General formula for coupling-loss characterization of
`single-mode fiber collimators by use of
`gradient-index rod lenses
`
`Shifu Yuan and Nabeel A. Riza
`
`A general formula for determining the coupling loss between two single-mode fiber collimators with the
`simultaneous existence of separation, lateral offset and angular tilt misalignments, and spot-size mis-
`match is theoretically derived by use of the Gaussian field approximation. Based on this general
`formula, the formulas for coupling losses that are due to the misalignment of insert separation, lateral
`offset, and angular tilt are given. The formula for the coupling loss that is due to Gaussian spot-size
`mismatch of two single-mode collimators is also given. Good agreement between these formulas and
`experimental results is demonstrated with gradient-index rod lens-based fiber collimators operating in
`the 1300-nm band. © 1999 Optical Society of America
`OCIS codes: 060.2310, 060.2340, 060.2430, 110.2760.
`
`Introduction
`1.
`Free-space-based fiber-optic components such as
`isolators, circulators, attenuators, switches, and
`wavelength-division multiplexers兾demultiplexers
`have become key devices for optical fiber communi-
`cations, optical fiber sensing, and radio-frequency
`photonics.1– 4
`In these free-space-based fiber-optic
`components, single-mode-fiber 共SMF兲 or multimode-
`fiber pigtailed fiber collimators have been widely
`used. This is so because the coupling between two
`fiber collimators has a large allowable separation dis-
`tance with a low loss5,6 that is critical for a practical
`free-space interconnected fiber-optic module or sub-
`system. However, to maintain low coupling loss re-
`quires that the separation distance between two fiber
`collimators be limited. An important engineering
`issue that is related to the use of fiber collimators is
`the excess loss performance of a pair of collimators
`
`When this research was performed, both authors were with the
`Center for Research and Education in Optics and Lasers and the
`School of Optics, University of Central Florida, P.O. Box 162700,
`4000 Central Florida Boulevard, Orlando, Florida 32816-2700. S.
`Yuan is now with Chorum Technologies, Incorporated, 1155 East
`Collins Boulevard, Suite 200, Richardson, Texas 75081. The
`e-mail address for N. A. Riza is riza@creol.ucf.edu; for S. Yuan it is
`shyuan@chorumtech.com.
`Received 15 October 1998; revised manuscript received 19 Feb-
`ruary 1999.
`0003-6935兾99兾153214-09$15.00兾0
`© 1999 Optical Society of America
`
`3214
`
`APPLIED OPTICS 兾 Vol. 38, No. 15 兾 20 May 1999
`
`owing to misalignment between the collimating and
`the focusing gradient-index 共GRIN兲 lenses. There
`are three types of misalignment that introduce inser-
`tion loss into the coupling, i.e., separation misalign-
`ment between the lens surfaces, offset misalignment
`between the longitudinal axes of the lenses, and an-
`gular tilt misalignment between the longitudinal
`axes of the lenses. Gaussian beam spot-size mis-
`match also causes insertion loss.
`One can determine the coupling loss of two multi-
`mode fiber collimators by calculating the overlap area
`of the output beams of the two collimators.5 For
`SMF collimators, one must use the Gaussian field
`approximation method7–9 to characterize the cou-
`pling loss rather than calculate the overlap area, and
`three formulas to describe the coupling loss that is
`due to only one of the three misalignments have been
`reported.6 Based on these formulas, a fiber array
`optical coupling design issue for a photonic beam for-
`mation system has been reported.10 However, those
`formulas are not suitable for describing the coupling-
`loss performance of fiber collimators with large sep-
`aration 共⬎10 cm兲 and large lateral offset 共⬎300 ␮m兲.
`In addition, they cannot characterize the coupling-
`loss performance when the three misalignments and
`spot-size mismatch exist simultaneously. Hence in
`this paper we analyze the coupling-loss characteris-
`tics of fiber collimators with simultaneous three mis-
`alignments and spot-size mismatch and give a
`general formula to describe the coupling-loss perfor-
`mance. Experimental results show that this for-
`
`Exhibit 1044, Page 1
`
`

`

`冤
`
`冥 ,
`
`(3)
`
`The GRIN lens has the ray matrix
`sin共冑AZ兲
`cos共冑AZ兲
`1
`冑A
`n0
`冑A sin共冑AZ兲
`cos共冑AZ兲
`⫺n0
`where Z is the length of the GRIN lens.12 The pitch
`of a GRIN lens is defined as p ⫽ 公A Z兾共2␲兲. For a
`quarter-pitch GRIN lens, i.e., p ⫽ 1兾4, we have 公A
`Z ⫽ ␲兾2. Thus the ray matrix of a quarter-pitch
`GRIN lens can be expressed as
`
`G ⫽冋A1 B1
`
`C1 D1
`
`册 ⫽冤
`
`冥 .
`
`1
`冑A
`0
`n0
`冑A
`0
`⫺n0
`Note that both GRIN lens 1 and GRIN lens 2 have the
`same ray matrix.
`The gap with a straight length of Z0 between GRIN
`lens 1 and GRIN lens 2 has a ray matrix11 of
`
`S ⫽冋A2 B2
`
`册 ⫽冋1 Z0
`
`册 .
`
`1
`0
`C2 D2
`It is well known that a Gaussian beam propagates in
`terms of the ABCD law11
`
`(4)
`
`(5)
`
`qi⫹1 ⫽
`
`.
`
`Aiqi ⫹ Bi
`Ciqi ⫹ Di
`Considering that the Gaussian optical field emitted
`from SMF1 has a modal field diameter of w0 with R0
`⫽ ⬁, and using the ABCD law 关i.e., Eq. 共6兲兴 and Eqs.
`共1兲, 共4兲, and 共5兲, we have
`
`(6)
`
`w1 ⫽
`
`␭
`␲nw0n0
`R1 ⫽ ⬁,
`
`,
`
`冑A
`w2 ⫽ w1冋1 ⫹冉 ␭Z0
`R2 ⫽ Z0冋1 ⫹冉␲nw1
`
`␭Z0
`
`␲nw1
`2
`
`(7)
`
`(8)
`
`(9)
`
`(10)
`
`,
`
`2冊2册1兾2
`冊2册 ,
`
`(11)
`
`(12)
`
`w3 ⫽
`
`n0
`
`R3 ⫽ ⫺
`
`␭
`
`⫽ w0,
`
`.
`
`冑A␲nw1
`1
`n0共冑A兲2Z0
`According to the derivations in Eqs. 共7兲–共12兲, we
`have arrived at unique results for the coupling prop-
`erty of SMF collimators compared with those de-
`scribed in Ref. 6. The light beam emitted from
`GRIN lens 1 is a Gaussian beam with its Gaussian
`waist at the surface of the lens. When the Gaussian
`beam propagates to the surface of GRIN lens 2, the
`beam spot size is w2 expanded with radius of curva-
`ture R2. For the beam at output surface P3 of GRIN
`lens 2, the spot size is w3 ⫽ w0; i.e., it has the same
`
`20 May 1999 兾 Vol. 38, No. 15 兾 APPLIED OPTICS
`
`3215
`
`Exhibit 1044, Page 2
`
`Fig. 1. Fiber coupling system using two quarter-pitch SMF colli-
`mators.
`
`mula is adequate for predicting the coupling losses
`that are due to the three kinds of misalignment.
`
`2. Gaussian Field Approximation of Light Propagation
`in Quarter-Pitch Gradient-Index Lenses
`Figure 1 shows the fiber coupling system with two
`SMF collimators. GRIN lens 1 is a transmitting
`lens, and GRIN lens 2 is a receiving lens or focusing
`lens.
`It is well known that the light-field distribu-
`tion of the fundamental mode in a SMF can be well
`approximated by a Gaussian profile.7–9 The Gauss-
`ian beam from SMF1 is collimated by GRIN lens 1
`and then focused into the output SMF2. Gaussian
`beam propagation in the fiber collimating system is
`shown schematically in Fig. 2. For a fiber collimator
`using a quarter-pitch GRIN lens the fiber is butt
`attached to the GRIN lens. The optical field emitted
`from SMF1 can be approximated by a Gaussian beam
`whose waist coincides with the end surface of SMF1.
`The fundamental-mode Gaussian beam can be de-
`scribed with its Gaussian spot size 共Gaussian beam
`radius兲 wi and radius of curvature Ri of equiphase
`surfaces. Combining the two parameters, we have a
`complex curvature parameter qi:
`
`␭
`␲nwi
`where ␭ is the wavelength of light in vacuum and n is
`the refractive index of the medium in the gap.11
`The GRIN rod lens has its largest refractive index
`along the longitudinal axis, and the refractive index
`decreases quadratically with radial distance. The
`refractive index can be expressed as
`
`2 ,
`
`⫺ j
`
`1 R
`
`i
`
`⫽
`
`1 q
`
`i
`
`n共r兲 ⫽ n0冉1 ⫺
`
`冊 ,
`
`Ar2
`2
`
`(1)
`
`(2)
`
`where n0 is the refractive index on the axis of the
`lens, 公A is the gradient constant, and r is the radial
`distance from the central axis.
`
`Fig. 2. Gaussian beam propagation in the fiber collimating sys-
`tem.
`
`

`

`Fig. 3. Output optical field distribution from GRIN lens 1.
`
`Fig. 4. Equivalent optical field distribution that can be perfectly
`coupled into the SMF2 by use of GRIN lens 2.
`
`spot size as the modal field radius of the fundamental
`mode in the SMF. The coupling loss results from the
`different radii of curvature; i.e., the perfectly aligned
`light beam coupled into the fiber should have a spot
`size w0 with a radius R ⫽ ⬁, whereas the beam from
`GRIN lens 2 has a spot size w0 with a radius R ⫽
`⫺1兾关n0共公A兲2Z0兴. When Z0 increases, 兩R兩 decreases,
`so collecting the beam to have good coupling requires
`a fiber with a bigger numerical aperture. Thus
`when we are considering such a fiber–fiber lens-
`coupling issue, we must take into account the radius
`of curvature of the Gaussian beam because consider-
`ing only the spot size cannot give us a good approxi-
`mation to determine coupling loss.
`
`3. Coupling-Loss Formula
`Consider the output field from GRIN lens 1 shown in
`Fig. 3. The Gaussian beam waist is denoted wT.
`The surface of GRIN lens 1 is the waist position.
`The x component Ex of the Gaussian light-field vector
`can be expressed as11
`
`Ex共x, y, z兲 ⫽ E1
`
`wT
`
`w共z兲 exp再⫺i关kz ⫺ ␩共z兲兴
`⫺ r2冋 1
`2R共z兲册冎 ,
`
`k
`
`⫹ i
`
`w2共z兲
`where E1 is the output field’s amplitude at the origin
`position 共x ⫽ 0, y ⫽ 0, z ⫽ 0兲, r is the radius from
`position 共x, y, z兲 to the z axis,
`k ⫽ 2␲n兾␭,
`
`(13)
`
`(14)
`
`(15)
`
`(16)
`
`2冊 ,
`␩共z兲 ⫽ tan⫺1冉 ␭z
`2冊2册 ,
`2冋1 ⫹冉 ␭z
`R共z兲 ⫽ z冋1 ⫹冉␲nwT
`冊2册 .
`
`w2共z兲 ⫽ wT
`
`␲nwT
`
`␲nwT
`2
`
`(17)
`
`␭z
`Note that the time factor exp共i␻t兲 is omitted from
`Eq. 共13兲. For coupling-loss analysis we must con-
`sider two fiber collimators because the loss is due to
`misalignment and mismatch of the two GRIN colli-
`mators. Although the light beam emitted from
`GRIN lens 1 goes through the gap and GRIN lens 2
`and reaches SMF2, only an effective beam equivalent
`
`to the fundamental mode in SMF2 can be coupled
`into the fiber, as shown in Fig. 4. Omitting the time
`factor exp共i␻t兲, we can express the x⬘ component Ex⬘
`of the light field of the effective equivalent beam in
`the gap as
`
`Ex⬘共x⬘, y⬘, z⬘兲 ⫽ E1
`
`wT
`
`k
`
`⫹ i
`
`w2共z⬘兲
`where r⬘ is the radius from position 共x⬘, y⬘, z⬘兲 to the
`z⬘ axis,
`
`w共z⬘兲 exp再⫺i关kz⬘ ⫺ ␩共z⬘兲兴
`⫺ r⬘2冋 1
`2R共z⬘兲册冎 ,
`2冊 ,
`␩共z⬘兲 ⫽ tan⫺1冉 ␭z⬘
`2冊2册 ,
`2冋1 ⫹冉 ␭z⬘
`
`R共z⬘兲 ⫽ z⬘冋1 ⫹冉␲nwR␭z⬘ 冊2册 .
`
`w2共z⬘兲 ⫽ wR
`
`␲nwR
`
`␲nwR
`2
`
`(18)
`
`(19)
`
`(20)
`
`(21)
`
`Note that in Eq. 共18兲 we have considered energy
`conservation for the coupling, so the expression has an
`E1 and a wT factor. The coupling loss between the
`two collimators can be determined by the coupling co-
`efficient between the two Gaussian beams. Coupling
`loss is determined by separation, offset, and angular
`misalignments of the two Gaussian beams, as shown
`in Figs. 5共a兲, 5共b兲, and 5共c兲, respectively.
`It should be
`pointed out the coupling loss that is due to these mis-
`alignments includes the loss that is due to the Gauss-
`ian beam spot-size mismatch because the two SMF
`collimators have different Gaussian beam radii. Fig-
`ure 6 shows a side view of two fiber collimators with
`the three combined misalignments and spot-size mis-
`match in two rectangular systems 共x, y, z兲 and 共x⬘, y⬘,
`z⬘兲. The coupling coefficient ␩c at z⬘ ⫽0 between the
`two Gaussian beams can be expressed as7,9
`
`␩c ⫽
`
`⫹⬁
`
`Ex共x, y, z兲兩z⬘⫽0
`
`⫺⬁
`
`⫺⬁
`
`⫹⬁兰
`2兰
`2
`␲E1
`2wT
`⫻ E*x⬘共x⬘, y⬘, z⬘兲兩z⬘⫽0dx⬘dy⬘.
`(22)
`Note that there is a factor 2兾共␲E1
`2兲 in Eq. 共22兲
`2wT
`whose function is to maintain ␩c ⫽ 1 共i.e., for energy
`
`3216
`
`APPLIED OPTICS 兾 Vol. 38, No. 15 兾 20 May 1999
`
`Exhibit 1044, Page 3
`
`

`

`conservation兲 when no misalignments exist. Al-
`though in Eq. 共12兲 we consider only the Ex component,
`the same equation is suitable for the Ey component,
`so this coupling coefficient applies for all polarization
`directions.
`It is readily shown from Fig. 6 that the
`two rectangular coordinate systems are related by
`
`x ⫽ x⬘ cos ␪ ⫺ z⬘ sin ␪ ⫹ X0,
`
`z ⫽ x⬘ sin ␪ ⫹ z⬘ cos ␪ ⫹ Z0,
`
`y ⫽ y⬘,
`
`(23)
`
`(24)
`
`(25)
`
`Fig. 6. Side view of two fiber collimators with three combined
`misalignments in two rectangular systems 关共x, y, z兲 and 共x⬘, y⬘, z⬘兲兴.
`
`r2 ⫽ x2 ⫹ y2 ⫽ 共x⬘ cos ␪ ⫺ z⬘ sin ␪ ⫹ X0兲2 ⫹ y⬘2.
`Considering cos ␪ ⬇ 1 when ␪ is very small 共e.g.,
`␪ ⱕ 0.3°, a typical case for GRIN lens misalignment
`with SMF coupling兲, we can replace z in w共z兲 and
`R共z兲 and ␩共z兲 by Z0 and put them into Eq. 共13兲. For
`z⬘ ⫽0, Eq.
`共13兲 can be rewritten as
`Ex共x, y, z兲兩z⬘⫽0 ⫽ Ex共x⬘ ⫹ X0, y⬘, x⬘ sin ␪ ⫹ Z0兲
`wT
`w共Z0兲 exp兵⫺i关kZ0 ⫺ ␩共Z0兲兴其
`2R共Z0兲册
`2R共Z0兲册 x⬘2 ⫹再冋 1
`⫻ exp(⫺再冋 1
`k
`⫻ 2X0 ⫹ ik sin ␪冎 x⬘ ⫹冋 1
`
`2R共Z0兲册X02冎)
`⫻ exp再⫺冋 1
`2R共Z0兲册 y⬘2冎 .
`
`k
`
`⫹ i
`
`(27)
`
`(26)
`
`⫽ E1
`
`⫹ i
`
`w2共Z0兲
`
`k
`
`⫹ i
`
`w2共Z0兲
`k
`
`⫹ i
`
`w2共Z0兲
`
`w2共Z0兲
`Replacing r⬘2 with x⬘2 ⫹ y⬘2 in Eq. 共18兲 and consider-
`ing the equivalent field distribution Ex⬘ at position z⬘
`⫽ 0, we have
`
`Ex⬘共x⬘, y⬘, z⬘兲兩z⬘⫽0 ⫽ E1
`
`wT
`wR
`
`exp冉⫺
`
`2冊exp冉⫺
`
`x⬘2
`wR
`
`2冊 .
`
`y⬘2
`wR
`
`(28)
`
`Putting Eqs. 共27兲 and 共28兲 into Eq. 共22兲, and using the
`following integral6:
`
`exp关⫺共ax2 ⫹ bx ⫹ c兲兴dx ⫽冑␲
`
`a
`
`兰
`
`⫹⬁
`
`⫺⬁
`
`
`
`exp冋共b2 ⫺ 4ac兲册 ,
`
`4a
`
`we can find the integral in Eq. 共22兲 to be9
`A共C ⫹ jH兲
`2B
`
`册exp共⫺j␺0兲,
`
`where
`
`,
`
`,
`
`F ⫽
`
`G ⫽
`
`exp冉⫺
`
`冊 .
`
`T ⫽
`
`(29)
`
`(30)
`
`(31a)
`
`(31b)
`
`(31c)
`(31d)
`
`(31e)
`(31f)
`
`(31g)
`
`(31h)
`
`(32)
`
`␩c ⫽ C0 exp冋⫺
`C0 ⫽冋4D册1兾2
`
`B
`␺0 ⫽ AG ⫺ tan⫺1 G
`D ⫹ 1
`A ⫽ 共kwT兲2兾2,
`B ⫽ G2 ⫹ 共D ⫹ 1兲2,
`C ⫽ 共D ⫹ 1兲F2 ⫹ 2DFG sin ␪
`⫹ D共G2 ⫹ D ⫹ 1兲sin2 ␪,
`D ⫽ 共wR兾wT兲2,
`2X0
`2 ,
`kwT
`2Z0
`2 ,
`kwT
`H ⫽ GF2 ⫺ 2D共D ⫹ 1兲F sin ␪ ⫺ GD2 sin ␪.
`(31i)
`The power transmission coefficient can be written as
`T ⫽ 兩␩c兩2 ⫽ ␩c␩c*. Putting Eqs. 共31兲 into Eq. 共30兲, we
`can calculate the power transmission coefficient T to be
`4D
`AC
`B
`B
`Thus the total coupling loss in decibels between two
`misaligned 共i.e., three simultaneous misalignments兲
`SMF collimators can be expressed as
`Ltot共X0, Z0, ␪兲 ⫽ ⫺10 log T
`
`共a兲 Separation misalignment between the two GRIN lens
`Fig. 5.
`共b兲 Offset misalignments between the longitudinal axes
`surfaces.
`共c兲 Angular misalignments between the lon-
`of the GRIN lenses.
`gitudinal axes of the GRIN lenses.
`
`⫽ ⫺10 log冋4D
`
`B
`
`exp冉⫺
`
`冊册 .
`
`AC
`B
`
`(33)
`
`20 May 1999 兾 Vol. 38, No. 15 兾 APPLIED OPTICS
`
`3217
`
`Exhibit 1044, Page 4
`
`

`

`From these results we conclude that the loss perfor-
`mance between two fiber collimators is similar to the
`loss performance of two SMF’s without fiber collima-
`tors.9
`Specifically, by replacing the modal field radius of
`the SMF with the Gaussian beam radius in the for-
`mula for SMF coupling loss derived in Ref. 9, we can
`easily obtain the coupling-loss formula shown in Eq.
`共33兲. Note that, when no misalignments exist 关i.e., X0
`⫽ 0, Z0 ⫽ 0, ␪ ⫽0, and wR ⫽ wT 共no spot size mis-
`match兲兴, from Eq. 共33兲 we have Ltot共X0 ⫽ 0, Z0 ⫽ 0, ␪ ⫽
`0兲 ⫽ 0 dB, which indicates the coupling loss owing to
`misalignments is 0 dB. This does not mean that in a
`perfect-alignment situation the coupling loss is 0 dB.
`In fact, some other factors such as the imperfection of
`GRIN lenses and backreflection of the lens surfaces
`cause additional coupling loss 共e.g., 0.2–0.3 dB兲. This
`additional loss is not included in Eq. 共33兲. To arrive at
`the total coupling loss of fiber collimators we must add
`this additional loss to Eq. 共33兲.
`Based on our general formula in Eq. 共33兲, we can
`derive the normalized loss performances for the three
`kinds of fiber collimator misalignment with different
`Gaussian spot-size mismatch. The separation mis-
`alignment condition is shown in Fig. 5共a兲. The nor-
`malized coupling loss that is due to separation
`misalignment is
`
`Ls ⫽ Ltot共X0 ⫽ 0, Z0, ␪ ⫽ 0兲 ⫺ Ltot共X0 ⫽ 0, Z0 ⫽ 0, ␪ ⫽ 0兲
`
`
`24wT2wR
`
`⫽ ⫺10 log冤
`
`␭2Z0
`2
`␲2n2
`
`2兲2冥 .
`
`
`
`⫹ 共wT2 ⫹ wR
`
`(34)
`
`The lateral offset misalignment condition is shown in
`Fig. 5共b兲. The normalized loss that is due to lateral
`offset misalignment at a specific separation distance
`Z0 can be written as
`
`Ll ⫽ Ltot共X0, Z0, ␪ ⫽ 0兲 ⫺ Ltot共X0 ⫽ 0, Z0, ␪ ⫽ 0兲
`2兲
`n2␲2共wT
`2 ⫹ wR
`20
`2 ⫹ ␲2n2共wT
`2 ⫹ wR
`ln 10
`
`⫽
`
`␭2Z0
`
`2兲2 X0
`
`2.
`
`(35)
`
`The angular tilt misalignment condition is shown in
`Fig. 5共c兲. The normalized loss that is due to angular
`offset misalignment at a specific separation distance
`Z0 can be written as
`
`La ⫽ Ltot共X0 ⫽ 0, Z0, ␪兲 ⫺ Ltot共X0 ⫽ 0, Z0, ␪ ⫽ 0兲
`
`␭ 冊2冋冉 ␭Z0
`冊2
`⫹冉wR
`2冊2
`冉n␲wR
`⫹ 1册
`⫹冋冉wR
`冊2
`冉 ␭Z0
`2冊2
`⫹ 1册2
`
`␲nwT
`
`wT
`
`␲nwT
`
`wT
`
`⫽
`
`20
`ln 10
`
`sin2 ␪.
`
`(36)
`
`Equation 共33兲 also describes the loss performance
`that is due to Gaussian spot-size mismatch.
`If no
`other misalignment exists, the normalized loss that is
`
`3218
`
`APPLIED OPTICS 兾 Vol. 38, No. 15 兾 20 May 1999
`
`Fig. 7. Changing curves of beam radius w共z兲 for several Gaussian
`waists wg.
`
`due only to spot-size mismatch 共different wT and wR兲
`can be derived as
`
`Lm ⫽ Ltot共X0 ⫽ 0, Z0 ⫽ 0, ␪ ⫽ 0, wT, wR兲
`⫺ Ltot共X0 ⫽ 0, Z0 ⫽ 0, ␪ ⫽ 0, wT, wR ⫽ wT兲
`4
`
`⫽ ⫺10 log
`
`冉wR
`
`wT
`
`⫹
`
`wT
`wR
`
`冊2 .
`
`(37)
`
`4. Numerical Analysis
`For GRIN lens free-space interconnections it is im-
`portant to know the tolerance distance between two
`GRIN rod lenses. The tolerance distance is the max-
`imum distance at which good coupling efficiency 共e.g.,
`⬍0.5 dB兲 can be achieved. This is especially true for
`free-space-based fiber-optic components such as
`fiber-optic circulators, isolators, matrix switches, and
`photonic delay lines. According to Eq. 共34兲, the tol-
`erance distance will be related to wavelength ␭ and
`the Gaussian beam widths at the output surface of
`the GRIN lens and to the refractive index n in the
`gap. When the beam width increases, the diver-
`gence of the beam decreases, so the tolerance distance
`becomes larger. With Eq. 共16兲, Fig. 7 shows curves
`for beam radius w共z兲 with several Gaussian waists wg
`at ␭ ⫽1.3 ␮m. From the figure we can clearly see
`that when waist wg becomes smaller, the divergence
`of the beam becomes larger. The divergence angle
`共half-apex angle兲 of a Gaussian beam can be given as
`␪d ⫽ tan⫺1关␭兾共␲wgn兲兴 ⬇ ␭兾共␲wgn兲.11 This relation-
`ship of divergence angle ␪d and Gaussian beam waist
`wg is also shown in Fig. 8. Suppose that the fiber
`collimators are identical and that they have the same
`Gaussian waists wT ⫽ wR ⫽ wg. Thus, using Eq.
`共34兲, we can write the insertion loss that is due to the
`separation distance as
`
`Ls共Z0兲 ⫽ 10log冉1 ⫹
`
`4冊 .
`
`2
`
`␭2Z0
`4␲2n2wg
`
`(38)
`
`Exhibit 1044, Page 5
`
`

`

`Fig. 8. Relationship of divergence angle ␪d and Gaussian beam
`waist wg.
`
`Fig. 10. Theoretical result of the normalized coupling loss that is
`due to the mismatch of the Gaussian beam spot sizes.
`
`This relation is plotted in Fig. 9, where n ⫽ 1 共air
`gap兲 and wg ranges from 170 to 250 ␮m.
`It is clearly
`shown that, for a bigger Gaussian waist at the end of
`the surface, the tolerance separation distance in-
`creases, given a certain fixed loss value. For exam-
`ple, when the beam waist is 170 ␮m, the tolerance
`distance Z0 for a 0.5-dB loss is ⬃4.9 cm, whereas
`when the beam Gaussian waist becomes 250 ␮m, the
`tolerance distance Z0 for a 0.5-dB loss is increased to
`⬃10.5 cm.
`It is well known that using a pair of
`GRIN rod lenses with a longer pitch greater than the
`quarter pitch can move the output beam waist to a
`distance D from the output GRIN surface.13,14 By
`using this method one can increase the tolerance dis-
`tance because this method moves the optimized cou-
`pling position from butt coupling to the condition that
`the two GRIN lenses have a separation of 2D.
`The coupling loss that is due to spot-size mismatch
`is not so sensitive as angular tilt and lateral offset
`misalignments. Because the Gaussian beam widths
`of SMF collimators can easily be selected to be ap-
`proximately the same, the coupling loss comes mainly
`from the three kinds of misalignment. Based on Eq.
`
`Fig. 9. Coupling-loss variation owing to changing of Gaussian
`waist wg.
`
`共37兲, we have plotted the relation of the normalized
`coupling loss to the changing value of wR兾wT, as
`shown in Fig. 10. From this figure we can find that,
`when 0.7 ⬍ wR兾wT ⬍ 1.4, the additional coupling loss
`that is due to the mismatch of the Gaussian spot size
`is less than 0.5 dB. Usually the difference between
`the Gaussian spot sizes wR and wT is quite small, so
`the coupling loss that is due to the mismatch of spot
`size is also small compared with the loss that is due
`to the other misalignments.
`
`5. Experimental Verification
`We conducted experiments to prove that our formula
`is adequate to characterize SMF collimator coupling.
`In our experiments, two SMF collimators with serial
`numbers 共SN’s兲 464584 and 464585 manufactured by
`Nippon Sheet Glass America, Inc., are used.12 The
`SMF used for the collimators is Corning SMF-28 fi-
`ber15 with a core diameter of 8.3 ␮m and a 125-␮m
`cladding. When it is used at the 1.3-␮m wavelength,
`the SMF has a 9.3-␮m mode field diameter 共which
`indicates that w0 ⫽ 4.65 ␮m兲 and a numerical aper-
`ture of 0.13. The quarter-pitch Selfoc GRIN lenses
`are 4.8 mm long and 1.8 mm in diameter, with a
`refractive index of n0 ⫽ 1.5916 on axis and a gradient
`constant 公A ⫽ 0.327 mm⫺1. The air gap refractive
`index is n ⫽ 1. An antireflection coating upon the
`GRIN lens surface makes the surface reflection less
`than 0.5%. Putting these data into Eq. 共7兲, we have
`w1 ⫽ ␭兾共␲nw0n0
`公A兲 ⫽171 ␮m. Thus the waist of
`the Gaussian beam for the Nippon Sheet Glass SMF
`collimator is theoretically calculated to be ⬃171 ␮m.
`To prove this theoretical prediction we measured the
`waist of Gaussian beam for both transmitting and
`receiving GRIN lenses, using the knife-edge profile
`method.
`In this technique a razor blade is placed in
`front of the Gaussian beam and, as the blade is moved
`across the beam, the power transmission is measured
`as a function of the blade position. For the power at
`a certain position for a knife blade across the Gauss-
`
`20 May 1999 兾 Vol. 38, No. 15 兾 APPLIED OPTICS
`
`3219
`
`Exhibit 1044, Page 6
`
`

`

`Fig. 12. Comparison of experimental and theoretical results for
`the normalized coupling loss that is due to the separation between
`the two fiber collimators.
`
`(40)
`
`the power transmitted with perfect alignment in the
`corresponding situation 共P0兲. The measured nor-
`malized coupling loss is denoted
`L ⫽ ⫺10log共Pm兾P0兲.
`Accordingly, the normalized coupling loss is not the
`practical insertion loss.
`It indicates the additional
`coupling loss that is due to the corresponding mis-
`alignment.
`In our experiments the fiber collimator with SN
`464585 is used as the transmission fiber and the fiber
`collimator with SN 464584 is used as the receiving
`fiber. This means that wT ⫽ 175.9 ␮m and wR ⫽
`162.1 ␮m.
`A comparison of theoretical values and actual mea-
`sured values for the normalized coupling losses that
`are due to the corresponding misalignments was
`performed. The experimental results show good
`agreement with the theoretical results. Our ex-
`perimental results match the experimental data
`shown in Ref. 6 for Z0 ⱕ 15 cm. Furthermore, we
`did additional experiments to show that our theo-
`retical results from our derived formulas agree with
`the experimental results for much larger values of
`Z0. Specifically, Fig. 12 shows this normalized
`coupling loss that is due to the separation between
`the two fiber collimators. The normalized coupling
`loss is not the total insertion loss because there is
`an additional loss that must be added to the nor-
`malized coupling loss when the collimators are per-
`fectly aligned but spaced a certain distance apart,
`as shown by the data in Fig. 12. The normalized
`coupling-loss data are shown for the condition that
`the separation is less than 15 cm.6,16 Unlike the
`formula given in Ref. 6, our theoretical results
`agree with the experimental results even when the
`distance of separation reaches 50 cm. Hence, for
`the first time to our knowledge, we have a more
`comprehensive picture of SMF collimator coupling–
`alignment.
`For lateral offset misalignment we measured two
`
`Exhibit 1044, Page 7
`
`Fig. 11. Measured results of the output power distribution with
`共a兲 162.1-␮m waist for the fiber collimator
`motion of the blade:
`with SN 464584, 共b兲 175.9-␮m waist for the fiber collimator with
`SN 464585.
`
`ian beam, the passing optical power can be given as
`共see Appendix A兲
`
`
`
`冋1 ⫺ erf冉冑2x冊册 ,
`
`Pb共x兲 ⫽
`
`P0
`2
`
`w
`
`(39)
`
`where P0 is the total energy of the beam, x is the
`position coordinate of the blade, and erf indicates an
`error function. Figure 11 gives the measured re-
`sults of the output power distribution with the motion
`of the blade for our two fiber collimators. A Laser-
`tron ␭ ⫽ 1300 nm fiber-optic link was used in our
`experiments. The optical power after the blade was
`measured with a Newport powermeter as the blade
`moved at various positions. Using the function
`shown in Eq. 共39兲 to fit the data, we can find that the
`waists of the output beams of the two collimators are
`175.9 ␮m 共SN 464585兲 and 162.1 ␮m 共SN 464584兲,
`which to within experimental tolerances agree with
`the theoretical results of 171 ␮m.
`The optical loss performance between the SMF col-
`limators is measured with the Lasertron semiconduc-
`tor laser diode as a power source. The normalized
`coupling loss was measured as a ratio of the power
`transmitted when the lenses were misaligned 共Pm兲 to
`
`3220
`
`APPLIED OPTICS 兾 Vol. 38, No. 15 兾 20 May 1999
`
`

`

`Fig. 13. Comparison of the experimental and theoretical results
`for the normalized coupling loss that is due to the lateral offset
`when 共a兲 Z0 ⫽ 0, i.e., butt-coupling situation, and 共b兲 Z0 ⫽ 10 cm.
`
`sets of normalized coupling loss, i.e., when the sepa-
`ration of the two fiber collimators was Z0 ⫽ 0 and Z0
`⫽ 10 cm. We measured the normalized coupling
`loss by comparing the detected light power in the
`lateral offset misalignment situations with the de-
`tected light power when collimators are aligned at a
`separation distance of 0 or 10 cm without lateral
`offset misalignment. Figure 13共a兲 shows the nor-
`malized coupling loss that is due to the lateral offset
`when Z0 ⫽ 0, i.e., the butt-coupling situation.
`In
`this figure the theoretical analysis agrees with the
`experimental results. Figure 13共b兲 shows the nor-
`malized coupling loss that is due to the lateral offset
`when Z0 ⫽ 10 cm. Comparing the theoretical result
`with the experimental results, we find that, when the
`lateral offset is ⬍300 ␮m, the theoretical results
`agree with the experimental results. However,
`there is a slight mismatch of curves when the lateral
`offset is larger than 300 ␮m, and the predicted loss
`with our approximate formulas is greater than the
`measured loss because we use the light-field Gauss-
`ian representation in the derivation to represent the
`light from the fiber collimator.
`In reality, this is a
`close approximation but not a perfect fit.
`For angular tilt misalignment we also measured
`two sets of normalized coupling loss when the sepa-
`
`Fig. 14. Comparison of experimental and theoretical results for
`the normalized coupling loss that is due to the angular tilt when 共a兲
`Z0 ⫽ 0, i.e., butt-coupling situation and 共b兲 Z0 ⫽ 10 cm.
`
`ration of the two fiber collimators was Z0 ⫽ 0 and Z0
`⫽ 10 cm. We measured the normalized coupling
`loss by comparing the detected light power in the
`angularly misaligned situations with the detected
`light power when the collimators were aligned at a
`separation distance of 0 or 10 cm without angular tilt
`misalignment. The experimental and theoretical
`results are given in Figs. 14共a兲 and 14共b兲 for distances
`of Z0 ⫽ 0 and Z0 ⫽ 10 cm, respectively. The theo-
`retical results are in good agreement with the exper-
`imental results when Z0 ⫽ 0 cm. When the
`separation distance is Z0 ⫽ 10 cm the theoretical
`values are also higher than the practical measured
`loss when the tilt angle is larger than 0.3°. This is so
`because of the approximation of the small angular tilt
`that we made in our derivations, in which we re-
`placed z in w共z兲, R共z兲 and ␩共z兲 with Z0.
`6. Conclusion
`In conclusion, we have theoretically derived a general
`formula for the coupling loss between two fiber colli
`mators with the simultaneous existence of separa-
`tion, lateral offset, and angular tilt misalignments
`and Gaussian beam spot-size mismatch by using the
`Gaussian field approximation. Based on this for-
`mula, the expressions for normalized coupling losses
`that are due to the separation misalignment, lateral
`offset misalignment, angular tilt misalignment, and
`
`20 May 1999 兾 Vol. 38, No. 15 兾 APPLIED OPTICS
`
`3221
`
`Exhibit 1044, Page 8
`
`

`

`w2冊dx
`exp冉⫺
`冑2
`␲ 兰
`
`冋1 ⫺ erf冉冑2x0冊册 ,
`
`2x2
`
`⬁
`
`x0
`
`w
`
`where erf denotes the error function. The above for-
`mula gives the power detected by an optical power-
`meter as the position of the knife blade changes.
`
`study is partially supported by grant
`This
`N000149510988 from the U.S. Office of Naval Re-
`search.
`
`References
`1. M. S. Borella, J. P. Jue, B. Ramamurthy, and B. Mukherjee,
`“Components for WDM lightwave networks,” Proc. IEEE 85,
`1274–1307 共1997兲.
`2. W. J. Tomlinson, “Application of GRIN-rod lenses in optical fiber
`communication systems,” Appl. Opt. 19, 1127–1138 共1980兲.
`3. N. A. Riza and S. Yuan, “Low optical interchannel crosstalk, fast
`switching time, polarization independent 2 ⫻ 2 fiber optic switch
`using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342
`共1998兲.
`“Directly modulated
`4. N. Madamopoulos and N. Riza,
`semiconductor-laser-fed photonic delay line with ferroelectric
`liquid crystals,” Appl. Opt. 37, 1407–1416 共1998兲.
`5. J. C. Palais, “Fiber coupling using graded-index rod lenses,”
`Appl. Opt. 19, 2011–2018 共1980兲.
`6. R. W. Gilsdorf and J. C. Palais, “Single-mode fiber coupling
`efficiency with graded-index rod lenses,” Appl. Opt. 33, 3440–
`3445 共1994兲.
`7. H. Kogelnik, “Coupling and conversion coefficients for optical
`modes,” in Proceedings of the Symposium on Quasi-Optics, J.
`Fox, ed., Vol. 14 of Polytechnic Institute Microwave Research
`Institute Symposia Series 共Polytechnic Brooklyn, Brooklyn,
`N.Y., 1964兲, pp. 335–347.
`8. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell
`Syst. Tech. J. 56, 703–719 共1977兲.
`9. S. Nemeto and T. Makimoto, “Analysis of splice loss in single-
`mode fibres using a Gaussian field approximation,” Opt. Quan-
`tum Electron. 11, 447–457 共1979兲.
`10. J. K. Kim and N. A. Riza, “Fiber array optical coupling design
`issues for photonic beamformers,” in Pacific Northwest Fiber
`Optic Sensor Workshop, E. Udd, ed., Proc. SPIE 2754, 271–282
`共1996兲.
`11. A. Yariv, Optical Electronics in Modern Communications, 5th
`ed. 共Oxford U. Press, New York, 1997兲.
`12. Selfoc Product Guide, manufacturer’s literature on fiber colli-
`mators 共NSG America, Inc., New Jersey, 1997兲.
`13. T. Sakamoto, “Coupling characteristic analysis of single-mode
`and multimode optical-fiber connectors using gradient-index-
`rod lenses,” Appl. Opt. 31, 5184–5190 共1992兲.
`14. W. S. Wu, “Reduction of coupling loss in many-to-many colli-
`mating system for optomechanical matrix switch,” Opt. Eng.
`37, 1834–1837 共1998兲.
`15. Opto-electronics Group, Corning, Inc., Corning Optical Fiber
`Product Information PI1044, on Corning SMF-28 single-mode
`optical fiber 共Corning, Inc., Corning, N.Y., 1996兲.
`16. N. A. Riza and S. Yuan, “Demonstration of a liquid-crystal
`adaptive alignment tweeker for high-speed infrared band
`fiber-fed free-space system,” Opt. Eng. 37, 1876–1880
`共1998兲.
`
`Exhibit 1044, Page 9
`
`⫽
`
`⫽
`
`P0
`w
`
`P0
`2
`
`beam spot-size mismatch are given. Experiments
`were conducted at the 1300-nm fiber-optic band with
`GRIN fiber lenses and SMF’s. Compared with ear-
`lier theory and experiments, our coupling-loss for-
`mula gives a more accurate picture of loss owing to
`separation misalignment, with experimental results
`agreeing with theory for an air gap of as much as 50
`cm. Derived coupling-loss formulas for lateral offset
`and angular tilt misalignments also agree with ex-
`periments but within the small angular tilt and
`Gaussian field approximations used to arrive at
`closed-form analytical results. Our newly derived
`and demonstrated formulas for GRIN-lens-based sin-
`gle mode fiber-optic coupling can give more compre-
`hensive insight
`to photonic engineers who are
`developing next-generation fiber-optic signal process-
`ing and communication systems.
`
`Appendix A. Knife-Edge Beam Profile Method
`For a Gaussian beam, the light-field amplitude can be
`expressed as
`
`exp冉⫺
`
`w2 冊 ,
`
`x2 ⫹ y2
`
`1 w
`
`w2冊 ⫽
`
`r2
`
`exp冉⫺
`
`1 w
`
`E共x, y兲 ⬀
`
`where x and y indicate the two coordinates and w is the
`Gaussian waist radius of the beam. The light inten-
`sity can be written as
`
`冊 .
`
`2 exp冉⫺
`
`1 w
`
`2x2 ⫹ 2y2
`w2
`Considering the total field power to be P0, i.e.,
`
`I共x, y兲 ⬀兩 E共x, y兲兩2 ⬀
`
`兰
`
`⬁ 兰
`
`⬁
`
`I共x, y兲dxdy ⫽ P0,
`
`⫺⬁
`⫺⬁
`We can rewrite the light intensity I共x, y兲 as
`
`冊 .
`
`␲w2 exp冉⫺
`
`2P0
`
`I共x, y兲 ⫽
`
`2x2 ⫹ 2y2
`w2
`In the knife-edge beam profile technique a razor
`blade is placed in front of the Gaussian beam. As the
`blade is moved across the beam, the power transmis-
`sion is measured as a function of the blade position.
`Suppose that the blade is moved along the x-axis di-
`rection and the position of blade is denoted x0. Look-
`ing at the power at position x0 for a knife blade across
`the Gaussian beam, the passing optical power can be
`given as
`
`⬁
`
`⫺⬁
`
`x0
`
`I共x, y兲dxdy
`
`⬁ 兰
`Pb共x0兲 ⫽兰
`␲w2兰
`␲w2兰
`
`2P0
`
`⫽
`
`2P0
`
`⫽
`
`⬁
`
`x0
`
`⫺⬁
`
`exp冉⫺
`⬁ 兰
`exp冉⫺
`w2冊dy兰
`
`2x2 ⫹ 2y2
`w2
`
`冊dxdy
`exp冉⫺
`w2冊dx
`
`2x2
`
`⬁
`
`x0
`
`⬁
`
`2y2
`
`⫺⬁
`
`3222
`
`APPLIED OPTICS 兾 Vol. 38, No. 15 兾 20 May 1999
`
`