Microlenses are used to connect light sources to de-tectors, modulators, and optical fibers. Common mi-crolens fabrication methods are reactive ion etching,melting, and photolysis.1,2 With the melting methodit is possible to make fine plano–convex lenses orarrays of them. The materials that have been usedwith this method are photoresist,3 fused silica,4,5 andsome polymers.6,7 Lenses made with photoresistshow good characteristics and can be transferred toother substrates by reactive ion etching. Althoughplano–convex lenses have good optical behavior,lenses with two spherical surfaces might be preferredfor some applications. In the past some attemptswere made to make biconvex microlenses.7,8
In this paper we describe the fabrication and test-ing of microlenses with two spherical surfaces. Wefabricated two types of microlenses: spherical androd. The spherical surfaces were produced by meltingwith a CO2 laser, as described in Section 2. In Section3 two studies are presented: One is a study of thesurfaces of lenses with a scanning-electron micro-
scope (SEM); the other is a study of the deviation ofthe microlens profile from a theoretical circle. Theresolution of the rod and spherical lenses, by imagingof a test chart, and Gaussian curves for the lenses areshown in Section 4. A theoretical study of the behav-ior of the lenses by means of ray tracing is describedin Section 5. Finally, in Section 6 some applicationsare presented.
2. Method of Fabrication
The processing of materials with CO2 lasers is basedon the melting or softening of the material. It is pos-sible to polish surfaces of glass of �80 mm diame-ter9,10 and small surfaces such as the tip of an opticalfiber.11 Plano–concave fused-silica lenslets, whichrange from 100 to 500 �m in diameter, have beenmade upon plane substrates by CO2 lasers.12 Plano–convex lenses also have been formed directly uponthe surfaces of glass plates by use of CO2 lasers.13 Anin-depth study of the heating–cooling cycle of opticalglasses by CO2 radiation was made by Veiko etal.14–18 They also described the fabrication of fiberconnectors, diode laser optics, microlens arrays,waveguides, and other components.
In 1664 Robert Hooke made small spherical lensesat the ends of thin strands of glass.19 These strandswere heated and, when they reached the melting tem-perature, the associated surface tension formedspherical lenses. More recently, spherical lenses atthe ends of optical fibers were made with CO2 lasers.This was done to increase the acceptance angle of theflat end of the fiber20 or to focus light.21 Based onthese methods we have made spherical lenses bymelting the ends of fibers. However, we have used our
S. Calixto ([email protected]) and F. J. Sanchez-Marin arewith the Centro de Investigaciones en Optica, Apartado Postal1-948, Leon, Guanajuato c.p. 37000, Mexico. M. Rosete-Aguilar iswith Centro de Ciencias Aplicadas y Desarrollo Tecnologico, Uni-versidad Nacional Autónoma de México, Apartado Postal 70-186,D.F. c.p. 04510, Mexico. L. Castañeda-Escobar is with the Insti-tuto Nacional de Astrofisica, Optica y Electronica, Apartado Postal51, Puebla, Puebla, Mexico.
Received 26 August 2004; revised manuscript received 17 De-cember 2004; accepted 18 December 2004.
lenses to form images, not to feed light into the fiberor to focus it.
A. Spherical Lenses
The fabrication of small spherical lenses has evolved,and now it is possible to find on the market sphericallenses with sizes from tens of micrometers to milli-meters.22,23 Problems related to these lenses includehandling, fixing, and positioning. The spherical mi-crolenses that we propose are attached to stems; thusthey are easy to handle and fix. The material that weused is fused silica.
Our method to fabricate microlenses with stems isas follows: Depending on the desired size of the lens,the diameter of a glass cylinder is chosen. We havetried fused-silica cylinders with diameters rangingfrom �30 to 200 �m. After melting, spherical lenseswith diameters from 35 to 300 �m were obtained. Thematerial source of the cylinders was optical fiber. Thethinnest available fiber is a single-mode fiber with adiameter of 125 �m. To obtain thinner cylinders, wepulled some single-mode fibers with a special instru-ment24 consisting of a torch and two step motors topull it. First the torch heats the fiber, following lon-gitudinal displacement. When the fiber attains a cer-tain temperature the step motors pull the fiber. Thesemovements, and the temperature detection, are con-trolled by a computer.
To make a spherical lens at the end of the cylinder,we used a germanium lens to focus light from a CO2laser onto the cylinder end (Fig. 1). When the glassattained the melting temperature, a spherical lenswas formed by surface tension. In all cases, the spher-ical microlenses remained attached to the cylinder.The main parameters in the fabrication process arethe chemical and physical characteristics of the glasscylinder, the power of the infrared beam, the focallength of the germanium lens, the distance betweenthe end face of the cylinder and the lens (defocus), andthe exposure time. By controlling these parameters,one can control the lens’s size and shape to someextent. In Fig. 2 two spherical microlenses with dif-ferent diameters are shown. The smaller lens has adiameter of �35 �m; the larger, a diameter of�300 �m.
The CO2 laser that we used was continuous wave�� � 10.6 �m�, and the output power ranged from 10to �30 W. The former value was used for the thinnercylinders (�30 �m in diameter); the latter values, forthe thicker cylinders ��1 mm�. The focal lengths of
the germanium lenses ranged from 10 to 12.5 cm.Because of the Gaussian intensity profile of the laserbeam, the focused laser’s spot diameter was chosen tobe �40% bigger than the cylinder diameter. Toachieve this spot size we placed the ends of the cyl-inders inside the focal point by �10% to �15% of thefocal length. Exposure times lasted fractions of a sec-ond when thin cylinders were melted and severalseconds for thick cylinders. We monitored the stabil-ity of the laser by measuring a small part of the beam.
B. Rod Lenses
To make rod lenses, first a cylinder of approximatelythe desired lens length was cut. Then the rod endswere illuminated (Fig. 1), one at a time, until meltingoccurred. Cylinders with diameters ranging from�0.9 to 1.2 mm were used. It was found that, afterexposure to the laser light, a cylinder’s flat ends be-came segments of a sphere. In Fig. 3(a), as an exam-ple, the end surfaces of two cylinders with differentdiameters are shown. It should be mentioned that, ifthe exposure time was short, if the power of the CO2beam was weak, or both, the flat end did not becomea segment of a sphere. It seems that the attainedtemperature was not enough to produce melting ofthe entire surface. An example of an underexposedend can be seen at the left in Fig. 3(b).
3. Surface Studies
The surfaces of our spherical lenses were studiedwith a SEM (see Fig. 4). To find out how close to acircular shape the profile of a spherical lens was, weused an image-processing procedure to segment (i.e.,to separate in a single image) the outer border of thelens. We took three points from the profile (i.e., theouter border) of the lens in Fig. 4 to obtain the radius
Fig. 1. Optical configuration used to melt the end surfaces of glasscylinders.
Fig. 2. (a) Glass cylinder before the melting step. The distancebetween two consecutive marks in the scale is 100 �m. (b) Cylinderwith a spherical lens. The parameters at melting time were thefollowing: time of exposure, 200 ms; laser power, 18 W; defocus,10% �f � 12.5 cm�. (c) Spherical lens with a diameter of �300 �m.
and the coordinates of the center of a theoretical cir-cle. We compared this theoretical circle with the ac-tual profile of the lens by subtracting the distancefrom the center to each point of the profile. In Fig. 5(a)the deviation of the profile from the best-fitting circleis plotted. The axis of the abscissa corresponds to thepixel number that belonged to the rim. From thisfigure it is possible to see that the deviation of thelens’s surface from a perfect circle in most of theselected profile is no more than �0.4 �m. It should bementioned that the profile of the lens in Fig. 5(a) isjust 1 pixel wide and that we took care that the errorintroduced by the procedure was no larger than 1pixel. Then, because in the image of Fig. 5(a) 1 pixel
corresponds to 0.097 �m, this value was the maxi-mum error of our measurements.
The profile study outlined above was also madewith a rod lens. This time the diameter of the lenswas �865 �m. Therefore we used an optical micro-scope, instead of the SEM, to investigate the lens’sprofile. The deviation of the rod lens’s profile from thetheoretical circle is plotted in Fig. 5(b). It is possibleto see that a deviation of no more than �20 �m oc-curred.
To evaluate whether a measured deviation of20 �m from a spherical profile is good, we evaluatethe variation in the wave-front aberration when theradius of curvature changes by 20 �m. Let us sup-pose that an axial point is located at an infinite dis-tance from the rod lens. The surfaces of the rod lensare spherical, with a radius of 448 �m (with a mea-sured deviation of 20 �m) and a diameter of 865 �m.The refractive index for the rod lens is n � 1.45851for � � 0.650 �m. By applying Rayleigh’s quarter-wavelength rule we determine whether this changecan be tolerated. According to Rayleigh’s criteria, ifthe wavefront deformation is less than a quarter of awavelength then the amount of aberration can be
Fig. 3. (a) Ends of rod cylinders with spherical shape. The diam-eter of the biggest cylinder is 1.2 mm. Parameters at recordingtime were as follows: time of exposure, 1 s; laser power, 30 W;defocus, 10% �f � 12.5 cm�. (b) The cylinder at the left shows asurface that is not spherical because of improper exposure to the IRbeam.
Fig. 4. SEM photograph of a spherical lens.
Fig. 5. (a) Deviation of a spherical lens profile with reference tothe best-fitting circle. (b) Deviation of a rod lens profile with ref-erence to the best-fitting circle.
tolerated. In this case, a quarter of a wavelengthcorresponds to 0.16 �m for a wavelength of 0.65 �m.For an axial point, the spherical aberration is theonly aberration that contributes to the primary wave-front aberration as
W � (1�8)SI,
where SI is a Seidel term given by
SI � �A2h�(u�n),
where A � n�u � hc� � n��u� � hc� and ��u�n�� u��n� � u�n. In these expressions u and u� are theangles of the incident and refracted rays, respec-tively, measured from the optical axis; h is the heightof the marginal ray at the surface; and n and n� arethe refractive indices before and after the surface,respectively.
Evaluating for an incident ray such that u � 0 andh � 432.5 �m yields, for the wave-front aberrationproduced by the first spherical surface with a radiusof curvature of 448 �m, W � 2.43 �m. Now, if theradius of curvature increases by 20 �m, the wave-front aberration is given by W � 2.13 �m. The dif-ference between the two wave-front aberrations isgiven by �W � 0.3 �m, which is larger than 0.16 �m,so a measured deviation of 20 �m in the radius of thesurface is not good. However, the evaluation aboveassumes that the full aperture of the microlens isbeing used. As we explain below in Section 4, a pin-hole can be placed in front of the rod lens to improvethe quality of the images. The pinhole could have adiameter of 600 �m. If only 600 �m of the diameter ofthe lens surface is going to be used, however, then thedeviation of the surface from a sphere also reduces to�10 �m.
So, evaluating for a marginal ray such that u � 0and h � 300 �m, and assuming an increase in theradius of curvature of 10 �m, we have that thechange in the wave-front aberration is given by �W� �1�8��SI � 0.16 �m. So the change in the wave-front aberration is within the tolerance of a quarter ofa wavelength.
In an attempt to get images that were less aber-rated, we used a ray-tracing program. It was foundthat rod lenses with longer focal distances will per-form better. Attempts were made to make rod lenseswith surfaces that had long radii of curvature. Toachieve this objective, short exposure times and weakinfrared beams were used. In Figure 6 two ends ofdifferent rod lenses can be seen. This photograph wastaken with a microscope. Illumination was providedwith the ends of optical fibers circularly arranged.The rod end in the upper part had a shorter radius ofcurvature than the end in the lower part. Also, it canbe seen that the reflection of the circular light sourceon the rod’s surface has a circular shape. At the bot-tom part of Fig. 6 the end surface of another rod lens,with a larger radius of curvature, shows the reflectionof the circular light source as a deformed circle. This
means that the lens had a nonspherical surface. Also,some debris can be seen on the surface. To explain thesource of the debris, one should remember that tomake a small rod lens, first we cut a cylinder from along glass rod. The ends of the small cylinder haduneven surfaces with sharp peaks. During melting,when the proper exposure time is used, the wholesurface heats until the glass becomes liquid, so theunevenness and sharp peaks disappear. Then aneven, good spherical surface is formed. However, ifthe exposure time is not long enough to heat thewhole surface, a spherical form will not be achievedand the sharp peaks will instead be rounded and willnot disappear: They will form debris such as thatshown on the surface of the lens.
4. Resolving Power of the Lenses
We tested imaging with our spherical lenses by usinga test target (USAF 1951) as the object. For example,a spherical lens with a diameter of �36 �m was usedto form a unitary image of element 5 of group 7 of thetest target. To achieve this image we used the follow-ing method: The lens was placed in contact with ele-ment 5. Then it was moved away from the test target,in steps. Above the lens we placed a microscope [Fig.7(a)] to see the image given by the lens. When theimage had the same size as the object [Fig. 7(b)] theimage-formation process ended. The results obtainedtell us that the spherical lens can resolve details of�5 �m �203 line pairs �lp��mm�. We measured theback focal distances of our spherical lenses by locat-ing the image of a distant object �43 cm�. These backfocal distances ranged from �35 to �90 �m.
The formation of images of objects larger than thelens was also analyzed. An object (a pair of pliers) of18 cm was placed 18 cm from a spherical lens of150 �m diameter. The image given by the lens, asinvestigated with a microscope, can be seen in Fig.7(c).
Imaging by a rod lens was also tested. For instance,
Fig. 6. Ends of two rod lenses illuminated by a light source com-posed of ends of optical fibers arranged in a circle. The rod lens inthe upper part had a spherical surface. The rod in the lower partdid not receive proper exposure during fabrication; this surface isnot spherical.
a 4.8 mm long lens was chosen. The USAF test targetwas used as the object. The image given by the rodlens can be seen in Fig. 7(d). In the center of thephotograph it is possible to see some elements ofgroups 6 and 7.
To improve the image produced by rod lenses, astudy was made with ray-tracing software. Moreabout this study is presented in Section 5 below. Theresult of the study was that the rod lenses could givebetter images if an iris were placed in front of theentrance surface. To test this result we performed theexperiment by choosing a lens with a diameter of1152 �m and a thickness of 5.7 mm. The diameter ofthe pinhole was �600 �m. The object was element 3of group 5 of a USAF test target. The distance be-tween the object and the lens was chosen to producean image of the same size as the object. A photographof the object taken with a microscope can be seen inFig. 8(a). The image given by the rod lens without apinhole is shown in Fig. 8(b). As can be seen, theimage is blurred. To improve the image we placed thepinhole in contact with the entrance surface of thelens. The result can be seen from Fig. 8(c). Notice thatthe contrast is better and that elements 5 and 6 ap-pear clearer, meaning that a better resolution wasachieved. With regard to distortion, the image in Fig.8(c) shows a negative distortion compared with Fig.8(b). We should mention that, at this stage, the rodlens was taken out of the optical configuration, and no
image was seen. This means that the pinhole by itselfdid not form an image.
To find out more about the geometrical character-istics of the lenses we made an experiment involvingthe Gauss law for lenses. A lens (spherical or rod), aUSAF test target (used as the object), and a micro-scope were placed in X–Y–Z stages. The objective ofthis experiment was to find the conjugate points, thatis, given a distance of the object from the lens, to findthe distance where the image was formed. Again, weused a microscope to find the image produced by themicrolens.
To do this experiment we chose a spherical lenswith a diameter of �149 �m. Its profile was studiedwith a SEM, and a plot similar to the one shown inFig. 5 was obtained. In this case it was seen that adeviation of �1 �m from the best-fitting circle wasobtained. The result of the Gauss law experiment canbe seen in Fig. 9(a). A theoretical curve, relating con-jugate points, was calculated with the help of theformulas presented in Appendix A. The refractiveindex of the fused-silica lens was taken as 1.45851,and the radii of curvature of both surfaces were takenas 74.5 �m. From the plot [Fig. 9(a)] it can be seenthat experimental and theoretical curves show a sim-ilar behavior. A disagreement of the curves occurredwhen the object was close to the lens. The cause ofthis disagreement could be differences in the radii of
Fig. 7. (a) Optical configuration used to study the image producedby a microlens. Components not to scale. (b) Photograph of theimage (element 5, group 7) produced by a spherical lens with35 �m diameter. (c) Photograph produced by a spherical lens(150 �m diameter) of a pair of pliers �18 cm long. (d) Photographof the image produced by a rod lens of groups 6 and 7 of a USAFtest target.
Fig. 8. (a) Photograph taken with a microscope of group 5 of aUSAF test target. (b) Photograph of group 5 obtained with a rodlens. (c) Photograph of the image of group 5 obtained with the rodlens when a pinhole of 600 �m diameter was placed before the lens.
the lenses. The radius of a lens as seen from abovewas slightly different from that when the lens wasseen from one side. The ratio of the two radii is�1.0601. This means that the lens was toroidal. An-other explanation of the disagreement of the theoret-ical and experimental data is given as follows: Let usconsider an axial point and its image formed by anoptical system with spherical aberration. Within theparaxial approximation the image point is formed atthe paraxial position; however, at full aperture thebest image is located not at the paraxial position butat the position of the circle of least confusion, locatedat half the distance between the paraxial ray and themarginal ray. When the experiments are done, theimage plane is located where the best image is ob-served, which is closer to the circle of least confusionthan to the paraxial plane. How far the best plane isfrom the paraxial plane depends on the longitudinalspherical aberration given by the optical system andon the particular object’s position. The expression forprimary longitudinal spherical aberration is given by
LSA �SI
2nu2,
where SI is the Seidel coefficient for primary spheri-cal aberration, which one can calculate in any opticalsystem by tracing a marginal paraxial ray throughthe system. n is the refractive index in image space,however, which in our case is air, so n � 1 and u is theconvergence angle in image space. For a fixed en-trance pupil size, SI does not vary appreciably but u
does vary when the position of the object is changed.As the object gets closer to the focus of the lens, itsimage distance increases and therefore the conver-gence angle, u, decreases. So, as the object gets closerto the lens, the longitudinal spherical aberration in-creases and the separation between the paraxialplane and the best plane increases. This is the reasonfor the disagreement between the experimental andtheoretical results in Fig. 9(a).
The Gauss law for rod lenses was investigated inthe following way: A rod lens �n � 1.45851� with athickness of 5.8 mm, a diameter of 1.152 mm, andradii of curvature of 674 �m was chosen for the test.We measured the radii of curvature of its surfaces bystudying them with a microscope. The result of theconjugate points experiment can be seen from Fig.9(b). Just a few experimental points were taken. Thereason for this is that the images were not as sharp asthe images given by spherical lenses. This result isdue, possibly, to the aberrations shown by the rodlens. Also, in Fig. 9(b) the theoretical curve, calcu-lated with the formulas given in Appendix A, isshown. We shall come back to this example in Section5.
5. Ray Tracing
In Section 4, an experiment to improve the imagegiven by a rod lens by using a pinhole was mentioned.The basis of that experiment is as follows: Let ussuppose that we have an object located at twice theeffective focal length of a rod lens. We will have a 1:1imagery situation. By analyzing the image formationwith ray tracing, we obtained the diagram shown inFig. 10. We noticed that rays that travel far from theoptical axis are totally internally reflected by thebody of the lens. Thus they produce a flare about theimage given by the lens. Rays that are not too close tothe periphery are refracted by the surface and pro-duce large aberrations. To avoid these rays we placedan iris in front of the rod lens such that only rays closeto the optical axis contribute to the image formation.
At the end of Section 4 we mentioned the Gausscurve for a rod lens. Now we present some character-istics of this curve. The lens has refractive index n� 1.45851, thickness 5.8 mm, diameter 1.152 mm,and radii of curvature 674 �m. The first-orderparameters (see Appendix A) for this lens aref � �2.08434 mm, h1 � �5.63863, and h2
Fig. 9. Behavior of the image distance as a function of objectdistance, from each vertex of the spherical surface (a) for a spher-ical lens and (b) for a rod lens. Circles represent experimental data.
Fig. 10. Schematic diagram showing the use of an iris to stopmarginal rays.
� 5.63863 mm. Normally a biconvex lens is identifiedas a positive lens with its principal planes locatedinside the lens. However, in a rod lens the thicknessof the lens is large enough that a parallel ray re-fracted on the first surface of the lens intersects theoptical axis before it reaches the second surface. As aresult, the effective focal length in a biconvex rod lensis negative and the principal planes are located out-side the lens.
An object located at 2f � �4.16868 mm from thefirst principal plane is located at 1.46995 mm fromthe first vertex of the lens; its image is real andinverted and has unit magnification. As we can seefrom Fig. 9(b), images near this point were observedexperimentally. In other words, we found experimen-tally that this lens works reasonably well when it isworking close to 1:1 imagery. This result was ex-pected because we know that a symmetrical system isfree of distortion, lateral color, and coma aberra-tions.25 A symmetrical system is one in which eachhalf of the system, including the object and imageplanes, is identical to the other half such that, if thefront half is rotated 180° about the center of the stop,it will coincide with the rear half. The rod lens undertest is not fully symmetrical because there is no stoplocated at the center of it. However, the aberrationsare small for positions close to 1:1 imagery, as isconfirmed when modulation transfer function (MTF)curves for three different positions of the object arecalculated [Fig. 11(a)]. When the object is close toposition 1.46995 mm, the MTF curves tell us that theimages should show good contrast. However, forother positions, contrast is weak.
A way to reduce the aberrations of the imageformed by rod lenses even further is to use two iden-
tical rod lenses. The optical configuration will be sym-metrical relative to the center plane of the object andimage planes. The identical rod lenses are separatedby the sum of their back focal distances, forming anafocal system. A pinhole is placed at half the distancebetween the two rod lenses such that the entranceand exit pupils of the system are located at infinity.With this arrangement, an object located at a dis-tance equal to the back focal length of the rod lensfrom its first vertex will have an image that is real,inverted, and with unit magnification. The aberra-tions for off-axis objects in the two-rod lens systemare greatly reduced compared with the aberrationsfor off-axis objects in a single-rod lens. The field ofview is increased. Relay lens systems used in endo-scope objectives26 have this configuration.
The images given by spherical lenses showed goodcontrast for most of the positions of the object withrespect to the vertex. This result is confirmed by theMTF results given by the design program in Fig.11(b). Notice that modulations are better than thoseshown in Fig. 11(a).
6. Applications
Because of the small sizes of our spherical lenses,they can be used to modulate light coming from smalllight sources. One of these sources can be a vertical-cavity surface-emitting laser (VCSEL). It is possiblethat in the future these sources could be used totransfer information from chip to chip or from a chipto a fiber.
To investigate the possibility of using a sphericallens with a VCSEL we performed the following ex-periment: A spherical microlens with a diameter of�152 �m was chosen [Fig. 12(a)]. A VCSEL with anoutput aperture of 13 �m was the light source [Fig.12(b)]. The lens was placed in contact with the aper-ture. Then it was moved, in steps, away from theVCSEL. At each step, the image given by the lens wasobserved with a microscope. When this image had thesame size as the aperture the process ended [Fig.12(c)]. The amplification was unitary. The photo-graph in Fig. 12(d) shows the aperture image whenthe VCSEL was in the ON state.
The feasibility of using a rod lens to form an imageof an array of VCSELs was also tested. A 1 � 6 arraywas used as the object. The rod lens had a length of�5.8 mm and a diameter of 1.152 mm. The unitaryimage of the array is shown in Fig. 12(e).
7. Conclusions
We have presented a method with which to fabricatespherical and rod lenses. Their surfaces were evalu-ated and their capability to form images has beenshown. Also, a method to improve the images of rodlenses has been presented.
With respect to resolution, spherical lenses exhibitbetter behavior than rod lenses. To compare the per-formance of the spherical lenses with that of plano–convex lenses reported elsewhere we foundinformation for a few examples in the literature. Ta-ble 1 lists the results. As can be seen, some lenses
Fig. 11. MTF behavior. The parameter is the distance betweenthe object and the first spherical surface of the lens. (a) MTF for arod lens, (b) MTF for a spherical lens.
have resolutions as high as 1200 lp�mm; others showmoderate performance �400 lp�mm� and, finally, ourspherical lens shown fair resolution [203 lp�mm; Fig.7(b)].
With respect to deviation of the profile from aspherical surface, Table 2 lists the parameters forseveral lenses made from different materials as re-ported in the literature. From this table one can seethat some of them exhibit a deviation as small as5 nm. Others have a deviation of �24 �m. Our spher-ical lens exhibits a moderate deviation of 0.4 �m; therod lens, a deviation of 20 �m.
Despite the facts shown in Tables 1 and 2 it isadvisable that the whole process, including design,fabrication, testing, and determination of the lens’spurpose and of the environment where the elementwill work, should be considered before a given opticalelement is chosen.
Appendix A
The effective focal length of a thick lens in air withrespect to the principal planes is given by
1f � (nL � 1)� 1
R1�
1R2
�(nL � 1)tnLR1R2
, (A1)
where nL is the refractive index of the lens, R1 and R2are the radii of curvature of the first and secondsurfaces of the lens, respectively, and t is the thick-ness of the lens.
The first and second principal plane positions mea-sured from the first and second vertices of the lens aregiven by
h1 � �f(nL � 1)t
R2nL, (A2)
h2 � �f(nL � 1)t
R1nL, (A3)
respectively, which are positive when the planes lie tothe right of their respective vertices.
The Gauss equation describes paraxial image for-mation as
1soh1
�1
sih2�
1f , (A4)
where soh1 and sih2 are the object and image distancesmeasured from the first and second principal planes,respectively.
It is more convenient to rewrite the Gauss equationin terms of distances measured from the vertices ofthe lens. Let soV1 and siV2 be the object and imagedistances measured from the first and second verticesof the lens, respectively. These distances are relatedto soh1 and sih2 as
soh1 � soV1 � h1, (A5)
sih2 � siV2 � h2. (A6)
Substituting into the Gauss equation yields the dis-tance from the second vertex of the lens to the image:
Table 1. Deviation of the Measured Cross-Sectional Profile from aSpherical Curve
Fig. 12. (a) Spherical lens used to collimate the light from theVCSEL in (b). (c) Image of the aperture of the VCSEL produced bythe spherical lens. Notice that the aperture image has approxi-mately the same size as the aperture in (b). (d) VCSEL apertureimage when the VCSEL was in the ON state. (e) Unitary imagegiven by a rod lens. Notice that one of the six VCSELs was in theOFF state.
Table 2. Image Resolution of Some Plano–Convex Lenses
If we consider a rod, a plot of siV2 versus soV1 [Eq. (A7)]can be seen in Fig. 13. When the object is located faraway from the rod lens, we are on the first branch ofthe hyperbola (the branch on the right-hand side ofthe figure). This branch shows the formation of erectreal images with magnification less than 1. A mag-nification equal to 1 is present when the object islocated at the first principal plane, soV1 � �h1 and islarger than 1 when the object is between the firstprincipal plane and the focal plane (the vertical as-ymptote). The second branch of the hyperbola showsthe formation of both real and virtual images. Forobjects located between the focal plane and the objectpoint, �fh2�f � h2� � h1, whose image is located at zerodistance from the second vertex, i.e., siV2 � 0, theimages are virtual; i.e., they are located to the left ofthe second vertex of the rod lens, inverted, and withmagnification greater than 1. For objects located be-tween the point �fh2�f � h2� � h1 and the first vertexof the lens, soV1 � 0, the images are real, inverted, andwith magnification decreasing as the object getscloser to the vertex passing through unit magnifica-tion.
We thank David Monzon, Ismael Torres, MonicaBueno Martinez, Carlos Perez-Lopez, Miguel AvalosBorja, I. Gradilla, S. Eitel, and N. C. Bruce for fruitfuldiscussions. This research was partly supported bythe Mexican National Council of Science and Tech-nology (project 40172-Y).
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