Coupling efficiency of optics in single-mode fiber
`components
`
`R. E. Wagner and W. J. To’mlinson
`
`Many single-mode fiber components include some form of optics, such as lenses or mirrors, for collecting
`light from a source fiber or laser and concentrating it on a receiving fiber. For such components there is a
`direct and simple relationship between coupling efficiency and optical aberrations. This paper combines
`fiber-coupling fundamentals, classical optics, and diffraction theory to provide a compact description of cou-
`pling efficiency that includes the effects of aberrations, fiber misalignments, and fiber-mode mismatch.
`
`l.
`
`Introduction
`
`In response to rapid advanCes in optical fiber tech-
`nology, workers around the world are developing a wide
`variety of passive optical components, such as couplers,
`switches, and wavelength multiplexers, for manipu-
`lating and processing the signals in fibers; and signifi-
`cant interest in single-mode fibers and long-wavelength
`laser diodes has created a need for such components
`that can be used with single-mode fibers. Most of these
`components include some form of optics, such as lenses
`or mirrors, for collecting light from a source fiber or a
`laser and concentrating it on a receiving fiber or fi-
`bers.
`
`These components always exhibit optical-coupling
`loss, which is completely determined by the degree to
`which the optics depart frOm ideal, that is, by the ab-
`errations and misalignments of the optical system.
`This means that a component designer, in the process
`of optimizing coupling efficiency, can make beneficial
`use of the enormous body of knowledge that already
`exists concerning classical optical imaging systems.
`This paper develops the relationship between optical
`aberrations and coupling efficiency for single-mode
`fiber components. The result provides a compact de-
`scription of coupling efficiency and includes the effects
`not only of aberrations but also of fiber misalignments
`
`The authors are with Bell Laboratories, Holmdel, New Jersey
`07733.
`Received 16 January 1982.
`0003-6935/82/152671-18$01.00/0.
`© 1982 Optical Society of America.
`
`It is equally applicable
`andof fiber-mode mismatch.
`to fiber-to-fiber coupling and to laser-to—fiber coupling.
`Although the present analysis is restricted to single-
`mode fiber components, similar techniques are appli-
`cable to multimode components.
`In Sec. II there is an overview of the optical-coupling
`problem including a general description of the problem,
`an outline of the steps involved in its solution, the basic
`result, and noteworthy features of the result. This is
`followed by a series of definitions that establish the
`similarities and differences between the coupling
`problem and classical optical imaging, and then by a
`more formal development that is based on rigorous
`diffraction theory. However, the formal development
`emphasizes a qualitative understanding of the problem
`and solution, leaving many of the details for an ap-
`pendix.
`In the remaining sections the basic results are used
`to calculate coupling efficiency for some practical sit-
`uations that arise in component design. These include
`the effects of fiber misalignments, third-order aberra-
`tions, and random wave front perturbations.
`In some
`cases the calculations involve numerical evaluation of
`an integral, but in several cases the solutions are in
`closed form.
`
`ll. Overview
`
`The basic problem, illustrated in Fig. 1, is to calculate
`the power-coupling efficiency between two fibers, or a
`laser and a fiber, coupled by an optical system. Light
`diverging from a source fiber (or laser), located at the
`left of the figure, passes through the optical system,
`which may consist of lenses and other elements. These
`convert the diverging beam into a converging beam that
`forms an aberrated image of the source fiber (or laser).
`A receiving fiber is located near the image, so some of
`the beam is coupled into the receiving fiber. The
`problem, then, is to relate the power-coupling efficiency
`
`1 August 1982 / Vol. 21, No. 15 / APPLIED OPTICS
`
`2671
`
`Petitioner Ciena Corp. et al.
`
`Exhibit 1037-1
`
`

`

`SOURCE FIBER
`(OR LASER)
`DNERGING
`
`
`
`BEAM
`
`
`OPTIC AXIS
`
`CONVERGING
`BEAM
`
`RECENING
`FIBER
`
`
`
`OPTICAL
`
`SYSTEM
`IMAGE 0F
`
` SOURCE FIBER
` EXIT
`PUPIL
`
`'Fig. 1. Coupling from a source fiber (or laser) to a receiving fiber
`through a general optical system. The receiving fiber lies near an
`image of the source fiber.
`
`to the fiber positions and characteristics and to the
`classical optical system properties.
`The analysis is independent of whether the source is
`a fiber or a laser so, for simplicity, the balance of this
`paper only refers to the source as a fiber.
`Although the fiber-coupling efficiency problem is
`similar in many respects to the classical optical imaging,
`problem, there is an important difference. This dif-
`ference is that the single-mode source fiber, lens, and
`receiving fiber together comprise a coherent system,
`rather than an incoherent system with a Lambertian
`source as is most often encountered in classical optics.
`This means that field distributions and the coherent
`optical transfer function are of primary concern, rather
`than irradiance distributions and the incoherent optical
`transfer function. Despite this difference, much of the
`insight gained from classical optics can be directly ap-
`plied to the coupling efficiency problem.
`'
`i
`In principle, it is possible to make a lossless passive
`component that will provide perfect coupling to the
`receiving fiber provided the source radiates into a single
`spatial mode of the radiation field. However, it is not
`possible to provide lossless coupling to a single-mode
`receiving fiber if the source radiates into more than one
`mode (the classical Lambertian source is an extreme
`example of this), or even if the source always radiates
`into a single mode but not always into the same spatial
`mode.
`Formally, the coupling efficiency relationships are
`developed in a series of steps that correspond to deter-
`mining the field distributions of the light at various
`planes in the optical system (see Fig. 2). The field
`distribution on the source plane, which is located at the
`end of the source fiber, is closely related to the mode
`pattern of the fiber. From this source field, diffraction
`theory is used to determine the field that would exist at
`an artificial plane within the optical system, called the
`entrance-pupil plane.
`The field distribution at a second such plane, called
`the exit-pupil plane, is then determined as a transfor-
`mation of the entrance-pupil field distribution. The
`transformation is governed by the ideal paraxial imag-
`ing properties of the optical system and by the coherent
`optical transfer function, a multiplicative term that
`
`2672
`
`APPLIED OPTICS / Vol. 21. No. 15 / 1 August 1982
`
`RECEIVING
`
`SOURCE
`
`
`
`
`SOURCE
`FLAME
`
`EXIT PUPIL
`PLANE
`
`OPTICAL IMAGING
`ENTRANCE
`PUPIL
`SYSTEM
`PLANE
`
`Fig. 2. Overview of the geometry involved in fiber-to-fiber (or
`laser-to-fiber) coupling through an optical imaging system. Four
`planes are important:
`the source and image planes containing the
`two fiber endfaces, and the entrance- and exit—pupil planes of the
`optical system.
`
`describes the imperfections, or aberrations, of the op-
`tical system.
`The exit-pupil field distribution produces an image
`field distribution, again calculated by diffraction theory,
`at the image plane located at the endface of the receiv—
`ing fiber. The power-coupling efficiency‘is then the
`squared modulus of the overlap integral of the image
`field distribution and the mode pattern of the receiving
`fiber.
`Conceptually, this description of coupling efficiency
`is\convenient because it corresponds directly to the
`propagation of light through the optical system from
`source fiber to receiving fiber.. The result, however,
`involves the evaluation of a triple integral—one for the
`overlap and one for each of the two diffraction calcula-
`tions. For computational purposes the overlap integral
`can be transformed to any other convenient plane in the
`optical system.
`If it is transformed to the exit pupil
`plane, the power-coupling efficiency T can be expressed
`as a single integral with an integrand that is the product
`of three simple terms:
`the far—field distribution of the
`source-fiber mode pattern ‘I/s, the far-field distribution
`of the receiving-fiber mode pattern \I/R, and the coher-
`ent transfer function of the optical system L.
`T = | f \IIZgLW'RdaP.
`.
`
`(1)
`
`In Eq. (1) the three terms ‘I/g, \I’};, and L are all directly
`measurable quantities and are related to fiber and op—
`tical system parameters in a familiar way. As a conse-
`quence the overlap integral expressed in the exit pupil
`most easily provides insight into the coupling efficiency
`problem and is most convenient for coupling efficiency
`calculations.
`' Several features of this result are particularly note-
`worthy. First, a single simple formulation accounts for
`lens aberrations, fiber misalignments, and mechanical
`alignments of the various device components. Even
`complicated effects such as nonflat fiber endfaces can
`be included if the appropriate far-field distribution is
`used. Second, when the calculations are performed at
`the exit pupil of the optical system, only one integration
`is required instead of three. This is because the two
`diffraction integrals that relate the source field distri-
`
`Petitioner Ciena Corp. et al.
`
`Exhibit 1037-2
`
`

`

` t—z—i
`.f‘
`
`RECEMNG
`FIBER
`
`Fig. 3. Details of the geometry for analysis of the coupling-efficiency problem illustrating the first-order properties of the optical system.
`Included are the source—fiber position S, its image position S’, and the nearby receiving-fiber position I. Also shown are the source plane
`(3), the entrance—pupil plane (e), the exit-pupil plane (2’), the receiving-fiber image plane (i), and appropriate points and coordinates in those
`‘
`planes.
`
`bution to the image field distribution are eliminated.
`Third, all three factors involved in the coupling effi-
`ciency integral are directly measurable experimentally,
`using various scanning and interferometric techniques.
`This permits a direct comparison between experimental
`and theoretical results. Fourth, the far-field distribu-
`tions, which are needed for the coupling efficiency cal-
`culations in the exit pupil, are easier to measure than
`the actual mode patterns of the fibers, which would be
`needed if the calculations were to be considered as a
`' triple integral at the image plane. Finally, the formu-
`lation presents a clear and simple understanding of how
`various imperfections will affect coupling efficiency.
`Ill. Definitions
`
`To be useful in the context of classical optics, the
`formal development which follows in Sec. IV must be
`based on geometrical constructions that are consistent
`with accepted conventions. These conventions, which
`involve the source plane, the entrance- and exit-pupil
`planes, the image plane, and the aberrations, are re-
`viewed in this section, emphasizing the similarities and
`differences between the fiber-coupling case and classical
`optics. A list of the symbols used in this paper, and
`their definitions, is given in Appendix B. For readers
`who are interested in a more complete description of
`classical optical analysis techniques we recommend the
`books by Born and Wolf,1 and Welford.2
`
`A.
`
`Source and Image Planes
`
`As a Visual aid for the definitions that follow, refer to
`
`the sketch of Fig. 3. This sketch contains two arbi-
`trarily positioned fibers and an optical system repre-
`sented schematically by its first and last surfaces with
`all internal surfaces omitted. The optic axis is AA’, and
`the center of the source- and receiving-fiber endfaces
`are S and I, respectively. The point S, located on the
`source—fiber endface, has a paraxial image 8’ which lies
`near the receiving fiber. The point I, located on the
`receiving fiber, does not necessarily coincide with the
`paraxial image point S ’, because the receiving fiber
`many not be perfectly aligned with the paraxial image
`8". The source plane 3 contains the point S, while the
`image plane i contains the point I; both planes are
`perpendicular to the optic axis AA’. The image plane
`is defined to contain the point I and not S’, because
`coupling calculations involve the field distribution that
`exists at the fiber endface.
`
`The displacements from the optic axis of the points
`S, S’, and I are represented by the vectors ii, 17’, and 1‘1;
`respectively, with corresponding magnitudes 1], 11’, and
`m.
`
`B. Aperture Stop
`
`The stop is important in classical imaging systems
`because it affects imaging properties of the system.
`In
`the fiber-coupling case some of this importance is lost,
`although the stop still serves as a useful reference for the
`definition of the entrance- and exit«pupil planes and the
`system aberrations.
`The stop is an aperture within the system that
`physically limits the size of the cone of light that is ac-
`1 August 1982 / Vol. 21, No. 15 / APPLIED OPTICS
`2673
`
`Petitioner Ciena Corp. et al.
`
`Exhibit 1037-3
`
`

`

`FIELD DISTRIBUTIONS
`
` e:
`iRECEIVING
`FIBER
`
`Fig. 4. Geometry for the fiber-coupling analysis, illustrating the source-fiber field distributions that occur in the entrance-pupil (e) and exit-pupil
`(e’) planes. These distributions may be de-centered by the amounts X” and 2;, if the source-fiber axis is not directed toward the center of
`the entrance pupil.
`
`In partic-
`cepted from any point on the source plane.
`ular, one of the lens components, or an aperture delib-
`erately introduced for this purpose, limits the light from
`the axial point 0, which is the intersection of the optic
`axis and the source plane. This particular element is
`called the stop.
`In classical optical systems, where the stop is smaller
`than the cone of light from the source, the size and lo-
`cation of the stop offer some measure of control on
`image quality, because these affect both optical system
`throughput and the aberrations. By contrast, the stop
`in single—mode fiber-coupling systems is ordinarily large
`enough that it does not physically restrict the beam,
`since the goal is to couple as much light as possible from
`one fiber to the other, and coupling efficiency cannot
`be increased by removing light.
`Since the fiber characteristics, rather than the stop
`itself, set the angular spread and hence the physical
`extent of the beam, the location of the stop is somewhat
`arbitrary. As defined, hOWever, it is consistent with the
`established convention in classical optics and, therefore,
`also useful for establishing the entrance- and exit-pupil
`planes and the system aberrations for the fiber-coupling
`case. Also, with this definition of the stop, its position
`does not depend on the characteristics or orientation of
`the input fiber, so that the aberrations of the optical
`system can be defined and measured without reference
`to the particular fiber configuration.
`
`C. Entrance- and Exit-Pupil Planes
`
`The entrance and exit pupils are two artificial planes
`that allow the imaging properties of the optical system
`
`2674
`
`APPLIED OPTICS / Vol. 21,No. 15 / 1 August 1982
`
`to be described without direct reference to the actual
`lens surfaces or the stop. The entrance and exit pupils
`are images of the stop as seen from the source and re-
`ceiving sides of the optical system, respectively. They
`are represented in Fig. 3 by the planes e and e’, and
`these two planes are scaled images of each other.
`Two rays, the marginal and the chief, define the
`connection between the source and entrance-pupil
`planes and an analogous connection between exit-pupil
`and image planes. The marginal ray OP, and its image
`P’O’, originates at the axial point 0 and passes through
`the edge of the stop. This ray defines the numerical
`aperture (N.A.) of the optical system and the pupil
`magnification. The chief ray, so called because it is the
`central ray for the cone of light accepted by the system,
`originates at an off-axis point, such as S, and passes
`through the center of the stop.
`(If the source fiber lo-
`cation S coincides with the axial point 0, some other
`point on the source plane must be chosen to define the
`chief ray. The choice is completely arbitrary.) This
`ray defines the lateral magnification of the system.
`Since the entrance and exit pupils are images of the
`stop, the chief ray also passes through their centers, Q
`and Q’. Ordinarily the distribution of light in the exit
`pupil is controlled by the stop, which is usually centered
`on the optic axis.
`In the fiber-coupling case, however,’
`the source fiber controls the distribution of light in the
`exit pupil as shown in Fig. 4, and this distribution is not
`necessarily centered in the exit pupil.
`It is well to remember that both the entrance-pupil
`plane e and the exit-pupil plane e’ are artificial con-
`structs. Their locations are fixed, however, by the ac-
`
`Petitioner Ciena Corp. et al.
`
`Exhibit 1037-4
`
`

`

`tual lens surfaces which determine the paths of rays
`through the system.
`In what follows, the first-order
`imaging characteristics and the optical aberrations are
`completely described in terms of these artificial planes
`e and e’ without direct reference to the stop position or
`actual lens surfaces.
`
`D. Coordinate System
`
`For diffraction calculations a coordiante system is
`established in which the z axis is the optic axis AA’.
`It
`is convenient to define four independent coordinate
`systems corresponding to each of the four planes, all
`with a common 2 axis. The vectors X5, X9, Xe, and X,-
`denote (x,y) coordinates in these systems, where the
`subscripts refer to the source plane, the entrance-pupil
`plane, the exit-pupil plane, and the image plane, re-
`spectively (see Fig. 3).
`
`E.
`
`First-Order Imaging Properties
`The optical properties of interest are the image and
`pupil magnifications, because these determine the lo—
`cation of the image and the extent of the converging
`wave in the exit pupil for a given source-fiber position
`and NA. The image magnification m is 11’/n, where n
`and 1)’ are the distances from S and S’ to the z axis, re-
`spectively (see Fig. 3). The pupil magnification me is
`h’/h, where h and h’ are the distances from the points
`P and P’ to the optic axis, respectively. The pupil
`magnification can alternately be expressed as me = 17/n’
`- z’/z = z’/(mz), where z and 2’ are the distances from
`plane 8 to e and from plane e’ to the paraxial image
`plane, respectively.
`
`F. Wave-FrontAberrations
`
`The wave-front aberrations of the optical system
`represent perturbations of the system from its first-
`order characteristics. These aberrations are expressed
`in the exit pupil as shownIn Fig. 5. The wave-front
`aberrations are defined as the optical path difference
`W(Xe) between the actual wave-front 2’ existing1n the
`exit pupil and an ideal spherical reference wave--front
`2 that converges toward point I. The vertex of the
`reference wave--front 21s taken at Q’, so the radius of
`the reference wave front15 R,= Q’I. The aberrations
`W(X,,) are measured on the reference wave——front 2, but
`at pupil coordinates Xe projected onto the exit pupil e’.
`This definition of wave--front aberrations1s consistent
`with that normally used in optical lens design programs,
`in which the reference sphere is taken to have infinite
`radius.3 Such consistency requires, however, that R,-
`be sufficiently large, a restriction imposed for other
`reasons in the coupling efficiency analysis [see Appendix
`A, Eq. (A17)] and satisfied by virtually all practical
`single-mode lens—coupling systems.
`For a perfect optical system a diverging spherical
`wave in the entrance pupil produces a converging
`spherical wave in the exit pupil. Even1n this case,
`however, the wave--front aberrations W(Xg ) may not be
`zero, because the center I of the reference wave-front
`2 does not necessarily coincide with the image point 8’
`toward which the actual spherical wave-front 2’ con-
`
`x,
`
`
`
`x.
`
`ACTUAL ABERRATED
`WAVEFRONT 2’
`REFERENCE WAVEFRONT
`2 (CENTERED ON 1),
`
`
`
`RECEIVING
`
`,
`.
`FIBER
`8
`I
`
`Image-space geometry for the coupling-efficiency problem
`Fig. 5.
`illustrating an ideal reference wave-front 2‘ that converges toward the
`receiving fiber (point I) and the actual wave-front 2’ emerging from
`the exit pupil. The difference between the two is called the wave-
`front aberration W(X;). Such aberrations cause a reduction in
`coupling efficiency because they represent an improper phase between
`the two distributions to be coupled.
`
`verges. Such aberrations are real in the sense that they
`cause coupling losses at the receiving fiber and are
`characterized as fiber misalignment aberrations.
`If the
`optical system contains wave-front aberrations of its
`own, these contribute to the total wave-front aberra-
`tions as well. Both optical system wave-front aberra-
`tions and misalignment aberrations are described in the
`exit pupil, allowing both coupling effects to be handled
`simultaneously in a manner consistent with accepted
`convention in classical optics.
`
`G. Coherent Optical Transfer Function
`
`The coherent optical transfer function L(Xe) relates
`the exit-pupil field distribution to the scaled en-
`trance-pupil distribution.
`In the geometlical optics
`approximation it is given as L = exp[—ik W(X;,)], where
`diffraction within the optics is neglected. The neglect
`of diffraction between the two pupil planes is the major
`simplifying assumption involved in this development.
`In many cases this approximation is sufficiently accu-
`rate, but in special cases, such as when the beam is ex—
`tremely small or when the stop is far from the lens sur-
`faces, diffraction within the optics must be consid—
`ered.
`
`H. Third-Order Aberrations
`
`In classical optics itIS quite often useful to express
`W(X,) as a polynomial expansion of theImage height
`17’ and of normalized exit-pupil coordinates Xe/h’. The
`normalized exit-pupil coordinates are usually repre-
`sented by polar coordinates (10,111), where d) is the angle
`measured from the plane that contains the optic axis
`AA’ and the source point S.
`For a symmetrical optical system the polynomial
`expansion for wave-front aberrations carried to the
`third-order is
`
`1 August 1982 / Vol. 21, No. 15 / APPLIED OPTICS
`
`2675
`
`Petitioner Ciena Corp. et al.
`
`Exhibit 1037-5
`
`

`

`W(P:¢) = (Wozopz)
`+ szofl'292
`+ W040!)4
`+ W22217'2112 005%
`+ (W011!) Simb)
`+ (Wow/1 COSdJ)
`+ Wailfl'sp coed)
`+ W131n’p3 cosdJ
`+....
`
`Focal shift (ER’)
`Field curvature
`Spherical aberration
`Astigmatism
`Lateral shift (63:)
`Lateral shift (6y)
`Distortion
`Coma
`
`'(2)
`
`In Eq. (2) the terms in parenthesis correspond to linear
`image shifts (fiber misalignments) and the remaining
`terms are the ordinary third-order aberrations.
`According to Eq. (1) the coupling efficiency T is ex-
`pressed as an integral in the exit pupil, so it is conve-
`nient to factor those terms in Eq. (2) that have identical
`exit—pupil dependences.
`It is also convenient to nor-
`malize the coordinates according to the size of the
`source-fiber far-field distribution in the exit-pupil h;
`(see Fig. 4), rather than to the size of the stop h’, for
`reasons that will become apparent in Secs. V and VI.
`The resulting polynomial expansion is
`
`W(Ye/h;) = W20r2 + Water“
`+ ler cosqS + W31r3 coso + Wlxr simt
`+ er2 code) + .
`.
`. ,
`
`(3)
`
`In thiscase the image height 11’ does
`where r = [Ye/h; | .
`not appear explicitly in the expansion but enters im-
`plicitly through the new coefficients W20, le, W31, and
`W22.
`A term-by-term comparison of Eqs. (2) and (3) indi-
`cates the correspondence between the two sets of coef—
`ficients:
`
`Wlx = Wont)“
`
`le = (Weiy + W311n’3)ps.
`
`W20 = (W020 + W22071’2)P§,
`
`W40 = Wowpg,
`
`W31 = Wial'fl'pg.
`
`W22 = W22211'2P3,
`
`(4)
`
`where p8 = h'S/h’. The coefficients from Eq. (3) all have
`dimensions of length, while some of those from Eq. (2)
`do not. The coefficients from Eq. (3) enter the coupling
`efficiency integral, Eq. (1), as parameters. According
`to Eq. (4), they depend on the ordinary third-order
`aberration coefficients from Eq. (2), the image height
`11’, and the size of the source distribution relative to the
`stop ps.
`
`Formal Development
`IV.
`This section establishes the relationship between
`power-coupling efficiency and the fiber and optical
`system properties.
`It is based on the scalar diffraction
`theory, with a geometry as defined in Sec. III. Readers
`who are interested in a thorough description of dif-
`fraction theory fundamentals should take advantage of
`the excellent texts by Goodman4 and Gaskill.5
`
`2676
`
`APPLIED OPTICS / Vol. 21, No. 15 / 1 August 1982
`
`A. Source-Plane Distribution
`
`The source-fiber mode pattern is denoted by ¢S(Y),
`and it i_s_norn_1_alized so that its total power is unity (i.e.,
`j' Ms (X) | 2dX = 1). The mode pattern (05 determines
`the field distribution at the source-fiber endface and
`thus at the source plane. For a flat fiber endface these
`two distributions differ only by a position_shift wax,
`- '71) and by a linear phase factor exp [ik (Xs -- T1) - fs]
`that depends on the angle between the fiber axis FS and
`the source plane [see Appendix A, Eq. (A2)]. Here f3
`is a unit vector parallel to the fiber axis FS (see Fig.
`4).
`
`B. Entrance-Pupil Distribution
`The light from the source fiber propagates to the
`entrance pupil, forming a diverging wave field there that
`is described by Fraunhofer diffraction of the source-
`plane field distribution ¢5(Z — 1'1) exp[ik(X, — 71) - f8].
`Theresulting field distribution in the entrance pupil,
`\I’S (Xe — X05), is the far-field distribution of the fiber.
`The distribution W5 is not necessarily centered in the
`entrance_pupil but in general is decentered by the
`amount X03 (see Fig. 4).
`For a flat fiber endface the entrance-pupil distribu-
`tion is simply the scaled Fourier transform of the mode
`pattern, but multiplied by a phase term that represents
`a spherical wave diverging from point S [see Appendix
`A, Eq. (AE3)]. This field distribution ‘I/sOTe -— KM) is
`positioned on the point in the entrance pupil where the
`fiber axis FS extended intersects the entrance—pupil
`plane e. This positioning is fixed in the diffraction
`equations by the linear phase factor exp [ik (Y, — 17) - f5]
`[see Appendix A, Eqs. (A2)—(A4)].
`C.
`Far-Field Measurements
`
`The far-field distribution ‘Ifs can usually, but not
`always, be measured directly. The far—field power
`distribution I ‘113 | 2 can always be measured directly by
`square-law detection, but such measurements destroy
`any phase variations that the far—field distribution
`‘I/s(Xe) may contain. The distinction is relatively
`unimportant, however, whenever the far-field distri-
`bution is real, such as when the fiber-mode pattern has
`radial symmetry [i.e., whe_r_1 ¢S(X) = ¢S(—m.
`If the
`far-field distribution ‘I’s(Xg) does contain phase vari-
`ations, these can be recovered using interferometric
`measurement techniques.6
`
`D.
`
`Exit-Pupil Distribution
`
`The far-field distribution in the entrance pupil pro-
`duces a corresponding distribution in the exit pupil (see
`Fig. 4), which then propagates to the image plane to
`form a field distributigl ther_e.
`In the exit pupil the
`field distribution \II'S(X'e — X;,) is the product of a
`scaled, but converging,_version of the entrance-pupil
`distribution ‘I/S(Xe — X03) and the coherent optical
`transfer function.
`As a result, the exit-pupil field distribution is the
`scaled far-field distribution of the source fiber multi-
`plied by two additional phase terms—one representing
`a spherical wave converging to the paraxial image S’ and
`
`Petitioner Ciena Corp. et al.
`
`Exhibit 1037-6
`
`

`

`the other equal to the coherent transfer function of the
`optical system alone [see Appendix A, Eq. (A6)]. To
`be consistent with previous comments concerning the
`proper image plane these two phase terms are adjusted
`so that the first term represents a spherical wave con-
`verging to the point I (see Fig. 5) and the second is a
`transfer function that includes both optical system and
`misalignment aberrations [see Appendix A, Eqs. (A13)
`and (A14)].
`
`E.
`
`Image-Plane Distribution
`
`The exit-pupil wave field, considered to converge on
`the_point I, forms an image-plane field distribution
`A,(Xi) that is described by Fresnel diffraction of the
`exit-pupil field distribution [see Appendix A, Eq. (A9),
`for the exact form of Ai(X,~)]. This image field is de-
`scribed by a double integral that results from the two
`diffraction calculations. TheImage--plane field dis-
`tribution A- determines the field distribution Af inci-
`dent on the receiving——fiber endface. For a flat fiber
`endface the two distributions Af and A- differ by only
`a linear phase factor exp [1k (X '— m)- f3], where f3'18
`a unit vector parallel to the receiving--fiber axis [see
`Appendix A, Eq. (A15)].
`
`F. Coupling to Receiving Fiber
`Although the image field contains all of the optical
`power radiated by the source fiber, not all this power is
`coupled to the receiving fiber unless the receivingjiber
`mode pattern it}; (X) is identical to the field Af(X) in-
`cident on the receiving—fiber endface.
`(The image field
`does not contain all the radiated power if a physical stop
`intercepts some of the light, or if there is absorptionIn
`the optical system.)
`If the two distributions the and A;
`are different, the power--coupling effi_ciency_TIS the
`squared modulus of the integral7 f A]: (XNR (X)dX [see
`Appendix A, Eq. (A16)].
`In this integral the receiv-
`ing-fiber mode pattern is normaflzed so that its total
`power is unity [i.e., f |¢R(X)|2dX— 1].
`The _coup1ing efficiency T, as represented by f
`AfbedXi ,_is actually a triple integral since A (X,- ), and
`hence Af(X ), is represented by a double integral. As
`shownIn Appendix A [see Eqs. (A16)— (A19), however,
`this triple integral can be most conveniently trans-
`formed to a single integral in the exit-pupil coordinates.
`The terms'1n this single integral involve the measurable
`quantities ‘115 (X —X,',s), ‘I’R (X Xar), and W(Xe ) [see
`Appendix A, Eq. (A19) and Figs. 4 and 5].
`G. Result
`
`The general coupling efficiency equation for optics
`in single—mode components is
`
`= If \P,S(Xc - XS.) expi-ikWfiLfl‘I’MZ - ENTER (5)
`
`Where ‘113 and ‘1’}; are the far-field distributions of the
`source and receiving fibers, respectively, referred to the
`exit pupil and appropr_iately scaled and positioned.
`The term exp[—ik W(Xe )]
`is the coherent optical
`transfer function L(Xe ) of the system, and the wave»
`front aberrations W(Xe ) include contributions from
`both the optical system and from fiber misalignments
`
`as discussed earlier. Equation (5) is subject to only
`minor restrictions that are satisfied by virtually all
`practical single-mode lens-coupling systems [see Ap—
`pendix A, Eq. (A20)] and neglects only the diffraction
`effects that occur between the two pupil planes.
`Both far-field distributions \IIS and ‘11}, are normal—
`ized so that
`their total power
`is unity [i.e., f
`|\I/S(Xe)|2dX;,= f I‘IIR(X;)|2dXe =1]. These distri-
`butions are positioned at the points X0, and X00, in the
`exit pupil where the respective fiber axes, as traced
`through the system, intersect the exit pupil plane (see
`Fig. 4).
`In the diffraction calculations of Appendix A
`the positioning X0, and X0, of these two distributions
`is set by_the linear phase factors exp [1k (X -—n)- f3] and
`exp[ik (Xi— m)-$3] and by the pupil magnification mg.
`The scale of the far-field distributions \IIS and ‘1’};1s that
`which would normally occur at the distances R’/m and
`Bi, respectively, where R’ is the distance from center of
`the exit-pupil plane Q’ to the paraxial image point 8’,
`m is the magnification of the optical system, and R, is
`the distance from Q’ to the receiving-fiber position I -
`(see Fig. 5).
`Although Eq. (5) is convenient because it relates
`coupling efficiency T to measurable quantities \II'R, \II'S,
`and W by a single integral, it is possible to relate the
`coupling efficiency directly to the fiber-mode patterns
`(b3 and 1&3. But this requires two more integrations,
`because the far-field distributions are related to the
`mode patterns by integral transformations as shown in
`Eq. (6):
`
`, —.
`__rr_z_
`—
`_.EE— .—1 —
`\rs(x.)— AR, fwsoisnxpi 1 R, x, XejdXSJ
`k
`(6)
`‘I’R(Xe)= —ARgfme) exp(-Ri— Xi- Xe) dXi.
`Equation (6), usedIn conjunction with Eq. (5), deter-
`mines coupling efficiency in terms of tbs and 1P3.
`Quadratic phase factors are absent from the expressions
`in Eq. (6) because these cancel in the process of ana-
`lyzing the coupling [see Appendix A, Eqs. (A8) and
`(A18)]. The normalization in Eq. (6) is such that the
`far-field distributions ‘1'3 and ‘11}; have unity power.
`
`H. Comments
`
`In the result, Eq. (5), the far-field distributions
`\IIS and ‘1’}, are not necessarily centered1n the exit pupil
`as is ordinarily the case in classical optics. For best
`coupling, however, the two distributions are usually
`positioned at the same point, and then it may be useful
`to consider the stop to be shifted from its originally
`defined position. Such sto