n modes. The schematic of the proposed system is shown in Figure 20(a), where two identical LPGs are required for mode conversion from the LP01 mode to the LP07 mode and back to the LP01 mode. Robust propagation of the LP07 mode with an effective mode area of 2070μm2 was demonstrated at 1600nm with the arrangement shown in Figure 19(c). The performance of the mode converter with high efficiency over broad bandwidth is shown in Figure 20(b). It was found that the LP07 mode suffered negligible bend loss at coil diameters down to 12cm. It was also found that modal stability increases with mode order. In a second paper [12], the effective mode areas of LP07 mode was simulated at various bend radii, showing stronger bend resistance than both LP03 and LP01 modes.

Recently, an erbium-doped HOM amplifier was demonstrated [41]. A small inner core was designed to have a LP01 mode MFD of 9μm, allowing for effective excitation of the LP01 mode when the HOM fiber was spliced to a single-mode fiber. Higher-order modes expanded to occupy the outer core. The LP010 mode had an effective mode area of 2700μm2 . Both the inner and the outer core were doped with erbium with absorption of ~30dB/m at 1530nm. The amplifier was both seeded at 1564nm and pumped at 1480nm in the LP010 mode. The pump was a Raman fiber laser.

minimize mode coupling between modes with larger effective mode index differences. The second requirement for mode coupling is that the overlap integral among the two modes and the perturbation needs to be non-zero. This requires the perturbation to break up the mode orthogonality. Although it was not explicitly spelled out in [11], it was assumed that the perturbations on fibers are mostly anti-symmetric. In this case, mode coupling dominates

**Figure 19.** Characteristics of an HOM fiber. The horizontal and vertical scales of the images are identical. (a) Near-field image of a fiber facet, showing the 86μm inner cladding. (b) Refractive-index profile of the HOM fiber, with a core similar to an SMF with an 86μm inner cladding and a down-doped outer trench. (c) Near-field image of the LP07 mode at 1600 nm after 12m propagation with a 4.5cm radius bend. (d) Intensity line scan of (c) and theoretical profile:

The effective mode index difference for LP01, LP04, LP05, LP06 and LP07 modes with their nearest neighbor anti-symmetric modes are shown versus effective mode area in Figure 18 [11]. The near-field images of the measured HOM modes are shown at the top of in Figure 18 and in Figure 19. It is clear that the effective mode index difference is significantly larger for the LP04- LP07 modes than for the LP01 mode with the same effective mode area. The effective mode index difference also increases slightly for higher order LP0n modes with the same effective mode area. For the same fiber, the effective mode area is actually larger for lower-mode-order LP0n modes. The schematic of the proposed system is shown in Figure 20(a), where two identical LPGs are required for mode conversion from the LP01 mode to the LP07 mode and back to the LP01 mode. Robust propagation of the LP07 mode with an effective mode area of 2070μm2

demonstrated at 1600nm with the arrangement shown in Figure 19(c). The performance of the mode converter with high efficiency over broad bandwidth is shown in Figure 20(b). It was found that the LP07 mode suffered negligible bend loss at coil diameters down to 12cm. It was also found that modal stability increases with mode order. In a second paper [12], the effective mode areas of LP07 mode was simulated at various bend radii, showing stronger bend

Recently, an erbium-doped HOM amplifier was demonstrated [41]. A small inner core was designed to have a LP01 mode MFD of 9μm, allowing for effective excitation of the LP01 mode when the HOM fiber was spliced to a single-mode fiber. Higher-order modes expanded to

and the outer core were doped with erbium with absorption of ~30dB/m at 1530nm. The amplifier was both seeded at 1564nm and pumped at 1480nm in the LP010 mode. The pump

occupy the outer core. The LP010 mode had an effective mode area of 2700μm2

was

. Both the inner

between LP0n and its anti-symmetric counterpart LP1n modes in optical fibers.

240 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

Aeff=~2100μm2

[11].

resistance than both LP03 and LP01 modes.

was a Raman fiber laser.

**Figure 20.** (a) Device schematic: light is coupled into and out of HOM with LPGs whose conversion efficiency being shown in (b). LPG with broadband coupling with efficiency >99% over 94 nm and with peak coupling efficiency as high as 99.93%. (c) Alternative schematic for characterizing HOM fibers: the cleave serves to fold the device propaga‐ tion path so that the single LPG acts as both the input and the output LPG. X, splice; OSA, optical spectrum analyzer [11].

The difference in effective index between nearest neighbor anti-symmetric modes at λ=1564 nm as a function of their effective area is plotted as points in Figure 21(a) for the LP01 through LP010 modes in the fiber. The LP02 and LP<sup>03</sup> modes have a large Aeff, but small mode spacing. As the mode order increases, mode spacing increases too, while Aeff decreases. Figure 21(b) shows the calculated intensity profiles at λ=1564nm for the LP01 and LP010 mode.

**Figure 21.** (a) Mode spacing as a function of effective area for the LP0n modes in the HOM fiber (points) compared to a conventional LP01 step-index fiber with V=5 (solid curve). (b) Intensity profiles of the LP01 and LP010 modes. These cal‐ culations were done at a wavelength of 1564 nm.

A narrow line width, external cavity laser was amplified to 50mW and combined with the high-power, single mode Raman fiber laser at 1480 nm in a single-mode pump/signal combin‐ er. The output of the pump/signal combiner was fusion-spliced to the HOM fiber. The length of the amplifier fiber after the LPG was 2.68 m. The measured slope efficiency at 1564nm was 43.2%. Over 20dB of gain was demonstrated by the amplifier.

#### **5. Chrially-coupled core fibers** Figure 21 (a) Mode spacing as a function of effective area for the LP0n modes in the HOM fiber (points) compared to a conventional LP01 step‐index fiber with V = 5 (solid curve). (b) Intensity profiles of the LP01 and LP010 modes.

area of 2700μm2

Raman fiber laser.

and LP010 mode.

A C*hirally-coupled-core* (CCC) fiber was first reported in 2007 [9]. The fiber had a large central core and a smaller side core wound around the central core in a helical fashion (Figure 22). The preform had two parallel cores and the fiber was spun during the draw to form the helical side core. In this first report, the central core had a diameter of 35μm and a NA of 0.07. The side core had a diameter of 12μm and NA of 0.09. Edge-to-edge core separation was 2μm. The helical pitch was 6.2mm. The fiber was measured to be single-mode at 1550nm over a short length of 25cm. It was multimode below 1500nm. The simulation predicted LP01 mode loss to be 0.3dB/m and all HOM loss to be >130dB/m for λ>1550nm. The fiber was also confirmed to be polarization-maintaining. A narrow line width, external cavity laser was amplified to 50mW and combined with the high‐power, single mode Raman fiber laser at 1480 nm in a single‐mode pump/signal combiner. The output of the pump/signal combiner was fusion‐spliced to the HOM fiber. The length of the amplifier fiber after the LPG was 2.68 m. The measured slope efficiency at 1564nm was 43.2%. Over 20dB of gain was demonstrated by the amplifier. 5 Chrially‐coupled core fibers A C*hirally‐coupled‐core* (CCC) fiber was first reported in 2007 [9]. The fiber had a large central core and a smaller side core wound around the central core in a helical fashion (Figure 22). The preform had two parallel cores and the fiber was spun during the draw to form the helical side core. In this first report, the central core had a diameter of 35µm and a NA of 0.07. The side core had a diameter of 12µm and NA of 0.09. Edge‐to‐edge core separation was 2µm. The helical pitch was 6.2mm. The fiber was measured to be single‐mode at 1550nm over a

These calculations were done at a wavelength of 1564 nm.

single‐mode fiber. Higher‐order modes expanded to occupy the outer core. The LP010 mode had an effective mode

1530nm. The amplifier was both seeded at 1564nm and pumped at 1480nm in the LP010 mode. The pump was a

The difference in effective index between nearest neighbor anti‐symmetric modes at λ = 1564 nm as a function of their effective area is plotted as points in Figure 21(a) for the LP01 through LP010 modes in the fiber. The LP02 and LP03 modes have a large Aeff, but small mode spacing. As the mode order increases, mode spacing increases too, while Aeff decreases. Figure 21(b) shows the calculated intensity profiles at λ =1564nm for the LP01

. Both the inner and the outer core were doped with erbium with absorption of ~30dB/m at

Figure 22 Structure of Chirally‐coupled‐core fiber [9].

short length of 25cm. It was multimode below 1500nm. The simulation predicted LP01 mode loss to be 0.3dB/m and all HOM loss to be >130dB/m for >1550nm. The fiber was also confirmed to be polarization‐maintaining.

The propagation of modes in the central core was affected by the coupling of modes between the central and **Figure 22.** Structure of Chirally-coupled-core fiber [9].

side cores. The fiber was designed to operate where there was no fundamental mode coupling with modes in the side core. Higher‐order modes in the central core were, however, coupled with the side core modes at the operating wavelength. The modes in the side core had high loss due to the tight bend from the helical The propagation of modes in the central core was affected by the coupling of modes between the central and side cores. The fiber was designed to operate where there was no fundamental mode coupling with modes in the side core. Higher-order modes in the central core were, however, coupled with the side core modes at the operating wavelength. The modes in the side core had high loss due to the tight bend from the helical arrangement. This led to high loss for the higher-order modes in the central core which were coupled to modes in the side core.

An ytterbium-doped CCC fiber was demonstrated in a subsequent paper [10]. The ytterbiumdoped central core had a 33μm diameter and 0.06 NA. The undoped side core had a 16μm diameter and 0.1 NA. The helical pitch was 7.4mm and the edge-to-edge core separation was 4μm. The low index coating provided a pump NA of 0.47. The pump guide had a 250μm diameter. The measured pump absorption was 2dB/m at 915nm. The fiber demonstrated 75% slope efficiency at 1066nm in a laser configuration.

In a more recent paper [42], a more detailed theoretical analysis of quasi-phase-matching (QPM) assisted by spin and orbital angular momentum was given. For two LP modes LP*l1m1* and LP*l2m2*, QPM is achieved when

$$
\beta\_{l1\,m1} \cdot \beta\_{l2\,m2} \sqrt{1 + K^2 R^2} \cdot \Delta mK = 0 \tag{2}
$$

arrangement. This led to high loss for the higher‐order modes in the central core which were coupled to modes in

core had a 33µm diameter and 0.06 NA. The undoped side core had a 16µm diameter and 0.1 NA. The helical pitch was 7.4mm and the edge‐to‐edge core separation was 4µm. The low index coating provided a pump NA of 0.47.

demonstrated 75% slope efficiency at 1066nm in a laser configuration.

An ytterbium‐doped CCC fiber was demonstrated in a subsequent paper [10]. The ytterbium‐doped central

**5. Chrially-coupled core fibers**

area of 2700μm2

Raman fiber laser.

and LP010 mode.

be polarization-maintaining.

**Figure 22.** Structure of Chirally-coupled-core fiber [9].

slope efficiency at 1066nm in a laser configuration.

βl1m1 βl2m<sup>2</sup> 1 + K2 R<sup>2</sup> - ΔmK

and LP*l2m2*, QPM is achieved when

5 Chrially‐coupled core fibers

core.

A C*hirally-coupled-core* (CCC) fiber was first reported in 2007 [9]. The fiber had a large central core and a smaller side core wound around the central core in a helical fashion (Figure 22). The preform had two parallel cores and the fiber was spun during the draw to form the helical side core. In this first report, the central core had a diameter of 35μm and a NA of 0.07. The side core had a diameter of 12μm and NA of 0.09. Edge-to-edge core separation was 2μm. The helical pitch was 6.2mm. The fiber was measured to be single-mode at 1550nm over a short length of 25cm. It was multimode below 1500nm. The simulation predicted LP01 mode loss to be 0.3dB/m and all HOM loss to be >130dB/m for λ>1550nm. The fiber was also confirmed to

A C*hirally‐coupled‐core* (CCC) fiber was first reported in 2007 [9]. The fiber had a large central core and a smaller side core wound around the central core in a helical fashion (Figure 22). The preform had two parallel cores and the fiber was spun during the draw to form the helical side core. In this first report, the central core had a diameter of 35µm and a NA of 0.07. The side core had a diameter of 12µm and NA of 0.09. Edge‐to‐edge core separation was 2µm. The helical pitch was 6.2mm. The fiber was measured to be single‐mode at 1550nm over a short length of 25cm. It was multimode below 1500nm. The simulation predicted LP01 mode loss to be 0.3dB/m and all HOM loss to be >130dB/m for >1550nm. The fiber was also confirmed to be polarization‐maintaining.

Figure 22 Structure of Chirally‐coupled‐core fiber [9]. The propagation of modes in the central core was affected by the coupling of modes between the central and side cores. The fiber was designed to operate where there was no fundamental mode coupling with modes in the side core. Higher‐order modes in the central core were, however, coupled with the side core modes at the operating wavelength. The modes in the side core had high loss due to the tight bend from the helical

The propagation of modes in the central core was affected by the coupling of modes between the central and side cores. The fiber was designed to operate where there was no fundamental mode coupling with modes in the side core. Higher-order modes in the central core were, however, coupled with the side core modes at the operating wavelength. The modes in the side core had high loss due to the tight bend from the helical arrangement. This led to high loss for the higher-order modes in the central core which were coupled to modes in the side

An ytterbium-doped CCC fiber was demonstrated in a subsequent paper [10]. The ytterbiumdoped central core had a 33μm diameter and 0.06 NA. The undoped side core had a 16μm diameter and 0.1 NA. The helical pitch was 7.4mm and the edge-to-edge core separation was 4μm. The low index coating provided a pump NA of 0.47. The pump guide had a 250μm diameter. The measured pump absorption was 2dB/m at 915nm. The fiber demonstrated 75%

In a more recent paper [42], a more detailed theoretical analysis of quasi-phase-matching (QPM) assisted by spin and orbital angular momentum was given. For two LP modes LP*l1m1*

=0 (2)

*m=l+s*; 

*l=±l1±l2*;

measured slope efficiency at 1564nm was 43.2%. Over 20dB of gain was demonstrated by the amplifier.

242 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

Figure 21 (a) Mode spacing as a function of effective area for the LP0n modes in the HOM fiber (points) compared to a conventional LP01 step‐index fiber with V = 5 (solid curve). (b) Intensity profiles of the LP01 and LP010 modes. These calculations were done at a wavelength of 1564 nm. A narrow line width, external cavity laser was amplified to 50mW and combined with the high‐power, single mode Raman fiber laser at 1480 nm in a single‐mode pump/signal combiner. The output of the pump/signal combiner was fusion‐spliced to the HOM fiber. The length of the amplifier fiber after the LPG was 2.68 m. The

single‐mode fiber. Higher‐order modes expanded to occupy the outer core. The LP010 mode had an effective mode

1530nm. The amplifier was both seeded at 1564nm and pumped at 1480nm in the LP010 mode. The pump was a

The difference in effective index between nearest neighbor anti‐symmetric modes at λ = 1564 nm as a function of their effective area is plotted as points in Figure 21(a) for the LP01 through LP010 modes in the fiber. The LP02 and LP03 modes have a large Aeff, but small mode spacing. As the mode order increases, mode spacing increases too, while Aeff decreases. Figure 21(b) shows the calculated intensity profiles at λ =1564nm for the LP01

. Both the inner and the outer core were doped with erbium with absorption of ~30dB/m at

the side core.

sample. (a) Calculated loss for side‐core LP11 and LP21 modes as a function of wavelength (top). (b) Calculated refractive indices of interacting modes and calculated QPM resonance positions. (c) Transmission of central core. (d) Simulated transmission of central core. **Figure 23.** Calculation and measurement of quasi-phase-matching (QPM) for 1.5m-long CCC fiber sample. (a) Calcu‐ lated loss for side-core LP11 and LP21 modes as a function of wavelength (top). (b) Calculated refractive indices of inter‐ acting modes and calculated QPM resonance positions. (c) Transmission of central core. (d) Simulated transmission of central core.

spin and orbital angular momentum was given. For two LP modes LP*l1m1* and LP*l2m2*, QPM is achieved when

����� � �����√������ �

where β*l1m1* and β*l1m2* are the respective propagation constants of the two modes; K=2π/; and is helical pitch.

plotted in Figure 23(a) showing high loss for these modes in certain wavelength regimes. The mode indices of the LP11 and LP21 modes of the side core and LP01 mode of the central core are plotted in Figure 23(b). It can be seen clearly that the LP11 mode and LP21 modes of the side core naturally phase‐match to the LP01 mode of the central core at ~1.22µm and ~0.81µm respectively. The QPM by angular momentum from the helical side core extends the

In a more recent paper [42], a more detailed theoretical analysis of quasi‐phase‐matching (QPM) assisted by

*s*=‐2, ‐1, 0, 1, 2. The loss versus wavelength for the LP11 and LP21 modes in the side core is

�� � � (2)

Figure 23 Calculation and measurement of quasi‐phase‐matching (QPM) for 1.5m‐long CCC fiber

where β*l1m1* and β*l1m2* are the respective propagation constants of the two modes; K=2π/Λ; and Λ is helical pitch. *Δm=Δl+Δs*; *Δl=±l1±l2*; *Δs*=-2,-1, 0, 1, 2. The loss versus wavelength for the LP11 and LP21 modes in the side core is plotted in Figure 23(a) showing high loss for these modes in certain wavelength regimes. The mode indices of the LP11 and LP21 modes of the side core and LP01 mode of the central core are plotted in Figure 23(b). It can be seen clearly that the LP11 mode and LP21 modes of the side core naturally phase-match to the LP01 mode of the central core at ~1.22μm and ~0.81μm respectively. The QPM by angular momentum from the helical side core extends the phase matching to a number of other wavelengths determined by equation 2. These phase matching wavelengths are plotted in Figure 23(b) as red dotted vertical lines for the LP11 mode of the side core and the LP01 mode of central core coupling and blue dotted vertical lines for the LP21 mode of side core and LP01 mode of central core coupling. The measured transmission of the central core is plotted in Figure 23(c). The analysis of the CCC fiber can be simplified significantly in a helical coordination. The Maxwell equations keep the same form in the new coordination system, but the tenor form of permittivity and permeability needs to be transformed [42]. The resulting tenor does not have any z-dependence, which significantly simplifies the analysis. The simulated loss of various modes is shown in Figure 23(d), demonstrating that the narrow peaks in the transmission arise from coupling between the LP01 mode in the central core and the LP11 and LP21 modes in the side core.

The high loss of the LP01 mode in the central core at λ>1.3μm was not explained in the paper. It may be due to the angular momentum assisted coupling between LP01 mode in the central core and LP01 mode in the side core. This long wavelength cut-off has been used for the suppression of stimulated Raman scattering in fibers [43].

The higher loss for the higher-order-modes in the central core between 1-1.1μm in Figure 23(d) was not clearly explained either. For the LP11 mode in the central core, one possible reason for its high loss is coupling to LP21 mode of the side core through angular momentum-assisted QPM. Recently, a 60μm core CCC fiber was also demonstrated [44].
