foundations for low loss grin fiber coupling

Foundations for low-loss fiber gradient-index lenspair coupling with the self-imaging mechanism

Martin van Buren and Nabeel A. Riza

A fiber-optic collimator that emits a Gaussian beam with its beam waist at a certain distance after theexit face of the lens is labeled a self-imaging collimator. For such a collimator, the waist of the emittedGaussian beam and its location are partly dependent on the properties of the gradient-index �GRIN� lens.Parameters for the self-imaging collimator are formulated in terms of the parameters of a GRIN lens �e.g.,pitch, core refractive index, gradient index, length� and the optical wavelength. Next, by use of theGaussian beam approximation, a general expression for the coupling power loss between two self-imaging-type single-mode fiber �SMF� collimators is, for the first time to our knowledge, derived as afunction of three types of misalignment, namely, separation, lateral offset, and angular tilt misalignment.A coupling experiment between two self-imaging collimators with changing separation distance is suc-cessfully performed and matches the proposed self-imaging mechanism coupling loss theory. In addi-tion, using a prism, lateral offset, as well as angular tilt, misalignments are experimentally simulated fora two self-imaging collimator coupling condition by a single collimator reflective test geometry. Exper-imental results agree well with the proposed loss formulas for self-imaging GRIN lenses. Hence, for thefirst time to our knowledge, the mathematical foundations are laid for employing self-imaging-type fibercollimators in SMF-based free-space systems allowing optimal design for ultra-low-loss coupling. © 2003Optical Society of America

OCIS codes: 110.2760, 060.2310.

1. Introduction

Coupling loss is a key issue for the design of bothintegrated-optic and fiber-optic components and sub-systems involving free-space light interaction. Ear-lier research that used mode mismatch analysis wasconducted for insertion loss determination for fiber-to-integrated-optic waveguide chip coupling by indi-vidual lenses such as a gradient-index �GRIN� lens.1Free-space-based fiber-optic components such as iso-lators, circulators, attenuators, switches, and wave-length division multiplexers and demultiplexers2–5

frequently make use of single-mode fiber �SMF� col-limators to couple light between fibers. The mainadvantage of these collimators is the moderate cou-pling power loss for large separation distance. The

The authors are with the Photonic Information Processing Sys-tems Laboratory, School of Optics, Center for Research and Edu-cation in Optics and Lasers, University of Central Florida, 4000Central Florida Boulevard, Orlando, Florida 32816-2700. Thee-mail address for N. A. Riza is [email protected].

Received 2 April 2002; revised manuscript received 3 September2002.

0003-6935�03�030550-16$15.00�0© 2003 Optical Society of America

550 APPLIED OPTICS � Vol. 42, No. 3 � 20 January 2003

coupling power loss is dependent on the distance be-tween the input GRIN lens and the output GRIN lensand the misalignment of these lenses. There arethree types of misalignment: separation, lateral off-set, and angular tilt misalignment. Recently, a gen-eral formula to determine the coupling loss of twoSMF collimators with these three misalignments wasgiven.6,7

The fundamental mode that is the only mode al-lowed to travel through the SMF propagates by ap-proximation as a Gaussian beam in free space. Aquarter-pitch GRIN lens that is directly connected toa SMF emits by approximation a Gaussian beamwith its beam waist on the exit face of the GRIN lens.8These lenses were used in Ref. 6. However, GRINlenses with pitches other than 0.25 are also used infree-space fiber-optic interconnections. Further-more, an air gap between the SMF and the GRIN lensis also typical as this prevents backreflection.9 An-other advantage of an air gap between the SMF andthe GRIN lens is that the magnitude of the gap can beadjusted to set the location of the beam waist at acertain distance after the exit plane of the GRIN lens.We call such a lens a self-imaging lens. There can bea minimum coupling power loss when these GRINlenses are separated by a certain distance that is

labeled the self-imaging distance.10 Ideally therewill be zero coupling power loss when two identicalGRIN lenses are used and the separation distancebetween the collimator end faces is twice the GRINlens beam-waist distance. We call this phenomenathe self-imaging mechanism because the symmetry ofthe optical components placement �i.e., fibers, GRINlenses, and free-space air gaps� forms an imaging-type situation leading to minimal coupling loss.Recently, this self-imaging effect has been experi-mentally confirmed.11 Note that the air gap can bereplaced by a tapered GRIN lens solid-optic SMFinterconnect design leading to the same beam char-acteristics that lead to the self-imaging mechanism.Also note that the optical beam characteristics ofwhat we call the self-imaging mechanism has beenknown and utilized in optical designs such as lasercavities12 and perhaps in other commercial free-space optics.

We begin this paper by providing the basics of agiven GRIN lens and related Gaussian beam charac-teristics that are required to lay out the proposedfoundations for GRIN lens free-space coupling loss.The separation distance for which the power loss be-tween the two coupled GRIN lenses is the least de-pends on the location of the beam waist. In thispaper, using ray matrix theory, we find an analyticalexpression for this beam waist location as a functionof the parameters of the GRIN lens and the SMF. Inaddition, the power loss that is due to misalignmentof two free-space coupled self-imaging collimators isanalyzed. The coupling power loss is determinedwith the aid of the coupling loss coefficient defined asthe overlap area of the electric field amplitude of thetwo collimator output beams.13 Considering thecommon case of small angular tilt misalignment �e.g.,�0.3 deg�, approximations are made leading to ananalytical expression for the power coupling loss thatis written as a function of the separation, lateraloffset, and angular tilt misalignment. Experimentsare performed to match the misalignment-based cou-pling loss formulas for self-imaging collimators.

2. Basics of a Gradient-Index Lens

A. Design and Coupling-Loss Analysis ofSelf-Imaging Collimators

The classic GRIN lens design, such as that by NipponSheet Glass Japan, used a radial refractive-indexvariation to realize the rod lens effect. Specifically,this GRIN lens has a parabolic-shaped refractive-index profile where this refractive index is expressedas

nr � n0�1 �Ar2

2 � , (1)

where n0 is the refractive index at the center of theGRIN lens, �A is the gradient constant, and r is thedistance to the central axis. More recently, anotherGRIN lens design called GRADIUM by LightPathTechnologies has emerged in which the refractive in-

dex varies in the axial direction. The specific axialindex change profile is, to our knowledge, not publicinformation. Because our earlier research6,7 usedradial-type GRIN lens theory, we continue in thispaper with radial GRIN lens analysis. Neverthe-less, the concepts and approach taken in this papercan be applied in general to any GRIN lens, includingthe GRADIUM lens.

In general, a SMF is not directly connected to theGRIN lens. Specifically, in a typical fiber collimatorpackage there is a small air gap between the SMFand the GRIN lens �of the order of 1 �m�, where thisgap also prevents backreflection.9 When the lighthas crossed the gap, it is captured and collimated bythe GRIN lens, finally traveling through the airagain. Figure 1 shows schematically how theGaussian beam propagates through the GRIN lens.The objective is to determine the location of the beamwaist of the Gaussian beam after the GRIN lens as afunction of the magnitude of the gap between the endof the fiber and the entrance face of the GRIN lensand some parameters of the GRIN lens.

To determine the state of a Gaussian beam after ithas passed through an optical system, e.g., a lens ora mirror, one can make use of ray matrices and theABCD law.8 If the gap between the end of the SMFand the GRIN lens has a length L, the ray matrix ofthis gap is14

Mgap � �1 L0 1� . (2)

Subsequently, the Gaussian beam will enter a GRINlens. The ray matrix of the GRIN lens is given by15

MGRIN � � cos��AZ�1

n0�Asin��AZ�

�n0�A sin��AZ� cos��AZ�� .

(3)

Here Z is the length of the lens. The pitch of aGRIN lens is defined as p � �AZ��2��. A quarter-pitch GRIN lens �p � 0.25� means that �AZ � ��2.A quarter-pitch GRIN lens will transform the lightemitted by a point source at an edge of the lens intoa parallel light beam at the other edge of the lens.However, in this paper we do not use a quarter-pitchlens to enable the proposed self-imaging collimator.

Fig. 1. Propagation of the Gaussian beam through a GRIN lens.The solid curves represent the beam width of the Gaussian beam.The complex radii of curvature at the end of the SMF, at the left-and right-hand side of the GRIN lens, and at the location of thebeam waist are denoted as q0, q1, q2, and q3, respectively.

20 January 2003 � Vol. 42, No. 3 � APPLIED OPTICS 551

Finally, after the light has left the GRIN lens, theGaussian beam travels through the air again. Theray matrix is similar to Eq. �2� and equals

Md � �1 d0 1� . (4)

Here, d is the distance the Gaussian beam has trav-eled after the GRIN lens. It is assumed that thelight has its beam waist at the end of the fiber.Hence the complex curvature parameter at the end ofthe fiber can be written as12,16

q0 � i�nw0

2

� iz0. (5)

where w0 is given by17

w0

a 0.65 � 1.619V�3�2 � 2.879V�6,

1.2 � V � 2.405. (6)

By applying the ABCD law with Eq. �2� as a matrix,we can determine the complex beam radius when theGaussian beam reaches the entrance face of theGRIN lens. The complex beam radius at this pointis denoted as q1. To obtain the complex beam radiusat a distance d after the GRIN lens, denoted as q3, weapply the ABCD law two more times using Eqs. �3�and �4� leading to

q3 ��

�2 � 2 � d � i��

�2 � 2 , (7)

where

� L cos��AZ� �1

n0�Asin��AZ�, (7a)

� � z0 cos��AZ�, (7b)

� � �n0�AL sin��AZ� � cos��AZ�, (7c)

� �n0�Az0 sin��AZ�. (7d)

The next objective is to determine the location afterthe GRIN lens where the Gaussian beam has itsbeam waist. The beam waist can be found at a cer-tain value for d such that the condition

Re�q3� � 0 (8)

is satisfied. This implies that the beam waist can befound at a distance

d � ��

�2 � 2 (9)

after the GRIN lens. Figure 2 shows a graph ofthis distance as function of the GRIN lens pitch.Here the gap between the fiber and the GRIN lenshas a value of 1 �m. The GRIN lens has a corerefractive index of n0 � 1.6 and gradient constant of�A � 0.3 � 103 m�1. Furthermore, a wavelength

of � 1550 nm is used in the calculations. It canbe observed that, for a pitch of approximately 0.25,the distance from the GRIN lens edge to the beamwaist becomes zero. The value of the pitch atwhich d becomes zero depends on the gap betweenthe fiber and the GRIN lens. A sharp maximum isrecognizable for a value of the pitch of approxi-mately 0.26.

Next the value of the beam waist is found. Thebeam waist is denoted as wT, where the T refers totransmitter, and is given by

w � wT � � Im�q3�

�n �1�2

� ��

�2 � 2

�n�

1�2

. (10)

B. Gradient-Index-Lens–Gradient-Index-Lens CouplingLoss in the Symmetric Self-Imaging Case

In this subsection, a formula is derived to predict thepower coupling loss between two misaligned GRINlenses. In Subsection 2.A the Gaussian beam ap-proximation and the propagation of a Gaussian beamthrough a GRIN lens were discussed. The parame-ters that were investigated in Subsection 2.A, e.g.,the beam waist and its location, are used in the the-ory of the coupling loss formula. To describe theelectric fields belonging to a Gaussian beam, a coor-dinate frame is introduced. The origin is located atthe center of the exit face of the GRIN lens, and thez axis coincides with the propagation axis of theGaussian beam �see Fig. 3�. If a fiber is directlycoupled to a quarter-pitch GRIN lens, the beam waistwould be found at the exit face of the GRIN lens.

Fig. 2. Distance d between the edge of the GRIN lens and thebeam waist as a function of the pitch of the GRIN lens according toEq. �9�. For the gap L between the SMF and the GRIN lens avalue of 1 �m was taken. For a pitch of approximately 0.26, thebeam-waist distance has its maximum value of 40 mm.


The x component of the electric field can be expressedas8

Ex� x, y, z� � E1

wT

w� z�exp�i�kz � �� z��

� r2� 1w2� z�

� ik

2 R� z�� , (11)

where E1 is the maximum electric field that is locatedat the origin �x � y � z � 0� and r is the distance tothe z axis,

k �2�n

, (12)

�� z� � tan�1� z�nwT

2� , (13)

w2� z� � wT2�1 � � z

�nwT2�2� , (14)

R� z� � z�1 � ��nwT2

z �2� . (15)

Consider now the general situation in which theGRIN lens is not a quarter-pitch lens and there mightbe a gap between the fiber and the GRIN lens. InSubsection 2.A a relationship for the location of thebeam waist was derived in terms of the magnitude ofthe gap between the fiber and GRIN lens and thepitch of the GRIN lens. If it is assumed that thedistance between the exit face of a certain GRIN lens1 and the location of the beam waist of the Gaussianbeam is equal to d1 �see Fig. 4�, then the coordinateframe is shifted a distance d1 to the left comparedwith the previous setup �see Fig. 3�. This frame iscalled coordinate frame 1. The shift implies that z

should be replaced by �z � d1� in Eq. �11�, and theelectric field in coordinate frame 1 can be described by

Ex� x, y, z� � Ex� x, y, z � d1�

� E1

wT

w� z � d1�exp�i�k� z � d1�

� �� z � d1�� r2� 1w2� z � d1�

� ik

2 R� z � d1�� . (16)

Because a typical transmissive interconnection ge-ometry uses two GRIN lenses, a second GRIN lens isintroduced for the coupling loss analysis. In general,this lens depicted as GRIN lens 2 in Fig. 5 will be ableto capture only a part of the light emitted by GRIN lens1 whereas the other part is lost. The magnitude ofthis coupling loss is determined by the overlap of theelectric field of the beam produced by GRIN lens 1 withthe beam equivalent of GRIN lens 2. This beamequivalent can be treated in the same way as the beamproduced by GRIN lens 1. In analogy with GRINlens 1, the entrance face of GRIN lens 2 is placed atthe origin of another coordinate frame, as shown inFig. 5. The new coordinates are distinguished withprimes �x�, y�, z�� and are called coordinate frame 2.

For initial analysis, a special case is considered,namely, a symmetric optical coupling setup. Inother words, the two GRIN lenses are identical, andthe gap between GRIN lens 2 and SMF2 is the sameas the gap between GRIN lens 1 and SMF1. Thiscauses the distance from the entrance face of GRINlens 2 to the beam waist to be d1 again. Below, inSubsection 2.C, this setup is generalized. Analo-gously to the beam produced by GRIN lens 1, in co-ordinate frame 2 the x� component of the electric fieldbelonging to the beam equivalent of GRIN lens 2 incoordinate frame 2 can be written as

Ex�� x�, y�, z�� E� x�, y�, z� � d1�

� E1

wT

w� z� � d1�exp�i�k� z� � d1�

� �� z� � d1�� r�2� 1w2� z� � d1�

� ik

2 R� z� � d1�� , (17)

Fig. 3. SMF directly coupled to a quarter-pitch lens and the beamwidth of the Gaussian beam. The beam waist is located at theedge �called the exit face� of the GRIN lens.

Fig. 5. GRIN lens 2 in coordinate frame 2. The location of thebeam waist is z� � d1.

Fig. 4. General case in which the beam waist does not coincidewith the edge of the GRIN lens. The beam waist is located at z �d1.


where r� represents the distance to the z� axis.As mentioned above, the coupling loss depends on

the overlap of the beam produced by GRIN lens 1with the beam equivalent of GRIN lens 2, or theoverlap of Eq. �16� with Eq. �17�. The coupling coef-ficient �c at z� � 0 takes the form18,19

�c �2

�E12wT

2 �� Ex� x, y, z��z��0

� Ex�*� x�, y�, z��z��0dx�dy�. (18)

If there is no loss, the coupling coefficient shouldequal one. The magnitude of the coupling coefficientis dependent on the position of GRIN lens 1 relativeto GRIN lens 2, or in other words, the position of the

coordinate frame 1 relative to coordinate frame 2.This relative position is determined by three types ofmisalignment between the two lenses. These mis-alignments are separation misalignment �Fig. 6�a��,lateral offset misalignment �Fig. 6�b��, and angulartilt misalignment �Fig. 6�c�� represented by the sym-

bols Z0, X0, and �, respectively. Because the funda-mental mode of the Gaussian beam is rotationallysymmetric around its axis of propagation, it is neithernecessary to distinguish a �x �in the x, z plane� and a�y �in the y, z plane� nor to distinguish a lateral offsetmisalignment in the x and y directions. First, theseparation misalignment is taken into consideration,and the lateral offset and angular tilt misalignmentare not considered. In this special case, the relationbetween the two coordinate frames is

x � x�, y � y�, z � z� � Z0, r � r�. (19)

With the aid of this coordinate transformation, Eq.�16� can be expressed in terms of x�, y�, and z�. Sub-stituting z� � 0 leads to

Ex� x, y, z��z��0 � E1

wT

w�Z0 � d1�exp�i�k�Z0 � d1�

� ��Z0 � d1�� r��2� 1w2�Z0 � d1�

� ik

2 R�Z0 � d1�� . (20)

Substituting Eqs. �17� and �20� into Eq. �18� and us-ing the standard integral

�� exp��r2�dxdy ��

, (21)

we obtain

The power transmission coefficient indicates whatpart of the power emitted by GRIN lens 1 is capturedby GRIN lens 2. This power transmission coefficientis written as T � ��c�

2 and becomes

For d1 � 0, this result is exactly the same as theformula found in earlier studies.6,20 From Eq. �23�,it is verified that there is no loss �T � 1� if theseparation distance equals 2 times d1. This is thecritical condition that implements the earlier pro-posed self-imaging technique for coupling between

Fig. 6. Three types of misalignment: �a� separation misalign-ment �Z0�, �b� lateral offset misalignment �X0�, �c� angular tiltmisalignment ��. d1 � d2 � 100 mm, wT � wR � 0.5 mm, �1550 nm.

�c �2 expi�k�2d1 � Z0� � ��Z0 � d1� � ��d1��

w�Z0 � d1�w�d1� 1w2�Z0 � d1�

�1

w2�d1��

ik2 � 1

R�Z0 � d1��

1R�d1�

�� . (22)

T �4

w2�d1�

w2�Z0 � d1�� 2 �

w2�Z0 � d1�

w2�d1�� w2�Z0 � d1�w

2�d1�k2

4 � 1R�Z0 � d1�

�1

R�d1��2 . (23)


two GRIN lenses.10 This calculated result isshown in Fig. 7. For this example, the beam waistof both GRIN lenses was wT � wR � 0.5 mm andtheir beam waist distance was d1 � d2 � 100 mm.Infrared light with a wavelength of 1550 nm isassumed in the simulation. It is convenient to ex-press the loss in decibels, where L�dB� � �10 log T.In Fig. 7 the coupling loss in decibels is plotted as afunction of Z0. Note that there will be a loss whenZ0 is equal to zero, or in other words when thelenses are directly next to each other. Also notethat there exists a mirror symmetry for the interval0 � Z0 � 4d1 with Z0 � 2d1 as the symmetry axis.

C. Gradient-Index-Lens–Gradient-Index-Lens CouplingLoss in a General Self-Imaging Case

In Subsection 2.B we discussed the special case ofthe symmetric setup where only the separation mis-alignment is examined. In this subsection, we con-sider the general case of a nonsymmetric setup.Furthermore, not only the separation misalignmentis taken into account, but both lateral offset as wellas angular tilt misalignments are investigated. InFig. 8 the three types of misalignment are com-bined. From Fig. 8 we can derive

x � x� cos � � z� sin � � X0, (24)

z � x� sin � � z� cos � � Z0, (25)

y � y�, (26)

r2 � x2 � y2 � � x� cos � � z� sin � � X0�2 � y2. (27)

Using Eqs. �24�–�27� we can express Eq. �16� in termsof x�, y�, and z�. To maintain an analytical solutionfor �c, we assume that sin � is small �e.g., � � 0.30, a

typical case for GRIN lens misalignment with SMFcoupling� and that

w12� x� sin � � Z0 � d1� w1

2�Z0 � d1�, (28)

R1� x� sin � � Z0 � d1� R1�Z0 � d1�, (29)

�1� x� sin � � Z0 � d1� �1�Z0 � d1�. (30)

Furthermore, the approximation cos � � 1 has beenmade, because � is small. These assumptions leadto an approximation for Ex at z� � 0 to be

Ex� x�, y�, z��z��0 E1

wT

w1�Z0 � d1�exp��i�k� x� sin �

� Z0 � d1� � �1�Z0 � d1��

� exp�� x� � X0�2�

� � 1w1

2�Z0 � d1�

� ik

2 R1�Z0 � d1�� . (31)

In approximations �28�–�30� a subscript 1 is added tothe functions w, �, and R to indicate that these func-tions belong to GRIN lens 1. Because the generalcase of a nonsymmetric setup is studied, we assumedthat the beam-waist distance of GRIN lens 2 equalsd2 instead of d1. Thus, instead of Eq. �17� for thesymmetric situation, we use

Ex�� x�, y�, z�� E1

wT

w2� z� � d2�exp�i�k� z� � d2�

� �2� z� � d2�� r�2� 1w2

2� z� � d2�

� ik

2 R2� z� � d2�� . (32)

Fig. 7. Relationship between the coupling loss and the separationmisalignment, where there exists a mirror symmetry in whichZ0 � 2d1. For a separation distance of Z0 � 2d1, there is theo-retically no loss. Lateral offset and angular tilt misalignmentsare assumed to be nonexistent.

Fig. 8. GRIN–GRIN coupling case in which we combined thethree types of misalignment, i.e., separation misalignment �Z0�,lateral offset misalignment �X0�, and angular tilt misalignment ��.


The subscripts 2 in the functions �2, w2, and R2means that wT in Eqs. �13�–�15� is replaced by wR.The subscript R refers to receiver. Hence

�2� z�� tan�1� z�

�nwR2� , (33)

w22� z�� wR

2�1 � � z�

�nwR2�2� , (34)

R2� z�� z��1 � ��nwR2

z� �2� . (35)

Substituting relation �31� and Eq. �32� into Eq. �18�and using the integral

��

��

exp��x2 � �x � ��dx � ��

exp��2 � 4�

4 � ,

(36)

we obtain

�c �2

w1�Z0 � d1�w2�d2� Fexp�i�1�exp�G2 � 4FH

4F � .

(37)

Here

F � Fr � iFi �1

w12�Z0 � d1�

�1

w22�d2�

� ik2 � 1

R1�Z0 � d1��

1R2�d2�

� , (38)

G � Gr � iGi �2X0

w12�Z0 � d1�

� ik� X0

R1�Z0 � d1�� sin �� , (39)

H � Hr � iHi �X0

2

w12�Z0 � d1�

� ikX0

2

2 R1�Z0 � d1�,

(40)

�1 � k�d1 � d2 � Z0� � �1�Z0 � d1� � �2�d2�. (41)

For F, G, and H, the subscripts r and i represent thereal and imaginary value of that number, respec-tively. Next, the argument of the last exponent can

be split up into real and imaginary parts, which aredenoted as J and �2, respectively. It follows that

J � Re�G2 � 4FH4F �

�Fr�Gr

2 � Gi2 � 4Fr Hr� � Fi�2Gr Gi � 4FiHr�

4�Fr2 � Fi

2�,

(42)

�2 � Im�G2 � 4FH4F �

��Fi�Gr

2 � Gi2 � 4FiHi� � Fr�2Gr Gi � 4Fr Hi�

4�Fr2 � Fi

2�.

(43)

Using Eqs. �42� and �43�, we can write the couplingcoefficient as

�c �2

w1�Z0 � d1�w2�d2� Fexp�i��1 � �2��exp� J�.

(44)

The power transmission coefficient can be calculatedwith T � ��c�

2 and the loss in decibels with L�dB� ��10 log T. This loss becomes

L � �10 log� 4 exp�2J�

w12�Z0 � d1�w2

2�d2��F�2� . (45)

For d1 � d2 � 0, Eq. �45� is in agreement with theresults found by earlier research.6,20 To comparethis relation with Eq. �23�, belonging to the symmet-ric setup, we assumed that d2 � d1 and that wR � wT.

In Fig. 9, Eq. �45� is plotted in absence of angulartilt misalignment for different values of the separa-

Fig. 9. Coupling loss as a function of the lateral offset misalign-ment for different values of the separation misalignment with noangular tilt misalignment. The plots belonging to Z0 � d1 andZ0 � 3d1 coincide. Also the plots belonging to Z0 � 0 and Z0 � 3d1

are exactly the same. The loss equals zero when Z0 � 2d1 andX0 � 0.


tion distance between the two GRIN lenses in termsof d1. Note that some plots in Fig. 9 coincide, e.g.,the graph of Z0 � d1 is exactly the same as the graphfor Z0 � 3d1. This is also the case for the graphsbelonging to Z0 � 0 and Z0 � 4d1 and is in agreementwith the symmetry of Fig. 7. Furthermore, it can beobserved that there is no loss for Z0 � 2d1 and X0 �0, which corresponds with the self-imaging condition.To make this graph, we assumed that the beam waistof both lenses was wT � wR � 0.3 mm and theirdistances to their beam waist d1 � d2 � 200 mm. Awavelength of � 1550 nm was assumed for thecalculation.

If there is no angular tilt misalignment, an expres-sion for the loss as a function of the lateral offset losscan be found. This loss, denoted as Loffset, can bewritten as

Loffset � L�Z0, X0, � � 0� � L�Z0, X0 � 0, � � 0�

� AoffsetX02, (46)

in which the offset coefficient Aoffset is equal to

Aoffset ��5

ln 101

�F�2 Fr� 4w1

4�Z0 � d1��

k2

R12�Z0 � d1�

�4Fr

w12�Z0 � d1�

�� Fi

4w1

2�Z0 � d1�� k

R1�Z0 � d1�� Fi�� .

(47)

In other words, the offset loss curve is a parabola andthe coefficient in front of the square term, the offsetcoefficient, is dependent on the separation distance Z0.

Figure 10 shows the coupling loss as a function ofthe angular tilt misalignment for different values ofthe separation distance Z0 according to Eq. �45�.Here X0 is fixed at zero. Note that the coupling lossis sensitive for angular tilt misalignment. Again, aloss of 0 dB is reached when there is no angular tiltmisalignment and Z0 is equal to 2 times d1. Finally,it can be observed that, if there is no angular tiltmisalignment, the loss for a separation distance ofZ0 � d1 is equal to the loss for a separation distanceof Z0 � 3d1. The loss is also the same for Z0 � 0 andZ0 � 4d1, if angular tilt misalignment is absent. Tomake the measurements shown in Fig. 10, the samevalues for d1, d2, wT, wR, and were used as in Fig. 9.

The proposed theory indicates that, dependent onthe accuracy of each type of misalignment, the bestsetup to reduce the loss is different. Although theideal separation distance is Z0 � 2d1 when there is nolateral offset and angular tilt misalignment, fromFig. 10 it can be concluded, for example, that oneshould prefer a separation distance of Z0 � d1 to aseparation distance of Z0 � 2d1, if the angular tiltmisalignment is larger than approximately 0.045deg. The point of intersection belonging to this caseis indicated with an arrow in Fig. 10.

Like a lateral offset loss Loffset, an angular tilt loss

Ltilt is introduced for analysis. This loss indicateswhat the influence is of the angular tilt in the absenceof lateral offset misalignment. This loss is equal to

Ltilt � L�Z0, X0 � 0, �� L�Z0, X0 � 0, � � 0�

� Atilt sin2 � Atilt�2, (48)

where the tilt coefficient can be written as

Atilt �5Fr k

2

ln 10�F�2. (49)

In addition to the three misalignments, there isanother possible source for the coupling loss, namely,spot-size mismatch. When the beam waists of thebeams produced by the GRIN lenses are not exactlythe same, it is impossible in theory to find a zero-losssetup. Assuming there is neither lateral offset norangular tilt misalignment, the least loss will be foundat a separation distance equal to the beam-waist dis-tance of the transmitting lens plus the beam-waistdistance of the receiving lens �Z0 � d1 � d2�. In thissetup the theoretical coupling power loss that is dueto spot-size mismatch is calculated to be

L � �10 log4

�wR

wT�

wT

wR�2 . (50)

This result is the same when uses two non-self-imaging GRIN lenses �d1 � d2 � 0� as described in anearlier study.6 Usually the difference between thebeam waists wT and wR is quite small, so the spot-sizemismatch loss is also small compared with the lossthat is due to misalignment. It follows that, for

Fig. 10. Loss as a function of the angular tilt misalignment fordifferent values of the separation misalignment. The lateral off-set misalignment is fixed at zero. When there is no angular tiltmisalignment, the loss for both the pairs Z0 � d1 and Z0 � 3d1 aswell Z0 � 0 and Z0 � 4d1 is the same. There is no loss when Z0 �2d1, and the angular tilt misalignment is equal to zero.


0.7 � wR�wT � 1.4, the loss that is due to spot-sizemismatch is less than 0.5 dB.

3. Experimental Verification of the Self-Imaging Theory

In this paper we have established the theory behindthe GRIN–GRIN coupling loss for the self-imagingcondition. To verify this theory, experiments areperformed. At first, some parameters of the Gauss-ian beam produced by the used GRIN lenses are de-termined. The location and magnitude of the beamwaist can be determined by means of a mirror exper-iment and a razor blade experiment, respectively.Second, the coupling loss for separation misalign-ment is verified experimentally. Finally, suitableexperiments are performed to verify the coupling losstheory for both lateral offset as well as angular tiltmisalignment. For these experiments we used atransmitting GRIN lens and a movable triangularglass prism, which created another imaginary receiv-ing GRIN lens with a certain misalignment relativeto the transmitting GRIN lens. By moving thisprism, we could change the artificial lateral offset orangular tilt. For the experiments a LightPath Tech-nologies GRIN lens T5100 series is used because itsdesign exhibits the self-imaging behavior, and othercommercial GRIN lens manufacturers such as NSGcurrently do not produce self-imaging-type lenses.We denote this lens as GRIN lens 1.

A. Identical Gradient-Index Lens Coupling Case

A GRIN lens emits a Gaussian beam with its beamwaist a certain distance d after the exit face of theGRIN lens. In this experiment this distance is de-termined. In Subsection 2.B a theoretical formulafor the loss as a function of the separation distance oftwo identical GRIN lenses was derived �see Eq. �23��.Here both lateral offset as well as angular tilt mis-alignment were assumed to be absent. A simulationof a setup in which two identical GRIN lenses arefacing each other without the two above-mentionedtypes of misalignment can be realized by means of amirror placed perpendicular to the Gaussian beam.This way the imaginary separation distance betweenthe two identical GRIN lenses is twice as large as thedistance from the GRIN lens to the mirror. A sche-matic overview of the experimental setup is shown inFig. 11 where we placed a horizontally movable mir-ror in front of the GRIN lens. The Gaussian beam

emitted by the GRIN lens is reflected by the mirrorand is captured by the same GRIN lens. The powerof the captured beam can be measured with a detec-tor connected to a powermeter by a circulator. Amaximum optical power will be detected when thedistance from the mirror to the GRIN lens is equal tothe beam-waist distance. By moving the mirror infront of the GRIN lens, we can find this beam-waistdistance. It is important that the mirror is perpen-dicular to the Gaussian beam. If this is not the case,one creates an undesired lateral offset as well as anangular tilt misalignment.

Two different mirror experiments were performed.At first, the mirror is aligned after each horizontalshift. In theory, this realignment is not necessary,but in practice the mirror can be misaligned by smallvibrations during the horizontal movement of themirror. The disadvantage of this method is that themirror at one distance could be better aligned than atanother distance. This makes the experiment lessreliable. Nevertheless, we can compensate thismeasurement error by taking many measurementpoints. For GRIN lens 1, the results of the experi-ment in which the mirror is realigned each time canbe observed in Fig. 12. To obtain reliable results,this experiment is repeated several times. The re-sults of the different measurements showed consis-tency. In this plot the loss of the connectors �0.1 dBfor each connector� and the loss of the circulator �1.1dB� are taken into account. According to this plot,

Fig. 11. Overview of the mirror experiment setup. A circulator is used to send the light from the source toward the GRIN lens and thereflected light toward the detector. This detector is connected to a powermeter that measures the reflected power. i�o, in�out.

Fig. 12. Coupling loss result of the mirror experiment when weadjust the mirror for GRIN lens 1. From these results it followsthat the beam-waist distance is equal to 4 cm.


the beam-waist distance of GRIN lens 1 is approxi-mately 4 cm. Note that this is an approximate dis-tance because the minimum of the plot is smooth.

Another way to perform the experiment is to alignthe mirror only once. This alignment is done atthe largest possible distance between the GRINlens and the mirror for the following reason. Thelarger the distance from the GRIN lens to the mir-ror, the more sensitive the detected optical power isfor misalignment. Hence, if the mirror is wellaligned at a large distance, it certainly is aligned forsmaller distances. To reduce vibrations, we use atranslational stage that is able to move smoothly.The results can be observed in Fig. 13. This ex-periment also indicates that the beam-waist dis-tance for GRIN lens 1 is �4 cm. The minimummeasured coupling loss for both experiments is ap-proximately 0.35 dB, where a 0.18-dB loss is due tothe mirror. Note that, for larger separation dis-tances between the mirror and the GRIN lens, asmall misalignment of the mirror leads to a largerundesired offset misalignment �see Fig. 14�. Thisleads to a higher loss for large separation distances,which is an inevitable consequence of the used ex-

perimental setup. In addition, the presence of ab-errations that is due to the GRIN lens makes itpractically difficult for insertion loss to go to zero.

The beam waist for the light emitted by the GRINlenses is determined with the knife-edge beam pro-file method.21 Figure 15 shows the result for GRINlens 1. The beam waist for GRIN lens 1 is equal to0.479 mm. This result is in good agreement withthe specifications.22 Theoretically, this experi-ment could be used to find the beam-waist distance.By repeating the experiment for several distancesbetween the GRIN lens and the plane of the razorblade, we can determine the minimum beamwidth—that is, the beam waist. The distancewhere this minimum beam width occurs is thebeam-waist distance. In practice, the divergenceis too small and the minimum is too smooth todetermine an accurate value for d.

B. Gradient-Index-Lens–Gradient-Index-Lens CouplingLoss with a Separation Misalignment

Two GRIN lenses were used for a self-imaging cou-pling loss experiment. GRIN lens 1 was the trans-mitting lens. As a receiving lens we used anotherT5100 series GRIN lens. This lens has a beamwaist of wR � 0.48 mm and a beam waist distanceof d2 � 9 cm. We use these parameters to comparethe experimental data with the theoretical couplingloss. The results of this separation coupling lossexperiment are shown in Fig. 16. Here the self-imaging mechanism is confirmed. To make a bet-ter comparison, we elevated the theoretical losscurve 0.35 dB. This value of 0.35 dB is the mini-mum experimental loss. This loss can be causedby the connectors in between the fibers and possiblesmall undesired angular tilt and lateral offset mis-alignments. More importantly, it can also becaused by aberrations in the GRIN lens. Theoret-ically, the minimum loss should take place at aseparation distance of Z0 � d1 � d2 � 13 cm. Us-ing the Fig. 16 plot for this pair of GRIN lenses, wecan obtain a minimum loss at a separation distanceof Z0 � 12.5 � 1 cm.

Fig. 13. Coupling loss result of the mirror experiment with noadjustment to the mirror. According to this experiment, GRINlens 1 has a beam-waist distance of 4 cm.

Fig. 14. �a� For a large separation distance a certain misalign-ment � of the mirror leads to a small overlap area or a large loss.�b� For a smaller separation distance the same misalignment �results in a larger overlap, thus a smaller loss.

Fig. 15. Results for the razor blade experiment for GRIN lens 1.For GRIN lens 1 the beam waist is 0.479 mm.


C. Coupling Loss with a Lateral Offset Misalignment

To verify the coupling loss formula for lateral offsetmisalignment, an offset experiment is performed.Using a movable glass triangular prism, we created anartificial offset misalignment. The principle of thisidea is based on two total internal reflections. To un-derstand the setup, we consider a straight-light beamentering the glass prism perpendicularly, exactly inthe middle of the hypotenuse plane, as is shown in Fig.17�a�. We are dealing with a Gaussian beam insteadof a straight beam. However, it can be assumed thatthe divergence of the Gaussian beam is small enoughrelative to the dimensions of the prism. For the re-flection by the prism, the beam can be treated as if itwere straight.� Here only the light paths of the upperand the lower beam borders are shown. After twototal internal reflections the upper beam border willcoincide exactly with the lower beam border and viceversa. In other words, the prism will reflect an in-coming beam upside down. For a symmetric beam,like the Gaussian beam, the prism will act as a mirror.Figure 17�b� shows that a horizontal shift over a dis-

tance X0 leads to a lateral offset misalignment of 2X0.In the lateral offset setup, the laser source, circulator,detector, and powermeter are in the same way con-nected to the GRIN lens as in the mirror experiment.The main advantage of this method is that only oneGRIN lens is needed, and thus the setup can be con-sidered as symmetric, i.e., the imaginary GRIN lens isidentical to the real GRIN lens.

The prism experiment is performed with GRIN lens1 for which we found d � 4 cm and wT � 0.479 mm.The experiment was repeated for several separationdistances between the GRIN lens and the prism. Theseparation distance Z0 equals twice the distance fromthe prism to the GRIN lens. Figure 18 shows theexperimental data for the separation distances of 8,110, and 150 cm. In the graphs the normalized offsetloss is plotted versus the lateral offset misalignment,which equals twice the offset of the prism relative tothe propagation axis of the Gaussian beam.

The normalized offset loss is defined as the lossthat is due to lateral offset only. This is the reasonwhy the minimum offset loss is set to zero. In realitythere is a certain minimum loss caused by the inter-nal reflections at the right-angle edges in the prism,imperfections of the prism, and possible small angu-lar misalignments of the prism. We could denotethe normalized loss as

L � �10 log�Pm

P0� , (51)

in which Pm is the measured power in a certain setupwith a lateral offset and P0 is the maximum measuredpower during the experiment, i.e., when the prism hasno lateral offset relative to the Gaussian beam.

The experimental data were compared with the the-oretical result as found above. The theoretical curveLoffset � AoffsetX0

2 is also plotted in each chart. Foreach separation distance the offset coefficient accord-ing to Eq. �47� was calculated. The value of Aoffsetbelonging to the separation distance is displayed in

Fig. 16. Comparison of the experimental data with our theoreti-cal result for separation coupling loss between two GRIN lenses.The theoretical curve was shifted 0.35 dB upward to make a bettercomparison. The experiment shows good agreement with theory.

Fig. 17. Prism is reflecting a light beam �a� without offset and �b� with offset. When there is no lateral offset, the reflected beam exactlycoincides with the incoming beam. The offset of the reflected beam is twice as large as the offset of the prism relative to the incomingbeam.


each graph in Fig. 18. Here the parameters forGRIN lens 1 were used, that is, wT � wR � 0.48 mmand d1 � d2 � 4 cm. In general, it can be concludedthat the experimental data match the theory.

D. Coupling Loss with Angular Tilt Misalignment

In addition to lateral offset loss, the loss that is due toangular tilt misalignment needs to be verified experi-mentally. The setup is partly the same as describedin Subsection 3.C. Now a convex lens is placed inbetween the prism and the GRIN lens. The geometryof the setup is shown in Fig. 19. The GRIN lens emitsa Gaussian beam with a beam waist wT at a distanced1 away from the GRIN lens. A convex lens is placedin front of the GRIN lens at exactly its focal distance f.The distance from the beam waist to the convex lens isdenoted as X and it equals X � f � d1. We place themovable prism at a variable distance L after the con-vex lens. As described in Subsection 3.C, the prismwill reflect the Gaussian beam with a lateral offsettwice as large as the offset of the prism. On its wayback the beam propagates through the lens for a sec-ond time. Because the convex lens is exactly its focaldistance separated from the GRIN lens, the Gaussianbeam will be focused exactly on the exit face of the

GRIN lens. This way an artificial angular tilt mis-alignment of

� � tan�1�2X0

f � (52)

is created. By moving the prism, we can change theangle �. Unlike the setups with the mirror and theprism, this lens-based design is not a symmetric

Fig. 18. Comparison of the experimental data with the theoretical results for the normalized coupling loss that is due to lateral offset fora separation distance of �a� 8 cm, �b� 110 cm, and �c� 150 cm. The experimental data match the theory.

Fig. 19. Geometry of the angular tilt setup. By placing a convexlens exactly at its focal distance f after the GRIN lens, we can focusthe reflected Gaussian beam on the exit face of the GRIN lensunder a certain angle �. The distance L between the convex lensand the prism can be adjusted to vary the imaginary separationdistance Z0.


setup because the Gaussian beam is distorted by theconvex lens. In other words, the imaginary receiv-ing GRIN lens is not identical to the transmittingGRIN lens as was the case in the above experiments.However, by means of ray matrix calculations, thebeam waist wR and the beam-waist location Ybw canbe calculated in terms of X, f, L, and wT. For aderivation of these expressions, see Appendix A.The location and magnitude of the beam waist are

Ybw � �B0 D � A0 Cz0

2

B1 D � A1 Cz02 , (53)

wR �

�nD2 � C2z0

2

��A1Ybw � A0� D � �B1 Ybw � B0�C�z02�1�2

,

(54)

respectively, where

A1 �2f �L

f� 1� , (55)

A0 � 1 �2Lf

, (56)

B1 � 1 �2�X � L�

f�

2LXf 2 , (57)

B0 � X � 2L �2LX

f, (58)

C �2f �L

f� 1� , (59)

D � 1 �2�X � L�

f�

2LXf 2 , (60)

z0 ��nwT

2

. (61)

Now L can be adjusted to vary the beam waist wR andits location Ybw. The imaginary receiving GRIN lensis assumed to be a quarter-pitch lens, i.e., it has itsbeam waist on its edge or d2 � 0. Furthermore, weassumed that � is small enough to make the approx-imation cos � � 1 and that the separation distancebetween the two GRIN lenses is equal to Z0 instead ofZ0 cos �. This assumption is justified for the maxi-mum angle of 0.20 for which the experiment wasexecuted. From Fig. 19 it follows that the separa-tion distance is equal to Z0 � f � Ybw. The way thelaser source, the circulator, the detector, and the pow-ermeter are connected to the GRIN lens is similar tothe above setup.

Several experiments were performed, each with adifferent value for L. A comparison of the theorywith the experimental data was made. The resultsfor L � 70 cm and L � 15 cm are shown in Figs.20�a� and 20�b�, respectively. A plano–convex lenswith a focal distance of f � 0.75 m was used �see Fig.19�. Furthermore, we assumed that the beamwaist of GRIN lens 1 was wT � 0.48 mm and d1 �4 cm as was concluded from the razor blade and themirror experiments. Inserting these values inEqs. �53� and �54� and using Z0 � f � d1, we obtainZ0 � 0.95 mm and wR � 0.48 mm for L � 70 cm andZ0 � 23 cm and wR � 0.35 mm for L � 15 cm. It isassumed that the imaginary receiving GRIN lenshas the beam waist of its beam equivalent at its exitface �quarter pitch� and thus d2 � 0. Subse-quently, it follows from Eq. �49� that Atilt � 6.34dB�mrad2 for L � 70 cm and Atilt � 3.75 dB�mrad2

for L � 15 cm. For these values the theoreticalcurve is also plotted in Figs. 20�a� and 20�b�. Thistheoretical curve matches the experimental data forL � 70 cm. Although the measured loss for L � 15cm is somewhat low compared with the theoreticalcurve, the experimental data generally obey thetheory.

Fig. 20. Comparison of the experimental data with the theoretical results for the normalized coupling loss that is due to angular tilt for�a� L � 70 cm and �b� L � 15 cm.


4. Conclusion

A GRIN lens with its beam waist at a certain distanceafter its exit face is referred to as a self-imagingcollimator. This collimator has the power to imple-ment the described low-loss free-space coupling be-tween GRIN lenses by the so-called self-imagingmechanism. The beam-waist magnitude and loca-tion for such a self-imaging collimator are calculatedin terms of the parameters of the radial-type GRINlens �i.e., pitch, core refractive index, gradient index,length�, the air gap between SMF and GRIN lens, andthe optical wavelength. For the first time to ourknowledge, theoretical misalignment coupling lossformulas have been derived to employ self-imaging-type fiber collimators in SMF-based free-space sys-tems. Experimental results described in this paperagreed well with the proposed mathematically derivedexpressions. For the first time to our knowledge, theprovided coupling loss theory in this paper allows pho-tonic engineers to improve designs for optimal ultra-low-loss coupling in fiber-based free-space systemssuch as dynamic wavelength equalizers, variableattenuators,23–25 and large three-dimensional opticalcross-connect switches.

Appendix A: Propagation of a Gaussian Beam throughthe Setup

In Subsection 3.D, a setup containing a triangularglass prism and a convex lens was proposed to createan artificial angular tilt misalignment. By varyingthe distance L between the prism and the convexlens, we can change the separation distance Z0 be-tween the GRIN lenses of the imaginary setup. Aninevitable consequence of the distortion of the convexlens is that the beam waist of the imaginary GRINlens will be changed. In this appendix a derivationof the expressions for the beam waist and its locationare given.

To determine the state of a Gaussian beam, after ithas passed an optical system, e.g., a lens, one canmake use of ray matrices and the ABCD law.8 Whena certain optical system k has the ray matrix

Mk � �Ak Bk

Ck Dk�

and the complex radius of curvature before theGaussian beam enters the optical system is equal toqk, then the complex radius of curvature after theoptical system equals

qk�1 �Ak qk � Bk

Ck qk � Dk.

To apply the ABCD law, the matrix of the wholeoptical system has to be found. This optical systemincludes a straight section through the air to theconvex lens, a distortion by the convex lens, a secondstraight section through the air to the prism, a re-flection by the prism, again a straight section throughthe air back to the lens, a second propagation throughthe lens, and finally the Gaussian beam will travel

through the air again. Next the overall ray matrixof the described optical system is calculated.

Consider a self-imaging GRIN lens producing aGaussian beam with a certain beam waist wT at acertain distance d1 from the GRIN lens exit face.We take the beam-waist location as the starting pointrather than the exit face of the GRIN lens. Here thecomplex radius of curvature equals

q0 � iz0, (A1)

where

z0 ��nwT

2

. (A2)

From this point, the Gaussian beam travels astraight section over a distance X. The ray matrixbelonging to this straight section is

M1 � �1 X0 1� . (A3)

Second, the Gaussian beam propagates through aconvex lens with focal distance f. For a thin lens theray matrix14

M2 � � 1 0

�1f

1� (A4)

holds. After the lens the light travels a distance Lbefore it reaches the prism, which reflects the Gauss-ian beam with a lateral offset of 2X0. Then the lighttravels back until it enters the focal lens for a secondtime. The prism and the lateral offset 2X0 are bothsmall compared with the length L. We assume thatthe prism acts as a mirror at the plane perpendicularto the Gaussian beam located at the straight angle ofthe mirror. So the ray matrix

M3 � �1 2L0 1 � (A5)

is applicable to the following sections: the path fromthe convex lens to the prism, the reflection by theprism, and the path back from the prism to the con-vex lens. Subsequently, the laser light is again fo-cused by the lens for which matrix M2 holds again.

Finally, the Gaussian beam will travel againthrough the air over a certain distance Y. Similar toEqs. �A3� and �A5�, for this section the ray matrix

M4 � �1 Y0 1� (A6)

is used.To determine the overall ray matrix of the optical

system, we multiply the matrices as follows:

M5 � M4 � M2 � M3 � M2 � M1 � �A BC D� , (A7)


in which

A �2f �L

f� 1�Y � 1 �

2Lf

, (A8)

B � �1 �2�X � L�

f�

2LXf 2 �Y � X � 2L �

2LXf

,

(A9)

C �2f �L

f� 1� , (A10)

D � 1 �2�X � L�

f�

2LXf 2 . (A11)

Using the ABCD law, we can write the complex beamparameter at distance Y to the left of the focal lens as

q1 �BD � ACz0

2

D2 � C2z02 � i

� AD � BC� z02

D2 � C2z02 . (A12)

At the beam waist the Gaussian beam has a planarwave front, or in other words 1�R � 0. Recalling thedefinition of the complex beam parameter,

1q

�1R

� i

�nw2 , (A13)

it can be seen that for the beam waist the complexbeam parameter is purely imaginary. To determinethe location of the beam waist of the reflected Gauss-ian beam, the value of Y for which the real part of q1is equal to zero has to be found. In terms of Y thisimplies

Re�q1� ��B1 Y � B0� D � � A1 Y � A0�Cz0

2

D2 � C2z02 � 0,

(A14)

in which

A1 �2f �L

f� 1� , (A15)

A0 � 1 �2Lf

, (A16)

B1 � 1 �2�X � L�

f�

2LXf 2 , (A17)

B0 � X � 2L �2LX

f. (A18)

The solution of this linear equation is denoted as Ybwand equals

Y � Ybw � �B0 D � A0 Cz0

2

B1 D � A1 Cz02 . (A19)

So, theoretically at this distance at the left of the lens,the beam waist will be found. Actually one has totake this distance along the light path, which makes

an angle � with the normal instead of this distanceparallel to the normal. However, it is assumed that� is small enough so that cos � � 1. This assumptionis justified for the maximum angle of 0.20 for whichthe experiment was executed.

We can obtain the magnitude of the beam waist,denoted as wR, by inserting Ybw into Eq. �A12�.From the definition of the complex radius of curva-ture, Eq. �A13�, it follows that

w � wR � ��

�n Im�1�q1��1�2

�

�nD2 � C2z0

2

�� A1 Ybw � A0� D � �B1 Ybw � B0�C�z02�1�2

.

(A20)

The authors acknowledge the experimental sup-port from Z. Yaqoob, M. A. Arain, and S. A. Khan.We also thank J. Wolter at the Technical Universityin Eindhoven, The Netherlands, for suggesting thatM. Van Buren conduct a research project with N. A.Riza’s laboratory at the Center for Research and Ed-ucation in Optics and Lasers.

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