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derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation
вот зачем линейки помад обновляют, чтобы я мучалась, какую теперь покупать? господину Мияке от меня поклон за то, что мою любимую воду с производства не снимает.) хотя, на фоне революций и войн это была бы уже не потеря. ненавижу рэволюционэров!
из изобрЕтений я с Делиманой не подружилась.
вот зачем линейки помад обновляют, чтобы я мучалась, какую теперь покупать? господину Мияке от меня поклон за то, что мою любимую воду с производства не снимает.) хотя, на фоне революций и войн это была бы уже не потеря. ненавижу рэволюционэров!
из изобрЕтений я с Делиманой не подружилась.
вот зачем линейки помад обновляют, чтобы я мучалась, какую теперь покупать? господину Мияке от меня поклон за то, что мою любимую воду с производства не снимает.) хотя, на фоне революций и войн это была бы уже не потеря. ненавижу рэволюционэров!
из изобрЕтений я с Делиманой не подружилась.
derivatives of wave functions, u2 = a2(k2
1 − β2), w2 =
a2(β2 − k2
2) – arguments of wave functions, k1 = n1ko >
β > k2 = n2ko, ko = 2π/λo – wave numbers, respectively
in the core, cladding and outside the fibre – in vacuum,
β = 2π/
in optical fibre, βn = β/k = neff – normalized propagation
constant, which is an effective modal refractive index,
ω = 2πf, V = u2 + w2 = a2k2
o(n2
1 ͨ
2) = (akNA)2 –
normalized frequency, (n2
1 − n2
2)1/2 = NA – numerical aperture
of optical fibre, μi – relative magnetic permeability of
i-th region, n1, n2 – refractive indices of the core and the
cladding in the fibre, generally they are not constant but are
functions of radius ni(r) or radius and angle ni(r,), m –
azimuthal modal number in circular and elliptical fibres and
transverse modal number in square and rectangular core fibre,
l – radial
For m = 1, and a fibre with circular or strip core, the
ideal Eol and Hol modes are not coupled and have only the
mentioned in their names field components in the direction
of propagation. Particular form of Eq. (1), and the kind of
wave functions F depend on the fibre geometry, mainly the
core. We assume that the fibre supports hybrid modes HE,
EH or, in approximat
Two expressions in parentheses on the left side of Eq. (1)
represent modes of characteristics combined with magnetic
permeability (without charges on the boundary, Ho1 modes
in cylindrical fibre) and core/cladding refractions. Right side
of the equation couples the modes into a hybrid form of both
fields E and H in the direction of propagation.
F have to possess a property of mutual transformation
between various kinds of tailored optical fibres. Strip
and cylindrical cores are special cases of elliptical cores, etc.
Part of modal power of m-th mode Pm is carried in the core
Pm
1 , an
2 . The expression
η(r) = P1/Pm, where Pm = P1 + P2 is a modal power
profile η = (neffng − n2
2)/NA2 [51].
In the angular coordinates – longitudinal axial, radial and
azimuthal (z, r, φ
functions F are quasi-periodic and quasi-exponential Bessel
functions Jm(u) and Km(w), and the eigenequation