Optical and Quantum Electronics 7 (1975) 281-288

Bulb-ended fibre coupling to LED sources B. S. K A W A S A K I , D. C. J O H N S O N Communications Research Centre, Department of Communications, Ottawa, Canada KIN 8T5

Received 15 January 1975 An optical fibre with a spherical-bulb end is shown to be an efficient lens structure for coupling light from a small Lambertian source into the fibre. The coupling efficiency is calculated by determining the irradiance distribution of light accepted by the fibre after passage through the bulb end. The theoretical calculations are verified using an experimental technique that simulates the coupling geometry. The experimental technique is a simple method that can be readily extended to the evaluation of more complex fibre-coupling configurations.

1. Introduction High-radiance light-emitting diodes (LED's) with a small emitting area are now becoming available for use in fibre-optic communication systems and methods for coupling these sources efficiently to optical fibres are required. When the LED emitting area is smaller than the crosssectional area of the fibre core, the use of collimating optics (i.e., a lens or reflector) can improve the coupling efficiency significantly over that obtained by a simple butt-joint of the fibre to the LED [1]. Kato [2-] demonstrated a simple method for producing small lenses for this purpose by melting the end of the fibre to produce a spherical bulb of high quality formed by surface tension. In this paper, we calculate the coupling efficiency between a small Lambertian source and a multimode fibre with a bulb end. We then describe an experimental technique that we used to verify the theoretical calculations. We show that a spherical bulb of the proper size melted on the end of the fibre forms a simple but effective method for coupling light from an LED into the fibre. Although we consider coupling to a single fibre in this paper, our results can be applied to the coupling of light from a source into a fibre bundle using a single spherical lens. 2. Theory Fig. 1 is a schematic diagram showing the geometry and co-ordinate system we used to calculate the coupling efficiency. The centre of the bulb is taken as the origin of the co-ordinate system. For simplicity, the source has been assumed to be circular and positioned symmetrically on the fibre axis so that the coupling geometry possesses circular symmetry. Shealy and Burkhard [3-] have derived analytical expressions for the flux density incident on a receiver surface when rays emanating from a point source are refracted by an intervening surface. This method is adapted here to permit the calculation of the coupling efficiency of light from an extended source to a bulb-ended fibre. The method consists of first determining at each point of the receiver that portion of the irradiance from the extended source which is accepted by the fibre, and then integrating over the irradiance distribution to obtain the coupling efficiency. 9 1975 Chapman and Hall Ltd. 281

B. S. Kawasaki, D. C. Johnson SOURCE

::

BULB

_

[r

/__/0So,Bo, Oo,,o, i

~

,

Figure I A schematic diagram of the source, bulb and fibre receiver used in making the coupling efficiency calculations foran extended source. The diagram shows the pertinent symbols and coordinate system used in making the calculations. The quantities a, A and N are respectively unit vectors in the direction of the incident bundle of rays, the refracted bundle of rays and the normal to the sphere. ~i, Cs are the angles of incidence and refraction for the rays, respectively.

FIBER

dS~(O,y,-p)

~

,

\~

[NDEX no

In Fig. 1, a pencil of rays is shown propagating from an arbitrary position (Ro, 00, r on the source in the direction of the unit vector a and illuminating an element of area dS~ on the spherical surface. After refraction, the pencil of rays propagates in the direction of the unit vector A and illuminates an area dS2 on the receiver. The flux incident on the element of area dS1 of the refracting sphere is given by Fi, = a cos q5i d S 1

(1)

where o- is the flux density of the incident pencil of rays and q5i is the angle of incidence of the beam upon the spherical surface. The flux per unit area on the receiver surface is equal to the flux incident on dS~ times the transmittance p, divided by the element of area dS2 on the receiver surface mapped out by the refracted rays from dS~. That is, dS1 dFds,_dS 2 = pc COS (~i dS 2

(2)

For a source with a Lambertian angular intensity distribution, the incident intensity at S~ is I 0 COS

~r -- ~

rl

dSo

(3)

where I o is the radiance of the source, c~is the angle between the ray direction a and the source surface normal, dSo is an element of area on the source and r~ is the distance between the two elements of area dSo and dS1. Hence the flux per unit area on the receiver is dFds,-dS: -- p

Io cos ~ cos r dS1 2 - - dSo 9 ra dS2

(4)

To find the total flux per unit area on the element dSz, the above expression must be integrated. To facifitate this integration it is advantageous to rewrite Equation 4 as dFds,_,lS: -- p

Io cos ~ cos r rl:

al ~o:dS1

dS~ 9 dS1

(5)

The quantities d S z / d S 1 and dSo/dS1 can be found with the aid of differential geometry. Equation 5 then becomes

dFas~-dS~ = 282

c o s O cos ~162 r~ 2 + 6~ r~ + 6ol rxZld2 r2 2 + A1 r2 + Aol dS1

(6)

Bulb-ended fibre coupling to L E D sources where Ao = cos ~bs A 1 = [~(1 + cos 2 C~s)/R + (no/n) (cos 2 ~bi + cos z d?s)/rl]/COS (o~ A 2 = [(f2/R) 2 + (no/n) f2(1 + cos 2 0 i ) / R r l + (no/n) 2 cos 2 0 j q 2 ] / c o s c~s f2 = (no/n) cos q~i + cos ~b~ ao = - cos ~bi al = - Eft(1 + cos z ~ ) / R + (n/no) (cos 2 ~bs + cos 2 (pi)/r2]/cos c~i 6 2 = - [-(D'/R) 2 + (n/no) f2'(1 + cos 2 ~s)/Rr2 + (n/no) 2 cos 2 dpjr2]/cos (oi ~' = (n/no) cos ~bS + cos ~bi . In Equation 6, no and n are respectively the index of refraction of the m e d i u m between the source and the sphere and the index of refraction of the bulb material. The angle of refraction at the sphere surface is q~s and q is the angle between the unit vector A and the n o r m a l to the receiver surface $2 located z = - p f r o m the bulb centre. The transmittance p and the source radiance Io have been taken as unity and t h r o u g h o u t the rest of the p a p e r distances are normalized to R = 1. The total irradiance at a point on the receiver is now found by integrating Equation 6 over the surface of the sphere S 1. In this integration we sum only that portion of the flux density corresponding to rays within the cone of acceptance of the fibre. Since the coupling g e o m e t r y has circular s y m m e t r y it is necessary to calculate only the radial distribution of the irradiance. The coupling efficiency is then given by 2 f"0o Fds~-dS2 (Y) Y dy , 11 = ~zcr2

(7)

where ao and r o are the radii of the fibre core and source respectively. 3.

Experimental

An optical system that simulates the light collecting properties of a bulb-ended fibre was constructed in order to verify the calculated intensity patterns experimentally. A diagram of this optical system is shown in Fig. 2. The bulb on the fibre end is simulated by the large glass bulb ( ~ 4 m m diameter) with a flat face. The optical system consists of two lenses separated by the sum of their focal lengths. The flat face on the bulb is located at the focus of the first lens and is imaged through the optical system onto a detector. An iris d i a p h r a g m is situated between the two lenses at their c o m m o n focus. The angular distribution of the light collected by the optical system can be controlled by the size of the opening in the iris diaphragm. Thus this optical system allows fibres of different light accepting properties (i.e., different numerical apertures) to be simulated. The numerical aperture (NA) of the system is given by the expression

N A = sin ( t a n -~ r )

,

where r is the radius of the aperture of the iris d i a p h r a g m a n d f t is the focal length of the lens closest to the bulb. Fibres with N A <~ 0.2 can be simulated with our particular simulation system. APERTURE LENS # I

LENS 4 2

Figure 2 Schematic diagram of experimentto simulate coupling from an extended source to a bulbended fibre with a given numerical aperture. fl

fl

fz

fz

283

B. S. Kawasaki, D. C. Johnson

Two sources of different size were used in the simulation. The smaller diameter source was a circular GaAs LED (diameter ~ 0.4 ram) and the larger source was a painted light bulb apertured to form a flat emitting surface of 1.98 mm diameter. The radiation patterns from both these sources were verified by measurement to be Lambertian. The radial irradiance pattern collected by the optical system was measured for each source by imaging the source through the optical system on the linear photodiode array of an optical multichannel analyzer (OMA). To determine the total light coupled through the optical system the OMA detector was replaced by a single large-area photodiode that intercepted the entire light distribution. By monitoring the amount of light coupled through the optical system as the source position was varied the relative coupling efficiency as a function of the LED - bulb separation was determined.

4. Results and discussion In this section we present analytical and experimental results for (1) the radial irradiance patterns of light accepted by a fibre with a spherical bulb end and (2) the coupling efficiencies associated with LED and bulb-ended fibre configuration. We also describe extensions of the experimental technique to the study of more complicated coupling geometries.

4.1. Intensity patterns Fig. 3a shows a comparison of experimental and calculated irradiance patterns. The experimental light irradiance patterns were obtained using the small LED and an optical system with a numerical aperture of 0.1 which simulated a LED-bulb-ended fibre coupling configuration for a fibre with a numerical aperture of 0.1 and a ratio of LED-to bulb-radius of 0.082. The measured irradiance pattern corresponds to that radial distribution that would exist in the fibre at a distance z = - 0.78R from the bulb centre. The intensity distributions in Fig. 3a are shown for a sequence of positions (D) of the LED. Using Equation 6 the corresponding

EXPERIMENT

THEORY

D 1.21 EXPERIMENT

THEORY

1.21

147

,.T3 2O9

2BI 2.50

z 88

278 3.82

3.40

392 (o)

284

-6

BULB RADIUS

l

L

6 BULB RADIUS

Figure 3 Experimentally and theoretically generated radial profiles of accepted light i n t e n s i t y for coupling from a small source (radius = 0.082R) to a bulb-ended fibre. The results a r e shown for various values of separation between source and bulb (normalized to the radius of the bulb). (a) and (b) show results for fibre numerical apertures 0.1 and 0.2 respectively. In (b), the vertical scale of the profiles labelled 1.73 and 2.09 are reduced to 0.4 in comparison with the other profiles. The scale below the theoretical results shows the bulb radius.

Bulb-ended fibre coupling to L E D sources EXPEMIMENT

THEORY

EXPERIMENT

THEORY

1,05

f

kY-

.... Figure 4 Experimentally and theoretically generated radial profiles of accepted light intensity for coupling from a large source (radius = 0.41R) to a bulb-ended fibre. The results are shown for various values of separation between source and bulb (normalized to the radius of the bulb), (a) and (b) show results for fibre numerical apertures 0.1 and 0,2 respectively, The scale below the theoretical results shows the bulb radius. o

[a]

BULB RADIUS

~b}

0 QULB RADIUS

theoretical distributions were obtained and the vertical scale was set so as to match that of the measured distributions. The asymmetry in the patterns obtained experimentally is mainly attributable to the non-uniform gain along the photodiode array of the OMA detector. When the source is close to the bulb, the radiation pattern accepted by the fibre consists of a narrow central lobe. As the LED-to-bulb separation increases, the central lobe widens and a narrow high-intensity ring of light forms and merges with the central lobe. This ring corresponds to the rays which would have escaped the fibre but for the focussing action of the bulb. From a knowledge of the extent of the irradiance pattern accepted by a fibre of given numerical aperture, the optimum ratio of bulb radius to fibre-core radius can be determined. The pattern labelled D = 2.25 corresponds approximately to the condition of optimum coupling efficiency. The fibre core should be just sufficiently large to accept the sharp intensity spikes in the radiation pattern. In the case of coupling an LED to a fibre bundle with a single lens this nonuniform radiation pattern should be considered. It is apparent that a judicious choice of fibre bundle geometry to take advantage of the peaks in the radiation pattern would enhance the overall coupling efficiency of the fibre bundle. Fig. 3b shows the intensity patterns for the same source size as Fig. 3a but with an optical setup that simulates a fibre of numerical aperture 0.2. The variation of the intensity pattern with LED-to-bulb separation (D) is similar in form to that shown by Fig. 3a. The main differences are the more pronounced ring and the occurrence of the peak efficiency at a smaller value of D. The peak efficiency here occurs at approximately D = 1.73. A similar set of experiments was carried out using a larger light source (diameter = 1.98 ram). The results are shown in Fig. 4. The ring pattern is less apparent here than in Fig. 3. Also the relative shapes of the intensity distributions are almost the same for numerical apertures of 0.1 and 0.2, but, as will be seen in the next section, the coupling efficiencies are quite different.

4,2. Coupling efficiency The relative coupling efficiency as a function of the source position was measured for the four coupling configurations treated in Figs. 3 and 4. The results are plotted in Figs. 5 and 6. For each configuration the absolute coupling efficiencies were calculated using Equation 7. The experimental results are relative only. The vertical scale of the experimental results was adjusted to provide a best fit to the theoretical results. The maximum coupling efficiency 285

B. S. Kawasaki, D. C. Johnson

15--

3--

I0-

B--

5-

I--

THEORETICAL MAXIMUM

(a)

- BUTT-JOINTING

uJ O

,-9

,= LU

o

I

1

I

~o

I

_~ 30-

[

~15-o ;~ ITHEORETICAL

oo

I

1

MAXIMUM

uJ

I0--

20-

5--~

(b]

I0-

-- BUTT-JOINTING

0

I

I

4 I

I

2 .3 5 SOURCE POSITION (BULB RADII UNITS)

0

I

1

I

I

I

2 5 4 5 SOURCE POSITION (BULB RADII UNITS)

6

Figure 5

Figure 6

Figure 5 Coupling efficiency versus source to bulb separation for small source (radius = O.O82R). (a) and (b) show results for fibre numerical apertures 0.1 and 0.2, respectively. Figure 6 Coupling efficiency versus source to bulb separation for large source (radius = 0.41R). (a) and (b) show results forfibre numerical apertures 0.1 and 0.2, respectively.

attainable by butting a Lambertian source against the end of a flat fibre is given by NA 2. This value is indicated in Figs. 5 and 6 by the lower arrow along the ordinate. It can be seen that the use of a bulb on the fibre end significantly improves the coupling for a wide range of LED-bulb separations. The maximum possible coupling efficiency can be determined using Liouville's theorem [1]. For a large source (ratio of source radius to fibre core radius >> NA) the maximum possible coupling efficiency is given by

~m.x

= (a~

\r--o~

where ro and ao are the source and fibre-core radii respectively. For the source size used in obtaining Fig. 6, the maximum coupling efficiencies were determined to be 2.91 and 14.5% for fibres of NA 0.1 and 0.2, respectively. In calculating the maximum efficiency, the fibre core radius ao was chosen to be just sufficiently large to collect the light distribution at maximum coupling efficiency as shown in Fig. 4. The maximum coupling efficiencies are indicated by the upper arrows in Fig. 6. It can be seen that for large sources (ro/a o >> NA), bulb coupling approaches closely the optimum coupling configuration. For small sources (ro/ao < NA), close to 100% coupling is possible. The bulb-ended fibre configuration provides less than 100% 286

Bulb-ended fibre coupling to LED sources

OFFSET

I0 0.052

0.9 0.8

/

~

OA04

0.7

~ 0,6 0.156

toO.5

io_i,

iO,

. , ,

OI

08

0.3

OFFSET (BULB RADII

Figure 7

I

04 0.5 UNITS)

I

0,6

Figure 8

Figure 7 Experimentally observed radial profiles of accepted light intensity for coupling from a small source (radius = 0.082R) to a bulb ended fibre, The bulb to source separation is D = 2.1 and the profiles are shown for various values of offset of the source (normalized to the bulb radius). Figure 8 Experimentally observed variation in the total coupling efficiency with offset of the source. The experimental configuration has D = 2,1 and the small source (radius = O.082R).

efficient coupling, however as seen in Fig. 5 its efficiency is considerably greater than the butttype coupling. The penalty paid for higher coupling efficiencies in the case of a small source is the critical dependence of efficiency on the LED-bulb separation. 4.3. Offset s o u r c e s

The theoretical calculation of the coupling efficiency for a source that is offset from the fibre axis can be exceedingly difficult because the irradiance distribution in the fibre no longer has circular symmetry. Hence it is difficult to calculate the variation of coupling efficiency as the source is offset from the fibre axis. The experimental determination however is straightforward using the simulation technique shown here. We have measured the intensity patterns as the small source was offset from the fibre axis. The results shown in Fig. 7 were measured for an LED-bulb separation corresponding to maximum efficiency when on axis. The patterns shown in Fig. 7 were recorded along the direction in which the diode was moved and they would be different in other radial directions. The coupling efficiency was also monitored as the source was offset from the fibre axis. The results are shown in Fig. 8. An offset of 0.2R (i.e., 0.27ao) reduces the coupling efficiency by 50~. 287

B. S. Kawasaki, D. C. Johnson

5. Conclusions We have demonstrated that, when the L E D is smaller than the core area of the optical fibre to which it is coupled, the efficiency of coupling can be significantly increased by forming a bulb on the end of the fibre. The optimum relative size of the bulb to the core for a given L E D size can be determined theoretically or by the simple experimental technique shown. The theoretical method used is general and applicable to any shape or L E D emission pattern as well as to multiple lens and reflector systems. However, for more complex configurations the computations become exceedingly tedious. On the other hand, the experimental method is straightforward and can be used to accurately characterize complex coupling configurations.

Acknowledgement The authors would like to thank K. O. Hill for his helpful technical suggestions.

References 1. 2. 3. 4.

M. C. HUDSON, Appl. Optics 13 (1974) 1029-1033. D. KATO, Y. Appl. Phys. 44 (1973) 2756-2758. D. L. SnEALY and D. ~. BURKHARD, Optica Acta 20 (1973) 287-301. D. C. JOHNSONand B. s. KAWaSAKI, Proceedings of the Electro-optics International 1974 Conference, Brighton, England, March 19-21, (1974) 95-102.

288