Abstract

We present a method for designing optical fibers that support field-flattened, ring-like higher order modes, and show that the effective and group indices of its modes can be tuned by adjusting the widths of the guide’s field-flattened layers or the average index of certain groups of layers. The approach provides a path to fibers that have simultaneously large mode areas and large separations between the propagation constants of their modes.

© 2013 OSA

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References

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  1. J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett.32(7), 748–750 (2007).
    [CrossRef] [PubMed]
  2. S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
    [CrossRef]
  3. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A17(4), 1448–1453 (1978).
    [CrossRef]
  4. A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE3666, 40–44 (1999).
    [CrossRef]
  5. J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
    [CrossRef]
  6. W. Torruellas, Y. Chen, B. McIntosh, J. Farroni, K. Tankala, S. Webster, D. Hagan, M. J. Soileau, M. Messerly, and J. Dawson. “High peak power ytterbium-doped fiber amplifiers,” Proc SPIE 6102, 61020N (2006).
  7. B. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE6453, 645307, 645307-9 (2007).
    [CrossRef]
  8. C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008).
    [CrossRef]
  9. N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys.20(2), 304–310 (2010).
    [CrossRef]
  10. D. Jain, C. Baskiotis, and J. K. Sahu, “Mode area scaling with multi-trench rod-type fibers,” Opt. Express21(2), 1448–1455 (2013).
    [CrossRef] [PubMed]
  11. A. Yariv, Optical Electronics, 3rd Edition, (Holt, Rinehart and Winston, 1985).
  12. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  13. A. W. Snyder and J. D. Love, Optical Waveguide Theory p.644 (Chapman and Hall Ltd, 1983).

2013 (1)

2010 (1)

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys.20(2), 304–310 (2010).
[CrossRef]

2008 (2)

C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008).
[CrossRef]

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

2007 (2)

J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett.32(7), 748–750 (2007).
[CrossRef] [PubMed]

B. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE6453, 645307, 645307-9 (2007).
[CrossRef]

2004 (1)

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

1999 (1)

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE3666, 40–44 (1999).
[CrossRef]

1978 (1)

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A17(4), 1448–1453 (1978).
[CrossRef]

Barty, C. P. J.

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

Baskiotis, C.

Beach, R.

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

Culpepper, M.

B. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE6453, 645307, 645307-9 (2007).
[CrossRef]

Dawson, J. W.

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

Elkin, N. N.

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys.20(2), 304–310 (2010).
[CrossRef]

Fan, D.

C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008).
[CrossRef]

Fini, J. M.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett.32(7), 748–750 (2007).
[CrossRef] [PubMed]

Ghalmi, S.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Ghatak, A. K.

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE3666, 40–44 (1999).
[CrossRef]

Goyal, I. C.

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE3666, 40–44 (1999).
[CrossRef]

Jain, D.

Jindal, R.

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE3666, 40–44 (1999).
[CrossRef]

Jovanovic, I.

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

Liao, Z.

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

Lin, C.

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A17(4), 1448–1453 (1978).
[CrossRef]

Mermelstein, M.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Napartovich, A. P.

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys.20(2), 304–310 (2010).
[CrossRef]

Nicholson, J. W.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Payne, S. A.

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

Ramachandran, S.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett.32(7), 748–750 (2007).
[CrossRef] [PubMed]

Robin, C.

B. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE6453, 645307, 645307-9 (2007).
[CrossRef]

Sahu, J. K.

Shen, L.

C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008).
[CrossRef]

Stolen, R. H.

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A17(4), 1448–1453 (1978).
[CrossRef]

Tang, Z.

C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008).
[CrossRef]

Troshchieva, V. N.

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys.20(2), 304–310 (2010).
[CrossRef]

Vysotsky, D. V.

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys.20(2), 304–310 (2010).
[CrossRef]

Ward, B.

B. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE6453, 645307, 645307-9 (2007).
[CrossRef]

Wattellier, B.

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

Yan, M. F.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Ye, Y.

C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008).
[CrossRef]

Zhao, C.

C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008).
[CrossRef]

Laser Photonics Rev. (1)

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008).
[CrossRef]

Laser Phys. (1)

N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys.20(2), 304–310 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (1)

C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008).
[CrossRef]

Phys. Rev. A (1)

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A17(4), 1448–1453 (1978).
[CrossRef]

Proc. SPIE (3)

A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE3666, 40–44 (1999).
[CrossRef]

J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004).
[CrossRef]

B. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE6453, 645307, 645307-9 (2007).
[CrossRef]

Other (4)

W. Torruellas, Y. Chen, B. McIntosh, J. Farroni, K. Tankala, S. Webster, D. Hagan, M. J. Soileau, M. Messerly, and J. Dawson. “High peak power ytterbium-doped fiber amplifiers,” Proc SPIE 6102, 61020N (2006).

A. Yariv, Optical Electronics, 3rd Edition, (Holt, Rinehart and Winston, 1985).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

A. W. Snyder and J. D. Love, Optical Waveguide Theory p.644 (Chapman and Hall Ltd, 1983).

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Figures (8)

Fig. 1
Fig. 1

Index profiles of portions of a flattened waveguide, illustrating half-wave stitching. The black curves represent scaled index, the grey curves represent the field of the flattened mode (arbitrary units), and the tables list the scaled thickness and index of each group’s layers. The left edge of the first layer of each group is placed at v = 0.5π and on both sides of all groups η = 1. On the left sides of all groups ψ = 1 and ψ´ = 0; on the right side of (a), ψ = −0.78 and ψ´ = 0, and on the right sides of (b) and (c), ψ = −1 and ψ´ = 0. All quantities are dimensionless.

Fig. 2
Fig. 2

Index profiles of portions of a flattened waveguide, illustrating full-wave stitching. The black curves represent scaled index, the grey curves represent the corresponding field of the flattened mode (arbitrary units), and the tables list the scaled thickness and index of each group’s layers. The left edge of the first layer of each group is placed at v = 0.5π and on both sides of all groups η = 1. On the left sides of all groups ψ = 1 and ψ´ = 0; on the right side of (a), ψ = 0.66 and ψ´ = 0, and on the right sides of (b) and (c), ψ = 1 and ψ´ = 0. The minimum fields for examples (a), (b) and (c) are −0.78, −0.78, and −0.71, respectively. All quantities are dimensionless.

Fig. 3
Fig. 3

Index profiles of portions of a flattened waveguide, illustrating fractional wave stitching. The black curves represent scaled index, the grey curves represent the field of the flattened mode (arbitrary units), and the tables list the scaled thickness and index of each group’s layers. The left edge of the first layer of each group is placed at v = 0.5π and on the right side of a) and both sides of (b) and (c), η = 1. On the left sides of (b) and (c), ψ = 1 and ψ´ = 0; on the right side of all groups, ψ = 1 and ψ´ = 0. All quantities are dimensionless.

Fig. 4
Fig. 4

Termination layers applied to three simple flattened modes. The black curves represent scaled index and the grey curves represent the scaled field. In (a) the scaled area is 47 (converted to effective area via Eq. (5)) and the scaled peak field is 1/√41 (converted to field via Eq. (6)); in (b) the scaled area is 46 and the scaled peak field is 1/√42; and in (c) the scaled area is 53 and the scaled peak field is 1/√46. All quantities are dimensionless.

Fig. 5
Fig. 5

Line-outs of the scaled index (dark lines) and field (grey lines) for the three designs; a), b), and c) correspond to Designs A, B, and C. All quantities are dimensionless.

Fig. 6
Fig. 6

Field (not irradiance) distributions for the LP03 and LP13 modes of the three example designs – two flattened-mode fibers and a step index fiber. The colors blue and red designate positive and negative polarities of the field and the depth of the color designates its relative amplitude. All figures are scaled as the one on the left, and all quantities are dimensionless.

Fig. 7
Fig. 7

Plots (a) through (c) show, as a function of the azimuthal order l, the size-spacing products for the effective indices of the modes of the three designs (Θeff is defined in Eq. (59)). The red circles designate the LP03 mode, which for A and B is the flattened mode and the red arrow in (b) highlights the relatively large spacing that has been created between the LP12 and LP22 modes of design B. For all of a design’s modes, the value of Aeff used to calculate its size-spacing products is the area of that design’s LP03 mode. The legend adjacent to (c) applies to all figures, and all quantities are dimensionless.

Fig. 8
Fig. 8

Plots (a) through (c) show, as a function of the azimuthal order l, the size-spacing products for the effective indices of the modes of the three designs (Θg is defined in Eq. (62)). The red circles designate the flattened mode, which for A and B is the flattened mode, and the red arrow in (b) highlights the significant increase in the spacing between the group velocities of the LP12 and LP22 modes by design B. For all of a design’s modes, the value of Aeff used to calculate its size-spacing products is the area of that design’s LP03 mode. The legend adjacent to (c) applies to all figures, and all quantities are dimensionless.

Tables (1)

Tables Icon

Table 1 Parameters for two three-ringed flattened mode designs (A and B) and a step-index design (C). All quantities are dimensionless.

Equations (62)

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N A flat = n flat 2 n clad 2
v= 2π λ rN A flat
η( v )= [ n 2 ( v ) n clad 2 ] / N A flat 2
η = group η i A i / group A i
A eff = ( λ/ 2π ) 2 N A flat 2 A eff scaled
ψ= 2π λ N A flat P 0 1 2 ψ scaled
ψ'= ψ / r
lim v 0 ( Δv η1 )=mπ
{ 2 r 2 + 1 r r l 2 r 2 + ( 2π λ ) 2 [ n 2 ( r ) n eff 2 ] }ψ( r )=0
v= 2π λ r n flat 2 n clad 2
η= n 2 ( v ) n clad 2 n flat 2 n clad 2
η eff = n eff 2 n clad 2 n flat 2 n clad 2
{ 2 v 2 + 1 v v l 2 v 2 +η( v ) η eff }ψ( v )=0
ζ=r ψ r =v ψ v
ψ( x )=A J l ( x )+B Y l ( x )     ( n> n eff )
x=v | η η eff |
A= π 2 [ x 1 Y l '( x 1 ) ψ 1 Y l ( x 1 ) ζ 1 ]
B= π 2 [ x 1 J l '( x 1 ) ψ 1 + J l ( x 1 ) ζ 1 ]
J l ( x )x Y l '( x )x J l '( x ) Y l ( x )=2/π
x J l '( x )=l J l ( x )x J l+1 ( x )
x Y l '( x )=l Y l ( x )x Y l+1 ( x )
ψ( x )=A I l ( x )+B K l ( x )     ( n< n eff )
A= x 1 K l '( x 1 ) ψ 1 + K l ( x 1 ) ζ 1
B= x 1 I l '( x 1 ) ψ 1 I l ( x 1 ) ζ 1
K l ( x )x I l '( x )x K l '( x ) I l ( x )=1
x I l '( x )=l I l ( x )+x I l+1 ( x )
x K l '( x )=l K l ( x )x K l+1 ( x )
{ 2 v 2 + 1 v v l 2 v 2 }ψ( v )=0
ψ=A v +l +B v l           ( n= n eff ,l 0 )
A= v 1 l 2l ( l ψ 1 + ζ 1 )
B= v 1 l 2l ( l ψ 1 ζ 1 )
ψ=A+Bln( v )        ( n= n eff ,l= 0 )
A= ψ 1 ζ 1 ln( v 1 )
B= ζ 1
ψ=A J l ( x )      ( η> η eff )
ψ=A I l ( x )      ( η< η eff )
ψ=A v l      ( η= η eff , l0 )
ψ=A      ( η= η eff , l=0 )
ψ=A K l ( x )
[ ψ 2 ζ 2 ]=M[ ψ 1 ζ 1 ]
M= m 1 ( x 2 )m( x 1 )
m( x )= π 2 [ x Y l '( x ) Y l ( x ) x J l '( x ) J l ( x ) ]
m( x )=[ x I l '( x ) I l ( x ) x K l '( x ) K l ( x ) ]
m( x )= 1 2 [ v l 1/ l v l l v l v l ]
m( x )=[ 1 ln( v ) 0 1 ]
M[ 1 Ω core ]=( const )[ 1 Ω clad ]
Ω=ζ/ψ
Ω clad = x K l '( x ) K l ( x ) | x= x clad
2π ( const ) 2 0 ψ 2 rdr = P 0
ψ 2 = ( 2π λ ) 2 ( n flat 2 n clad 2 ) P 0 ψ scaled 2
2π ( const ) 2 0 ψ scaled 2 vdv= 1
2π ψ 2 vdv =π ζ 2 l 2 ψ 2 η η eff +π v 2 ψ 2
2π ψ 2 vdv = π v 2 2 ( ζ 2 2ψζ )+π v 2 ψ 2
2π ψ 2 vdv =π ( v 2 ) 2 [ 1 2 ( ζ+ψ ) 2 +2 ( ζψ ) 2 ln( v )2( ζ 2 + ψ 2 ) ]+π v 2 ψ 2
2π ψ 2 vdv =π ( v 2l ) 2 [ 1 l+1 ( ζ+lψ ) 2 1 l1 ( ζlψ ) 2 2( ζ 2 + l 2 ψ 2 ) ]+π v 2 ψ 2
π [ v 2 ψ 2 ] 0 v 1 +π [ v 2 ψ 2 ] v 1 v 2 +...+π [ v 2 ψ 2 ] v clad
A eff =2π ( ψ 2 rdr ) 2 ψ 4 rdr = ( λ/ 2π ) 2 n flat 2 n clad 2 A eff scaled
A eff scaled =2π ( ψ 2 vdv ) 2 ψ 4 vdv
Θ eff = η eff A eff scaled = A eff λ 2 ( n eff 2 n clad 2 )
n eff n g n clad 2 n flat 2 n clad 2 = η ψ 2 vdv ψ 2 vdv
Θ eff,g = A eff λ 2 ( n eff n g n clad 2 )
Θ g = A eff λ 2 ( n g 2 n clad 2 )=2 Θ eff,g Θ eff + λ 2 n eff 2 A eff ( Θ eff,g Θ eff ) 2

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