One of the most important parameters of multimode fi-ber lightguides �FLs� is the directional pattern �DP� of theradiation, which depends both on the parameters of the light-guide itself—in particular, on the scattering properties of theend surfaces—and on the modal composition. However, theangular intensity distribution of the output radiation, i.e., theDP, cannot be rigorously calculated based only on theoreticalconcepts of the propagation of electromagnetic radiation inideal cylindrical dielectric lightguides. This is because, first,the transfer functions1 of each individual waveguide mode,taking into account the differential modal quenching and en-ergy exchange with the other modes, are not known, and,second, the indicated problem is too laborious, and its solu-tion would require a long time to be spent on the calcula-tions. Literature sources, for example, Ref. 2, contain solu-tions only for certain particular cases. This paper thereforepresents a method based on a quasi-ray model that possessesthe simplicity of the ray approach but makes it possible totake into account the angular transfer responses �ATRs� ofactual FLs, the DP of the source, and, if necessary, the scat-tering properties of the end surfaces.
The technique presented below is applicable to any typeof FL, but FLs with a stepped refractive-index profile �RIP�are apparently of the greatest interest, since, even when thelightguide is short, the influence of the internal imperfectionsneeds to be taken into account to obtain the correct result inthis case.
I. THE QUASI-RAY MODEL
The classical ray model3 assumes that a ray that is inci-dent on the input end of a lightguide in the core area at anangle of � relative to the FL axis less than the aperture angle��c�, after being partially reflected from the end, then propa-gates in the lightguide. The total power introduced into theFL from the radiation source is determined by integratingover the radius and the angles of a cylindrical coordinatesystem.3 However, taking into account the imperfections ofthe lightguide and the differential modal damping presentsserious difficulties in such an approach. Therefore, to sim-plify the calculations of the DPs of the lightguides, it is ex-pedient to match each input angle � to the experimentallydetermined intensity distribution I�� ,��, where I is the inten-
592 J. Opt. Technol. 74 �9�, September 2007 1070-9762/2007/
sity of the output radiation into unit solid angle, and � is theoutput angle relative to the axis. The resulting modal distri-bution also depends on the width w of the incident ray. How-ever, for practical calculations, it is possible to consider thetwo most typical cases: w�dc, where dc is the cladding di-ameter of the FL, and w�d0, where d0 is the core diameter.Studies of various lightguides having a stepped RIP withdiameter d0 from 10 �m to 1 mm showed that, when thelength of the FL is 0.2 m or more, the intensity distributionof the output radiation can be considered azimuthally homo-geneous, and the dependence on the azimuthal angle � canaccordingly be neglected. The normalized I��� dependencecharacterizes the DP for a given input angle, while the inte-gral
p��� = 2��0
�/2
I��,��sin �d� �1�
is the total power of the output radiation. In practice, it canbe difficult to measure the absolute intensity; therefore it isexpedient to convert to the normalized distribution, for ex-ample, normalized to the intensity at �=0 and �=0:
In��,�� = I��,��/I�� = 0,� = 0� . �2�
Similarly,
Pn��� = P���/P�� = 0� . �3�
Normalized power Pn��� is a dimensionless quantity,used in subsequent calculations as a factor that depends onthe input angle and determined, as a rule, experimentally.Therefore, it is expedient to introduce the notation Tn���� Pn���. The Tn��� dependence should be regarded as thenormalized integral ATR of the input, describing the proper-ties of a specific lightguide and actually taking into accountthe differential mode damping and the interconnection of themodes. Then, using the ATR, the calculation of the DP of theoutput radiation, In���, for an arbitrary DP of the radiationsource, S�g�, reduces to integration over the input angle:
In��� � �0
�/2
Tn���S���In��,��sin �d� . �4�
Taking into account that �c�� /2 for most FLs, it can beassumed that sin ���, while the integration can be carried
out with the limits 0–�, and this makes it possible in certaincases to obtain solutions in the form of simple analyticalexpressions. If there is any way to determine the power P0 ofthe radiation incident on the core area, the total quenchingfactor in the given section of the fiber can be determinedfrom
= �P0 − P�� = 0��/P0.
Accordingly, using the normalized ATR T��� or Eqs.�2�–�4� and calculating the necessary normalizing factors forEq. �4�, I��� can be determined in photometric units of illu-minance or luminous intensity. It should also be pointed outthat, when the FL is represented in the form of two or severalsections of length L1 ,L2 , . . . ,Ln with the correspondingtransfer responses for both the normalized and the unnormal-ized ATR of the entire FL, the expression T�� ,�
nLn�
=�nT�� ,Ln� is valid only if �n
Ln�Ls, where Ls is the length
at which the steady-state modal distribution is established.As will be shown below, both the differential and the
integral ATR for various FLs can be approximated in mostcases by known mathematical functions with a small numberof variable parameters. If the parameters of the approximat-ing functions and the DP of the radiation source are known,there is a possibility of determining the intensity distributionof the radiation coming out of the lightguide. Thus, it issufficient to specify only a few parameters in order to calcu-late the DP of the radiation of the lightguides. The approachused here makes it possible, on one hand, to avoid the com-plex computations associated with the eigenfunctions of thewaveguide modes and, on the other hand, to actually takeinto account the real differential mode damping and the in-terconnection of the modes when radiation propagates alongthe lightguide.
To correctly approximate the angular dependences, theactual physical processes that occur when radiation propa-gates along the lightguides should be taken into account. Thebest known and all-purpose model is the so-called diffusionmodel, which describes the change of the angular distribu-tion of the propagating radiation in accordance with the dif-fusion equation.3–5 In a cylindrical coordinate system, thesolution of a diffusion equation with initial conditions in theform of a function analogous to Gaussian noise is the gen-eralized Rayleigh �Rayleigh–Rice� distribution,6
W��� =�
�2 exp�−�2 + 2
2�2 I0� �
�2 ,
where I0 is the modified zeroth-order Bessel function, and �, , and � are the distribution parameters. In the small-angleapproximation, the intensity distribution with unit power canthen be written
R��,�,�� =1
2��F2���
exp�−�2 + �2
2�F2���I0� ��
�F2��� , �5�
where �F��� is the half-width of the angular distributionwhen the input angle is �. However, for problems of fiberoptics, it is more convenient to use a dimensionless functionwith different normalization:
593 J. Opt. Technol. 74 �9�, September 2007
R��,�,�� = exp�−�2 + �2
�F2��� I0� 2��
�F2��� . �6�
The radiation power in a given solid angle when the inputangle is � can then be computed from
P =P0
��F2�
�1
�2 ��1
�2
R��,�,���d�d� ,
where P0 is the total radiation power, and �1, �2, �1, and �2
are the initial and final limits of integration.In accordance with the laws that characterize the diffu-
sion model, �F increases as the FL gets longer. Theoretically,if the modes do not interact for the model under consider-ation, in the absence of diffusion, the width of the angulardistribution of the output radiation must correspond to thediffraction divergence ��d of the radiation at an aperturewith diameter equal to the diameter d0 of the FL core.7 How-ever, as shown by experimental studies, even on the shortestsections of lightguides �in fact, when L→0�, half-width �F isat least several times greater than the theoretical value. Thisresult corresponds to the results of other papers, for example,Ref. 8. Therefore, the possibility of using the indicatedmodel requires experimental verification.
II. EXPERIMENTAL STUDY OF ANGULAR TRANSFERRESPONSES
An apparatus similar to that of Ref. 9 was used for anexperimental study of the ATR. The beam of a He−Ne laserwith wavelength �=0.6328 �m illuminated the input end ofan FL fastened to a goniometric table. The technique of Refs.9 and 10 was used to adjust the device, making it possible toset the input angle of the radiation close to zero in a planeperpendicular to the plane of rotation of the goniometrictable. The preliminary adjustment was carried out with backreflection of the beam from the input end. If the plane of theinput end is not perpendicular to the axis of the lightguide,by zero input angle we shall understand the input angle atwhich the beam refracted at the input surface of the FL isparallel to the axis of the lightguide. Accordingly, the anglesare read from the indicated direction. To determine the ATR,we varied the input angle � and measured the power of theoutput radiation, whereas, to determine the angular distribu-tion I��� of the output radiation, where � is the angle relativeto the axis of the FL, we used a scattering screen located atdistance L from its output end. The resulting images wererecorded by an array-type video camera and were transmittedto the computer for processing. The distance L was chosen sothat, for the FLs of interest, the condition of the far-field ofdiffraction �FFD� was satisfied. To reduce the effect of noisecaused by the speckle structure, the distributions were aver-aged over the azimuthal angle when measurements weremade of I���. Examples of the measured ATRs are shown inFig. 1, and examples of the I��� distributions are shown inFig. 2. If the variation of the power of the output radiation asthe inputangle varies ����c� is caused by the coupled-modeeffect, calling to mind the expression
where erf is the error function, it can be assumed that, withGaussian angular distribution of the propagating radiation,the best approximation for normalized ATRs is a function ofthe form
T��� =1
2�1 − erf�� − �e
�G , �8�
where �e and �G are the approximation parameters. Param-eter �G determines the slope of the falloff of the ATR as �→�e. When �G→�, the value of �e corresponds to the clas-sical definition of the aperture angle. When the normalizationof Eq. �5� is used, the value of �G will be less than in Eq. �8�by a factor of 2. A drawback of using the approximation ofEq. �8� for applied purposes is the impossibility of obtaininganalytical solutions for most problems of fiber optics. There-fore, in practice it is more convenient to use a simpler ap-proximation in the form of the Gaussian function
FIG. 1. Integral input ATRs. 1 and 2—Experiment, 1—for an FL with d0
=440 �m, L=2 m, 2—for a fiber–optic bundle with core diameter of thesingle lightguides d0=22 �m; L=0.25 m; 3 and 4—approximation using thefunction in Eq. �8�.
TABLE I. Approximation factors of the integral angular transfer fuof 0.6328 �m.
594 J. Opt. Technol. 74 �9�, September 2007
T��� = exp�− �2/�g2� . �9�
Although the aperture angle of the lightguide is determinedby the difference of the refractive indices of the core and thecladding, the values of �e and �g in Eqs. �8� and �9� candiffer from the value of �a=n1 arcsin�NA /n1�, where n1 is therefractive index of the core of the FL, and NA is the numeri-cal aperture. When the higher modal groups are damped sub-stantially more than the lower groups, the values of �e and �g
can be less than �a, and, in the presence of cladding modesand evanescent modes, they can be greater than �a. The pa-rameters of the approximating functions �e, �g, and �G of theexperimentally measured normalized ATRs of the lightguideswere determined by an optimization method. The relativerms error was calculated from
a = ��i=1
n
�Pi − G��i��2/�n − 1�1/2��n−1�i=1
n
Pi ,
where Pi is the normalized power of the output radiationwhen the input angle is �i, and n is the number of experi-mental points. The resulting values of the approximation pa-rameters for a polymeric optical lightguide �POL� and afiber-optic bundle �FOB� are shown in Table I. Thus, forexample, for a 20-m POL of type HFBR-RNS020, the rated
FIG. 2. Differential ATRs for various radiation-input angles for an FL withd0=440 �m, L=2 m. 1–3—Experiment, 4–6—approximation; 1 and 4—�=0°, 2 and 5—�=10°, 3 and 6—�=20°.
ns of FLs and the relative error of approximation at a wavelength
nctio
594D. V. Kisevetter
value of �a was 27.4°−30°, while the measured value was�e=23.1°. It follows from the resulting data �see Table I� thatEq. �8� gives a relative error of approximation a factor of1.5–2.5 less than when Eq. �9� is used. The only exception isan FL of great length �1 km�. It should be pointed out thatthe possibility of using the approximation of Eq. �8� for anFL with a steady-state modal distribution requires experi-mental verification in each specific case.
Typical In��� distributions for short sections of an FLwith a stepped RIP are shown in Fig. 2. Starting from Eq.�6�, by optimizing over two parameters—the normalizingfactor and �F—the dependence of �F on the input angle wascalculated �Fig. 3�. The relative rms error of approximationof the In��� dependences for 0����c was 15–30%. In par-ticular, for the dependences shown in Fig. 3, 21% for �=0°,20% for �=10°, and 28% for �=20°. The effect of T��� onthe form of the In��=const,�� dependences was neglected inthe given approximation. Therefore, the smaller is �F withrespect to �c, the smaller the approximation error. It shouldbe pointed out that the character of the �F��� dependencecan differ substantially for different lightguides. The �F���function can be either increasing or decreasing. There is nofunction that accurately describes �F��� for any lightguide,but the following approximation can be used as the simplestestimate:
�F��� = �0�1 + � exp�− �2/��2�� , �10�
where �0, ��, and � are experimentally determined param-eters. The quantity �0 is the limit of �F as �→�, and � isconnected with the other parameters by the relationship �=�F��=0� /�0−1. Depending on the sign of ����−1�, Eq.�10� makes it possible to obtain either �F�����0 or �F�����0. Thus, for example, for the dependence shown in Fig. 3,the possible approximation factors are �0=20°, ��=27°, and�=−0.685. Similar dependences were obtained for other FLs.The smallest divergence of �F was obtained for an FOB. Inparticular, for an FOB 0.5 m long with single light-guidingcores 22 �m in diameter, �F varied from 3° to 6° as theinput angle increased from 0° to 15°. When �=0°, the ex-perimental divergence of �F differed by only a small factorfrom the theoretical value for a defect-free FL. For all FLsstudied here, the experimental In��� dependences differedfrom the approximation in Eq. �6� by greater asymmetry and
FIG. 3. Half-width �F of the intensity distribution of the output radiation vsinput angle � for an FL with d0=440 �m, L=2 m. 1—Experiment,2—calculation.
595 J. Opt. Technol. 74 �9�, September 2007
by an excess, as well as by increased radiation intensity closeto the axis when ���F. However, for most applied prob-lems, the approximation of Eq. �6� makes it possible to ob-tain sufficient accuracy of the calculation of the DP. As willbe shown below, the form of the approximating function canbe made more accurate when necessary by using a moreaccurate model of the scattering when radiation propagatesalong the lightguide. In this case, the rms error of approxi-mation of In��� can be reduced severalfold.
The I��� dependence was also experimentally studied forvarious input angles of the radiation for the shortest possiblelightguide sections of 0.1−0.01 m. A detailed considerationof the intensity distributions of the output radiation made itpossible to detect the cause of the diffusion effect in anglespace. It is established that, regardless of the length of theshort sections of FLs with selective excitation of the wave-guide modes by an oblique beam in the far zone of diffrac-tion, besides an annular structure of the outgoing radiationI���, regions of increased illuminance are observed, shapedlike an angular fragment of a spiral �Fig. 4�. In some cases,the indicated regions are formed by a group of bright spots ofthe speckle structure. The angular width ��s for the FLs ofinterest was 0.1°–0.8°, which corresponded to 2–100 timesthe diffraction divergence ��d=� /d0 or more, depending onthe quality of the FL, the core diameter, and the input angle�. The angular width ��p in most cases corresponded to thediffraction divergence ��d, i.e., to the angular size of a singlespeckle spot with coherent radiation. In some cases, therewas a dependence of ��p on the azimuthal angle �. Theangular size over the azimuth �� was � /4−3� /2. The indi-cated regularities are most characteristic of quartz-polymer
FIG. 4. Radiation-intensity distribution coming from a lightguide: �a� dia-gram showing notation, �b� metallic cylindrical lightguide 0.03 m long at�=4°, �c� polymeric FL with d0=1000 �m and L=0.025 m at �=5°.
595D. V. Kisevetter
lightguides with d0�400 �m, but they show up to a greateror lesser degree in any fiber lightguides.
From the viewpoint of a ray interpretation, the appear-ance of the indicated regions can be explained by reflectionof the ray from defects of the core-cladding boundary in theform of waviness �r�� ,z�, where z is the axis of a cylindri-cal coordinate system that coincides with the axis of thelightguide. For the regions indicated above to appear in theFFD, the spatial length of such defects must be substantiallygreater than the radiation wavelength, while the defect itselfmust lie close to the output end at a distance less than thelength at which the steady-state distribution over the crosssection of the FL core is established. Nonfulfillment of theformer condition results in scattering in a wide range ofangles and accordingly increases ��s to amounts that may becommensurable with the aperture angle of the FL and re-duces the contrast of the indicated region. Nonfulfillment ofthe latter condition causes the spatial distributions to be av-eraged over the azimuthal angle and consequently also re-duces the contrast.
A similar effect can be observed in a short metallic light-guide made from aluminum foil. In this case, scattering onthe surface waviness also causes azimuthally inhomogeneousregions of intensity to appear in both the near and far zonesof diffraction. An example of the I��� distribution for a me-tallic lightguide of length 0.03 m and diameter 2 mm when�=4° is shown in Fig. 4c. The given experiment demon-strated all the effects mentioned above that characterize FLs:the appearance of a region of azimuthally inhomogeneousintensity even for a single reflection of the ray from the innersurface of the lightguide, the dependence of the scattering onthe angle of incidence of the ray, and an increase of theangular divergence ��s when there is an additional artificialmicrodeformation of the lightguide’s surface.
A fundamentally different character of the effect of im-perfections of the lightguide on the intensity distribution ofthe output radiation was observed in short sections of a poly-meric lightguide with r0=500 �m in a rigid protective clad-ding of length 0.025 and 0.15 m, with connectors securingthe FL. It was established that different regions had differentpolarization directions in both the near zone and the far zoneof diffraction. The intensity distribution in the near zone,observed through the analyzer, in general was similar to theimages shown in Refs. 2 and 11 for the case of large trans-verse loads. The angular sizes ��s and ��p were substan-tially greater than for quartz–polymer lightguides. The domi-nant influence of induced birefringence is evidenced by thespatial inhomogeneity of the polarization direction and ac-cordingly by the decrease of the integral degree of planepolarization pa by comparison with the theoretical value.11,12
For defect-free lightguides of the indicated length, the theo-retical estimate of Ref. 12 in both cases gives pa�1, but theexperimental values of pa were 0.44 and 0.2, or 0.8 and 0.4close to the axis. Thus, broadening of the angular divergenceof the output radiation can be caused not only by waviness ofthe core-cladding boundary of the waveguide but also byinduced birefringence.
596 J. Opt. Technol. 74 �9�, September 2007
If only the intensity distribution of the output radiation,i.e., the DP, needs to be calculated for applied purposes, andif the polarization or frequency characteristics can be ne-glected, there is no substantial significance in what causesthe broadening of the angular divergence on the initial sec-tion of the FL. If ��s��c, the classical diffusion model canbe used to calculate I��� in both cases for most existing FLsmore than 0.5 m long. However, it can be fundamentallyimportant to use the correct physical model of the intercon-nection of the modes when calculating the characteristics as-sociated with the phases of the wavy modes.
III. THE ANGULAR RESPONSES OF LIGHTGUIDES WITHROUGH ENDS
One more factor that can affect the DP of the radiation ofmultimode FLs is the roughness or optical inhomogeneity ofthe end surfaces. Scattering at the surface of the input endalters the modal composition of the input radiation by com-parison with input through a nonscattering surface. Themodal distribution is accordingly altered at the output fromthe lightguide, except for the case of steady-state modal dis-tribution in long FLs. Scattering at the output end also altersthe intensity distribution of the output radiation. To estimatethe influence of the roughness of the input end, it is expedi-ent to use the technique of Refs. 13 and 14, which actuallyconsists of integrating the power of the scattered radiation,taking into account the angular transfer response of the FL,as well as the approximation of the scattering indices �SIs� oflight of the directed and diffusely scattered components. Infirst approximation, the normalized SI function can be ap-proximated by the sum of a function and a Gaussianfunction,15,16 which in a cylindrical coordinate systemgives14
f��,�,��,��� = kn�� − ����� − ��� +kd
��S2
�exp�− ��2 + ����2 − 2 cos�� − �������/�S2� ,
�11�
where �, �, ��, and �� are the azimuthal and radial angles ofthe incident and scattered waves; kn and kd are the coeffi-cients of the directed and diffuse transmission; and �S is thehalf-width of the SI. Accordingly, the power transmitted tothe lightguide modes can also be conventionally separatedinto two components Pn and Pd, caused by the directed anddiffuse components of the SI. When it is averaged over azi-muthal angle ��, the function in Eq. �11� is transformed intothe Rayleigh–Rice distribution function of Eq. �6�, with anadditional term in the form of a function. If it is assumedthat all the power is scattered in accordance with the distri-bution of Eq. �11�, we can write
kn + kd = 1. �12�
However, for actual SIs, the power scattered into angles ��� �5–10��S, exceeds by a large factor not only the quantitycorresponding to the Gaussian approximation, but also thequantity obtained by a more rigorous calculation using theBeckmann formula.15,16 This is explained by the fact that the
596D. V. Kisevetter
height distribution of the profile of actual rough surfacesdiffers from a Gaussian distribution17 and by the effect ofsurface scattering.18,19 In this case, kn+kd�1. The simplestestimate from above for the value of kd is the total transmis-sion coefficient ks of a plane-parallel plate with a rough inputsurface and a refractive index equal to the refractive index ofthe core of an FL.14 By multiplying Pd by the factor ks, it ispossible to take into account the non-Gaussian nature of theform of the SI and thereby to make the calculations moreaccurate.
In a more accurate approximation, the SI of an opticallyinhomogeneous surface can be approximated by a sum ofGaussian functions with different half-widths and corre-sponding weighting factors ki. The Beckmann formula15,16
with �=0 can be regarded as a particular case of such anapproximation, obtained on the basis of theoretical conceptsfor Gaussian surface roughness. The division of the SI intodirected and diffuse components is valid when rc��, whererc is the correlation distance of the surface roughness,20 andthis actually means taking into account only two terms, in-cluding the zeroth order, in the Beckmann formula. Whenrc��, the accuracy of the approximation of Eq. �11� is re-duced as a consequence of the increase of the higher-orderterms. An approximation of the SI in the form of three termsis more universal than Eq. �11� but simpler for practical useby comparison with Ref. 15:
f��,�,��,��� = �i=1
3
ki exp�− ��2 + ����2
− cos�� − �������/�i2�/��i
2. �13�
Moreover, the normalization condition must be satisfied forcoefficients ki:
�ki = 1. �14�
Assuming �1��c, �2��c, and �3��c, the first term in thecalculation of the efficiency of the input and the DP can beregarded as the function in Eq. �11�, while the third term isregarded as scattered background, weakly dependent on thescattering angle. If �3
2��22 and �c�� /2, the fraction of
power P3 will be negligible by comparison with P1+ P2. Thisterm can therefore be neglected. However, coefficient k1 inthis case can be commensurate with k1 and k2. Accordingly,k3 needs to be taken into account when normalizing. Thus,even when rc��, it is formally possible to use the approxi-mation of Eq. �11� after correcting coefficient kd. A similarmethod can be used to take into account the non-Gaussiancharacter of the scattering at internal imperfections of thelightguide.
Using Eq. �6� for the I��� distribution at various inputangles �, we get
597 J. Opt. Technol. 74 �9�, September 2007
I��� �knT�����F
2 R��,�,�F�
+kd
��S2�
0
� T�����F
2����R��,��,�S�R���,�,�F���d��.
�15�
A similar expression is valid for the case of scattering at arough surface on the output end:
I��� �knT�����F
2 R��,�,�F�
+kdT�����S
2�F2�
0
�
R��,��,�F�R���,�,�S���d��, �16�
where � is the axial scattering angle. Equations �15� and �16�correspond to unit power P0, and therefore the proportions ofEqs. �15� and �16� can be represented in the form of equali-ties by multiplying the right-hand sides of the equations bythe quantity P0. With diffuse scattering kn→0, the desiredDP function is determined in the form of a convolution of thetwo Rayleigh–Rice distribution functions determined by theinternal and surface scattering. If it is assumed that T���=1,which is possible when �c��S, �F���= const, and, assum-ing that there are no optical losses in the FL, Eqs. �15� and�16� can be represented as
I��� �kn
��F2 R��,�,�F� +
kd
��A2 R��,�,�A� , �17�
where �A= ��S2+�F
2�1/2. In this approximation, the scatteringat a rough surface on the input and output ends has the sameeffect on the DP of the radiation. Examples of the calculatedI��� / Imax dependences for the parameters �F, �c, and �S thatcharacterize actual lightguides with a rough input surface areshown in Fig. 5. The calculations used the formula15,16,20
kn = exp�− �R�2 cos2���/�2�, kd = 1 − kn, �18�
where R� is the rms height of the roughness �Ra�0.8R��,and ��9.8 is a proportionality factor. As follows from the
FIG. 5. Theoretical normalized intensity distributions of the output radiationfor a rough input end of the lightguide at �n=0.3, �F /�c=�S /�c=0.15.1—distribution for a nonscattering input end �kd=0�, 2—distribution withexclusively diffuse scattering �kd=1, kn=0�, 3–7—distribution for variousrelative roughness heights R� /�: 3—0.5, 4—0.6, 5—0.7, 6—0.8, 7—0.85.
597D. V. Kisevetter
results of the calculation and from general relationships, R�
→0 implies that kn→0, and the I��� dependence accordinglycorresponds to an initial Rayleigh–Rice distribution withhalf-width �F. When R� /��1, it follows that Pn�Pd, andtherefore the distribution of the output radiation has half-width �A. When 0�R���, the I��� dependence is an inter-mediate case between the two cases considered earlier.
However, for actual FLs, with a more accurate treatment,identical surface roughness on the input and output ends cre-ates similar but different I��� distributions. The fundamentaldifference of Eqs. �15� and �16� consists of the factorT���� /�F
2���� under the integral of Eq. �15�. The factor T����actually limits the input angle to the value of the apertureangle, and radiation scattered at an angle substantiallygreater than the aperture angle accordingly has virtually noeffect on the DP at the output. There is no such limitation forscattering at the output end of the FL, and theoretically theradiation can be scattered within 0°–90°. The dependence of�F on �� affects the normalization of the distributions andhas the effect that annular structures accompanying selectiveexcitation of the FL modes with scattering by the input andoutput ends will differ even when Tn���= const.
Experimental studies were carried out on various FLswith various forms of optical inhomogeneities of the endsurfaces. It is known from Refs. 13 and 14 that, when theinput end surface of an FL is rough, a calculation of the inputefficiency and the angular characteristics using the approxi-mation in Eq. �11� gives good agreement with experiment.Moreover, the smaller the height of the roughness obtainedby grinding with free abrasive, the more accurately is the SIapproximated by a function of the form of Eq. �11� and,accordingly, the smaller the error when calculating the angu-lar characteristics. A similar result is obtained in this paper.The measured In��� distributions corresponded to the calcu-lated ones and confirmed the regularities described above.Thus, for example, when the height R� of the roughness ofthe end is 0.19 �m for FLs with d0=440 and 200 �m, thevariations of the contrast of the annular structure of the out-put radiation corresponded to the theoretical estimate towithin 10–20%. The hardest case to calculate is that in whichlight-scattering varnish is used on the end surfaces of thelightguide, since, even for a thin layer ��5 �m�, such a scat-terer cannot be regarded as a thin phase screen. Experimentalstudies were carried out using AK5192 varnish.21 The thick-ness h1 of the deposited layer was about 30 �m. The SI ofthe light-scattering varnish was measured on glass plateswith the same thickness of the deposited layer. The half-width of the resulting SI in the approximation of Gaussianfunctions was 5°–7°. Such an approximation underestimatedthe intensity by about 20% in the range of angles 0°–1° andby more than a factor of 2 for angles greater than 10°. How-ever, because of small values of In���10° � with respect tothe maximum value Imax, the error of the approximation was5–10%. Examples of the experimental and calculatedI��� / Imax dependences at a wavelength of 0.6328 �m for FLshaving d0=440 �m with scattering by the input or outputsurface are shown in Fig. 6. Because the maximum intensityI decreases as the input angle increases, the normalized
max
598 J. Opt. Technol. 74 �9�, September 2007
value of the scattered background I��=0� / Imax increases inthis case. Equations �15� and �17� and the approximation inEq. �10� with the parameters indicated above were used forthe calculations. It can be concluded by comparing the re-sulting data that, for the example considered here, the calcu-lated dependences correctly describe the main regularities, inparticular, the disappearance of the annular structure of theI��� distribution. Substantial differences of the calculatedand measured dependences are observed only when ���e,and this is associated with the inaccurate approximation ofthe indices when the scattering angles are greater than 10°.The accuracy of the calculations can be substantially in-creased by using the approximation in Eq. �13�, but this com-plicates the calculations and is not required in practice inmost cases.
CONCLUSIONS
It has been shown that the angular transfer responses ofactual FLs can be approximated by functions associated withsolutions of the diffusion equation: the integral ATR is afunction of the form of Eq. �8�, while the angular distributionof the output radiation power for FLs with a stepped RIP isthe Rayleigh–Rice distribution function. The directional pat-tern of the radiation of the FL can accordingly be computedvia the parameters �F, �e, and �G.
It is expedient to document the approximation param-eters of the ATR for FLs of a definite length in order toincrease the accuracy of the calculations of the parameters ofthe introduced radiation by comparison with existing tech-niques.
It has been experimentally established that the well-known effect of an abrupt increase of the angular divergenceof the output radiation when the lightguide is short is causedby imperfections of the core and can be observed even witha single reflection of the ray from the interface of the core–cladding media. Therefore, even for short sections of FLswith stepped RIP, the interconnection of the waveguidemodes needs to be taken into account when calculating theDP.
FIG. 6. Normalized intensity distribution of the output radiation for FLswith d0=440 �m and L=2 m. 1—With nonscattering ends �experiment�; 2,4, and 5—diffusely scattering input end; 3 and 6—diffusely scattering out-put end; 2 and 3—experiment; 4–6—calculation; 4—with �S=5°, 5 and6—with �S=6°.
598D. V. Kisevetter
The DP function of radiation from a scattering end sur-face of a lightguide can be specified in the form of twoterms, associated with directed and diffuse transmission. Theformer is the original DP for a nonscattering end, attenuatedby a factor of kn, and the latter is a convolution of twoRayleigh–Rice distribution functions, caused by internal andsurface scattering.
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