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Insights into Optical Fiber

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• When does NA provide a good estimate of the fiber's acceptance angle?
• Why is MFD an important coupling parameter for single mode fibers?
• Does NA provide a good estimate of beam divergence from a single mode fiber?

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When does NA provide a good estimate of the fiber's acceptance angle?

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Figure 192A  Rays incident at angles ≤θ_max will be captured by the cores of multimode fiber, since these rays experience total internal reflection at the interface between core and cladding. 

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Figure 192B  The behavior of the ray at the boundary between the core and cladding, which depends on their refractive indices, determines whether the ray incident on the end face is coupled into the core. The equation for NA can be found using geometry and the two equations noted at the top of this figure. 

Numerical aperture (NA) provides a good estimate of the maximum acceptance angle for most multimode fibers, as shown in Figure 192A. This relationship should not be used for single mode fibers. 

NA and Acceptance Angle
Incident light is modeled as rays to obtain the relationship between NA and the maximum acceptance angle (θ_max ), which describes the fiber's ability to gather light from off-axis sources. The equation at the top of Figure 192A can be used to determine whether rays traced from different light sources will be coupled into the fiber's core. 

Rays with an angle of incidence ≤θ_max  are totally internally reflected (TIR) at the boundary between the fiber's core and cladding. As these rays propagate down the fiber, they remain trapped in the core.

Rays with angles of incidence larger than θ_max  refract at the interface between core and cladding, and this light is eventually lost from the fiber. 

Geometry Defines the Relationship
The relationship among NA, θ_max , and the refractive indices of the core and cladding, n_core  and n_clad , respectively, can be found using the geometry diagrammed in Figure 192B. This geometry illustrates the most extreme conditions under which TIR will occur at the boundary between the core and cladding.

The equations at the top of Figure 192B are expressions of Snell's law and describe the rays' behavior at both interfaces. Note that the simplification sin(90°) = 1 has been used. Only the indices of the core and cladding limit the value of θ_max .

Angles of Incidence and Fiber Modes
When the angle of incidence is ≤θ_max , the incident light ray is coupled into one of the multimode fiber's guided modes. Generally speaking, the lower the angle of incidence, the lower the order of the excited fiber mode. Lower-order modes concentrate most of their intensity near the center of the core. The lowest order mode is excited by rays incident normally on the end face.

Single Mode Fibers are Different
In the case of single mode fibers, the ray model in Figure 192B is not useful, and the calculated NA (acceptance angle) does not equal the maximum angle of incidence or describe the fiber's light gathering ability.

Single mode fibers have only one guided mode, the lowest order mode, which is excited by rays with 0° angles of incidence. However, calculating the NA results in a nonzero value. The ray model also does not accurately predict the divergence angles of the light beams successfully coupled into and emitted from single mode fibers. The beam divergence occurs due to diffraction effects, which are not taken into account by the ray model but can be described using the wave optics model. The Gaussian beam propagation model can be used to calculate beam divergence with high accuracy.

Date of Last Edit: Jan. 20, 2020

Why is MFD an important coupling parameter for single mode fibers?

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Figure 192C  For maximum coupling efficiency into single mode fibers, the light should be an on-axis Gaussian beam with its waist located at the fiber's end face, and the waist diameter should equal the MFD. The beam output by the fiber also resembles a Gaussian with these characteristics. In the case of single mode fibers, the ray optics model and NA are inadequate for determining coupling conditions. The mode intensity (I ) profile across the radius ( ρ ) is illustrated.

As light propagates down a single mode fiber, the beam maintains a cross sectional profile that is nearly Gaussian in shape. The mode field diameter (MFD) describes the width of this intensity profile. The better an incident beam matches this intensity profile, the larger the fraction of light coupled into the fiber. An incident Gaussian beam with a beam waist equal to the MFD can achieve particularly high coupling efficiency.

Using the MFD as the beam waist in the Gaussian beam propagation model can provide highly accurate incident beam parameters, as well as the output beam's divergence.

Determining Coupling Requirements
A benefit of optical fibers is that light carried by the fibers' guided mode(s) does not spread out radially and is minimally attenuated as it propagates. Coupling light into one of a fiber's guided modes requires matching the characteristics of the incident light to those of the mode. Light that is not coupled into a guided mode radiates out of the fiber and is lost. This light is said to leak out of the fiber. 

Single mode fibers have one guided mode, and wave optics analysis reveals the mode to be described by a Bessel function. The amplitude profiles of Gaussian and Bessel functions closely resemble one another, which is convenient since using a Gaussian function as a substitute simplifies the modeling the fiber's mode while providing accurate results (Kowalevicz). 

Figure 192C illustrates the single mode fiber's mode intensity cross section, which the incident light must match in order to couple into the guided mode. The intensity (I ) profile is a near-Gaussian function of radial distance ( ρ ). The MFD, which is constant along the fiber's length, is the width measured at an intensity equal to the product of e^-2 and the peak intensity. The MFD encloses ~86% of the beam's power.

Since lasers emitting only the lowest-order transverse mode provide Gaussian beams, this laser light can be efficiently coupled into single mode fibers.

Coupling Light into the Single Mode Fiber
To efficiently couple light into the core of a single-mode fiber, the waist of the incident Gaussian beam should be located at the fiber's end face. The intensity profile of the beam's waist should overlap and match the characteristics of the mode intensity cross section. The required incident beam parameters can be calculated using the fiber's MFD with the Gaussian beam propagation model.

The coupling efficiency will be reduced if the beam waist is a different diameter than the MFD, the cross-sectional profile of the beam is distorted or shifted with respect to the modal spot at the end face, and / or if the light is not directed along the fiber's axis.

References
Kowalevicz A and Bucholtz F, "Beam Divergence from an SMF-28 Optical Fiber (NRL/MR/5650--06-8996)." Naval Research Laboratory, 2006.

Date of Last Edit: Feb. 28, 2020

Why is MFD an important coupling parameter for single mode fibers?

Significant error can result when the numerical aperture (NA) is used to estimate the cone of light emitted from, or that can be coupled into, a single mode fiber. A better estimate is obtained using the Gaussian beam propagation model to calculate the divergence angle. This model allows the divergence angle to be calculated for whatever beam spot size best suits the application.

Since the mode field diameter (MFD) specified for single mode optical fibers encloses ~86% of the beam power, this definition of spot size is often appropriate when collimating light from and focusing light into a single mode fiber. In this case, to a first approximation and when measured in the far field,

,

is the divergence or acceptance angle (θ_SM ), in radians. This is half the full angular extent of the beam, it is wavelength () dependent, and the beam's waist diameter has been set equal to the fiber's MFD (Kowalevicz).

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		Rayleigh Range:
		Beam Radius at Distance z:
Figure 194D  These curves illustrate the consequence of using NA to calculate the divergence (θSM ) of light output from a single mode fiber. Significant error in beam spot diameter can be avoided by using the Gaussian beam propagation model.  This plot models a beam from SM980-5.8-125. The values used for NA and MFD were 0.13 and 6.4 µm, respectively. The operating wavelength was 980 nm, and the Rayleigh range was 32.8 µm.

Gaussian Beam Approach
Although a diverging cone of light is emitted from the end face of a single mode optical fiber, this light does not behave as multiple rays travelling at different angles to the fiber's axis.

Instead, this light resembles and can be modeled as a single Gaussian beam. The emitted light propagates similarly to a Gaussian beam since the guided fiber mode that carried the light has near-Gaussian characteristics.

The divergence angle of a Gaussian beam can differ substantially from the angle calculated by assuming the light behaves as rays. Using the ray model, the divergence angle would equal sin^-1(NA). However, the relationship between NA and divergence angle is only valid for highly multimode fibers.

Figure 194D illustrates that using the NA to estimate the divergence angle can result in significant error. In this case, the divergence angle was needed for a point on the circle enclosing 86% of the beam's optical power. The intensity of a point on this circle is a factor of 1/e² lower than the peak intensity.

Equations (2) and (3) were used to accurately model the divergence of the beam emitted from the single mode fiber's end face. The values used to complete the calculations, including the fiber's MFD, NA, and operating wavelength are given in Figure 194D's caption. This rate of beam divergence assumes a beam size defined by the 1/e² radius, is nonlinear for distances z < z_R, and is approximately linear in the far field (z >> z_R).

The angles noted on the plot were calculated from each curve's respective slope. When the far field approximation given by Equation (1) is used, the calculated divergence angle is 0.098 radians (5.61°).

References
Kowalevicz A and Bucholtz F, "Beam Divergence from an SMF-28 Optical Fiber (NRL/MR/5650--06-8996)." Naval Research Laboratory, 2006.

Date of Last Edit: Feb. 28, 2020
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