60
2 Optical Fiber Structures and Light Guiding Principles
2.4.2 Cutoff Wavelength and V Number
An important parameter connected with the cutoff condition is the normalized
frequency V (also called the V number or V parameter) defined by
V =
2πa
λ c
n
2
1 − n
2
2
1/2 ≈
2πa
λ
n 1
√
2
(2.27)
where the approximation on the right-hand side comes from Eq. (2.23).
This parameter is a dimensionless number that is related to the wavelength and
the numerical aperture and determines how many modes a fiber can support. The
number of modes that can exist in a waveguide as a function of V may be conveniently
represented in terms of a normalized propagation constant b defined by Gloge [19]
b =
(β/k)
2
− n
2
2
n
2
1 − n
2
2
(2.28)
Figure 2.20 gives a plot of b (in terms of β/k) as a function of V for a few of
the low-order modes. This figure shows that except for the lowest-order HE 11 mode,
each mode can exist only for values of V that exceed a certain limiting value (with
each mode having a different V limit). The modes are cut off when β = n 2 k. The
wavelength at which all higher-order modes are cut off is called the cutoff wavelength
λ c . The HE 11 mode has no cutoff and ceases to exist only when the core diameter is
zero. This is the principle on which single-mode fibers are based. By appropriately
choosing a, n 1 , and n 2 so that
V =
2πa
λ
n
2
1 − n
2
2
1/2 ≤ 2.405
(2.29)
then all modes except the HE 11 mode are cut off.
The ITU-T Recommendation G.652 states that an effective cutoff wavelength
should be less than or equal to 1260 nm for single-mode fiber operation in the
1310 nm wavelength region [20]. This specification also ensures that the fiber is
single-mode for operation in the 1550 nm region.
Drill Problem 2.4 Consider a step-index fiber that has a 5 μm core radius,
a core index n 1 = 1.480, and a cladding index n 2 = 1.477. (a) Show that at
820 nm the fiber has V = 3.514. (b) From Fig. 2.20 find the four modes that
exist at 820 nm. (c) Verify that at 1310 nm only the HE 11 mode exists.
Example 2.7 A step-index fiber has a normalized frequency V = 26.6 at a 1300 nm
wavelength. If the core radius is 25 μm, what is the numerical aperture?
2 Optical Fiber Structures and Light Guiding Principles
2.4.2 Cutoff Wavelength and V Number
An important parameter connected with the cutoff condition is the normalized
frequency V (also called the V number or V parameter) defined by
V =
2πa
λ c
n
2
1 − n
2
2
1/2 ≈
2πa
λ
n 1
√
2
(2.27)
where the approximation on the right-hand side comes from Eq. (2.23).
This parameter is a dimensionless number that is related to the wavelength and
the numerical aperture and determines how many modes a fiber can support. The
number of modes that can exist in a waveguide as a function of V may be conveniently
represented in terms of a normalized propagation constant b defined by Gloge [19]
b =
(β/k)
2
− n
2
2
n
2
1 − n
2
2
(2.28)
Figure 2.20 gives a plot of b (in terms of β/k) as a function of V for a few of
the low-order modes. This figure shows that except for the lowest-order HE 11 mode,
each mode can exist only for values of V that exceed a certain limiting value (with
each mode having a different V limit). The modes are cut off when β = n 2 k. The
wavelength at which all higher-order modes are cut off is called the cutoff wavelength
λ c . The HE 11 mode has no cutoff and ceases to exist only when the core diameter is
zero. This is the principle on which single-mode fibers are based. By appropriately
choosing a, n 1 , and n 2 so that
V =
2πa
λ
n
2
1 − n
2
2
1/2 ≤ 2.405
(2.29)
then all modes except the HE 11 mode are cut off.
The ITU-T Recommendation G.652 states that an effective cutoff wavelength
should be less than or equal to 1260 nm for single-mode fiber operation in the
1310 nm wavelength region [20]. This specification also ensures that the fiber is
single-mode for operation in the 1550 nm region.
Drill Problem 2.4 Consider a step-index fiber that has a 5 μm core radius,
a core index n 1 = 1.480, and a cladding index n 2 = 1.477. (a) Show that at
820 nm the fiber has V = 3.514. (b) From Fig. 2.20 find the four modes that
exist at 820 nm. (c) Verify that at 1310 nm only the HE 11 mode exists.
Example 2.7 A step-index fiber has a normalized frequency V = 26.6 at a 1300 nm
wavelength. If the core radius is 25 μm, what is the numerical aperture?