In the disk geometry without a lens, the maximum coupled light power of 36 ,uW can be achieved. With a lens the LED junction area can be reduced so that the ...
Calculation of Coupling Losses Between Light Emitting Diodes and Low-Loss Optical Fibers K. H. Yang and J. D. Kingsley
We have investigated the problem of coupling light emitting diodes to contemporary low-loss multimode We have calculated the maximum coupled light optical fibers (NA - 0.14) for optical communications. power and the coupling efficiencies for a disk geometry and a strip geometry with and without a lens. In 2 our calculations, we assumed that the homostructure LED has a radiance of 30 W/sr cm and requires an 2 electrical power input JV or 2 X 104 W/cm . In the disk geometry without a lens, the maximum coupled light power of 36 ,uW can be achieved. With a lens the LED junction area can be reduced so that the coupled light power of 31 uW, which is 0.1% of the input electric power, is possible. In the strip geometry with a lens a coupled light power of 260 ,uW is achievable.
Introduction
The interest in optical communication has increased substantially with the development of the technology for the fabrication of cladded glass fibers that have losses less than 20.dB/km and dispersions
We are principally interested in systems employing high order multimode fibers so that a ray-optics approach can be used to evaluate the coupling since the relevant source and fiber dimensions involved are much greater than the wavelength of the radiation.
low enough to permit communications bandwidths in excess of 100 MHz. Light emitting diodes (LED's)
Disk Sources
are generally regarded as the most attractive radiation sources for use with these fibers in an optical communications system in that they are inexpensive, easily modulated, and are capable of half-lives approaching 100,000h. The characteristics of contemporary optical communications fibers and related matters were reviewed recently by Miller et al. 1 Typical fiber core diameters are only 50-75 im, and a plane optical wave entering the end of a fiber will propagate in the fibers having the lowest attenuations
only if the angle of incidence is relatively
small, typically less than 100 to the fiber axis. Since LED dimensions are usually greater than 200 gim, and they radiate into either
2
7r or 47r sr, it is clear
from fundamental optical principles that only a small fraction, the order of a few percent, of the radiation emitted by the LED can be coupled into the fiber, no matter what auxiliary optical elements are employed. It is our purpose here to quantify the magnitude of the LED to fiber coupling loss and the power level achievable from several configurations and, hopefully, remove some of the misconceptions
concerning
this problem that seem to be prevalent.
The authors are with General Electric Corporate Research & Development, P.O. Box 8, Schenectady, New York 12301. Received 29 July 1974. 288
APPLIED OPTICS / Vol. 14, No. 2 / February 1975
We initially represent the light emitting diode as a circular plane source with radius R and a constant radiance B (radiant power per unit solid angle per unit area from the source). The optical fiber consists of a core with diameter D and index of refraction n 1
and a cladding with index of refraction n2. Figure 1(a) depicts the case with the planar source normal to the fiber axis, where there is a uniform dielectric medium with index of refraction no between the optical fiber and the light emitting area. This is essentially the geometry employed by Burrus2 in his experiments with high radiance diodes, and we will refer to it as the normal geometry. The optical fiber in this case has a maximum acceptance angle 0 given by the
following relation: sinO = (ni2
-
n 2 2)l/2/no,
n > n2.
(1)
Hence, any point on the end of the optical fiber will have an effective acceptance solid angle: 2 = 27r( - cosO) = 4
sin 2 ( 0/2).
(2)
This effective acceptance solid angle is the maximum useful solid angle into which a small area dS around point S on the LED surface will radiate. When the distance between S and the optical fiber is smaller than D/2 cotO,the amount of radiation coupled into the optical fiber from the source area dS is always constant and equal to B 9a dS. The differential
D2 + (2R 2
TOB[
c1 (a) S r
-
-
-
> n2
_
RD(1
-
l-L
-1
t 0, -T'
\
C
c
(IDEAL EFFICIENCYFROM SMALL SOURCEAND LENS SYSTEM)
o
Im ax
TS2aB7r
I1:
2w7
'D2 sin-l
]
D, will increase the effective source area, but also diminish the effective solid angle in such a way that Ic does not exceed
LED junction plane, as shown in Fig. 1(a). The dahsed curves show the ideal coupled light power and the ideal coupling efficiency by using a lens system for comparison.
System Implications
C/
~rI2
0
D
R
98w)r
I max
To illustrate the significance of the coupling problem we will use numerical values typical of low-loss
optical fibers3 and high radiance; planar homostructure LED sources.2 We assume ni + n 2 = 3.0, n
source area dS' at S' which is at distance r from S couples less light within the solid angle Qainto the optical fiber.
It can be easily shown that in the ge-
ometry of Fig. 1(a), the light power coupled into the optical fiber can be written as
-
2
= 0.0065, D = 50
gim,no = 3.5, B = 370
W/sr cm2 , T = 1.4 The maximum power coupled into the fiber is then 36 AW, whereas the total externally radiated power is 37 mW, assuming the source is isotropic. Consequently, only 1% of the emitted power is coupled into the fiber, or the coupling loss is 20 dB.
It can be shown' that with correct detector/amplifier design the SNR is quantum noise limited for I = TB /4 [D2r cos' (D) - r2 (D2 - rI)112] powers greater than a few tenths of a microwatt. Then, SNR = I/(4h vFdb), where X is the quantum D cotO 2 2 2 efficiency of the detector (0.5), I is the modulated (D cot0 + r )1 dr optical power incident on the detector, h v is the phofor L = (D/2) cotO, ton energy, Fd is the degradation in SNR introduced where T is the transmission coefficient, assumed to by the multiplication process in the detector, and b is be independent of 0 within Q£a.The factor D cotO! the signal bandwidth. With h v = 2.4 X 10-19 J, F =
J
(D2 cot2O + r2 )1 /2 varies from 1 to 0.9996 as r varies from 0 to D using values typical of low-loss glass fibers ( 2.3°, for no = 3.5, and D = 50 gim). We
2.6,
can, therefore, neglect this factor in carrying out the integration. The evaluation of Eq. (3) gives us
It would appear, therefore, that the fiber could introduce a factor of 7 attenuation (8.7 dB) before the
7 = 1/2,the
SNR is given by SNR = 2
To achieve a SNR = 104 and b
= 108
X 1017
(I/b).
Hz, a power I
=
5 AW is required.
February
1975 / Vol. 14, No. 2 / APPLIED OPTICS
289
bandwidth or SNR would have to be compromised. Unfortunately, a radiance Is great as 30 W/sr cm2 has been achieved so far only with devices whose operating lives were limited to a few thousand hours. More conventional LED's can have lives of 100,000 h but
have radiances limited by heating to a few tenths of a W/sr cm2 . It appears that a better compromise between radiance and device life must be achieved through proper design. The power at the detector could obviously be increased by having parallel sources and transmission paths. This has the added advantage of redundancy but the disadvantage of greater cost. Instead of having one LED for each fiber, one can simply have a
larger LED, but the radiance of a larger area source will almost certainly be lower than a small source, and the coupling efficiency is further reduced since the packing fraction of even a hexagonal close packed 2 array is only (r/2)(31/2)(core diam/outside diam) or 50% for a 3-mil core and a 7-mil o.d. In principle one could fabricate a monolithic array of sources on a sin-
gle chip, but this adds to the cost of the source and aggravates the fiber bundle-to-source alignment problem.
100% with an ideal optical arrangement when the source area is very small, but the power coupled into the fiber never exceeds I, max in the geometry of Fig. 1(a).
The turning points on the dashed curves in Fig.
1(b) correspond to R equal to D( 0 a/87r)1/2. The
transmission coefficient T contained in Imax can be maximized by optical impedance matching between the dielectric media of indices of refraction no and n1, respectively. The product of the effective solid angle and fiber diameter cannot be increased by impedance matching, however. The conclusions of the previous discussion apply to any geometry having a planar LED source perpendicular to the fiber axis. LED's having encapsulating lenses, hemispherical domes, Weierstrass spheres, or other shapes can be analyzed by considering the imaging effect of the lenses etc. between the junction and the fiber. The maximum power that can be coupled into the fiber in the normal geometry is given by Eq. (4). The imaging optics permits one to reduce the junction area and increase the over-all efficiency, but the coupled power increases only if the source ra-_ diance increases as a consequence of the reduction in junction area.
Use of Imaging Optics
A lens or other optical system cannot be used to couple more energy into optical fiber if R > D since, as shown in Eq. (4), I, depends only on the radiance
B, which is invariant for the object and its image under the use of any passive imaging system. In the region of R • D, an optical system can be utilized to increase the efficiency with which energy is coupled
into the fiber as is shown by the dashed curves in Fig. 1(b).
Assume that the fiber and planar source are arranged as shown in Fig. 1, except a lens with diameter
D is interposed between the source and the fiber at a distance from the source equal to its focal length F. Radiation from points on the source displaced from the fiber and lens axis a distance y impinges on the end of the fiber at an angle
A
Since this
= tan'ly/F.
radiation propagates into the fiber only if a3 • 01, where 01 is given by Eq. (1) with no = 1, the effective
source radius is F tan 1 . The effective solid angle is
that subtended by the lens, 27r1 - [F2 /(F2 + D2 / 4)] 1/2).
Consequently the power launched into the fiber is IclZ =
27 2 BT(F
tn
01
2tan2o/.( F =I sin'01/2 D
( F2 + D2/4)
(6)
2 F( ) \)/2] (\E }
For 01small and F ' D, I,11
F2 +
I,,.
DI/4)
With 01= 8, F
Strip Sources
The optics that would be interspersed between the source and the fiber are reversible so that one can visualize the area of the source that contributes to the power in the fiber by considering the area illuminated by radiation exiting the fiber and falling on the source. It is clear, therefore, that it may be possible to utilize a larger source and hence couple more power, if the fiber axis lies in the plane of the junction. The effective solid angle of points in the junction further than D/2 cot0 from the fiber end is less than 4ir sin 2 (0/2), but since cotO >> 1, the effective
junction area, (D 2 /2) cot0, is substantially greater than the effective area for the case shown in Fig. 1 (7r/4)D2. Internal absorption may reduce the effective area, however. This coplanar geometry is illustrated in Fig. 2. The LED is again represented by a planar source, and we assume that each elemental area of the junction radiates isotropically. This is almost certainly a good approximation for the distribution of radiation inside the diode. (The external distribution is determined by the shape of the diode and internal absorption.) With this assumption the effective source width for this geometry is much smaller than for the geometry of Fig. 1. In the limit of nearly zero solid
angle, the source width is of the order of the sum of the minority carrier diffusion lengths, or only a few microns. Consequently, since we have assumed that each elemental volume of the junction radiates isotropically, the effective radiance in the plane of the
= D = 50 gm, B 1 = 30 W/sr cm2 , T = 1, we find Ic 1l = 31 gW, and the total radiated power = 27-2B (F tan0) 2 = 292 gW, so the coupling loss is 9.7 dB.
junction is very large, as can be determined
The lens has thus increased the coupling efficiency 10 times, but the coupled power is about 15%less. In principle, the coupling efficiency can be increased to
diance (W/unit solid angle, cm2) is much greater in
290
APPLIED OPTICS / Vol. 14, No. 2 / February 1975
by direct
observation of an LED chip. Note that the radiant intensity (W/unit solid angle) is isotropic, but the ra-
Obviously the power coupled into the fiber can be increased still more by making > (D/2) cot@. The additonal contribution is given by
x x Z tanG+D/2
I
I~~~~~~~~~ z
_-
_-
-
_
-
-
-
--
-
Z .
-
I
D
n2
_0
D
no
~~~~~~no
z tan
`ŽIC2
J
[1
2ITBT D[l
IL
- D/2
TBD2f
OPTICAL FIBER
-
LED
Fig. 2. The LED junction plane is confined within the region between x = z tanO + D/2 and x = z tang - D/2. The n o, n , and n 2 are the indices of refraction of the LED dielectric, the fiber core, and cladding, respectively.
the plane of the junction than perpendicular to this plane. For a nonzero acceptance solid angle the effective source width is increased and the effective radiance, or power launched into the fiber, must be determined by a calculation based on the model of Fig. 2. The optical fiber has an acceptance solid angle given by Eqs. (1) and (2), with n = the refractive index of the LED material. The source regions I and II as shown in Fig. 2 couple no useful light into the optical fiber and region III, confined between the lines x = z tan0 + D/2 and x = z tan0 - D/2, is the only useful area. For simplicity, we consider a rectangular junction with length and width D which is in direct contact with the optical fiber as shown in Fig. 2 in dashed lines. The light coupled into the optical fiber can be approximated by 1C2(=)
f
[D2 cos. l
D /2
x(D 2
[(D coto)
+
2 11 2
x ]
dx
>
7BT D 2 tanO/2.
(9)
Equations (8) and (9) indicate that a very long strip junction LED can couple up to 27 times the power into a fiber in a coplanar geometry than can be coupled in the normal geometry. For the parameters previously assumed, the launched power approaches 1 mW. An LED with a junction much longer than D /2 cotO
630 gm is probably impractical, however,
because of the power dissipation implied and because internal absorption limits the effective length. Burrus achieved radiances 30 W/cm2 sr in homostructure diodes by using electrical power inputs -2 X 104 W/cm 2 . For a junction 50 gm X 630 gm this re-
quires -6-W input for a launched power of 0.4 mW. As with the normal geometry, the coupling efficiency can be increased in the coplanar geometry by using a lens or mirror to image the radiation from the LED onto the fiber, but with some decrease in the maximum power that can be coupled. This has the added benefit of alleviating the requirement on internal absorption. We again assume that a lens having the same di-
-X2)1/2
42
sco)].
D2
BT
2adZf
-
D
In the limit of I >>D, A'&2
I
(2 + D2)1/2
Z2-]dz
for 0
1 ' D/2
JUNCTION cot0.
(7)
As before, we neglect the x dependence of the term
D cot0/[(D cot0) 2 + x 21,1/2 since 0 is small. For (D /2) cot0 the evaluation of Eq. (7) gives I
BD(1 -
cos0)
[47ff+ 16 _ 3(3)1/2]
< a ) nLEDI
(8)
Equation (8) indicates that the power coupled into the fiber is proportional to the junction area D if I < (D/2) cot0. If = 7r(D/4), the areas of the geometries of Figs. 1 and 2 are equal, but I2 is only 69% of I1, indicating that the coplanar geometry of Fig. 2 is somewhat less efficient. With 1 = (D /2) cot0, I 2 = (2.16)BTD2 cos0 tan (0/2), and for 0 = 2.280, the
power launched into the fiber is increased to 11 times that achievable from the normal geometry. The coplanar geometry couples more power into the fiber than the normal geometry, provided 1 > 1.14D. This requires that the absorption coefficient
a$ (1/D)- 200cm-1 .
b) n> I
Fig. 3. It shows the insertion of a lens between the fiber and the LED in strip geometry, where W is the LED junction width and 01 is the fiber collecting half-angle referred to air. February 1975 / Vol. 14, No. 2 / APPLIED OPTICS
291
Summary of Results of Analyses for Disk and Strip Geometries With and Without a Lensa
Table I.
Power Into Fiber
Source Geometry Disc,
2
= D
dia
72 B1TD
2
Elec. Power Required
1
2
T
sin (2 )
Fiber normal to 22
disc, dia = 1
Focused 2F tan
0.4W
36pW
junction
Lens, dia = D and focal length = F. Lens and fiber co-
2
axial with junction normal
B1 TF
F1l
x
2
tan
2
7TJV F
01
Fiber
2 2 If2 01 tan B1TD
2 2.16 B 1 TD n0
0. 031W JV 2- n
cotO 1
axis in center
6.2W
0.4mW B T n F D tan
(
(
1
1
01 01
2F tan
2 = 2F tan
2
3r1
Focused rectangular width
tan(0 1' 2)
1
of strip
strip
01
= T
Rectangular strip 2 cot
tan
F2] F2 +D2 /4j
31pW width = D length = n
JV D
D
2
JV-8 n
F tan2
)
length =
0 4nF2 tan 01 2F tan D
D (1 Lens,
0.26mW
0. 28W
1
fiber coaxial
with center of strip a
The numerical values of the coupled powers and the electrical power input required assume a radiance of 0 watts/steradian-cm2 for a power input, JV, of 2 - 1 0 watts/cm , and a fiber having
a numerical
aperture
of 0.14, or an acceptance
half angle of 80.
ameter as the fiber core D and focal length F is placed close to the fiber, and the LED source is located so as to be imaged into the fiber by the lens, as shown in Fig. 3.
We are interested in estimating the greatest dimensions the LED junction can have and still be imaged into the fiber. Consider first points in the LED on a line collinear with the fiber and lens axes. Rays emanating from the focal point are refracted by the lens so that they enter the fiber parallel to the axis. Rays from points on the axis closer to the lens than 292
APPLIED OPTICS / Vol. 14, No. 2
/
February 1975
the focal point enter the fiber diverging away from the axis. If these rays are to propagate, the angle of divergence must be less than the value of 01 implied by Eq. (1) with no = 1. It follows that points closer to the lens than F/{1 + [(2F tan01 )/D ]}cannot utilize the full aperture of the lens. We assume, therefore, that the lens and LED are separated by F/I1 + [(2F tanol)/D]I. Radiation from points further from the lens than the focal points enters the fiber converging toward the axis so we find that an optimum LED junction length is
4noF
/tan [ I - ( 2F tan ) 2]
in the direction of the fiber axis, where n 0 is the refractive index of the LED material. In the focal plane, points further from the axis than F tan01 cannot be coupled into the fiber. Consequently, we assume that the junction width (perpendicular to the fiber axis) is 2F tan0k. The junction area is, therefore, A = 8nF3
tan2/[1
_ (2F
tanG0 )2]
(10)
The effective solid angle subtended by the lens (measured external to the LED) is given approxi- (r/4) (D 2 /F 2 ) for points on the axis
mately by
near the focal point. points away from the junction is reduced by an acceptable estimate (37r/6)
(D 2 IF 2
The effective solid angle for axis at the extremes of the -2 times, so we assume that of the effective solid angle is
).
Our estimate of the power coupled into the fiber from this geometry is, therefore, = 3v BTnOFD tan 2 O1 1c3 = 2 {1 - [(2F tan0)/D] 2 }
.1
(11)
Again we take B 1 = 30 W/cm 2 sr, T = 1, no = 3.5, F = D = 50gm, and 0 = 8. We find I3 = 0.26
mW compared to 0.4 mW for a strip junction source without focusing. Whereas 6 W of electrical power input were required with no lens, only about 0.28 W is required if focusing is employed. Since the length of the junction implied by this geometry is 10-2 cm, the absorption coefficient should be
