Abstract

The loss resulting from roughness scattering at hole interfaces within solid core photonic crystal fibers is theoretically analyzed and compared with measurements on fabricated fibers. It is found that a model roughness spectrum corresponding to frozen in capillary waves gives results in reasonably good agreement with experiments on small core fibers. In particular, the roughness scattering loss is shown to be only weakly dependent on wavelength. Agreement at a larger core size requires a long length-scale cut-off to be introduced to the roughness spectrum. Due to the long range nature of the roughness correlations, the scattering is non Rayleigh in character and cannot be interpreted in terms of a local photon density of states.

© 2005 Optical Society of America

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References

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  1. J. Schroeder, �??Light scattering in glass,�?? in Treatise on Materials Science and Technology Vol. 12, M. Tomozawa and R. H. Doremus, eds. (Academic Press, 1977).
  2. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight and P. St. J. Russell, �??Ultimate low loss of hollow-core photonic crystal fibers,�?? Opt. Express 13, 236-244 (2005) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236</a>.
    [CrossRef] [PubMed]
  3. B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea, H. Sabert, T. A. Birks, J. C. Knight and P. St. J. Russell, "Low loss (1.7 dB/km) hollow core photonic band gap fibre," in Proc. Optical Fiber Commun. Conf. (Los Angeles, 2004), post-deadline paper PDP24.
  4. P. J. Roberts, F. Couny, T. A. Birks, J. C. Knight, P. St.J. Russell, B. J. Mangan, H. Sabert, D. P. Williams and L. Farr, �??Achieving low loss and low nonlinearity in hollow core photonic crystal fibers,�?? in Proc. CLEO 2005 (Baltimore, 2005), paper CWA7.
  5. J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1485-1496 (2004) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485</a>.
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  6. T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, "Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss," Opt. Express 9, 681-686 (2001) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681</a>.
    [CrossRef] [PubMed]
  7. L. Farr, J. C. Knight, B. J. Mangan and P. J. Roberts, �??Low loss photonic crystal fiber,�?? in European Conference on Optical Communication (Copenhagen, 2002), post-deadline paper PD13.
  8. F. Couny, H. Sabert, P. J. Roberts, D. P. Williams, A. Tomlinson, B. J. Mangan, L. Farr, J. C. Knight, T. A. Birks and P. St. J. Russell, �??Visualization of the photonic band gap in hollow core photonic crystal fibers using side scattering,�?? Opt. Express 13, 558-563 (2005) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-558">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-558</a>.
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  14. S. G. Johnson, M. Ibanescu, M. A. Skorobogatty, O. Weisberg, J. D. Joannopoulos and Y. Fink, �??Perturbation theory for Maxwell�??s equation with shifting material boundaries,�?? Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  15. S. G. Johnson, M. Ibanescu, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, D. Roundy, Y. Fink and J. D. Joannopoulos, �??Anomalous loss and propagation in photonic-crystal waveguides,�?? presented at PECS-VI, Aghia Pelaghia, Crete, June 2005.
  16. S. Hughes, L. Ramunno, J. F. Young and J. E. Sipe, �??Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,�?? Phys. Rev. Lett. 94, 033903 (2005).
    [CrossRef] [PubMed]
  17. F. Seydou, O. Ramahi, R. Duraiswami and T. Seppanen, �??Computation of Green's Function for Finite-Size Photonic Crystals by Boundary Element Method,�?? in IEEE Antennas and Propagation Society Symposium (Monterey, USA, June 20-25, 2004), Vol. 4, pp. 4320-4323.
  18. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke and L. C. Botten, �??Multipole method for microstructured optical fibers: 1. Formulation,�?? J. Opt. Soc. Am. B 19, 2322-2330 (2002).
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  19. D. P. Fussell, R. C. McPhedran, C. M. de Sterke, �??Three-dimensional Green's tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders,�?? Phys. Rev. E 70, 066608 (2004).
    [CrossRef]
  20. R. F. Cregan, J. C. Knight, P. St. J. Russell and P. J. Roberts 1999, �??Distribution of spontaneous emission from an Er3+-doped photonic crystal fiber,�?? J. Lightwave Tech. 17, 2138-2141 (1999).
    [CrossRef]
  21. P. J. Roberts and T. J. Shepherd, �??The guidance properties of multi-core photonic crystal fibres,�?? J. Optics A: Pure and applied 3, S133-S140 (2001).
    [CrossRef]
  22. J.C. Knight, T.A. Birks, P.St.J. Russell and J.G. Rarity, "Bragg scattering from an obliquely illuminated photonic crystal fiber," Appl. Opt. 37, 449-452 (1998).
    [CrossRef]

Appl. Opt. (1)

CLEO 2005 (1)

P. J. Roberts, F. Couny, T. A. Birks, J. C. Knight, P. St.J. Russell, B. J. Mangan, H. Sabert, D. P. Williams and L. Farr, �??Achieving low loss and low nonlinearity in hollow core photonic crystal fibers,�?? in Proc. CLEO 2005 (Baltimore, 2005), paper CWA7.

Conference on Optical Communication (1)

L. Farr, J. C. Knight, B. J. Mangan and P. J. Roberts, �??Low loss photonic crystal fiber,�?? in European Conference on Optical Communication (Copenhagen, 2002), post-deadline paper PD13.

IEEE Antennas and Propagation Soc. Sym. (1)

F. Seydou, O. Ramahi, R. Duraiswami and T. Seppanen, �??Computation of Green's Function for Finite-Size Photonic Crystals by Boundary Element Method,�?? in IEEE Antennas and Propagation Society Symposium (Monterey, USA, June 20-25, 2004), Vol. 4, pp. 4320-4323.

J. Lightwave Tech. (1)

R. F. Cregan, J. C. Knight, P. St. J. Russell and P. J. Roberts 1999, �??Distribution of spontaneous emission from an Er3+-doped photonic crystal fiber,�?? J. Lightwave Tech. 17, 2138-2141 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Optics A: Pure and applied (1)

P. J. Roberts and T. J. Shepherd, �??The guidance properties of multi-core photonic crystal fibres,�?? J. Optics A: Pure and applied 3, S133-S140 (2001).
[CrossRef]

J. Phys. France (1)

J. Meunier, �??Liquids interfaces: role of the fluctuations and analysis of ellipsometry and reflectivity measurements,�?? J. Phys. France 48, 1819-1831 (1987).
[CrossRef]

J. Phys.: Condens. Matter (1)

J. Jäckle and K. Kawasaki, �??Intrinsic roughness of glass surfaces,�?? J. Phys.: Condens. Matter 7, 4351-4358 (1995).
[CrossRef]

Opt. Express (5)

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, "Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss," Opt. Express 9, 681-686 (2001) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681</a>.
[CrossRef] [PubMed]

M. Skorobogatiy, S. A. Jacobs, S. G. Johnson, and Y. Fink, "Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates," Opt. Express 10, 1227-1243 (2002) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227</a>.
[PubMed]

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1485-1496 (2004) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485</a>.
[CrossRef] [PubMed]

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight and P. St. J. Russell, �??Ultimate low loss of hollow-core photonic crystal fibers,�?? Opt. Express 13, 236-244 (2005) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236</a>.
[CrossRef] [PubMed]

F. Couny, H. Sabert, P. J. Roberts, D. P. Williams, A. Tomlinson, B. J. Mangan, L. Farr, J. C. Knight, T. A. Birks and P. St. J. Russell, �??Visualization of the photonic band gap in hollow core photonic crystal fibers using side scattering,�?? Opt. Express 13, 558-563 (2005) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-558">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-558</a>.
[CrossRef] [PubMed]

Optical Fiber Commun. Conf. (1)

B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea, H. Sabert, T. A. Birks, J. C. Knight and P. St. J. Russell, "Low loss (1.7 dB/km) hollow core photonic band gap fibre," in Proc. Optical Fiber Commun. Conf. (Los Angeles, 2004), post-deadline paper PDP24.

PECS-VI (1)

S. G. Johnson, M. Ibanescu, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, D. Roundy, Y. Fink and J. D. Joannopoulos, �??Anomalous loss and propagation in photonic-crystal waveguides,�?? presented at PECS-VI, Aghia Pelaghia, Crete, June 2005.

Phys. Rev. E (2)

D. P. Fussell, R. C. McPhedran, C. M. de Sterke, �??Three-dimensional Green's tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders,�?? Phys. Rev. E 70, 066608 (2004).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatty, O. Weisberg, J. D. Joannopoulos and Y. Fink, �??Perturbation theory for Maxwell�??s equation with shifting material boundaries,�?? Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. Hughes, L. Ramunno, J. F. Young and J. E. Sipe, �??Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,�?? Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Treatise on Materials Science and Tech. (1)

J. Schroeder, �??Light scattering in glass,�?? in Treatise on Materials Science and Technology Vol. 12, M. Tomozawa and R. H. Doremus, eds. (Academic Press, 1977).

Other (2)

L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Oxford: Pergamon, 1970).

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

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Figures (5)

Fig. 1.
Fig. 1.

(a) The measured scattered power as a function of effective index n for a PCF with pitch approximately 2.85 μm and core diameter about 2.8 μm at a wavelength of 1.55 μm. A SEM of the structure are included as an inset. In (b), the calculated scattered power due to SCW roughness is shown for a PCF with pitch 2.85 μm composed of circular holes of diameter 2.74 μm. The wavelength is again 1.55 μm.

Fig 2.
Fig 2.

Calculated angular distribution of power scattered due to surface roughness from a solid core PCF with pitch Λ=2.0 μm and hole diameter 1.96 μm. The wavelength is 1.55 μm.

Fig. 3.
Fig. 3.

The wavelength dependence of roughness loss calculated for a solid core PCF with pitch 1.5μm and circular holes of diameter 1.44 μm. A curve with 1/λ 4 dependence, approximately 50 times the strength of the bulk Rayleigh scattering contribution, is included for comparison.

Fig. 4.
Fig. 4.

The measured loss vs. wavelength for a fabricated PCF with a core diameter of around 1.5 μm. A SEM of the structure is included as an inset. The dashed curve has a dependence of λ -1.24.

Fig. 5.
Fig. 5.

A plot of the complex plane schematically showing the analytic structure of the integrand in Eq. (5). The poles associated with the Green tensor are shown by crosses and those of the roughness function Ψ͂(s,s';β - β 1) by dots. The Green tensor poles due to forward propagating modes 1 and 2 are shown in grey and have been subtracted. The thick continuous lines indicate branch cuts. β cl is the β-value of the cladding cut-off and β g=n g k. The path of integration is deformed from the real β-axis to the dashed curve shown.

Equations (39)

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E = 1 4 A 2 σ k SCW 2 ,
Ψ ͂ ( k SCW ) = k B T g σ 1 k SCW 2
Ψ ͂ m ( β ) = k B T g σ 1 S 1 [ k s ( m ) 2 + β 2 ] ,
h ( s , z ) h ( s + Δ s , z + Δ z ) = 1 2 π m = exp ( i m Δ s ) d β exp ( i β Δ z ) Ψ m ( β ) ,
γ 1 = k 3 4 π ( ε 0 μ 0 ) 1 2 ( n g 2 1 ) 2 Im [ j = 1 N holes
j 'th hole perimeter d s j 'th hole perimeter d s ' d β e ̂ 1 H ( r j ( s ' ) ) G 2 D ( r j + ( s ' ) , r j + ( s ) ; k , β ) e ̂ 1 ( r j ( s ) ) Ψ ͂ j ( s , s ' ; β 1 β ) ] ,
1 2 A d 2 r [ e ̂ 1 ( r ) × h ̂ 1 * ( r ) ] · z ̂ = 1 ,
Ψ ͂ j ( s , s ' ; Δ β ) = k B T g σ 1 S j m = exp [ i k s ( m ) ( s s ' ) ] [ k s ( m ) 2 + Δ β 2 ]
Ψ ͂ ( s , s ' ; Δ β ) = k B T g σ 1 2 Δ β exp ( Δ β ( s s ' ) ) .
β = n m k cos θ ,
Ψ ' ͂ m ( β ) = k B T g σ 1 S 1 [ k s ( m ) 2 + β 2 + k cut 2 ] ,
d b p d z i β p b p = q ζ p C pq b q ,
C pq ( z ) = k 4 ( ε 0 μ 0 ) 1 2 1 Δ β pq x section d 2 r { [ e ̂ p * · e ̂ q ( e ̂ p * · n ̂ ) ( e ̂ q · n ̂ ) ] n 2 z ( n 2 e ̂ p * · n ̂ ) ( n 2 e ̂ q · n ̂ ) z ( 1 n 2 ) } ,
b 1 ( L ) = b 1 ( 0 ) exp ( i β 1 L ) { 1 + q 0 L d z ' z ' L z ' d z d η q ( z d ) C 1 q ( z ' ) C q 1 ( z ' z d ) exp [ i ( β q β 1 ) z d ] } .
γ 1 = 1 L ( 1 b 1 2 ( L ) 2 b 1 ( 0 ) 2 ) = 2 L Re [ q 0 L d z ' z ' d z d η q ( z d ) C 1 q ( z ' ) C q 1 ( z ' z d ) exp [ i ( β q β 1 ) z d ] ] ,
γ 1 = 2 Re [ q d z d η q ( z d ) C 1 q ( 0 ) C q 1 ( z d ) exp [ i ( β q β 1 ) z d ] ] .
n p ( r , z ) z = n 0 p ( r ) r = r + u ( r , z ) · u ( r , z ) z
( n g p 1 ) j = 1 N holes j th hole interface d s δ ( r r j ( s ) ) h j ( s , z ) z ( p = 2 , 2 ) ,
C pq ( z ) = k 4 ( ε 0 μ 0 ) 1 2 1 Δ β pq j = 1 N holes j th hole interface d s { ( n g 2 1 ) [ e ̂ p * · e ̂ q ( e ̂ p * · n ̂ j ( s ) ) ( e ̂ q · n ̂ j ( s ) ) ] ( n g 2 1 ) ( n 0 2 e ̂ p * · n ̂ j ( s ) ) ( n 0 2 e ̂ q · n ̂ j ( s ) ) } h j ( s , z ) z .
{ ( n g 2 1 ) [ e ̂ p * · e ̂ q ( e ̂ p * · n ̂ j ( s ) ) ( e ̂ q · n ̂ j ( s ) ) ] ( n g 2 1 ) ( n 0 2 e ̂ p * · n ̂ j ( s ) ) ( n 0 2 e ̂ q · n ̂ j ( s ) ) } = ( n g 2 1 ) I j ( pq ) ( s ) ,
I j ( pq ) ( s ) = { e ̂ p * ( r j + ( s ) ) · e ̂ q ( r j + ( s ) ) + [ e ̂ p * ( r j ( s ) ) · n ̂ j ( s ) e ̂ p * ( r j + ( s ) ) · n ̂ j ( s ) ] [ e ̂ q ( r j + ( s ) ) · n ̂ j ( s ) ] } .
γ 1 = k 2 8 ( ε 0 μ 0 ) ( n g 2 1 ) 2 q 1 Δ β 1 q 2 Re [ j = 1 N holes k = 1 N holes j 'th hole perimeter d s I j ( 1 q ) ( s ) k 'th hole perimeter d s I k ( q 1 ) ( s ) ×
d z d η q ( z d ) exp ( i Δ β q 1 z d ) 2 2 z d Ψ jk ( s , s ; z d ) ] ,
Ψ jk ( s , s ; z d ) = h j ( s , 0 ) h k ( s , z d )
h j ( s , z ) z z = 0 h k ( s , z ) z z = z d = 2 2 z d Ψ jk ( s , s ; z d ) .
Ψ jk ( s , s ; z d ) = Ψ j ( s , s ; z d ) δ jk ,
γ 1 = k 2 8 ( ε 0 μ 0 ) ( n g 2 1 ) 2 Re [ j = 1 N holes j 'th hole perimeter d s j 'th hole perimeter d s d z d
q I j ( 1 q ) ( s ) I j ( q 1 ) ( s ) exp ( i Δ β q 1 z d ) η q ( z d ) Ψ j ( s , s ; z d ) ] .
q η q ( z d ) e ̂ q ( r ) e ̂ q H ( r ) exp ( i β q z d ) = 4 i k ( μ 0 ε 0 ) 1 2 G 3 D ( r + z d z ̂ , r + 0 z ̂ ; k )
q I j ( 1 q ) ( s ) I j ( q 1 ) ( s ) η q ( z d ) exp ( i β q z q ) = 4 i k ( μ 0 ε 0 ) 1 2 ×
e ̂ 1 H ( r j ( s ) ) G 3 D ( r j + ( s ) + z d z ̂ , r j + ( s ) + 0 z ̂ ; k ) e ̂ 1 ( r j ( s ) ) .
γ 1 = k 3 4 π ( ε 0 μ 0 ) 1 2 ( n g 2 1 ) 2 Im [ j = 1 N holes
j 'th hole perimeter d s j 'th hole perimeter d s d β e ̂ 1 H ( r j ( s ) ) G 2 D ( r j + ( s ) , r j + ( s ) ; k , β ) e ̂ 1 ( r j ( s ) ) Ψ ͂ j ( s , s , β 1 β ) ] ,
G 2 D ( r , r ; k , β ) = d z d exp ( β z d ) G 3 D ( r + z d z ̂ , r + 0 z ̂ ; k )
Ψ ̂ j ( s , s ; Δ β ) = d z d exp ( i Δ β z d ) Ψ j ( s , s ; z d ) .
γ 1 ( θ , ϕ ) = 1 4 π ( ε 0 μ 0 ) 1 2 k 5 n g ( n g 2 1 ) 2 sin θ Re lim R R Re j j 'th hole perimeter d s j 'th hole perimeter d s Ψ ͂ ( s , s ; β 1 β ) ×
e ̂ 1 H ( r j ( s ) ) G 2 D H ( R o ̂ , r j + ( s ) ; k , β ) G 2 D ( R o ̂ , r j + ( s ) ; k , β ) e ̂ 1 ( r j ( s ) )
β = β 1 + i 2 π m S j ,
G 2 D ( r , r ; k ; β ) G 2 D ( r , r ; k , β ) ( ε 0 μ 0 ) 1 2 1 4 π q = 1 2 e ̂ q ( r ) e ̂ q H ( r ) β q β .

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