Ultrathin four-reflection imagerpsilab.ucsd.edu/research/Origami Optics/files...Ultrathin four-reflection imager Eric J. Tremblay,1,* Ronald A. Stack,2 Rick L. Morrison,2 Jason H

Ultrathin four-reflection imager

Eric J. Tremblay,1,* Ronald A. Stack,2 Rick L. Morrison,2

Jason H. Karp,1 and Joseph E. Ford1

1Department of Electrical and Computer Engineering, University of California, San Diego,La Jolla, California 92093, USA

2Distant Focus Corporation, Champaign, Illinois 61820-7460, USA

*Corresponding author: [email protected]

Received 22 September 2008; revised 3 December 2008; accepted 4 December 2008;posted 5 December 2008 (Doc. ID 101823); published 7 January 2009

We present the design and experimental demonstration of an ultrathin four-reflection imager. TheF=1:15 prototype imager achieves a focal length of 18:6mm in a track length of just 5:5mm, providinga 17° field of view over 1.92 megapixels of a color image sensor with 3 μm pixels. We also present thedesign and experimental results of pupil-phase encoding and postprocessing, which were applied to ex-tend the depth of field and compensate a small amount of axial chromatic aberration present in the four-reflection imager prototype. © 2009 Optical Society of America

OCIS codes: 080.3620, 110.0110, 100.2000, 220.1920.

1. Introduction

The need for compact visible-light imagers has led toextensive use of short focal length, small aperturelenses that meet the strict space and weight con-straints required for many commercial and militaryapplications. While these miniature refractive lensesoften perform well in good lighting and within asmall range of optical magnification, resolution andlight collection are limited by the focal length andphysical track length of the lens. One method of sig-nificantly increasing focal length and magnificationwithout a corresponding increase in track length con-sists of reflecting the optical path multiple timeswith concentric reflectors, thus constraining theoptical propagation to occur within a thin opticalelement. This concept is based on an extension of tra-ditional reflective telescope design [1,2] with addi-tional reflectors to minimize track length and anenlarged diameter to maintain light collection [3].This thin multiple concentric reflector approach

leads to an optical lens design with large surfacepowers, a large obscuration ratio, and tight fabrica-tion tolerances. These challenges can be met by

(1) the use of multiple aspheric reflectors optimizedto correct the aberrations caused by the geometry,(2) an enlarged diameter to allow multiple concentricreflections, improve light collection, and offset thediffraction effects caused by the annular aperture,and (3) self-alignment of multiple surfaces throughuse of single-point diamond turning (SPDT) to fabri-cate concentric annular aspheric reflectors withminimal rechucking. An eight-reflection imager waspreviously demonstrated experimentally in [3] bydiamond turning a single side of a calcium fluoride(CaF2) lens blank and coating the substrate with pat-terned silver reflectors. The eight-reflection imagerof [3] achieved an effective focal length (EFL) of35mm in a physical track of just 5mm, providinga well-corrected image over a field of view (FOV)of 6:67°.

The steep ray angles present in a highly obscuredmultiple-reflector design create a shallow depth offield (DOF) [4]. Several hybrid optical–digital ap-proaches have been investigated for increasing theDOF of imaging systems [5–11]. These approachesoptimize the amplitude and/or phase of the imagingsystem’s pupil function to create optical systems thatare invariant with respect to defocus over a givenrange. Digital image restoration techniques are thenused to restore image contrast lost by the pupil

0003-6935/09/020343-12$15.00/0© 2009 Optical Society of America

10 January 2009 / Vol. 48, No. 2 / APPLIED OPTICS 343

modification. Wavefront coding, the hybrid optical–digital approach of Dowski and Cathey [5], utilizeda cubic-phase pupil function for extended DOF. Thisapproach is attractive since it is compatible withincoherent illumination and its phase-only modifica-tion of the pupil function maintains light throughputbetter than pupil function apodization approaches.In recent years, more general phase-only masksand several design approaches utilizing different me-trics in the frequency and spatial domain have beenintroduced for defocus invariance [9–11]. Previouslywe examined the application of an annular cosine-form pupil-phase function optimized in the frequencydomain for use with an eight-reflection lens design torelax the image-sensor alignment tolerances [12].Although experimental fabrication errors caused sig-nificant deviation from the design, we measured amodest improvement in focal plane alignment toler-ance and depth of focus.In this paper we present the design and experi-

mental demonstration of an ultrathin four-reflectionimager prototype with a novel and sensitive methodof refocus. Compared to the previously demonstratedeight-reflection imager of [3], this four-reflectionimager demonstrates a less extreme focal lengthextension; however, it offers a smaller diameterand volume, smaller obscuration, larger FOV, andlarger DOF. These characteristics make it suitablefor a wider range of applications. In this paper, Sec-tion 2 describes the design and simulation of ourfour-reflection imager. Section 3 describes the fabri-cation, integration, and experimental demonstrationof the four-reflection imager. Section 4 describes theapplication of pupil-phase encoding (PPE) to the fullaperture four-reflection design and compares theperformance to the baseline four-reflection imagerdesign. Finally, Section 5 contains our conclusions.

2. Four-Reflection Imager Design

Here we describe a four-reflection imager design in-tended for use as an ultrathin visible-light camera.The design achieves a FOVof 17° over 1.92 megapix-els of a color image sensor with 3 μm pixels in just5:5mm of total track.

A. General Considerations and Design Space

Figure 1 depicts the general design concept andgeometry for a thin multiple-reflection lens andcompares a thin paraxial lens of focal length F toan obscured multiple-reflection version using a sin-gle thin (paraxial) powered mirror at the first reflec-tion. With this simplified thin reflector geometry ofan arbitrary even number of reflections (e.g., 2, 4,6), we can examine and relate the important geo-metric parameters: FOV, aperture, focal length, dia-meter, thickness, and number of reflections. Thisapproach is similar to one described in [13], however,the analysis given here describes results given by apowered reflector rather than a refractive lens.The focal length of the multiple-reflection lens

shown in Fig. 1(b) will be F ¼ NT=ns, where F is

the focal length, N is the number of reflections, Tis the thickness of the lens, and ns is the refractiveindex of the image-side medium. For small FOV,we find that, to avoid vignetting losses, the width ofthe annular aperture is limited by the outermost ob-lique ray position at the second reflector. This condi-tion allows us to express the size of the aperture as afunction of the other geometric parameters as

w ¼ D2N

− tanðFOV=2Þ · F ·ð2 − 1=NÞ

N; ð1Þ

where w is the width of the annular aperture, and Dis the diameter. For FOV larger than

FOV ¼ 2tan−1

�DN

2FðN2− 1Þ

�; ð2Þ

and for four or more reflections, we find that thewidth of the aperture is no longer limited at thesecond reflector, but at the second to last reflector,assuming the image plane is designed to be in con-tact with the back of the lens. For these large valuesof FOV, the width of the annulus is given by

w ¼ D2N

− tanðFOV=2Þ · F · ðN − 1þ 1=NÞ: ð3Þ

A value of w ¼ 0 in Eq. (3) determines the maximumFOV possible where the aperture size has gone tozero. Figure 2 shows a plot of w versus FOV usingEqs. (1) and (3), where a sharp drop in aperture sizecan be seen once the FOV of Eq. (2) has been met.With N ¼ 2, the common two-reflector scheme, theaperture will be reduced to zero at the FOV ofEq. (2) (the max FOV), and Eq. (1) alone is sufficientto describe the relationship between parameters.

These relationships do not include diffraction ef-fects or aberrations but are useful in providing gen-eral geometric relationships to define potentially

Fig. 1. Design concept and geometry: (a) thin (paraxial) lens offocal length F and (b) four-reflection obscured version using a sin-gle thin (paraxial) reflector of focal length F.

344 APPLIED OPTICS / Vol. 48, No. 2 / 10 January 2009

useful design areas. As in all reflective systems, FOVis limited compared to a comparable refractive lens.Equations (1)–(3) indicate that as the number or re-flections (N), focal length (F), or thickness (T) are in-creased, maximum FOV decreases. The maximumachievable FOV is also increased if an internal di-electric substrate (ns > 1) is used, since the increasedindex of refraction reduces the focal length (with re-spect to a system in air), image height, and ray an-gles within the lens in the same way that focusing alens into a higher refractive-index medium reducesthese properties. In addition, as FOV is increased,the annular aperture size, w must decrease to pre-vent overlap of the annular reflectors. Finally, asthe diameter, D, is increased, both the FOV and wincrease. This last point is important since it statesthat we need an enlarged diameter to fit a number ofreflections into a thin design with reasonable FOVand aperture. With an enlarged diameter and sev-eral reflections, we can expect very large surfacepower, Φ, and surface “work,” Φy, where y is theray height off the optical axis on the first reflector.This large surface work will provide significant aber-rations that must be corrected for. To do this, eachsubsequent reflection of a multiple-reflector lenscan be used for aberration correction and/or to extendthe focal length of the system. Our present designshave required these surfaces to be aspheric due torequired surface powers and the severe constraintson thickness, FOV, and EFL. To illustrate this point,the root mean square geometric spot size of the four-reflection lens described in this paper was reduced bya factor of more than 300× using aspheric surfacescompared to the same constrained design using onlyspherical surfaces. Aspheric surfaces create addi-tional difficulties with tolerances but can be fabri-cated with little additional effort or cost using SPDT.Diffraction effects from the highly obscured

annular aperture are important since we expect ob-scuration ratios (defined as the diameter of theobscuration divided by the diameter of the outer

diameter) in excess of 0.7. Large obscuration ratiossignificantly reduce midspatial frequency values ofthe modulation transfer function (MTF) [14] and arenot usually acceptable in reflective telescopes due totheir effect on image contrast. In our case we canaccept the form of the highly obscured MTF sinceour diameter is scaled up enough to correct for theotherwise low contrast of the midspatial frequencies.We show in Section 2.B that the Nyquist frequency ofa typical small (2–3 μm pitch) pixel image sensor ismuch smaller than the cutoff frequency of the reflec-tive lens’s diffraction-limited MTF. For these spatialfrequencies of interest, the diffraction-limited MTFwill be sufficient for good contrast in our design.

B. Four-Reflection Imager Design: Specific

Our objective in the design of an ultrathin visible-light imager was to create a reflective lens designcapable of a 17° FOV, illuminating 1600 × 1200 3 μmpixels of a color image sensor in a physical tracklength close to 5mm. To meet these specifications, afour-reflection lens design in CaF2 was chosen andoptimized in ZEMAX EE, a commercially availableray-tracing program. Our four-reflection lens design,shown in Fig. 3(a), is a thin annular reflective lensutilizing four concentric aspheric reflectors to focusincoherent light to an image plane in near alignmentwith the back side of the lens. The solid geometry (so-lid dielectric as opposed to air) was chosen to increasethe achievable FOV, which is limited by the reflectivegeometry as described in Subsection 2.A. CaF2 was

Fig. 2. Example curves of annular aperture width (w) versusFOV for various diameters using paraxial geometric calculations(N ¼ 4, F ¼ 20mm, and ns ¼ 1 in this example). As aperture sizein increased, FOV decreases. As diameter is increased while main-taining fixed thickness, number of reflections, and focal length,both aperture and FOV increase.

Fig. 3. (a) Layout drawing (3=4 section) of the four-reflection im-ager and (b) monochromatic MTFat the nominal design object dis-tance of 10m. OD, object distance; ID, image distance; OTF, opticaltransfer function.


chosen as the substrate material for relatively lowdispersion but primarily due to compatibility withSPDT (with CaF2, an optical quality finish is possiblewithout postpolishing). In contrast to the eight-reflection lens in [3], the front and back sides of thefour-reflection lens are fabricated separately as twoplano-aspheric lens elements that are subsequentlyaligned and joined with index-matching gel betweenthe elements. Focusing is done by varying the nom-inal 300 μmpart spacing between these two elementsas very large changes in refocus are available with agap-spacing adjustment of the order of micrometers.Fabrication of the two lens elements separately alsoavoids a critical alignment in rechucking a single ele-ment during fabrication in favor of adjustable align-ment in assembly. This arrangement does, however,require the mechanical package to support the focusadjustment while maintaining the critical alignmentbetween the two lens elements.The four-reflection lens design shown in Fig. 3(a)

has an EFL of 18:6mm and a numerical aperture(NA) of 0.72. The effective F=# (defined as the focallength divided by the equivalent diameter of an un-obscured aperture of the same aperture area) is 1.15.The design is 28mm in diameter and 5:5mm thickwith an obscuration ratio of 0.81. Its 17° FOV fills a1600 × 1200 pixel color image sensor with 3 μmpixels. At the extremes of the field, near the cornersof the image sensor, relative illumination is 87%, andpincushion distortion is limited to 7%.The incoherent modulation transfer function

(MTF) is shown in Fig. 3(b). The design performs wellwith incoherent monochromatic light across the fullFOV up to the Nyquist frequency, 167 line pairs permillimeter (lp=mm) of the image sensor. Across thefull visible spectrum (486 to 656nm), up to 10 μmof lateral color aberration (at edge of field) is presentfrom dispersion as light enters the CaF2 lens mate-rial. In addition, larger axial color aberration thanexpected was found experimentally due to dispersionin the commercially available Cargille 0608 index-matching gel. This dispersion has been accountedfor in the current simulations and results in a chro-matic focal shift of 7 μm.Chromatic aberration is not intrinsic to concentric

multireflection lens designs in general, since an air-spaced catoptric lens with first-surface reflectorswould have no chromatic dependences. Even introdu-cing a flat cover glass on an air-spaced designdoes not introduce significant chromatic aberration.However, the first refraction at the interface of a di-electric-filled lens introduces lateral color aberrationthat cannot be easily compensated. If packagingconstraints or FOV make a dielectric-filled catadiop-tric lens desirable, it may be possible to omit theindex-matching gel in favor of a small adjustableair-gap for focus. In this arrangement, the additionof planar low-power surface-relief diffractiveelements fabricated with SPDT can be used to correctthe chromatic aberrations incurred from the air-dielectric refractions. To help correct the chromatic

focal shift present in our current four-reflectionimager design, we show in Section 4 simulatedand experimental results for the application of PPEto help correct the axial color aberration describedearlier by increasing the imaging system’s defocustolerance [15].

C. Image-Sensor Selection

The steep incidence angles present at the imageplane in this four-reflection lens design can createa problem with detection efficiency with conven-tional complementary metal-oxide semiconductor(CMOS) image sensors, especially those with smallpixels. This issue, often referred to as pixel vignet-ting, occurs due to shadowing from the metal inter-connect layers that surround each pixel [16].Microlenses are usually employed to help focus lightonto the active sensor area. However, microlenses donot function as intended with large NA designs [17],including our reflective lens as well as conventionallow F=# refractive lenses. In our prototype four-reflection imager, the microlenses are therefore cov-ered with a thin layer of index-matching gel, fillingthe void between the image sensor and four-reflection lens. To reduce loss caused by pixel vignet-ting, we have chosen a 1.93 megapixel (1600 × 1200)color CMOS image sensor with 3 micropixels madeby Forza/Sunplus. This image sensor, made by theIBM copper-CMOS process, has a relatively small in-terconnect stack height of 2 μm. With this image sen-sor, we found a 70% improvement in marginal rayefficiency as compared to a more conventional CMOSimage sensor with 3:6 μm aluminum interconnects.Further improvement is possible with the use ofback-side illuminated (BSI) image sensors, whichcompletely eliminate the obstruction caused by me-tal interconnects on the image sensor’s surface. BSIimage sensors are currently being developed to im-prove performance of high pixel count CMOS imagesensors with low F=# refractive optics [18].

D. Packaging

A mechanical package and support structure wasdesigned to meet the requirements of the four-reflection imager. The package was designed to holdthe front and back lens elements in precise align-ment in order to maintain optical performance whileallowing the gap spacing between them to be ad-justed for refocus as shown in the cutaway view inFig. 4. The material used for the package was 303stainless steel, which is similar in thermal expansioncharacteristics to the CaF2 lens material. This re-duces unwanted strain on the lens elements thatcould degrade optical performance. The front lenselement is designed to be glued into the main enclo-sure body while the back lens element is designed tobe glued into a back lens carrier. The back lenscarrier also serves as a mounting point for theimage-sensor circuit board and provides a moderateamount of tip/tilt adjustment for the circuit board aswell. A cover snaps onto the back of main enclosure


body to protect internals. The back lens carrier slidesinto a precision-machined bore in the main enclosurebody to maintain the 5 μm alignment needed be-tween the two elements. To set the position of theback lens carrier, and thus the gap spacing betweenthe two lens elements, a 168-tooth (120 pitch) gearadjustment ring is threaded onto precision threadscut into the back lens carrier. The adjustment ringrides on a machined surface located in the mainenclosure body. A small 12-tooth pinion allows forrotation of the adjustment ring using a small screw-driver, where one rotation of the pinion screw corre-sponds to an adjustment in the gap spacing of 28 μm.Four springs clamp the back lens carrier into themain enclosure body and provide the clamping forceneeded to expel the index-matching gel when adjust-ing lens elements closer together.

E. Refocus

The adjustable gap between the two lens elements ofthe four-reflection imager allows focus to be adjustedfor different object ranges via small adjustmentsaround the nominal 300 μm lens element gap spa-cing. MTF curves for the imager focused at 4m and1km are shown in Figs. 5(a) and 5(b), respectively.Some degradation of the tangential component of themost extreme field angles can be seen in Fig. 5(a)for close object distance; however, results are gener-ally excellent across a very large range of refocus.The total adjustment for this refocus is 14 μm. Thesmall travel of this focus adjustment method maybe especially suitable for use with a precision actua-tor such as a piezoelectric transducer. For our experi-mental purposes and to simplify the design, we havechosen a mechanical adjustment using a pinion-gearreduction assembly to adjust the spacing betweenthe two lens elements as described in Subsection 2.D.The main drawback of our chosen method is the tight5 μm (0:000200) bore tolerance between the main en-closure body and the back lens carrier that needs tobe maintained. This turned out to be very difficult forthe machine shop to consistently achieve, even withmatched sets of the parts. In order to ensure the backlens carrier slides smoothly in its machined boreduring a refocus, a certain amount of clearancewas required, and in practice, this clearance waslarger than 5 μm. This ultimately led to problemswith misalignment when a refocus was performed

due to rear lens carrier shifting in the bore. Futuredesigns will most likely use a flexure style focusingmechanism actuated with a piezoelectric transducer.

F. Fabrication and Assembly Tolerances

The four-reflection lens was intended to be fabricatedas two plano-aspheric lens elements to allow for focusadjustment by varying the gap spacing between thefront and back lens elements. While this geometryrelieves the critical alignment involved with re-chucking a single double-sided lens during SPDT aswell as the critical part thickness tolerance presentin the previously reported eight-reflection lens of [3],it presents additional difficulties with the assemblyof the two lens elements once they have been fabri-cated. The tolerances of the shifts and tilts of theaspheric surfaces are similar to those of the eight-reflection lens with minimum values of �10 μm forsurface shift (along the optic axis) and�0:01° surfacetilt. The overall lens element thickness tolerance is�50 μm, and the tolerable departure from the asphe-ric surface equation is �0:25λ (at 546nm) withineach annular reflector zone. Once fabricated, thetwo lens elements must be carefully aligned in themechanical package for centration and tilt. Theassembled centration and tilt tolerances are 5 μm

Fig. 4. Cross-section drawing of the mechanical lens package.

Fig. 5. Simulated refocus performance of the four-reflection lensdesign. Monochromatic MTF curves at object distances of (a) 4mand (b) 1km. A 14 μm gap adjustment is required to focus fromposition (a) to position (b). OTF, optical transfer function.


(radial) and �0:02°, respectively. These toleranceswere calculated using ZEMAX’s sensitivity analysisto maintain an MTF greater than 10% at167 cycles=mm across the full field at the nominalobject distance of 10m.

3. Four-Reflection Prototype

Here we describe the fabrication and assembly of afour-reflection imager for experimental demonstra-tion. Once the lens elements were fabricated usingSPDT, we aligned and glued the elements intoa mechanical package designed to allow for mechan-ical focus adjustment. Experimental results of theassembled imager are presented and discussed.

A. Fabrication and Assembly

ISP Optics [19] diamond turned the front and backplano-aspheric CaF2 lens elements and coated themwith patterned dielectric reflectors as shown inFig. 6(a). The specified fabrication tolerances forthese elements were achieved on the first pass. Oncefabricated, we aligned and glued the two reflectivelens elements with UV epoxy into the mechanicalpackage shown in Fig. 6(b). This process was carriedout on a measuring microscope to carefully align theelements for centricity and tilt. The front lens ele-ment was first centered and epoxied into the mainenclosure body. The completed front lens assemblywas then mounted to the microscope with blockingglue, and the rear lens element was placed directlyon top of front lens assembly. The direct contact re-sulted in zero gap spacing between the lens element’splanar sides to ensure minimal tilt. The back lenscarrier was then placed into position with the adjust-ment ring set appropriately. We then aligned the cen-ters of the front and back lens elements to within thecentration tolerance of 5 μm with the measuring mi-croscope and affixed the rear lens element to the backlens carrier with epoxy. The assemblies were thendisassembled from the alignment station and reas-sembled with index-matching gel filling the nominal300 μm gap spacing between the two lens elements.Finally, we aligned and mounted the image-sensor

circuit board onto the back lens carrier assembly. Thespacing between the transmissive aperture of the

rear lens element and the image-sensor surface wasset to 0:6mm. This spacing was constrained primar-ily by the necessary chip package clearance. Index-matching gel was used to fill the gap between therear lens element and the image sensor to removethe effect of the microlenses. The imager assemblyconnects to a computer through a ribbon cableconnected to the USB interface as shown in Fig. 6(b).Image-sensor settings, image capture, and proces-sing are controlled by MDOSim, a custom interactivecamera programming environment written in Py-thon and developed by Distant Focus Corporation.

B. Measured Performance

A full-field 1600 × 1200 image at an object distance of3:9m is shown in Fig. 7(a). The exposure time for theimage is 35ms under typical fluorescent laboratoryillumination. Examining the United States Air Force(USAF) resolution chart, which can be seen moreclosely in Fig. 7(b), we find the limit of resolutionto be group ð−1; 4Þ or 0:707 lp=mm in object spaceat this range. This corresponds to a measured angu-lar resolution of 0:363mrad.

Figure 7(c) shows the measured contrast transferfunction (CTF) (lensþ sensor) for the four-reflectionimager at 3:9m. CTF measurements were made byfirst carefully white balancing the image of a resolu-tion chart and then measuring the various bar-chartcontrast values referenced to the white and black le-vels at low spatial frequency. The image sensor’s rawBayer image data were used to avoid any additionalloss of contrast or resolution from the default colorinterpolation process. The CTF of the four-reflectionimager shows a resolution cutoff of approximately150 lp=mm in image space. For comparison, the CTFof a conventional refractive Sanyo SVCL-CS550VMF=1:4 zoom lens mounted to the same image sensoris shown as well in Fig. 7(c). The zoom lens focallength has been set to 19mm to match the focallength of the four-reflection imager lens. The Sanyolens resolution is image-sensor limited with a mea-sured resolution of ∼175 lp=mm. This measuredcutoff frequency is larger than the Nyquist frequencyof the image sensor due to additional bandwidtharound the fundamental frequency of the three-barimage [20].

The measured CTF results of the four-reflectionimager lens are close to those predicted by opticalsimulation when the dispersion of the index-matching gel and measured decentration of the finalassembled imager are included. The measured 5 μmdecentration between the two assembled lens ele-ments corresponds to the limit or tolerance, wheredetrimental effects on image quality at best focusstart to become significant.

With the same image-sensor settings and averagedpixel gains as the four-reflection imager, the conven-tional comparison lens and image sensor required anexposure of 17ms to image the resolution targetswith similar levels. The larger exposure required bythe four-reflection imager can be attributed to losses

Fig. 6. Four-reflection imager assembly: (a) diamond turned andcoated optical parts and (b) image of the assembled four-reflectionimager with USB interface PCB.


of the dielectric reflectors and residual pixel vignet-ting at the image sensor. Transmission through thefour-reflection imager lens was measured with acollimated green laser and large area silicon detectorto be 38%, which is significantly lower than the de-signed transmission of >90%. Assuming a secondfabrication run would achieve the design perfor-mance in mirror reflectivity, the sensitivity of thefour-reflection imager will be identical to thatachieved with the long conventional refractive lens.Figure 7(d) shows the measured DOF of the four-

reflection imager lens and the conventional compar-ison lens. With both lenses focused at 3:9m, imagesof the USAF resolution charts were taken at variousdistances to gauge the fall off in resolution as a func-tion of distance. As measured, the DOF of the four-reflection imager lens is roughly 3× smaller thanthe conventional lens. This difference is smaller thanthe approximate DOF difference predicted by 1=NA2

due to the large obscuration and, in part, to the nat-ure of the measurement. The large obscurationcauses an annular blur, which, for small values ofdefocus, does not affect resolution as significantlyas an unobscured aperture of the same NA. However,for large values of defocus, the annular blur causes a

rapid and sudden falloff in resolution and perceivedimage quality due to the annular bokeh [21].

C. Stray Light

Reflective imaging systems are extremely suscepti-ble to stray light since rays may encounter reflectivesurfaces nonsequentially and appear as noise at theimage sensor. Reflective telescopes typically uselarge hoods and internal field stops as primaryand secondary baffles to limit the angular extentof the light reaching the image sensor [22,23]. Thesestructures are often long compared to the physicalsize of the lens and are not desirable when creatingan ultrathin imager. A complete analysis of the ori-gins of stray light in the four-reflection imager is ne-cessary to develop appropriate baffling techniques.

We performed systematic stray light calculationsusing ZEMAX nonsequential analysis to identify cri-tical surfaces that can be seen directly by the imagesensor. Tracing rays in reverse from the image sensorto the input aperture is helpful to recognize specificpaths where light can directly reach the image sensor[24]. Multiple reflections occurring within thin reflec-tive lens designs lead to extreme ray angles that mayenter the system and strike incorrect facets or skip

Fig. 7. Performance at 3:9m: (a) full image, (b) enlarged and cropped 1951USAF resolution chart, (c) measured CTF (lensþ sensor) of thefour-reflection imager compared to a conventional F=1:4 lens of the same focal length and sensor, and (d) image space resolution versusobject distance for the four-reflection imager and conventional F=1:4 comparison imager.


surfaces altogether. Skew rays tend to migratearound the exit pupil and do not contribute to signif-icant levels of stray light.We confirmed the stray light simulations experi-

mentally by rotating the four-reflection imager withrespect to a fixed light source. The four-reflection im-ager has a designed 8:5° half-angle FOV. Significantamounts of light at incidence angles beyond 12° offaxis reach the image sensor. These errant paths ap-pear as bright bands at specific locations within thefield and can lead to complete image loss. An exam-ple of bright oblique stray light imaged by the imagesensor is shown in Fig. 8(a). Additionally, an annulargap exists between the two surfaces cut into the frontlens element and provides a direct ray path to thedetector without encountering any of the appropriatesurfaces. This effect is shown in Fig. 8(b). The reflec-tor of the front lens element must be covered to pre-vent light leaking through this front surface onto theimage sensor. Figure 8(c) shows simulated and mea-sured normalized intensity versus incidence anglefor the four-reflection imager, indicating the mostproblematic stray light paths to the image sensor.A simple hood placed on the front of the four-

reflection imager would have to extend 190mm tolimit incident rays to the design FOV. This excessivelength can be shortened by dividing the full 28mmaperture into small subapertures, each limiting theFOV. We found a thin, commercial honeycomb bafflemade by Tenebraex to be a suitable angle-selective

baffle for our four-reflection imager. The baffle hasa �12° angular cutoff and reduces the overall bafflelength to 6:3mm. Placing the honeycomb in front ofthe aperture causes a 40% drop in on-axis transmis-sion since vanes now block portions of the entrancepupil. Off-axis illumination is linearly attenuateddown to the cutoff angle. A central obscuration isplaced within the honeycomb to block the direct straylight path through the annular gap. By controllingstray light paths to the image sensor with this baffle,we have been able to use the camera in brightly litsituations, including direct sunlight.

Other aspects of image degradation associatedwith intense illumination are seen within the four-reflection imager. Chromatic flare from the dielectricmirror coating can be seen at some stray light angles,especially for rays entering the annular gap. Scatterfrom the honeycomb vanes can also reduce the con-trast of the image. Very bright spots formed on theimage sensor produce surface reflections that reenterthe folded track. These rays may reflect from the lastaspheric reflector back to the image sensor, leadingto a narcissuslike effect of thermal imagers, wherethe camera sees itself.

The honeycomb baffle works well for stray lightsuppression; however, despite its reduced thickness,it is still thicker than the four-reflection imager lens.More extreme versions of the subaperture bafflingconcept are being explored using capillary arraysand even high-resolution holographic film to exposethe desired baffling structure. The resulting devicescan be only micrometers thick, although care will benecessary to minimize scatter and loss.

4. Application of Pupil-Phase Encoding andPostprocessing

Here we describe the design and experimental imple-mentation of PPE and postprocessing for the four-reflection imager. We investigate this technique forextending the DOF of the four-reflection imagerwhile simultaneously correcting axial chromaticaberration.

A. Design of Phase-Modulated Surface

We applied PPE to the four-reflection imager bymodifying the nominally flat annular aperture onthe front side of the four-reflection lens. The largeobscuration ratio presents challenges for the applica-tion of the phase-modulated surface since pupil-phase modulation is limited along the radialcoordinate. It has been previously shown [12] thatthe cosine form of phase modulation is well suitedto such optical designs with large obscuration ratio.This is due to significant modulation possible in thecosinusoidal angular component along with a smal-ler radial component. The cosine form is also well sui-ted to fabrication with SPDT with a fast-tool servo toshape the periodic structure of the rotationally asym-metric surface. The general shape of the cosine formis described as

Fig. 8. Stray light in the four-reflection imager: (a) large angleoblique rays skip reflectors and arrive at the sensor as stray light,(b) axial light may enter through gaps in the front reflectors if acentral block is not used, and (c) simulated and measured normal-ized intensity versus incidence angle.


Sagðr; θÞ ¼Xmi¼1

airbi cosðwiθ þ φiÞ; ð4Þ

where r, θ, and Sagðr; θÞ specify the surface positionand shape in cylindrical coordinates. The weight oneach term is given by ai, the powers of the radialterms are given by bi, the radian frequency is givenby wi, and the phase terms are given by φi. In ourfour-reflection lens design, we optimized the specificsurface figure given by

Sagðr; θÞ ¼X8i¼1

aiðr − 0:81Þi cosð3θÞ

for f0:81 ≤ r ≤ 1:0g;ð5Þ

where the range of r values represent a scaled radiusfrom the limiting radius of obscuration through theouter aperture radius. This surface profile is appliedto the nominally planar front annular aperture of thefour-reflection lens.The goal of our optimization was to extend the

depth of focus of our four-reflection imager to�10 μm, thus increasing the depth of focus of thefour-reflection imager by a factor of approximately3× and creating a defocus tolerance large enough tocompensate for the axial color aberration present inour design. We carried out the optimization using acombination of commercial ray-tracing software,ZEMAX, and commercial numerical computing soft-ware, MATLAB. An intermediate software link pro-vided us with the flexibility to incorporate our ownimage-sensor modeling, postprocessing, and optimi-zation routines in MATLAB, with calls to ZEMAXto perform the necessary ray traces through ourfour-reflection lens design. We defined a custom mer-it function to optimize the optical system such that asingle restoration filter could be applied to increasethe depth of focus. The goals of the merit functionwere (1) to maximize the average MTF values acrossthe fields, focal plane positions, and wavelengths ofthe design up to the Nyquist frequency of the in-tended image sensor, (2) to reduce the variation be-tween MTFs across the fields, focal plane positions,and wavelengths, and (3) to maintain a minimumthreshold MTF for all fields, focal plane positions,and wavelengths up to the Nyquist frequency of theintended image sensor. Our merit function to beminimized for optimum system performance canbe described as

merit value ¼ offset − α1Aþ α2Bþ α3C; ð6Þ

where α1, α2, α3 are the weights on the mean,standard deviation, and minimum threshold terms,respectively, and offset is a user-specified term tomaintain a positive merit function value. The meanterm A is defined as

A ¼ meanc;f ;λðmeanux;uyðMTFc;f ;λðux;uyÞÞÞ; ð7Þ

where c is the configuration (focal plane position), f isthe field position, λ is the wavelength, and ux and uyare the spatial-frequency coordinates. Because ofsymmetry, only the first quadrant of the MTFc;f ;λðux;uyÞ function is needed in the optimization calcu-lations. In Eq. (7), mean indicates the arithmeticmean. Since a large mean value is desirable, thisterm is assigned negative value in themerit function,Eq. (6). The standard deviation term, B, in Eq. (6) isdescribed as

B ¼ meanux;uyðstdevc;f ;λðMTFc;f ;λðux;uyÞÞÞ: ð8Þ

Here MTF values are compared across the configura-tions, fields, and wavelengths to enforce similaritybetween optimized MTF values. The arithmeticmean is used to average the standard deviationvalues over the spatial-frequency coordinates to givea single value to the merit value equation. Finally,the threshold term, C, in Eq. (6) is described as

C ¼ meanc;f ;λðβc;f ;λÞ; ð9Þ

where

βc;f ;λ ¼8<:

�δthresh−MTFmin;c;f ;λ

ðδthresh=10Þ

�n

if�

δthresh−MTFmin;c;f ;λðδthresh=10Þ

�> 0

0 otherwise:

ð10Þ

In Eq. (10), δthresh is the chosen minimumMTF valuethreshold, n is the exponent power on the thresholdterm, and MTFmin;c;f ;λ is the calculated minimumMTF value within the first quadrant for each config-uration, field position, and wavelength up to theNyquist frequency of the image sensor. Dependingon the value of the minimum MTF value belowthreshold, Eq. (10) is designed to give positive valuesbetween zero and ten raised to the nth power. Thisterm in the merit value therefore enforces a mini-mum MTF across the configurations, field positions,and wavelengths for image restoration and penalizeszero crossings where image information is lost andphase inversion occurs.

We carried out optimization of the PPE surfaceusing the merit value described by Eqs. (6)–(10), uti-lizing several different optimization routines such asthe Nelder–Mead simplex (direct search) method,gradient/finite difference, and simulated annealingto investigate solutions offered by the differentmethods [25–27]. The weights of the terms inEq. (6) were adjusted to investigate their influenceon the results of optimization. In general, the stan-dard deviation term, B, and the minimum thresholdterm, C, received much stronger weighting than themean term, A. The threshold term in the merit valueequation was chosen to maintain a modulation ofgreater than 10% up to the Nyquist frequency ofour sensor (167 cycles=mm) across the operatingrange. Typically, a modulation of 10% is sufficient


for effective image restoration with postprocessing.We found the simplex method in general to be themost useful and reliable in terms of convergenceand speed; however, significant interaction was re-quired, and a large number of optimization startingpoints were tried to steer the optimization toward ausable solution. Modeled results of our final solutionare displayed in Fig. 9, where the axial point spreadfunction (PSF) over�10 μm defocus is shown for boththe unmodified and PPE designs. A simple inversefilter is used to generate the processed PSFs shownin the bottom row of Fig. 9 [28]. For display purposes,the intensity scale on the unmodified PSFs (top row)and the PPE PSFs (middle and bottom rows) differ bya factor of ∼4×. This value represents the trade-offbetween signal-to-noise ratio (SNR) and depth offocus with this technique. The trade-off is minimizedas much as possible in the optimization by maximiz-ing the average MTF across configurations, fields,and wavelengths; however, the significant variationbetween field positions in the baseline four-reflectionlens design required a relatively strong phase mod-ulation, lower average MTF, and more significanttrade-off of SNR to achieve our desired spatialinvariance over the image volume (depth of focusand DOF) [29,28].

B. Experimental Results

To demonstrate the optimized PPE surface, wehad the front element of the four-reflection lens re-fabricated with the optimized PPE surface profilediamond turned onto the previously flat annularaperture. The modified front element was then as-sembled with the standard rear element as described

in Section 3 to make a PPE four-reflection imager. Totest the addition of the PPE surface, we set up a com-parison test with the PPE four-reflection imager andthe unmodified four-reflection imager at the same ob-ject distance. Figure 10 shows axial PSFs measuredwith both the unmodified four-reflection imager andthe PPE four-reflection imager. We measured thePSF by imaging a 25 μm pinhole illuminated by abright white-light LED positioned 3:9m away fromthe two imagers. Best focus was found by adjustingthe pinion screw that varies the separation of thefront and back optical elements until the mostcompact PSF was found. We measured PSFs at bestfocus and �0:3m to characterize the PSF variationsthrough focus for the two imagers. The measured ob-ject depth of 0:6m corresponds to a depth of approxi-mately 20 μm (or �3:5 waves of defocus at 550nm) inimage space. Examining the PSFs for the unmodifiedimager, we found that, although the imager forms asmall bright PSF at best focus, the out-of-focus mea-sured PSFs display an asymmetry associated withthe measured decentration of 5 μm between the lenselements. In this case, we find the PSF at 3:6m to bebetter than the simulated results shown in Fig. 9.The asymmetry in the PSF leads to significant powerconcentration on a small PSF, as the object distanceis reduced from best focus. However, this behavior isnot symmetric through focus increasing the objectdistance from best focus caused the PSF to spreadmore quickly on the far side of best focus. The PSFs

Fig. 9. Simulated PSFs at best focus and �10 μm defocus (�3:5waves defocus at 550nm). Unmodified PSFs (top row), PPE PSFsbefore filtering (middle row), and inverse-filtered PPE PSFs (bot-tom row).

Fig. 10. Measured PSFs at 3:9m (best focus) and �0:3m (∼�3:5waves defocus at 550nm). Unmodified PSFs (top row), PPE PSFsbefore filtering (middle row), and inverse-filtered PPE PSFs (bot-tom row).


for the PPE four-reflection imager also show someasymmetry from a measured decentration of 11 μm(assembly error) between the lens elements. This de-centration is large enough to cause significant degra-dation in image quality in the PPE four-reflectionimager. The decentration tolerance for the PPE four-reflection lens is approximately 8 μm. Using matchedpixel gains and sensor settings, the exposure timesused for the PSF measurements of Fig. 10 are0.78 and 3:3ms for the unmodified and PPE four-reflection imagers, respectively. The bottom row inFig. 10 shows filtered results of the PPE four-reflection imager where a common Wiener filter hasbeen applied to restore contrast [28]. This Wiener fil-ter is based on the measured axial PSFat 3:9m, withits threshold and SNR balance parameters tweakedto subjectively provide the best-looking images basedon a human observer’s viewpoint.Figure 11 shows comparison images of a USAF

1951 resolution chart acquired with the unmodifiedand PPE four-reflection imagers at best focus (3:9m).The color images of the PPE imager were whitebalanced and converted to luminance-bandwidth-chrominance image data before the restoration filterwas applied to the Y channel only. At best focus, theresolution limit is 0:707 lp=mm in the horizontal andvertical directions for the unmodified imager and is0.63 and 0:561 lp=mm in the vertical and horizontaldirections, respectively, for the PPE imager. TheWiener filter was able to restore much of the resolu-tion of the PPE system; however, the significant mis-alignment of the lens elements resulted in someresolution loss in the horizontal direction. The noiseevident in Fig. 11(b) is introduced by the restorationfilter due to the significant noise gain of the filter.Figure 12 shows another comparison of the

unmodified and PPE four-reflection imagers with theobject distance reduced by 30 cm to 3:6m. At thisposition, the resolution of the unmodified imager isfound to have fallen to 0:561 lp=mm in the verticaldirection and 0:500 lp=mm in the horizontal direc-tion. The PPE imager, however, shows little perceiva-ble resolution loss over this distance, still resolving0:63 lp=mm in the vertical direction and 0:561 lp=mmin the horizontal direction.

Increasing the object distance by 30 cm to 4:2mcreated a more pronounced difference between thetwo four-reflection imagers. At 4:2m, the unmodifiedimager’s quality is degraded, resolving 0:315 lp=mmin the vertical direction and 0:353 lp=mm in thehorizontal direction. In contrast, the PPE imagershows a smaller amount of degradation resolv-ing 0:561 lp=mm in the vertical direction and0:445 lp=mm in the horizontal direction.

With the variations in assembly alignment of thetwo imager systems, it is difficult to directly quantifythe improvement gained with PPE and postproces-sing in this case. However, even with a measurablylarger decentration in the PPE four-reflection imager(11 μm) compared to the unmodified four-reflectionimager (5 μm), we find some improvement in theDOF. Additional effort is required to improve align-ment accuracy to more clearly examine the benefit ofPPE and postprocessing.

5. Conclusion and Discussion

We presented the design and successful experimen-tal demonstration of an ultrathin four-reflection im-ager that achieves a focal length of 18:6mm andeffective F=# of 1.15 in a track length of just5:5mm. This reflective lens provides a 17° FOV over1600 × 1200 3 μm pixels of a color CMOS sensorwhich is integrated with USB connection to compu-ter. The four-reflection lens and CMOS sensor aremated together within a stainless steel package thatprovides a sensitive focus adjustment with smallchanges of the order of a fewmicrometers to the nom-inal 300 μm spacing between the front and back lenselements. Despite a small chromatic focal shiftcaused by commercially available index-matchinggel and imperfect alignment in the optical assembly,the self-contained and packaged four-reflectionimager performed close to the optical design perfor-mance under laboratory conditions. Direct compari-son using an identical image sensor mated to a largeconventional lens of the same focal length showedthat resolution and subjective image quality are com-parable between the two lenses despite the dramati-cally reduced track length of the four-reflectionimager lens.

Fig. 11. Experimental comparison images at 3:9m (best focus):(a) unmodified imager and (b) processed PPE imager.

Fig. 12. Experimental comparison images at 3:6m (∼3:5 wavesdefocus at 550nm): (a) unmodified imager and (b) processedPPE imager.


We also investigated the application of PPEand postprocessing for extending the DOF and alle-viating the axial chromatic aberration of thefour-reflection imager. Although we expected apronounced improvement in defocus invariance fromdesign and simulation, we found the experimentalPSFs for the PPE imager to be significantly differentfrom simulation due to misalignment of the opticalassembly. Nevertheless, we found a discernable im-provement in DOF in the processed PPE imagescompared to those of the unmodified four-reflectionimager.Looking forward, such multiple-reflection lenses

may be suitable for inexpensive mass productionusing a glass molding process provided in part thatthe design include additional registration surfaces toaid in the alignment during assembly. As shown,centration is of particular importance and must bestrictly controlled to within a few micrometers forgood optical performance.

The authors acknowledge M. Stenner andM. Neifeld (University of Arizona) and R. Athaleand D. Healy (Darpa) for many useful technical dis-cussions. We acknowledge ISP Optics for fabricationservices. This research was supported by the DefenseAdvanced Research Projects Agency (DARPA) via theMONTAGE program, grant HR0011-04-I-0045 andby the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) through a graduate stu-dent scholarship.

References1. W. J. Smith, Modern Lens Design (McGraw-Hill, 2005),

Chap. 18.2. D. Korsch, Reflective Optics (Academic, 1991).3. E. J. Tremblay, R. A. Stack, R. L. Morrison, and J. E. Ford,

“Ultra-thin cameras using annular folded optics,” Appl. Opt.46, 463–471 (2007).

4. J. Hall, “F-number, numerical aperture, and depth of focus,” inEncyclopedia of Optical Engineering (Marcel Dekker, 2003),pp. 556–559.

5. E. R. Dowski, Jr. and W. T. Cathey, “Extended depth of fieldthrough wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).

6. W. T. Cathey and E. Dowski, “A new paradigm for imagingsystems,” Appl. Opt. 41, 6080–6092 (2002).

7. K. Kubala, E. Dowski, J. Kobus, and R. Brown, “Aberrationand error invariant space telescope systems,” Proc. SPIE5524, 54–65 (2004).

8. K. Kubala, E. Dowski, andW. T. Cathey, “Reducing complexityin computational imaging systems,” Opt. Express 11,2102–2108 (2003).

9. W. Chi and N. George, “Electronic imaging using a logarithmicasphere,” Opt. Lett. 26, 875–877 (2001).

10. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, andJ. van der Gracht, “Engineering the pupil phase to improveimage quality,” Proc. SPIE 5108, 1–12 (2003).

11. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, andJ. van der Gracht, “Pupil-phase optimization for extended fo-cus, aberration corrected imaging systems,” Proc. SPIE 5559,335–345 (2004).

12. E. J. Tremblay, J. Rutkowski, I. Tamayo, P. E. X. Silveira,R. A. Stack, R. L. Morrison, M. A. Neifeld, Y. Fainman, andJ. E. Ford, “Relaxing the alignment and fabrication tolerancesof thin annular folded imaging systems using wavefront cod-ing,” Appl. Opt. 46, 6751–6758 (2007).

13. M. W. Haney, “Performance scaling in flat imagers,” Appl. Opt.45, 2901–2910 (2006).

14. V. N. Mahajan, “Imaging with obscured pupils,” Opt. Lett. 1,128–129 (1977).

15. H. B. Wach, E. R. Dowski, Jr., and W. T. Cathey, “Control ofchromatic focal shift through wavefront coding,” Appl. Opt.37, 5359–5367 (1998).

16. P. B. Catrysse and B. A. Wandell, “Optical efficiency ofimage sensor pixels,” J. Opt. Soc. Am. A 19, 1610–1620(2002).

17. G. Agranov, V. Berezin, and R. H. Tsai, “Crosstalk andmicrolens study in a color CMOS image sensor,” IEEE Trans.Electron Devices 50, 4–11 (2003).

18. M. Dirjish, “BSI technology flips digital imaging upsidedown,” http://electronicdesign.com/Articles/Index.cfm?AD=1&ArticleID=19160.

19. http://www.ispoptics.com/.20. D. H. Kelly, “Spatial frequency, bandwidth, and resolution,”

Appl. Opt. 4, 435–435 (1965).21. T. Ang, Dictionary of Photography and Digital Imaging:

the Essential Reference for the Modern Photographer(Watson-Guptill, 2002).

22. R. Prescott, “Cassegrainian baffle design,” Appl. Opt. 7,479–481 (1968).

23. C. Leinert and D. Klüppelberg, “Stray light suppressionin optical space experiments,” Appl. Opt. 13, 556–564(1974).

24. G. Peterson, “Stray light calculation methods with optical raytrace software,” Proc. SPIE 3780, 132–137 (1999).

25. J. C. Lagarias, J. A. Reeds, M. H. Wright, andP. E. Wright, “Convergence properties of the Nelder–Meadsimplex method in low dimensions,” SIAM J. Optim. 9,112–147 (1998).

26. T. F. Coleman and Y. Li, “An interior, trust region approach fornonlinear minimization subject to bounds,” SIAM J. Optim. 6,418–445 (1996).

27. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimizationby simulated annealing,” Science 220, 671–680 (1983).

28. H. C. Andrews and B. R. Hunt, Digital Image Restoration(Prentice-Hall, 1977), Chap. 8, pp. 147–152.

29. B. R. Frieden, “Image enhancement and restoration,” inTopics in Applied Physics, Vol. 6 of Picture Processing andDigital Filtering, T. S. Huang, ed. (Springer-Verlag, 1979),pp. 177–248.


Documents

Ultrathin four-reflection imagerpsilab.ucsd.edu/research/Origami Optics/files...Ultrathin four-reflection imager Eric J. Tremblay,1,* Ronald A. Stack,2 Rick L. Morrison,2 Jason H