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Geometrical theory of multimode optical fiber-to-fiber connectors
Volume
7, number
March
OPTICS COMMUNICATIONS
3
GEOMETRICAL
THEORY
OF MULTIMODE
OPTICAL FIBER-TO-FIBER
1973
CONNECTORS
M. YOUNG Division
of Electrophysics
and Electronic
Engineering,
Received
Rensselaer
27 November
Polytechnic
Institute,
Troy, New
York 12181,
USA
1972
For optical communication to become a reality, it will be necessary to develop both permanent splices and detachable connectors for optical fibers. In this paper, I develop a simple, one-dimensional, geometrical theory with which to calculate losses introduced at a connection by alignment errors or fiber-end separation. I assume the fibers to be illuminated uniformly across the diameter of the core and within the acceptance angle. The calculation is verified with experiment. For typical fibers, the theory shows, not surprisingly, that radial misalignment of the fiber axes is by far the most severe problem. Even fibers with core diameters as large as 100 pm will have to be aligned more precisely than * 10 nm.
1. Introduction Joining to be one
optical
fibers
of the important
efficiently problems
end-to-end in optical
may
prove
com-
For optical communications to become a reality, it will be necessary to develop techniques for making permanent splices as well as convenient detachable connectors. The problem is particularly difficult if we envision a space division multiplexed communication system, in which each fiber carries a separate channel and must be joined singly, not as one member of a bundle. Fiber connectors have received relatively little attention in the literature. Krumpholz and his colleagues have made a complicated, detachable connector for individual single-mode fibers 111,21. Bisbee has succeeded in welding individual fibers together after aligning them under a microscope [3]. At RPI, we arc attempting to make individual connectors for multimode fibers of the type that might be used with light emitting diode sources of modest bandwidth. 1 have developed a one-dimensional geometrical theory that very simply describes the coupling efficiency of multimode fibers when one of several alignment errors is introduced. These errors are (i) radial misalignment 6 between the fiber axes, (ii) angular misalignment 4, and (iii) separation s between the fiber ends. I assume multimode fibers illuminated uniformlv across the core and within the acceptance half angle 0, and treat only munications.
Fig. 1. Misalignment
error 6 and angular
misalignment
@.
the case where the fibers are joined with index matching fluid or cement. Figs. 1 and 2 show the alignment errors schematically.’
2. Theory 2.1. Radial misalignment Since the fiber is taken to be illuminated uniformly across the face, the transmission in the onedimensional case is, by inspection, simply T6 = 1 ~ 6/b,
where b is the width of the core (the diameter in real fibers). 2.2. Angular misalignment Let us take the acceptance
half-angle of the fiber to 253
Volume
7, number
3
OPTICS
COMMUNICATIONS
March
1973
e
-------rllr -Nodx ------
dP
$b
eff dx
x
S
0.5t
1
; Fig. 2. Separations between index matched fiber ends. Each point on the face of the first fiber is assumed to radiate uniformly through angle 0.
be 0. (That is, the numerical aperture NA is equal to PZ sin 19,where n is the index of the core.) If the second fiber is inclined at angle @to the first, then certain of the rays emitted from the first fiber will fall on the second with angles of incidence greater than 6. Given the assumption of uniform illumination with a full angle of 20, this implies that T@=l
-G/20
is the transmission dimension.
of the connection,
calculated
in one
2.3. Separation
When the fibers are aligned but separated, the formula for the transmission of the connection cannot be found by inspection. This case is shown in fig. 2. In accord wit1 the assumption of uniform illumination, we assume that each point on the exit face of the fiber has radiance N=N,,
=0 ,
angles < 0 ; angles > 0.
Thus, the power dP emitted from a differential at x is
area dx
is suppressed in the one-
x + s tan 0
x/b
Eta”@ i 0.5
Fig. 3. Differential power coupled to the second fiber, as a function of normalized distance x/b from the axis of the first fiber.
dp
eff
=
s_tan-!fN2)- x
dp
2s tan 19
is transferred to the second fiber. This is linear in x and falls to a value of 0.5 when x reaches its maximum value b/2. Fig. 3 is a plot of dPeff/dx versus x, normalized to unity when the full power dP is transmitted to the core of the second fiber. The total transmission of the connection is the integral of dP,,/dx with respect to x (normalized so that the value of the integral is unity when s = 0). This is just equal to the area of the rectangle in fig. 3, minus the upper right-hand corner. Thus, T, = 1 -~~(s tan 0)/2b is the transmission of the connection as a function of s. When two or all three defects exist, the transmission is found by taking the appropriate product. For a very long fiber with significant loss, the illumination is possibly not uniform, either across the fiber or over the acceptance angle. If that is so, the calculation here represents the worst case.
3. Experiment
dPa No dx. where the second dimension dimensional approximation. Provided that
0
I conducted two series of experiments to determine the validity of the theory. I began with several 1 m lengths of plastic fiber, coupled to a HeNe laser with a microscope objective that had the proper numerical aperture (about 0.6). After measuring the relative power transmitted through the fiber, I broke the fiber in the middle and filed the newly broken ends flat, approximately perpendicular to the fiber axis. Translation stages joined the fiber ends physically, and a drop of index matching
Volume
7, number
OPTICS
3
March
COMMUNICATIONS
--0
0
0.5
1.0
0
s/b
0
Fig. 4. Transmission of the connection versus normalized separation s/b between the fiber ends. Solid line, theory; discrete points, average of several experiments,
fluid held on by surface tension joined them optically. It was relatively easy to attain transmission well over 0.9, although the surfaces were not polished and the matching fluid (dibutyl phthalate) was chosen arbitrarily. Figs. 4 and 5 show experimental results (similar to those of Bisbee [3] for transmission as a function of relative offset 6 and separation S, respectively. The quantities are normalized to the diameter of the fiber core. Agreement with theory is surprisingly good in view of the nature of the one dimensional approximation. To be certain the agreement was not fortuitous, I compared the theory with Bisbee’s experimental results as well. This is shown in fig. 6, which demonstrates the
I
1.0 I\
J\
=b
I
1973
Theory Experiment I 5
I 10 s/b
Fig. 6. Comparison of geometrical theory with experiment of ref. [3 ] : transmission of connection versus normalized separation s/b. Solid line, theory; broken curve, experiment. solid line is omitted to show broken curve clearly.)
accuracy
of the geometrical
(Part of
theory over a larger range
of s.
4. Conclusion The theory given above allows us to estimate the precision with which arbitrary fibers must be joined to make an acceptable connector. As an example, suppose we demand transmission in excess of 0.9. We take fibers with diameter of 100 pm and numerical aperture corresponding to a 10” halfangle. We may estimate that To falls to 0.90 when @is x 2”. T, will exceed 0.90, provided s be held less than 100 pm (one fiber diameter). These should not be difficult tolerances to exceed. On the other hand, the offset 6 will have to be held precise (for these fibers) to 10 pm or better, even if there is no error in the other parameters. Clearly radial alignment accuracy will limit the performance of most direct fiber to fiber connectors. With larger fibers, radial misalignment will naturally be a less severe problem.
References
[l] 0. Krumpholz,
Fig. 5. Transmission of the connection versus normalized misalignment error S/b. Solid line and discrete points as in previous figure.
paper reported by R.V. Pole, S.E. Miller, J.H. Harris and P.K. Tien, Appl. Opt. 1 I (1972) 1675. [2] M. Bomer, D. Cruchmann, J. Guttmann, 0. Krumpholz and W. Loffler, Elektronik und Ubertragungstechnik 26 (1972) 288. [3] D.L. Bisbeem
Bell System
Tech. J. 50 (1971)
3153,
3159.
25.5
7, number
March
OPTICS COMMUNICATIONS
3
GEOMETRICAL
THEORY
OF MULTIMODE
OPTICAL FIBER-TO-FIBER
1973
CONNECTORS
M. YOUNG Division
of Electrophysics
and Electronic
Engineering,
Received
Rensselaer
27 November
Polytechnic
Institute,
Troy, New
York 12181,
USA
1972
For optical communication to become a reality, it will be necessary to develop both permanent splices and detachable connectors for optical fibers. In this paper, I develop a simple, one-dimensional, geometrical theory with which to calculate losses introduced at a connection by alignment errors or fiber-end separation. I assume the fibers to be illuminated uniformly across the diameter of the core and within the acceptance angle. The calculation is verified with experiment. For typical fibers, the theory shows, not surprisingly, that radial misalignment of the fiber axes is by far the most severe problem. Even fibers with core diameters as large as 100 pm will have to be aligned more precisely than * 10 nm.
1. Introduction Joining to be one
optical
fibers
of the important
efficiently problems
end-to-end in optical
may
prove
com-
For optical communications to become a reality, it will be necessary to develop techniques for making permanent splices as well as convenient detachable connectors. The problem is particularly difficult if we envision a space division multiplexed communication system, in which each fiber carries a separate channel and must be joined singly, not as one member of a bundle. Fiber connectors have received relatively little attention in the literature. Krumpholz and his colleagues have made a complicated, detachable connector for individual single-mode fibers 111,21. Bisbee has succeeded in welding individual fibers together after aligning them under a microscope [3]. At RPI, we arc attempting to make individual connectors for multimode fibers of the type that might be used with light emitting diode sources of modest bandwidth. 1 have developed a one-dimensional geometrical theory that very simply describes the coupling efficiency of multimode fibers when one of several alignment errors is introduced. These errors are (i) radial misalignment 6 between the fiber axes, (ii) angular misalignment 4, and (iii) separation s between the fiber ends. I assume multimode fibers illuminated uniformlv across the core and within the acceptance half angle 0, and treat only munications.
Fig. 1. Misalignment
error 6 and angular
misalignment
@.
the case where the fibers are joined with index matching fluid or cement. Figs. 1 and 2 show the alignment errors schematically.’
2. Theory 2.1. Radial misalignment Since the fiber is taken to be illuminated uniformly across the face, the transmission in the onedimensional case is, by inspection, simply T6 = 1 ~ 6/b,
where b is the width of the core (the diameter in real fibers). 2.2. Angular misalignment Let us take the acceptance
half-angle of the fiber to 253
Volume
7, number
3
OPTICS
COMMUNICATIONS
March
1973
e
-------rllr -Nodx ------
dP
$b
eff dx
x
S
0.5t
1
; Fig. 2. Separations between index matched fiber ends. Each point on the face of the first fiber is assumed to radiate uniformly through angle 0.
be 0. (That is, the numerical aperture NA is equal to PZ sin 19,where n is the index of the core.) If the second fiber is inclined at angle @to the first, then certain of the rays emitted from the first fiber will fall on the second with angles of incidence greater than 6. Given the assumption of uniform illumination with a full angle of 20, this implies that T@=l
-G/20
is the transmission dimension.
of the connection,
calculated
in one
2.3. Separation
When the fibers are aligned but separated, the formula for the transmission of the connection cannot be found by inspection. This case is shown in fig. 2. In accord wit1 the assumption of uniform illumination, we assume that each point on the exit face of the fiber has radiance N=N,,
=0 ,
angles < 0 ; angles > 0.
Thus, the power dP emitted from a differential at x is
area dx
is suppressed in the one-
x + s tan 0
x/b
Eta”@ i 0.5
Fig. 3. Differential power coupled to the second fiber, as a function of normalized distance x/b from the axis of the first fiber.
dp
eff
=
s_tan-!fN2)- x
dp
2s tan 19
is transferred to the second fiber. This is linear in x and falls to a value of 0.5 when x reaches its maximum value b/2. Fig. 3 is a plot of dPeff/dx versus x, normalized to unity when the full power dP is transmitted to the core of the second fiber. The total transmission of the connection is the integral of dP,,/dx with respect to x (normalized so that the value of the integral is unity when s = 0). This is just equal to the area of the rectangle in fig. 3, minus the upper right-hand corner. Thus, T, = 1 -~~(s tan 0)/2b is the transmission of the connection as a function of s. When two or all three defects exist, the transmission is found by taking the appropriate product. For a very long fiber with significant loss, the illumination is possibly not uniform, either across the fiber or over the acceptance angle. If that is so, the calculation here represents the worst case.
3. Experiment
dPa No dx. where the second dimension dimensional approximation. Provided that
0
I conducted two series of experiments to determine the validity of the theory. I began with several 1 m lengths of plastic fiber, coupled to a HeNe laser with a microscope objective that had the proper numerical aperture (about 0.6). After measuring the relative power transmitted through the fiber, I broke the fiber in the middle and filed the newly broken ends flat, approximately perpendicular to the fiber axis. Translation stages joined the fiber ends physically, and a drop of index matching
Volume
7, number
OPTICS
3
March
COMMUNICATIONS
--0
0
0.5
1.0
0
s/b
0
Fig. 4. Transmission of the connection versus normalized separation s/b between the fiber ends. Solid line, theory; discrete points, average of several experiments,
fluid held on by surface tension joined them optically. It was relatively easy to attain transmission well over 0.9, although the surfaces were not polished and the matching fluid (dibutyl phthalate) was chosen arbitrarily. Figs. 4 and 5 show experimental results (similar to those of Bisbee [3] for transmission as a function of relative offset 6 and separation S, respectively. The quantities are normalized to the diameter of the fiber core. Agreement with theory is surprisingly good in view of the nature of the one dimensional approximation. To be certain the agreement was not fortuitous, I compared the theory with Bisbee’s experimental results as well. This is shown in fig. 6, which demonstrates the
I
1.0 I\
J\
=b
I
1973
Theory Experiment I 5
I 10 s/b
Fig. 6. Comparison of geometrical theory with experiment of ref. [3 ] : transmission of connection versus normalized separation s/b. Solid line, theory; broken curve, experiment. solid line is omitted to show broken curve clearly.)
accuracy
of the geometrical
(Part of
theory over a larger range
of s.
4. Conclusion The theory given above allows us to estimate the precision with which arbitrary fibers must be joined to make an acceptable connector. As an example, suppose we demand transmission in excess of 0.9. We take fibers with diameter of 100 pm and numerical aperture corresponding to a 10” halfangle. We may estimate that To falls to 0.90 when @is x 2”. T, will exceed 0.90, provided s be held less than 100 pm (one fiber diameter). These should not be difficult tolerances to exceed. On the other hand, the offset 6 will have to be held precise (for these fibers) to 10 pm or better, even if there is no error in the other parameters. Clearly radial alignment accuracy will limit the performance of most direct fiber to fiber connectors. With larger fibers, radial misalignment will naturally be a less severe problem.
References
[l] 0. Krumpholz,
Fig. 5. Transmission of the connection versus normalized misalignment error S/b. Solid line and discrete points as in previous figure.
paper reported by R.V. Pole, S.E. Miller, J.H. Harris and P.K. Tien, Appl. Opt. 1 I (1972) 1675. [2] M. Bomer, D. Cruchmann, J. Guttmann, 0. Krumpholz and W. Loffler, Elektronik und Ubertragungstechnik 26 (1972) 288. [3] D.L. Bisbeem
Bell System
Tech. J. 50 (1971)
3153,
3159.
25.5