Abstract

We investigate the polarization evolution and dispersive properties of the eigenmodes of birefringent media with arbitrarily twisted axes of birefringence. Analytical and numerical methods based on a transfer matrix approach are developed and used to study specifically helically twisted structures and the Bloch modes of periodically twisted media, as represented in particular by structural “rocking” filters inscribed in highly birefringent photonic crystal fibers. The presence of periodically twisted birefringence axes causes the group velocity dispersion curves to separate strongly from each other in the vicinity of the anti-crossing wavelength, where the inter-polarization beat-length equals an integer multiple of the rocking period. The maximum separation between these curves and the bandwidth of the splitting depend on the amplitude of the rocking angle. We also show that suitably designed adiabatic transitions, formed by chirping the rocking period, allow a broadband conversion between a linearly polarized fiber eigenmode and a single Bloch mode of a uniform rocking filter. The widely controllable dispersive properties provided by rocking filters may be useful for manipulating the phase-matching conditions in nonlinear optical processes such as four-wave mixing, supercontinuum generation, and the generation of resonant radiation from solitons.

© 2010 Optical Society of America

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  1. R. H. Stolen, A. Ashkin, W. Pleibel, and J. M. Dziedzic, “In-line fiber-polarization-rocking rotator and filter,” Opt. Lett. 9, 300–302 (1984).
    [CrossRef] [PubMed]
  2. P. St. J. Russell and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
    [CrossRef]
  3. K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27, 1548–1550 (1991).
    [CrossRef]
  4. D. C. Psaila, F. Ouellette, and C. Martijn de Sterke, “Characterization of photoinduced birefringence change in optical fiber rocking filters,” Appl. Phys. Lett. 68, 900–902 (1996).
    [CrossRef]
  5. M. Berwick, C. N. Pannell, P. St. J. Russell, and D. A. Jackson, “Demonstration of birefringent optical fibre frequency shifter employing torsional acoustic waves,” Electron. Lett. 27, 713–715 (1991).
    [CrossRef]
  6. K. J. Lee, H. C. Park, H. S. Park, and B. Y. Kim, “Highly efficient all-fiber tunable polarization filter using torsional acoustic wave,” Opt. Express 15, 12362–12367 (2007).
    [CrossRef] [PubMed]
  7. G. Kakarantzas, A. Ortigosa-Blanch, T. A. Birks, P. St. J. Russell, L. Farr, F. Couny, and B. J. Mangan, “Structural rocking filters in highly birefringent photonic crystal fiber,” Opt. Lett. 28, 158–160 (2003).
    [CrossRef] [PubMed]
  8. G. Statkiewicz-Barabach, A. Anuszkiewicz, W. Urbanczyk, and J. Wojcik, “Sensing characteristics of rocking filter fabricated in microstructured birefringent fiber using fusion arc splicer,” Opt. Express 16, 17258–17268 (2008).
    [CrossRef] [PubMed]
  9. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25, 1325–1327 (2000).
    [CrossRef]
  10. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
    [CrossRef]
  11. W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2008 (1)

2007 (1)

2006 (2)

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

2005 (1)

G. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

2003 (3)

G. Kakarantzas, A. Ortigosa-Blanch, T. A. Birks, P. St. J. Russell, L. Farr, F. Couny, and B. J. Mangan, “Structural rocking filters in highly birefringent photonic crystal fiber,” Opt. Lett. 28, 158–160 (2003).
[CrossRef] [PubMed]

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 2003).

2000 (1)

1997 (2)

D. C. Psaila, C. Martijn de Sterke, and F. Ouellette, “Comb filters based on superstructure rocking filters in photosensitive optical fibers,” Opt. Commun. 141, 75–82 (1997).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

1996 (1)

D. C. Psaila, F. Ouellette, and C. Martijn de Sterke, “Characterization of photoinduced birefringence change in optical fiber rocking filters,” Appl. Phys. Lett. 68, 900–902 (1996).
[CrossRef]

1991 (2)

M. Berwick, C. N. Pannell, P. St. J. Russell, and D. A. Jackson, “Demonstration of birefringent optical fibre frequency shifter employing torsional acoustic waves,” Electron. Lett. 27, 713–715 (1991).
[CrossRef]

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27, 1548–1550 (1991).
[CrossRef]

1990 (1)

P. St. J. Russell and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
[CrossRef]

1989 (1)

1984 (1)

1980 (1)

M. Monerie and L. Jeunhomme, “Polarization mode coupling in long single-mode fibres,” Opt. Quantum Electron. 12, 449–461 (1980).
[CrossRef]

1979 (1)

1978 (1)

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

1958 (1)

Anuszkiewicz, A.

Arriaga, J.

Ashkin, A.

Berwick, M.

M. Berwick, C. N. Pannell, P. St. J. Russell, and D. A. Jackson, “Demonstration of birefringent optical fibre frequency shifter employing torsional acoustic waves,” Electron. Lett. 27, 713–715 (1991).
[CrossRef]

Biancalana, F.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

Bilodeau, F.

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27, 1548–1550 (1991).
[CrossRef]

Bird, D. M.

G. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Birks, T. A.

Chinn, S. R.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Couny, F.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Dziedzic, J. M.

Efimov, A.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

Erdogan, T.

Evans, J. W.

Farr, L.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Hand, D. P.

P. St. J. Russell and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
[CrossRef]

Hedley, T. D.

G. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Hill, K. O.

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27, 1548–1550 (1991).
[CrossRef]

Jackson, D. A.

M. Berwick, C. N. Pannell, P. St. J. Russell, and D. A. Jackson, “Demonstration of birefringent optical fibre frequency shifter employing torsional acoustic waves,” Electron. Lett. 27, 713–715 (1991).
[CrossRef]

Jeunhomme, L.

M. Monerie and L. Jeunhomme, “Polarization mode coupling in long single-mode fibres,” Opt. Quantum Electron. 12, 449–461 (1980).
[CrossRef]

Johnson, D. C.

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27, 1548–1550 (1991).
[CrossRef]

Kakarantzas, G.

Kim, B. Y.

Knight, J. C.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25, 1325–1327 (2000).
[CrossRef]

Lee, K. J.

Malo, B.

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27, 1548–1550 (1991).
[CrossRef]

Mangan, B. J.

Martijn de Sterke, C.

D. C. Psaila, C. Martijn de Sterke, and F. Ouellette, “Comb filters based on superstructure rocking filters in photosensitive optical fibers,” Opt. Commun. 141, 75–82 (1997).
[CrossRef]

D. C. Psaila, F. Ouellette, and C. Martijn de Sterke, “Characterization of photoinduced birefringence change in optical fiber rocking filters,” Appl. Phys. Lett. 68, 900–902 (1996).
[CrossRef]

McIntyre, P.

Monerie, M.

M. Monerie and L. Jeunhomme, “Polarization mode coupling in long single-mode fibres,” Opt. Quantum Electron. 12, 449–461 (1980).
[CrossRef]

Omenetto, F. G.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

Ortigosa-Blanch, A.

Ouellette, F.

D. C. Psaila, C. Martijn de Sterke, and F. Ouellette, “Comb filters based on superstructure rocking filters in photosensitive optical fibers,” Opt. Commun. 141, 75–82 (1997).
[CrossRef]

D. C. Psaila, F. Ouellette, and C. Martijn de Sterke, “Characterization of photoinduced birefringence change in optical fiber rocking filters,” Appl. Phys. Lett. 68, 900–902 (1996).
[CrossRef]

Pannell, C. N.

M. Berwick, C. N. Pannell, P. St. J. Russell, and D. A. Jackson, “Demonstration of birefringent optical fibre frequency shifter employing torsional acoustic waves,” Electron. Lett. 27, 713–715 (1991).
[CrossRef]

Park, H. C.

Park, H. S.

Pearce, G.

G. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Pleibel, W.

Psaila, D. C.

D. C. Psaila, C. Martijn de Sterke, and F. Ouellette, “Comb filters based on superstructure rocking filters in photosensitive optical fibers,” Opt. Commun. 141, 75–82 (1997).
[CrossRef]

D. C. Psaila, F. Ouellette, and C. Martijn de Sterke, “Characterization of photoinduced birefringence change in optical fiber rocking filters,” Appl. Phys. Lett. 68, 900–902 (1996).
[CrossRef]

Reeves, W. H.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

Russell, P. St. J.

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
[CrossRef]

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

G. Kakarantzas, A. Ortigosa-Blanch, T. A. Birks, P. St. J. Russell, L. Farr, F. Couny, and B. J. Mangan, “Structural rocking filters in highly birefringent photonic crystal fiber,” Opt. Lett. 28, 158–160 (2003).
[CrossRef] [PubMed]

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25, 1325–1327 (2000).
[CrossRef]

M. Berwick, C. N. Pannell, P. St. J. Russell, and D. A. Jackson, “Demonstration of birefringent optical fibre frequency shifter employing torsional acoustic waves,” Electron. Lett. 27, 713–715 (1991).
[CrossRef]

P. St. J. Russell and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
[CrossRef]

Simon, A.

Skryabin, D. V.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

Snyder, A. W.

Statkiewicz-Barabach, G.

Stolen, R. H.

Taylor, A. J.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

Ulrich, R.

Urbanczyk, W.

Wadsworth, W. J.

Wojcik, J.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 2003).

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 2003).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. C. Psaila, F. Ouellette, and C. Martijn de Sterke, “Characterization of photoinduced birefringence change in optical fiber rocking filters,” Appl. Phys. Lett. 68, 900–902 (1996).
[CrossRef]

Electron. Lett. (3)

M. Berwick, C. N. Pannell, P. St. J. Russell, and D. A. Jackson, “Demonstration of birefringent optical fibre frequency shifter employing torsional acoustic waves,” Electron. Lett. 27, 713–715 (1991).
[CrossRef]

P. St. J. Russell and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
[CrossRef]

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27, 1548–1550 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (2)

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

Nature (1)

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

D. C. Psaila, C. Martijn de Sterke, and F. Ouellette, “Comb filters based on superstructure rocking filters in photosensitive optical fibers,” Opt. Commun. 141, 75–82 (1997).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

M. Monerie and L. Jeunhomme, “Polarization mode coupling in long single-mode fibres,” Opt. Quantum Electron. 12, 449–461 (1980).
[CrossRef]

Phys. Rev. B (1)

G. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Other (1)

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 2003).

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Figures (11)

Fig. 1
Fig. 1

Dispersion of the normal modes of a helically twisted medium. (a) Normalized wavevector (black curve with left arrow) and group index (blue curve with right arrow). (b) Normalized GVD in the cases of Δ n g = 0 (black curve with left arrow) and Δ β 2 = 0 (blue curve with right arrow).

Fig. 2
Fig. 2

(a) Schematic diagram of a structural rocking filter inscribed in a HiBi PCF—an example of a periodically twisted birefringent medium; θ m and Λ are the rocking angle amplitude and period. (b)–(d) Examples of rocking profiles: (b) rectangular, (c) triangular, and (d) sinusoidal.

Fig. 3
Fig. 3

(a),(b) Dispersion diagrams for the Bloch modes in a periodically twisted birefringent medium. The dark-blue solid lines (F and S) represent the modes of the unperturbed medium and the dark-blue dashed lines ( F and S ) are their replicas, spaced apart by the Brillouin zone width 2 π / Λ ; ω 0 is the angular frequency at resonance. The orange and light-blue solid curves ( F B and S B ) represent the Bloch modes. In (a), the slow mode of the unperturbed medium has a lower group velocity than the fast mode, while in (b) the fast mode has a lower group velocity—the case for the PCF discussed in this paper. (c) Extended dispersion diagram from (b), showing both anti-crossings and crossings at higher-order resonances where q = m 2 π / Λ with m 2 . It is assumed here that anti-crossings and crossings take place at odd- and even-order resonances, respectively.

Fig. 4
Fig. 4

(a),(b) Amplitudes of the partial waves in (a) the fast Bloch mode and (b) the slow one in a periodically rocked medium with a triangular profile and θ m = 60 ° . The amplitudes are normalized to unit power. (c) Evolution of the polarization state in the rocking filter, when linearly polarized light is launched along the fast axis at resonance. The red (normalized power 1.0 at position 0) and blue curves (normalized power 0.0 at postion 0) represent the power in the fast and slow axes, respectively. 100% conversion is far from being achieved because of the presence of significant higher-order partial waves, caused by the large rocking angle.

Fig. 5
Fig. 5

(a) Calculated intensity profiles and electric field directions (black arrows) of the two polarization eigenmodes of the PCF at a wavelength of 1553 nm. The first ring of hollow channels around the core is also shown on the mode profiles (white circles), and the white scale bar represents 1 μ m . (b) Calculated polarization beat-length of the fiber, plotted as a function of wavelength. (c) Calculated group refractive indices of the two polarization eigenmodes versus wavelength.

Fig. 6
Fig. 6

Poincaré sphere showing the polarization evolution in a rocking filter at resonance (1553 nm), when a fast fiber mode (green circle) is launched at the input. The rocking filter has a sinusoidal rocking profile with a rocking angle of θ m = 1 ° . The electric fields are normalized so that | E f | 2 + | E s | 2 = 1 .

Fig. 7
Fig. 7

(a) Poincaré spheres showing polarization evolution of the fast (left, red) and slow (right, blue) Bloch modes in a rocking filter which has a sinusoidal rocking profile with θ m = 1 ° . The green circles indicate the input polarization states. (b) Evolution of the Stokes parameter S 1 = | E f | 2 | E s | 2 at resonance (1553 nm) during propagation over one rocking period for rocking angles of 1°, 2°, and 3° ( ± 0.013 , ± 0.026 , and ± 0.039 , at position 0.0, respectively). The solid and dashed lines correspond to the fast and slow Bloch modes, respectively.

Fig. 8
Fig. 8

(a) Group refractive indices and (b) GVDs of the fiber eigenmodes and the Bloch modes in a rocking filter, plotted against wavelength. The rocking filter has a sinusoidal rocking profile with θ m = 1 ° and Λ = 603 μ m . (c) GVD profiles of the Bloch modes of rocking filters with rocking angles of 1°, 2°, and 3° for a fixed resonant wavelength of 1.553 μ m . (d) GVD profiles of the Bloch modes for different rocking periods— 494 μ m (resonance at 1.80 μ m ), 603 μ m (resonance at 1.553 μ m ), and 794 μ m (resonance at 1.30 μ m )—for a fixed rocking angle of 1°. In both (c) and (d), the solid and dashed lines correspond, respectively, to the fast and slow Bloch modes, and the rocking profiles of all the rocking filters are sinusoidal.

Fig. 9
Fig. 9

Dispersion of the Bloch modes including higher-order resonances. (a) Group refractive indices and (b) GVDs of the fiber eigenmodes and the Bloch modes in a rocking filter, plotted against wavelength. The rocking filter has a sinusoidal rocking profile with θ m = 1 ° and Λ = 2068 μ m (fundamental resonance at 0.750 μ m ). Higher-order resonances appear at 1.109 μ m ( m = 2 ) , 1.422 μ m ( m = 3 ) , and 1.736 μ m ( m = 4 ) . Among them, only the m = 3 resonance shows an anti-crossing, yielding a large change in GVD. (c) GVD profiles of the Bloch modes of rocking filters for a fixed rocking period of Λ = 2068 μ m , with rocking angles from 30° to 100° in steps of 10°; the m = 3 and m = 5 resonances are seen. The solid and dashed lines correspond, respectively, to the fast and slow Bloch modes, and the rocking profiles in all cases are sinusoidal.

Fig. 10
Fig. 10

(a) Sketch of two types of graded rocking filter transitions with pitch decreasing (−) and increasing ( + ) . Λ 0 is the rocking period of the uniform rocking filter. (b) Dispersion diagram illustrating the operating principle of adiabatic conversion in a (−) transition. The resonant frequency of the uniform rocking filter is ω 0 . Along the length of the transition, the fiber dispersion moves from F 1 F 2 F 3 and S 1 S 2 S 3 , and the Bloch mode dispersion curve goes from blue green red (gradually to the left-/right-hand side for the fast/slow Bloch mode). As a result, the input fast/slow fiber mode (orange circles, on the fiber dispersion curves F and S) adiabatically evolves into the fast/slow Bloch mode along the gray circles at ω 0 . Adiabatic conversion is achieved over the yellow spectral region (below horizontal solid line which is close to ω = ω 0 .

Fig. 11
Fig. 11

(a) Period profiles of two transitions suitable for broadband conversion from a fiber eigenmode into a Bloch mode of the uniform rocking filter. The gray horizontal dashed line marks the rocking period of the uniform rocking filter ( 603 μ m ) . (b),(c) Correlations | α f | and | α s | between the polarization state in the uniform rocking filter ( E out ) and the Bloch modes ( E B s and E B f ) when a fast fiber mode is launched at the input over the whole wavelength range. (b) and (c) correspond to (−) and ( + ) transitions, respectively, as shown in (a). If the slow fiber mode is launched at the input, the two curves in each plot swap positions. The vertical green dashed line corresponds to the resonant wavelength of the uniform rocking filter ( 1.553 μ m ) . The yellow region (to the left of the vertical solid line in (b) and to the right of the vertical solid line in (c)) indicates the wavelength range over which adiabaticity is preserved.

Equations (19)

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d E ( z ) d z = d d z [ E f ( z ) E s ( z ) ] = [ i k f ( 1 ρ ) Ω ( z ) ( 1 + ρ ) Ω ( z ) i k s ] [ E f ( z ) E s ( z ) ] ,
E ( z ) = [ E f ( z ) E s ( z ) ] = [ exp ( i k f z ) 0 0 exp ( i k s z ) ] [ e f ( z ) e s ( z ) ] = m p ( z ) e ( z ) .
d e d z = d d z [ e f e s ] = [ 0 Ω ( z ) exp ( i q z ) Ω ( z ) exp ( i q z ) 0 ] e ( z ) = m ( z ) e ( z ) ,
E ( z + L ) = [ m p ( z + L ) M e m p ( z ) ] E ( z ) = M E ( z ) ,
M e = i = 1 N ( I + m ( z i ) Δ z ) ,
k ± = k ¯ ± [ ( q / 2 ) 2 + Ω 2 ] 1 / 2 ,
n g ± = c d k ± d ω = n ¯ g ± Δ n g 2 1 [ 1 + ( 2 Ω / q ) 2 ] 1 / 2 ,
β 2 ± = d 2 k ± d ω 2 = β ¯ 2 ± { Δ β 2 2 1 [ 1 + ( 2 Ω / q ) 2 ] 1 / 2 + ( Δ n g / 2 c ) 2 Ω 2 [ ( q / 2 ) 2 + Ω 2 ] 3 / 2 } ,
E B ( z + Λ ) = M E B ( z ) = exp ( i k B Λ ) E B ( z ) ,
( E B f ( z ) E B s ( z ) ) = E B ( z ) = l E l   exp ( i k l z ) = exp ( i k B z ) P ( z ) ,
E l = 1 Λ 0 Λ P ( z ) exp ( i l 2 π Λ z ) d z .
k B ± = k ¯ 0 ± 1 Λ arccos ( sin 2 ( 2 θ m ) + cos 2 ( 2 θ m ) cos ( q Λ / 2 ) ) ,
n g B ± = n ¯ g ± Δ n g 2 cos 2 2 θ m   sin ( q Λ / 2 ) 1 [ sin 2 2 θ m + cos 2 2 θ m   cos ( q Λ / 2 ) ] 2 .
β 2 B ± = β ¯ 2 Λ 2   tan   2 θ m ( Δ n g 2 c ) 2 ,
k B ± = k ¯ 0 ± 1 Λ arccos [ Ω 2 + ( q / 2 ) 2 cos ( Λ Ω 2 + ( q / 2 ) 2 ) Ω 2 + ( q / 2 ) 2 ] ,
N p = π Λ | k B + k B | = π 2 [ π arccos ( ( Ω Λ ) 2 + π 2   cos ( Ω Λ ) 2 + π 2 ( Ω Λ ) 2 + π 2 ) ] 1 .
n g B ± = n ¯ g ± Δ n g 2 2 Ω 2   sin ( p Λ / 2 ) + ( q / 2 ) 2 p Λ   cos ( p Λ / 2 ) p 2 Λ Ω 2 + ( q / 2 ) 2 cos 2 ( p Λ / 2 ) ,
E ̂ ( z ) = α f E ̂ B f ( z ) + α s E ̂ B s ( z ) ,
| α f , s | 2 = | E ̂ B f , s ( z ) E ̂ ( z ) | 2 .

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