- Email: [email protected]
Estimation of scattered light on the surface of unclad optical fiber tips: a new approach
1 July 1998
Optics Communications 152 Ž1998. 307–312
Estimation of scattered light on the surface of unclad optical fiber tips: a new approach HanQun Shangguan a , Lee W. Casperson b
b,)
a Oregon Medical Laser Center, Portland, OR 97225, USA Rochester Theory Center for Optical Science and Engineering, and The Institute of Optics, UniÕersity of Rochester, Rochester, NY 14627, USA
Received 8 January 1998; revised 24 March 1998; accepted 25 March 1998
Abstract Unclad multimode optical fiber tips are often the basis for diffusing tips in photodynamic therapy or scalpels in laser surgery. In this study, we report an approximated Gaussian-beam model for the calculation of the light distribution along the surface of unclad fiber tips. The results are compared with detailed experimental measurements. The model yields quite good qualitative agreement with experimental results. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Photodynamic therapy; Fiber tips; Laser surgery
1. Introduction Optical fibers have been widely used to deliver laser radiation to the treatment sites for photodynamic therapy w1x or surgical cutting w2x. For example, an unclad fiber tip can be employed as a diffuser for photodynamic therapy when coated with a scattering material w3–5x or as a scalpel for surgical cutting when coated with an absorbing material w6x. In a recent article, we discussed the fabrication of cylindrical diffusers by coating scattering material on an unclad modified fiber tip w5x. The modified tip was made by covering the area where the intensity peak is located with hydrofluoric-acid-resistant materials during a chemical etching process. Verdaasdonk et al. demonstrated that accumulation of thermally coagulated tissue proteins and carbon on the tip surface is necessary to increase the absorption of radiation at the tip surface and achieve initial tissue ablation w7x. Thus, an absorbing coating for an
)
Permanent address: Department of Electrical and Computer Engineering, Portland State University, Portland, OR 97207, USA. E-mail: [email protected]
appropriate wavelength Že.g., Nd:YAG. on an unclad fiber tip may enhance the thermal effect w6x. This effect leads to the cutting of tissue with accompanying homeostasis. Furthermore, coating the quartz fiber tip with a thermally resistant material Ži.e., the material has a higher melting point than the quartz. at a hot spot may be an alternative method to reduce the thermal damage to the scalpel. Thus, a successful understanding of laser radiation along the bare core surface of an optical fiber tip is important for the design and fabrication of fiber-optic diffusers or scalpels. Unlike previous studies w4,8x, where ray tracing was used based on geometric optics, we first propose a new approach using a simplified model to calculate the irradiance distribution on the surface of unclad fiber tips. This model is based on an approximated Gaussian-beam formalism. The rigorous analysis of light propagation in multimode waveguides can be quite complicated. In a multimode optical fiber the output field distribution may depend sensitively on the input field and on the length and other parameters of the fiber. This complexity is in contrast to the single mode fiber, where the output distribution is essentially independent of the input and fiber length. Because of the difficulty of an exact treatment of multimode
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 1 7 1 - 0
H. Shangguan, L.W. Caspersonr Optics Communications 152 (1998) 307–312
308
fiber systems, there is value in developing more approximate models which might serve to describe at least the essential features of the multimode fiber output. The simplest physical optics solutions that one might seek to apply are those based on the Gaussian beam formalism. Also, most commercial lasers produce beams having Gaussian-shaped profiles w9x. However, in the near field a Gaussian beam would not vanish as quickly outside the fiber core as would the actual multimode field solutions. Thus, it is of interest to approximate the Gaussian beam in such a way that it can more closely represent the fields in the near-field region. The results are simple formulas which provide good qualitative agreement with measurements of intensity distributions along fiber tips. The basic equations governing the field distribution along a fiber tip are developed in Section 2. In their most practical form these equations give the intensity along the tip as a function of the core diameter and of the far-field diffraction angle of the unscattered beam. In Section 3 are reported experimental measurements of the light scattered from fiber tips, and the experimental and theoretical results are compared and discussed in Section 4. The approximate formulas are found to provide good agreement to within the accuracy of our experimental scattering measurements.
2. Model For any study of light propagation the fundamental starting point is the Maxwell-Heaviside equations. These equations can be combined to yield general wave equations which govern the field components of a propagating electromagnetic beam. In the past, solutions of these equations for optical fiber applications have sometimes been based on ray optics approximations. In this study we will instead start with analytic Gaussian beam solutions of the paraxial wave equation in a uniform medium and approximate these solutions to describe the field on the surface of an unclad fiber tip. The intensity distribution of a Gaussian beam in a homogeneous medium can be written as w10x I Ž r , z . s I0
w 02 2
w Ž z.
ž
exp y
2r2 w2 Ž z .
/
,
Ž1.
where I0 is the on-axis intensity at the waist of the beam and w Ž z . is the spot size or 1re amplitude radius. The spot size varies with the propagation distance z according to the relationship w Ž z . s w 0 1 q Ž zrz 0 .
2 1r2
,
Ž2.
where w 0 is the spot size at the beam waist and z 0 is the Rayleigh length. The Rayleigh length in turn is related to
the index of refraction n, the waist spot size w 0 , and the vacuum wavelength l by z 0 s np w 02rl. Eqs. Ž1. and Ž2. may be combined to obtain the explicit intensity formula IŽ r, z. s
I0 1 q Ž zrz 0 .
2
ž
exp y
2r2 w 02 1 q Ž zrz 0 .
/
2
.
Ž3.
If one is especially interested in the intensity distribution near the center of the beam, it is useful to replace the exponential in Eq. Ž3. by the first terms in a power series expansion. This type of approximation has long been used in studies of the interaction of Gaussian beams with nonlinear media w11x. Thus, we may write Eq. Ž3. approximately as IŽ r, z. s
I0 1 q Ž zrz 0 .
2
ž
2r2
1y
w 02 1 q Ž zrz 0 .
2
/
.
Ž4.
For a finite value of r this result becomes increasingly accurate as z increases, i.e., for z 4 z 0 . For small z it has the important property of going to zero at the radius r s w 0r '2 . This property is useful in finding an approximate representation for the intensity in the vicinity of a fiber tip, because that field also goes almost to zero at the cladding boundary. If the tip consists of the unclad fiber core in a medium having the same index of refraction, we can tentatively represent the intensity on the tip surface by the simple formula IŽ r, z. s
I0 1 q Ž zrz 0 .
s I0
2
1
1y
1 q Ž zrz 0 .
Ž zrz 0 . 2 1 q Ž zrz 0 .
2 2
.
2
Ž5.
To a first approximation, this formula should also represent the z dependence of the scattered intensity when the core is coated with appropriate scatterers. It may be noted that other established beam modes have an intrinsic field zero at a finite beam radius without the need for a quadratic beam approximation. For example, the next higher radial order of the Laguerre-Gaussian beam family also has a zero at the radius r s w 0r '2 . Similarly, the J0 Bessel-Gaussian beam has zeroes at distinct radii w12,13x. Such beams could be imagined to match an input field distribution over the finite fiber core without any approximation of the beam profile. On the other hand, in their truncated form these other field distributions do not propagate as beam modes, so a rigorous study based on such input beams would be more complicated. For this reason it has seemed appropriate here to use the simplest possible physical optics approximation as represented by Eqs. Ž4. and Ž5.. For reasons that will become clear, it is also useful to express the intensity distribution in terms of the diffraction
H. Shangguan, L.W. Caspersonr Optics Communications 152 (1998) 307–312
309
3. Experiments
Fig. 1. Schematic of unclad fiber tips with a cylindrical geometry Ža. and a conical shape Žb..
angle of the output beam. Using Eq. Ž2., the diffraction angle of a Gaussian beam can be written as
l
dw
ud s lim
s
Ž z™` .
dz
np w 0
.
Ž6.
With this substitution the Rayleigh length is w 0 s z 0 u . But the diffraction angle, ud , at the output is equal to the convergence angle, u at the input, and thus Eq. Ž5. can be written as
Ž zurw 0 . 2
I Ž r , z . s I0
1 q Ž zurw 0 .
2 2
s I0
2 Ž zur'2 r . 2 1 q Ž zur'2 r .
2
2
s I0
Ž '2 zurd . 2 1 q Ž '2 zurd .
,
2
Ž7.
To verify the working of the model described above, two experimental protocols were performed: Ži. irradiance distributions along the bare core surface of fiber tips with cylindrical geometry were measured using a CCD camera system and Žii. an isotropical ball-scattering probe was used for the conical geometry. The experimental results were then compared with the calculations. Step-index multimode silica-clad silica-core fibers ŽNA s 0.22. with 300–600 mm core diameters and 2 m lengths were used. One of the distal faces served as the input end and was wet-polished using standard fiber polishing techniques. At the other end, the protective buffer was thermally removed over a length of 20 mm. The tip was then fabricated into a cylindrical or conical taper using a chemical etching technique w5x. The geometry of the tip was measured using optical microscopy. The experimental setup for the cylindrical geometry is shown in Fig. 2. The tip was attached to a diffusive plane which was made of a piece of quartz glass Ž100 mm thick. coated with UV epoxy ŽLitetak 376, Loctite Co.. mixed with Al 2 O 3 powder Žthe weight ratio of UV epoxy to powder was 10 to 3.. The thickness of the layer was about 50 mm. For ease of adjusting the convergence angle, we used fiber–fiber coupling. Both distal faces of the coupling fiber, a silica optical fiber with 400 mm core diameter, were wet-polished flat. The various convergence angles were obtained by adjusting the distance between the two fibers. A narrow bandpass filter Ž l s 404.7 nm. was used in this experiment and an arc lamp source ŽOriel Co.. was used as the light source. The beam profile was similar to that of a conventional Gaussian beam. An index matching fluid ŽR.P. Cargille Laboratories, Inc.., whose refractive index was very close to that of the fiber core, was used in this experiment. According to Eq. Ž43. in Ref. w14x, the transmissivity was almost equal to one, and the second reflection is negligible. The large viscosity and surface tension of the index matching fluid permitted us to directly drop the liquid on a glass diffuser, and then the fiber could
where r s w 0r '2 represents the fiber core radius discussed above and d s 2 r is the core diameter. This geometry is illustrated in Fig. 1. For comparison with experimental data, it is convenient to normalize the intensity to its maximum value. Setting d Ird z to zero, one finds that the maximum occurs at the distance z s dr '2 u , at which distance the intensity is I0r4. Thus, we introduce the normalized intensity X
I Ž z. s4
Iz I0
s
4 Ž '2 zurd .
2
1 q Ž '2 zurd .
2 2
.
Ž8.
This formula is now in an appropriate form for comparison with experimental data.
Fig. 2. Experimental setup for measuring the light distribution on the bare core surface of a fiber tip with cylindrical geometry.
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H. Shangguan, L.W. Caspersonr Optics Communications 152 (1998) 307–312
Fig. 3. Experimental setup for measuring the light distribution on the bare core surface of a fiber tip with conical geometry.
fiber size. This suggests that the effects of the input coupling and fiber size on the distribution of the scattered light on the surface of bare fiber tips should be taken into the account for the purposes mentioned in Section 1. For the tip with conical geometry, the calculation using Eq. Ž8. by inserting the known z-dependence of the fiber radius Ž d s d 0 y 2 tan a z . indicates that the intensity is almost independent of the angle a Ždata not shown.. This suggests that the coupling condition and the fiber size determine the peak position when the light is directly scattered from bare fiber tips without the second reflection on the surface. To verify the model described in Section 2, experiments were performed on unclad fiber tips with cylindrical or conical geometry under various conditions such as different convergence angle and fiber size. A typical comparison between the measurements and calculations for tips with cylindrical geometry is presented in Fig. 5. Fig. 6 represents the case of a conical tip. Note that the calculated intensity in Fig. 6 was obtained in a qualitative way from
be inserted from above without the liquid leaking through the diffuser. The resulting pattern was intercepted with the diffusive plane and then captured by the CCD camera ŽXC-57, Sony. and image processing system. The output of this system represented the light distribution on the tip surface, since the diffusive plane was a good diffusing surface. The experimental setup for the conical geometry is shown in Fig. 3. The irradiance on the conical bare core surface of the fiber tip was intercepted with an isotropic fiber optic detector connected to a power meter ŽModel 351B, UDT Instruments.. The output of this meter represented a measure of the power level around the near field of the bare tip. The irradiance distribution can be determined by measuring the power level at a series of points along the surface. The fiber scattering probe was made from a 400 mm core fused silica fiber with a 0.5 mm diameter scattering probe and possessed an approximately isotropic response. The scattering material of the probe was a UV epoxyrpowder mixture. The probe was controlled by a translator ŽLine Tool Co.. with three-dimensional positioning capability. The irradiance distribution was measured on three ‘‘sides’’ with respect to rotation.
4. Results and discussion Two series of the calculated intensity using Eq. Ž8. for the tips with cylindrical geometry are presented in Fig. 4. The effect of the convergence angle on the intensity is shown in Fig. 4Ža. Ži.e., a constant fiber size with various convergence angles., while the effect of the fiber size is shown in Fig. 4Žb. Ži.e., a constant convergence angle with different fiber sizes.. It is clear that the peak position of the intensity moves forward to the distal end of the fiber tip with either decreasing convergence angle or increasing
Fig. 4. Intensity from Eq. Ž8. as a function of position z Ža. for a fiber with diameter ds600 mm and various values of the convergence angle u and Žb. for a fiber with various diameters and a convergence angle u s 0.1 rad.
H. Shangguan, L.W. Caspersonr Optics Communications 152 (1998) 307–312
Eq. Ž8. by inserting the known z-dependence of the fiber radius Ž d s d 0 y 2 tan a z, with d 0 s 0.3 mm and a s 1.25 = 10y3 rad.. Although the model is quite simple, it evidently provides good agreement with experimental data. Two techniques were used to measure the intensity along the bare fiber tips with different geometries: Ži. CCD imaging technique and Žii. isotropic probe technique. Use of the CCD imaging technique has some advantages over the isotropic probe technique. First, it made the measurements easier, for otherwise it was difficult to position the probe very close to the surface without touching it because the core was transparent. Second, the measurements were quite reproducible compared with those using the probe. The variation of the measured data with the CCD imaging technique was less than 5% of the mean, while larger error bars were obtained using the probe technique, which are drawn in Fig. 6 presenting the standard deviation from the mean of three measurements. However, the CCD imaging
311
Fig. 6. Measured and calculated intensity normalized to the maximum value as a function of position along an unclad tapered fiber tip. The tip specifications are u s 0.087 rad, ds 300 mm, and a s1.25=10y3 rad. The error bars represent the standard deviation from the mean of three measurements.
technique was limited for measuring bare fiber tips with conical geometry due to the difficulty of orienting the tips. In this study we developed a theoretical model for calculation of the intensity distribution along the surface of unclad multimode fiber tips with various geometries and input coupling conditions. This model is based on the Gaussian-beam formalism. Although it is not intended to yield quantitative accuracy, it does yield quite good agreement with experimental results. A model of this type would be useful for the design and fabrication of fiberoptic devices used as diffusing tips for photodynamic therapy or scalpels in laser surgery. It is possible that generalizations of the model might improve its accuracy in the near field or make it better applicable for tapered configurations. Generalizations would also be required in circumstances in which the coatings or refractive index changes lead to multiple internal reflections.
Acknowledgements We thank Dr. Kenton W. Gregory for his generous support of this project. This work was also sponsored in part by the National Science Foundation under Grant No. PHY94-15583. One of the authors ŽLWC. also expressed his appreciation to members of the Rochester Theory Center for Optical Science and Engineering and The Institute of Optics at the University of Rochester for valuable discussions and hospitality during his sabbatical visit. Fig. 5. Measured and calculated intensity normalized to the maximum value as a function of position along an unclad fiber tip. The open and filled circles are the measured data, and the dashed and solid lines are the corresponding calculations using Eq. Ž8.. The fiber diameters and the convergence angles are labeled. The convergence angle in Ža. was u s 0.16 and in Žb. a fiber tip of 500 mm diameter was used.
References w1x M.S. Mcphee, C.W. Thorndyke, G. Thomas, J. Tulip, D. Chapman, W.H. Lakey, Lasers Surg. Med. 4 Ž1984. 93. w2x S.N. Joffe, Y. Oguro, Advances in Nd:YAG Laser Surgery, Springer, New York, 1987.
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w3x J.T. Allardice, A.C. Rowland, N.S. Williams, C.P. Swain, Gastrointest. Endosc. 35 Ž1989. 548. w4x L. Hasselgren, S. Galt, S. Hrad, Appl. Optics 29 Ž1990. 4481. w5x H. Shangguan, T.W. Haw, K.W. Gregory, L.W. Casperson, Novel cylindrical illuminator tip for ultraviolet light delivery, in: G.S. Abela ŽEd.., SPIE Proceedings of Diagnostics and Therapeutic Cardiovascular Interventions III, vol. 1878, 1993, pp. 167–182. w6x D.D. Royston, J.H. Torres, S. Thomsen, P.S. Sriram, A.J. Welch, Lasers Surg. Med. 16 Ž1995. 189. w7x R.M. Verdaasdonk, F.C. Holstege, D.E. Jasen, C. Borst, Lasers Surg. Med. 11 Ž1991. 213.
w8x R.M. Verdaasdonk, C. Borst, Appl. Optics 30 Ž1991. 2172. w9x A. Sagi, A. Shitzer, A. Katzir, S. Akselrod, Opt. Eng. 31 Ž1992. 1417. w10x H. Kogelnik, T. Li, Appl. Optics 5 Ž1966. 1550. w11x V.I. Talanov, Self focusing of wave beams in nonlinear media, Appl. Optics 2 Ž1965.. w12x C.J.R. Sheppard, T. Wilson, Microwaves Opt. Acoust. 2 Ž1978. 105. w13x F. Gori, G. Guattari, C. Padovani, Optics Comm. 64 Ž1987. 491. w14x M. Born, E. Wolf, Principles of Optics, Pergamon Press, New York, 1980.
Optics Communications 152 Ž1998. 307–312
Estimation of scattered light on the surface of unclad optical fiber tips: a new approach HanQun Shangguan a , Lee W. Casperson b
b,)
a Oregon Medical Laser Center, Portland, OR 97225, USA Rochester Theory Center for Optical Science and Engineering, and The Institute of Optics, UniÕersity of Rochester, Rochester, NY 14627, USA
Received 8 January 1998; revised 24 March 1998; accepted 25 March 1998
Abstract Unclad multimode optical fiber tips are often the basis for diffusing tips in photodynamic therapy or scalpels in laser surgery. In this study, we report an approximated Gaussian-beam model for the calculation of the light distribution along the surface of unclad fiber tips. The results are compared with detailed experimental measurements. The model yields quite good qualitative agreement with experimental results. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Photodynamic therapy; Fiber tips; Laser surgery
1. Introduction Optical fibers have been widely used to deliver laser radiation to the treatment sites for photodynamic therapy w1x or surgical cutting w2x. For example, an unclad fiber tip can be employed as a diffuser for photodynamic therapy when coated with a scattering material w3–5x or as a scalpel for surgical cutting when coated with an absorbing material w6x. In a recent article, we discussed the fabrication of cylindrical diffusers by coating scattering material on an unclad modified fiber tip w5x. The modified tip was made by covering the area where the intensity peak is located with hydrofluoric-acid-resistant materials during a chemical etching process. Verdaasdonk et al. demonstrated that accumulation of thermally coagulated tissue proteins and carbon on the tip surface is necessary to increase the absorption of radiation at the tip surface and achieve initial tissue ablation w7x. Thus, an absorbing coating for an
)
Permanent address: Department of Electrical and Computer Engineering, Portland State University, Portland, OR 97207, USA. E-mail: [email protected]
appropriate wavelength Že.g., Nd:YAG. on an unclad fiber tip may enhance the thermal effect w6x. This effect leads to the cutting of tissue with accompanying homeostasis. Furthermore, coating the quartz fiber tip with a thermally resistant material Ži.e., the material has a higher melting point than the quartz. at a hot spot may be an alternative method to reduce the thermal damage to the scalpel. Thus, a successful understanding of laser radiation along the bare core surface of an optical fiber tip is important for the design and fabrication of fiber-optic diffusers or scalpels. Unlike previous studies w4,8x, where ray tracing was used based on geometric optics, we first propose a new approach using a simplified model to calculate the irradiance distribution on the surface of unclad fiber tips. This model is based on an approximated Gaussian-beam formalism. The rigorous analysis of light propagation in multimode waveguides can be quite complicated. In a multimode optical fiber the output field distribution may depend sensitively on the input field and on the length and other parameters of the fiber. This complexity is in contrast to the single mode fiber, where the output distribution is essentially independent of the input and fiber length. Because of the difficulty of an exact treatment of multimode
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 1 7 1 - 0
H. Shangguan, L.W. Caspersonr Optics Communications 152 (1998) 307–312
308
fiber systems, there is value in developing more approximate models which might serve to describe at least the essential features of the multimode fiber output. The simplest physical optics solutions that one might seek to apply are those based on the Gaussian beam formalism. Also, most commercial lasers produce beams having Gaussian-shaped profiles w9x. However, in the near field a Gaussian beam would not vanish as quickly outside the fiber core as would the actual multimode field solutions. Thus, it is of interest to approximate the Gaussian beam in such a way that it can more closely represent the fields in the near-field region. The results are simple formulas which provide good qualitative agreement with measurements of intensity distributions along fiber tips. The basic equations governing the field distribution along a fiber tip are developed in Section 2. In their most practical form these equations give the intensity along the tip as a function of the core diameter and of the far-field diffraction angle of the unscattered beam. In Section 3 are reported experimental measurements of the light scattered from fiber tips, and the experimental and theoretical results are compared and discussed in Section 4. The approximate formulas are found to provide good agreement to within the accuracy of our experimental scattering measurements.
2. Model For any study of light propagation the fundamental starting point is the Maxwell-Heaviside equations. These equations can be combined to yield general wave equations which govern the field components of a propagating electromagnetic beam. In the past, solutions of these equations for optical fiber applications have sometimes been based on ray optics approximations. In this study we will instead start with analytic Gaussian beam solutions of the paraxial wave equation in a uniform medium and approximate these solutions to describe the field on the surface of an unclad fiber tip. The intensity distribution of a Gaussian beam in a homogeneous medium can be written as w10x I Ž r , z . s I0
w 02 2
w Ž z.
ž
exp y
2r2 w2 Ž z .
/
,
Ž1.
where I0 is the on-axis intensity at the waist of the beam and w Ž z . is the spot size or 1re amplitude radius. The spot size varies with the propagation distance z according to the relationship w Ž z . s w 0 1 q Ž zrz 0 .
2 1r2
,
Ž2.
where w 0 is the spot size at the beam waist and z 0 is the Rayleigh length. The Rayleigh length in turn is related to
the index of refraction n, the waist spot size w 0 , and the vacuum wavelength l by z 0 s np w 02rl. Eqs. Ž1. and Ž2. may be combined to obtain the explicit intensity formula IŽ r, z. s
I0 1 q Ž zrz 0 .
2
ž
exp y
2r2 w 02 1 q Ž zrz 0 .
/
2
.
Ž3.
If one is especially interested in the intensity distribution near the center of the beam, it is useful to replace the exponential in Eq. Ž3. by the first terms in a power series expansion. This type of approximation has long been used in studies of the interaction of Gaussian beams with nonlinear media w11x. Thus, we may write Eq. Ž3. approximately as IŽ r, z. s
I0 1 q Ž zrz 0 .
2
ž
2r2
1y
w 02 1 q Ž zrz 0 .
2
/
.
Ž4.
For a finite value of r this result becomes increasingly accurate as z increases, i.e., for z 4 z 0 . For small z it has the important property of going to zero at the radius r s w 0r '2 . This property is useful in finding an approximate representation for the intensity in the vicinity of a fiber tip, because that field also goes almost to zero at the cladding boundary. If the tip consists of the unclad fiber core in a medium having the same index of refraction, we can tentatively represent the intensity on the tip surface by the simple formula IŽ r, z. s
I0 1 q Ž zrz 0 .
s I0
2
1
1y
1 q Ž zrz 0 .
Ž zrz 0 . 2 1 q Ž zrz 0 .
2 2
.
2
Ž5.
To a first approximation, this formula should also represent the z dependence of the scattered intensity when the core is coated with appropriate scatterers. It may be noted that other established beam modes have an intrinsic field zero at a finite beam radius without the need for a quadratic beam approximation. For example, the next higher radial order of the Laguerre-Gaussian beam family also has a zero at the radius r s w 0r '2 . Similarly, the J0 Bessel-Gaussian beam has zeroes at distinct radii w12,13x. Such beams could be imagined to match an input field distribution over the finite fiber core without any approximation of the beam profile. On the other hand, in their truncated form these other field distributions do not propagate as beam modes, so a rigorous study based on such input beams would be more complicated. For this reason it has seemed appropriate here to use the simplest possible physical optics approximation as represented by Eqs. Ž4. and Ž5.. For reasons that will become clear, it is also useful to express the intensity distribution in terms of the diffraction
H. Shangguan, L.W. Caspersonr Optics Communications 152 (1998) 307–312
309
3. Experiments
Fig. 1. Schematic of unclad fiber tips with a cylindrical geometry Ža. and a conical shape Žb..
angle of the output beam. Using Eq. Ž2., the diffraction angle of a Gaussian beam can be written as
l
dw
ud s lim
s
Ž z™` .
dz
np w 0
.
Ž6.
With this substitution the Rayleigh length is w 0 s z 0 u . But the diffraction angle, ud , at the output is equal to the convergence angle, u at the input, and thus Eq. Ž5. can be written as
Ž zurw 0 . 2
I Ž r , z . s I0
1 q Ž zurw 0 .
2 2
s I0
2 Ž zur'2 r . 2 1 q Ž zur'2 r .
2
2
s I0
Ž '2 zurd . 2 1 q Ž '2 zurd .
,
2
Ž7.
To verify the working of the model described above, two experimental protocols were performed: Ži. irradiance distributions along the bare core surface of fiber tips with cylindrical geometry were measured using a CCD camera system and Žii. an isotropical ball-scattering probe was used for the conical geometry. The experimental results were then compared with the calculations. Step-index multimode silica-clad silica-core fibers ŽNA s 0.22. with 300–600 mm core diameters and 2 m lengths were used. One of the distal faces served as the input end and was wet-polished using standard fiber polishing techniques. At the other end, the protective buffer was thermally removed over a length of 20 mm. The tip was then fabricated into a cylindrical or conical taper using a chemical etching technique w5x. The geometry of the tip was measured using optical microscopy. The experimental setup for the cylindrical geometry is shown in Fig. 2. The tip was attached to a diffusive plane which was made of a piece of quartz glass Ž100 mm thick. coated with UV epoxy ŽLitetak 376, Loctite Co.. mixed with Al 2 O 3 powder Žthe weight ratio of UV epoxy to powder was 10 to 3.. The thickness of the layer was about 50 mm. For ease of adjusting the convergence angle, we used fiber–fiber coupling. Both distal faces of the coupling fiber, a silica optical fiber with 400 mm core diameter, were wet-polished flat. The various convergence angles were obtained by adjusting the distance between the two fibers. A narrow bandpass filter Ž l s 404.7 nm. was used in this experiment and an arc lamp source ŽOriel Co.. was used as the light source. The beam profile was similar to that of a conventional Gaussian beam. An index matching fluid ŽR.P. Cargille Laboratories, Inc.., whose refractive index was very close to that of the fiber core, was used in this experiment. According to Eq. Ž43. in Ref. w14x, the transmissivity was almost equal to one, and the second reflection is negligible. The large viscosity and surface tension of the index matching fluid permitted us to directly drop the liquid on a glass diffuser, and then the fiber could
where r s w 0r '2 represents the fiber core radius discussed above and d s 2 r is the core diameter. This geometry is illustrated in Fig. 1. For comparison with experimental data, it is convenient to normalize the intensity to its maximum value. Setting d Ird z to zero, one finds that the maximum occurs at the distance z s dr '2 u , at which distance the intensity is I0r4. Thus, we introduce the normalized intensity X
I Ž z. s4
Iz I0
s
4 Ž '2 zurd .
2
1 q Ž '2 zurd .
2 2
.
Ž8.
This formula is now in an appropriate form for comparison with experimental data.
Fig. 2. Experimental setup for measuring the light distribution on the bare core surface of a fiber tip with cylindrical geometry.
310
H. Shangguan, L.W. Caspersonr Optics Communications 152 (1998) 307–312
Fig. 3. Experimental setup for measuring the light distribution on the bare core surface of a fiber tip with conical geometry.
fiber size. This suggests that the effects of the input coupling and fiber size on the distribution of the scattered light on the surface of bare fiber tips should be taken into the account for the purposes mentioned in Section 1. For the tip with conical geometry, the calculation using Eq. Ž8. by inserting the known z-dependence of the fiber radius Ž d s d 0 y 2 tan a z . indicates that the intensity is almost independent of the angle a Ždata not shown.. This suggests that the coupling condition and the fiber size determine the peak position when the light is directly scattered from bare fiber tips without the second reflection on the surface. To verify the model described in Section 2, experiments were performed on unclad fiber tips with cylindrical or conical geometry under various conditions such as different convergence angle and fiber size. A typical comparison between the measurements and calculations for tips with cylindrical geometry is presented in Fig. 5. Fig. 6 represents the case of a conical tip. Note that the calculated intensity in Fig. 6 was obtained in a qualitative way from
be inserted from above without the liquid leaking through the diffuser. The resulting pattern was intercepted with the diffusive plane and then captured by the CCD camera ŽXC-57, Sony. and image processing system. The output of this system represented the light distribution on the tip surface, since the diffusive plane was a good diffusing surface. The experimental setup for the conical geometry is shown in Fig. 3. The irradiance on the conical bare core surface of the fiber tip was intercepted with an isotropic fiber optic detector connected to a power meter ŽModel 351B, UDT Instruments.. The output of this meter represented a measure of the power level around the near field of the bare tip. The irradiance distribution can be determined by measuring the power level at a series of points along the surface. The fiber scattering probe was made from a 400 mm core fused silica fiber with a 0.5 mm diameter scattering probe and possessed an approximately isotropic response. The scattering material of the probe was a UV epoxyrpowder mixture. The probe was controlled by a translator ŽLine Tool Co.. with three-dimensional positioning capability. The irradiance distribution was measured on three ‘‘sides’’ with respect to rotation.
4. Results and discussion Two series of the calculated intensity using Eq. Ž8. for the tips with cylindrical geometry are presented in Fig. 4. The effect of the convergence angle on the intensity is shown in Fig. 4Ža. Ži.e., a constant fiber size with various convergence angles., while the effect of the fiber size is shown in Fig. 4Žb. Ži.e., a constant convergence angle with different fiber sizes.. It is clear that the peak position of the intensity moves forward to the distal end of the fiber tip with either decreasing convergence angle or increasing
Fig. 4. Intensity from Eq. Ž8. as a function of position z Ža. for a fiber with diameter ds600 mm and various values of the convergence angle u and Žb. for a fiber with various diameters and a convergence angle u s 0.1 rad.
H. Shangguan, L.W. Caspersonr Optics Communications 152 (1998) 307–312
Eq. Ž8. by inserting the known z-dependence of the fiber radius Ž d s d 0 y 2 tan a z, with d 0 s 0.3 mm and a s 1.25 = 10y3 rad.. Although the model is quite simple, it evidently provides good agreement with experimental data. Two techniques were used to measure the intensity along the bare fiber tips with different geometries: Ži. CCD imaging technique and Žii. isotropic probe technique. Use of the CCD imaging technique has some advantages over the isotropic probe technique. First, it made the measurements easier, for otherwise it was difficult to position the probe very close to the surface without touching it because the core was transparent. Second, the measurements were quite reproducible compared with those using the probe. The variation of the measured data with the CCD imaging technique was less than 5% of the mean, while larger error bars were obtained using the probe technique, which are drawn in Fig. 6 presenting the standard deviation from the mean of three measurements. However, the CCD imaging
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Fig. 6. Measured and calculated intensity normalized to the maximum value as a function of position along an unclad tapered fiber tip. The tip specifications are u s 0.087 rad, ds 300 mm, and a s1.25=10y3 rad. The error bars represent the standard deviation from the mean of three measurements.
technique was limited for measuring bare fiber tips with conical geometry due to the difficulty of orienting the tips. In this study we developed a theoretical model for calculation of the intensity distribution along the surface of unclad multimode fiber tips with various geometries and input coupling conditions. This model is based on the Gaussian-beam formalism. Although it is not intended to yield quantitative accuracy, it does yield quite good agreement with experimental results. A model of this type would be useful for the design and fabrication of fiberoptic devices used as diffusing tips for photodynamic therapy or scalpels in laser surgery. It is possible that generalizations of the model might improve its accuracy in the near field or make it better applicable for tapered configurations. Generalizations would also be required in circumstances in which the coatings or refractive index changes lead to multiple internal reflections.
Acknowledgements We thank Dr. Kenton W. Gregory for his generous support of this project. This work was also sponsored in part by the National Science Foundation under Grant No. PHY94-15583. One of the authors ŽLWC. also expressed his appreciation to members of the Rochester Theory Center for Optical Science and Engineering and The Institute of Optics at the University of Rochester for valuable discussions and hospitality during his sabbatical visit. Fig. 5. Measured and calculated intensity normalized to the maximum value as a function of position along an unclad fiber tip. The open and filled circles are the measured data, and the dashed and solid lines are the corresponding calculations using Eq. Ž8.. The fiber diameters and the convergence angles are labeled. The convergence angle in Ža. was u s 0.16 and in Žb. a fiber tip of 500 mm diameter was used.
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