E216
`
`Vol. 54, No. 28 / October 1 2015 / Applied Optics
`
`Research Article
`
`Optical analysis of miniature lenses
`with curved imaging surfaces
`DMITRY RESHIDKO* AND JOSE SASIAN
`College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA
`*Corresponding author: dmitry@optics.arizona.edu
`
`Received 18 May 2015; revised 13 August 2015; accepted 5 September 2015; posted 9 September 2015 (Doc. ID 241176);
`published 23 September 2015
`
`Miniature cameras for consumer electronics and mobile phones have been, and continue to be, in fast develop-
`ment. The system level requirements, such as manufacturing cost, packaging, and sensor characteristics, impose
`unique challenges for optical designers. In this paper, we discuss the potential optical benefits of having a curved
`image surface rather than a flat one. We show that curved sensor technology allows for optically faster lens
`solutions. We discuss trade-offs of several relevant characteristics, such as packaging, chief ray angle, image quality,
`and tolerance sensitivity. A comparison of a benchmark flat field lens, and an evaluation design imaging on a curved
`surface and working at f ∕1.6, provides useful specific insights. For a given image quality, departing from a flat
`imaging surface does not allow significantly reducing the total length of a lens.
`© 2015 Optical Society of America
`
`OCIS codes: (220.3620) Lens system design; (220.1000) Aberration compensation; (220.1010) Aberrations (global).
`
`http://dx.doi.org/10.1364/AO.54.00E216
`
`1. INTRODUCTION
`Miniature camera lens modules are integrated in a variety of
`portable devices, including cell phones, tablets, laptops and spy
`cameras, to mention common applications. The photographic
`performance of these tiny, always ready-for-action lenses, is re-
`markable as it rivals that of single-lens reflex cameras. Mobile
`camera technology and devices is a very fast growing field in the
`imaging market and is impacting the industry by decreasing the
`production of larger photographic cameras.
`The design and packaging of a miniature camera lens mod-
`ule imposes optical design challenges. A traditional objective
`lens can not be simply scaled down as a lens solution due to
`fabrication constraints, materials properties, manufacturing proc-
`ess, light diffraction and geometrical aberrations. The increasing
`demand for thinner, lighter, and low-cost mobile cameras has
`thus forced the development of new manufacturing technologies
`and of lens design solutions with high performance.
`The optical advantages of using a curved image sensor have
`been already discussed in the literature. A general conclusion is
`that a curved image surface allows designing simpler, more com-
`pact, and lower-cost optics [1–3]. Recently several researchers
`have made progress toward developing curved sensors [4–8].
`The curved sensor technology for the format which is suited
`for mobile cameras is near to being commercialized. On the
`2014 VLSI Symposia, Sony officially released two curved sen-
`sors. One has a diagonal of 43 mm, which is equivalent to a
`full-frame sensor. The other has a diagonal of 11 mm, which
`corresponds to a 2/3 in. sensor format [9]. For reference, a flat
`
`2/3 in. sensor has been used in the Nokia Lumia 1020 smart
`phone camera module.
`In this paper, we analyze how allowing for a curved imaging
`surface impacts the lens design of compact miniature lenses for
`mobile applications. In Section 2, we highlight typical optical
`design specifications of state-of-the-art mobile camera lenses.
`We derive design requirements by comparing products in the
`market, from patent data, and from publications in the mobile
`platform optical design and fabrication sector. We review the
`first-order imaging properties and discuss aberration correction
`in this class of miniature lenses. In Section 3, we examine how a
`curved image surface can benefit the lens design of mobile cam-
`eras. In Section 4, we show lens design examples for both flat
`and curved image surfaces. Section 5 provides a detailed com-
`parison of the designs presented in the previous section. We
`find that a curved image surface allows producing an equiva-
`lently performing design with faster f ∕# than a conventional
`design. We also show that other relevant characteristics, such as
`aberration balancing, image quality, and chief ray incidence
`angle on the sensor, are favorably impacted. In addition, we
`discuss distortion aberration as it relates to a curved imaging
`surface. Finally, we demonstrate improved sensitivity to manu-
`facturing tolerances. Section 6 concludes this paper.
`
`2. REVIEW OF MODERN MOBILE CAMERA
`LENSES
`In this section we present typical optical design requirements
`and trade-offs of state-of-the-art mobile camera lenses. First,
`
`1559-128X/15/28E216-08$15/0$15.00 © 2015 Optical Society of America
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`the camera specifications of smart phones from major brands
`(Apple, Samsung, Nokia, Sony, LG, and HTC) were studied.
`We found a few trends in the products available on the market:
`1. Five lens elements are often used.
`2. The image sensor pixel
`size varies between 1.1
`and 2.0 μm.
`3. Typical image sensor format is 1/3 in. (6 mm diagonal).
`The output image resolution is usually 8–12MP depending on
`the pixel size.
`4. The field of view (FOV) of the mobile camera is large.
`Common FOV values are 65–75 deg.
`5. The mobile camera lenses are designed being optically
`fast. The f ∕# varies between 2.0 and 2.2.
`Peter Clark provides a historical overview of patented designs
`and points at several interesting characteristics of miniature cam-
`era lens designs [10,11]. Here we focus our patent search on
`lenses that have specifications similar to the smart-phone camera
`specifications outlined above. The patent database of compact
`imaging lenses is extensive. In Subsections 2.A and 2.B we dis-
`cuss first-order imaging properties of these mobile camera lenses
`and analyze how aberrations are corrected in patented designs.
`
`A. First-Order Properties
`There are some important first-order properties to highlight.
`These are the total track to focal length ratio, the stop position,
`the working distance, the relative illumination, and the chief
`ray angle of incidence on the sensor.
`The practically achievable ratio of the total length to the focal
`length in lenses for mobile cameras is between 1.15 and 1.3 [12].
`Decreasing this ratio and making the lens a telephoto would be
`desirable; however, more optical power would be introduced in
`the individual lenses and aberration residuals would be large.
`The chief ray incidence angle (CRA) depends on the stop
`position and on the amount of pupil spherical aberration. The
`CRA impacts the relative illumination, which often is set to
`50% at the sensor corners. In order to avoid cross talk between
`adjacent pixels, the CRA is usually limited to no more than
`30 deg [10]. Thus lens telecentricity in image space is desirable,
`but actual lenses are not telecentric. For better CRA control,
`the aperture stop is placed close to the front of the lens, away
`from the image plane. The aperture stop position and the
`strong aspheric next to the image plane generate exit pupil
`spherical aberration, which reduces the CRA. As illustrated in
`Fig. 1(a), the chief ray for different field points crosses the op-
`tical axis further away from the sensor and thus the CRA is
`ingeniously reduced. As also shown in Fig. 1(b), the beam foot-
`prints from different field points shift at the exit pupil due to
`the substantial pupil spherical aberration.
`
`B. Aberration Correction
`From the aberration correction point of view a mobile lens can
`be divided into two groups of lenses, as shown in Fig. 2. The
`front group consists of two or three lens elements. This group
`provides effective degree of freedom for correcting spherical and
`coma aberrations, as well as for correcting chromatic change of
`focus and chromatic change of magnification. The first lens
`usually carries a substantial amount of positive optical power
`and therefore its shape determines substantially the amount
`of coma in the system; as the stop is near or at the first lens
`
`Fig. 1. U.S. patent application 20130258499: layout and spot dia-
`grams at the exit pupil showing substantial pupil spherical aberration.
`
`Fig. 2. U.S. patent application 20130258499.
`
`spherical aberration is controlled by the asphericity of at least
`one of the lens surfaces. To optically relax the lens a minimum
`amount of optical power is favorable for the first element and
`then to correct for chromatic change of focus the Abbe number
`of the first element is maximized, the Abbe number of the
`second element is minimized, or both. The lens small size also
`contributes to mitigate the effects of aberration as the wave-
`length does not scale down with the lens.
`The rear group consists of one or two lenses that are strongly
`aspheric and serve to some extent as field lenses; they correct
`field curvature and astigmatism of the front group. The rear
`group also contributes substantial distortion which cancels the
`distortion from the front group which lacks symmetry about
`the stop position.
`Mobile lenses are notorious by the extensive use of aspheric
`surfaces. The interaction of multiple aspherics within the de-
`sign allows for effectively controlling aberrations. Particularly,
`sharp imaging on a flat surface can be achieved without satisfying
`the classical requirement of having a Petzval sum nearly zero. The
`aspheric optical elements located close to the image plane con-
`tribute higher-order field curvature and astigmatism. Different
`orders of the field curvature and astigmatism are balanced to
`compensate for any residual Petzval curvature [13]. For example,
`U.S. patent application 20140300975 has a focal
`length of
`3.48 mm and a Petzval radius of about 12 mm. The multiple
`crossings of the sagittal and tangential field curves indicate the
`presence of higher-order field curvature and astigmatism, as
`shown in Fig. 3(b). The last lens is optically weak and has little
`contribution to the Petzval sum. However, this lens helps flatten
`the field by introducing higher-order astigmatism and field
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`Research Article
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`designed symmetrically around the aperture stop [14]. In a sym-
`metrical optical system all odd aberrations tend to cancel out
`permitting a higher level of aberration correction.
`
`C. Field Curvature and F -Number
`In a flat field lens field curvature correction is obtained by intro-
`ducing negative optical power, and this leads to more overall
`optical power. Field curvature correction optically stresses a lens
`and aberration residuals become larger. For a given image qual-
`ity a lens with a curved imaging surface can have a faster optical
`speed due to a reduced optical stress. Field curvature aberration
`is compensated by the curved sensor. An improvement of a one
`f -stop is a typical gain in speed for comparable imaging per-
`formance with respect to a flat field lens.
`
`D. Total Length
`The total length of a mobile lens is an important design param-
`eter. It would be attractive if curved sensor technology would
`allow designing a shorter lens, such as a telephoto lens. However,
`a shorter lens would require optically stressing the lens and de-
`parting more from symmetry. As the aberrations substantially
`increase with lens stress, the optical speed or FOV would have
`to be decreased to achieve the same aberration correction. Thus
`in practice no substantial reduction in length can be obtained
`from using a curved imaging surface.
`
`4. DESIGN EXAMPLES
`For comparison purposes in this section we present two design
`examples of miniature lenses: one with a flat imaging surface
`called the benchmark lens, and another with a curved imaging
`surface called the evaluation lens.
`
`A. Specification
`Following the discussion in Section 2, a set of reasonable lens
`specifications is provided. Assuming 1/3 in. sensor format (6 mm
`in diagonal) and a FOV of 70 deg, we choose a focal length of
`4.5 mm and limit the total length of the lens to 5.5 mm. We
`design for the visible light from 430 to 656 nm. The f -number
`is set at f ∕2.2. The specifications are summarized in Table 1.
`B. Benchmark Lens
`We develop a benchmark lens as follows. The U.S. patent
`application 20130258499 provides a lens with specifications
`
`Table 1. Design Specifications
`
`Requirement
`Sensor format
`f ∕#
`FOV [deg]
`f
`Total length [mm]
`Distortion
`Number of lenses
`Materials
`Edge thickness [mm]
`Center thickness [mm]
`Air gap [mm]
`Surface slope [deg]
`Element aspect ratio
`IR cut filter [mm]
`
`Value
`1∕300
`2.2
`70
`4.5
`<5.5
`<1%
`5
`COC, OKP4
`>0.1
`>0.3
`>0.1
`<55
`<1∶5
`0.2
`
`Fig. 3. U.S. patent application 20140300975: (a) Layout, (b) field
`curvature (scale 0.16 mm), and (c) field curvature if lens five is re-
`moved (scale 1.6 mm).
`
`curvature. To illustrate this the last lens element in Fig. 3(c) has
`been removed, the Petzval radius is nearly the same, and the large
`astigmatism and residual field curvature are then observed.
`It is typical to find in the literature excessive, redundant, or
`illogical use of aspheric coefficients and number of significant
`figures which can lead to prescription errors and difficulty in
`analyzing these miniature lens systems.
`
`3. DESIGN ADVANTAGES OF LENSES WITH
`CURVED IMAGING SURFACES
`In this section we mention how a curved imaging surface can
`benefit the lens design of a miniature mobile camera.
`
`A. Chief Ray Incidence Angle on the Sensor
`Since the image plane is curved and the CRA is calculated
`relative to the image surface normal, the chief ray incidence
`angle can be significantly reduced, which results in advantage.
`The chief ray incidence angle still needs to be limited to no
`more than 30 deg. However, this angle corresponds to a much
`larger angle in relation to a flat image plane. Consequently, the
`aperture stop, which is placed close to the front of the lens in a
`conventional mobile camera, can be shifted.
`
`B. Aperture Stop Position
`For a mobile camera with a curved imaging surface the stop lo-
`cation has more flexibility and can be moved to make the lens
`system to be less unsymmetrical about the stop. Symmetrical or
`nearly symmetrical lens design forms allow for a more balanced
`and better aberration correction. Many classical objectives are
`
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`Research Article
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`Vol. 54, No. 28 / October 1 2015 / Applied Optics
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`E219
`
`3.28E−07
`−6.73E−05
`−2.23E−05
`−4.93E−04
`1.05E−04
`1.58E−05
`−2.54E−04
`−1.05E−04
`
`−3.76E−06
`6.42E−04
`1.18E−04
`1.92E−03
`−6.13E−04
`−3.92E−05
`2.38E−03
`5.52E−04
`
`−4.04E−05
`−1.76E−03
`−1.23E−04
`1.73E−04
`1.00E−03
`−3.32E−05
`−9.61E−03
`−5.53E−04
`
`9.85E−04
`2.31E−04
`1.74E−03
`−1.28E−02
`8.19E−04
`−1.59E−04
`1.96E−02
`−1.57E−03
`
`−8.28E−03
`5.26E−04
`−1.30E−02
`1.93E−02
`−5.04E−03
`7.73E−04
`−2.26E−02
`5.88E−03
`
`3.69E−02
`3.04E−02
`3.01E−02
`−1.83E−02
`3.97E−03
`2.87E−03
`2.67E−02
`−4.55E−03
`
`−1.43E−01
`−1.65E−01
`−7.03E−02
`−3.90E−02
`−1.47E−02
`−1.52E−02
`−3.85E−02
`−1.24E−02
`
`−5.17E−05
`−1.86E−04
`
`−1.19E−05
`4.84E−04
`
`5.87E−05
`−4.35E−04
`
`1.20E−04
`−1.40E−03
`
`−3.52E−04
`2.11E−03
`
`−7.39E−03
`−5.59E−03
`
`1.33E−02
`1.77E−03
`
`16th
`
`14th
`
`12th
`
`10th
`
`8th
`
`6th
`
`4th
`
`0.1400
`
`Conic
`
`1.550,55.0
`
`F52R
`
`OKP4
`
`F52R
`
`OKP4
`
`F52R
`
`0
`
`Glass
`
`5.8702
`0.4500
`0.2000
`0.4820
`0.4499
`0.4696
`0.3906
`1.0341
`0.8691
`0.1845
`0.3472
`0.0496
`0.1703
`0.4100
`
`Inf
`Thic
`
`−10.8615
`
`2.5782
`3.3880
`8.5320
`14.4908
`−8.0687
`3.0512
`2.5203
`6.9383
`
`7.1311
`2.2494
`
`Inf
`Radi
`
`Inf
`
`Inf
`Inf
`
`STO
`
`OBJ
`Surf
`
`1
`2
`
`IMA
`13
`12
`11
`10
`
`7
`4
`8
`5
`9
`6
`
`Table3.PrescriptionfortheEvaluationDesign
`
`7.67E−07
`−2.42E−05
`
`−1.76E−05
`1.49E−04
`
`1.23E−04
`4.96E−04
`
`−7.35E−04
`−6.79E−04
`3.11E−04
`
`6.87E−03
`5.14E−03
`−2.71E−03
`
`−2.62E−02
`−1.26E−02
`2.63E−03
`
`7.28E−05
`−4.67E−03
`1.25E−03
`5.38E−02
`1.85E−02
`1.32E−02
`5.00E−03
`7.62E−03
`−1.71E−03
`−5.37E−03
`
`−5.03E−03
`6.20E−03
`−7.86E−03
`−7.00E−02
`−1.10E−02
`−1.60E−02
`−2.04E−02
`−1.92E−02
`2.56E−03
`2.41E−03
`
`2.71E−02
`2.04E−02
`−1.26E−03
`4.63E−02
`6.21E−03
`1.26E−02
`5.05E−02
`3.83E−02
`−1.91E−02
`−8.89E−03
`
`−8.67E−02
`−1.29E−01
`1.15E−02
`−5.18E−02
`−4.82E−02
`−2.87E−02
`−1.23E−01
`−8.88E−02
`2.84E−03
`−5.43E−03
`
`−3.2055
`3.9004
`−54.19
`
`1.517,64.2
`
`F52R
`
`F52R
`
`F52R
`
`OKP4
`
`F52R
`
`16th
`
`14th
`
`12th
`
`10th
`
`8th
`
`6th
`
`4th
`
`Conic
`
`Glass
`
`0.5063
`0.2000
`0.4000
`0.8486
`0.5929
`0.4901
`0.5740
`0.6860
`0.3272
`0.2453
`0.1018
`0.5487
`−0.1742
`
`Inf
`Thic
`
`1.5027
`2.2306
`3.5651
`2.8303
`521.22
`6.3429
`1.7092
`3.3031
`−46.17
`2.3150
`
`Inf
`Inf
`Radi
`
`Inf
`Inf
`Inf
`
`STO
`OBJ
`Surf
`
`IMA
`13
`12
`11
`10
`
`4
`5
`
`67
`3
`2
`8
`9
`
`Table2.PrescriptionfortheBenchmarkDesign
`
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`Research Article
`
`close to the required in Table 1 and is a good starting point for
`such a benchmark lens design. The lens has a focal length of
`4.109 mm and a FOV of 69.78 deg operating at f ∕2.2. We
`scaled up the lens and replaced the model materials from the
`patent application with real materials. We used Zeonex F52R
`(nd ˆ 1.5333 vd ˆ 53.5) and OKP4 (nd ˆ 1.6059 vd ˆ
`26.9). The lens was optimized while preserving the initial de-
`sign form. The final layout of the benchmark lenses is shown
`on top of Fig. 4. The lens prescription is given in Table 2.
`
`C. Evaluation Lens
`We considered and optimized a variety of lens forms and
`arrived at the lens we call the evaluation lens. In our optimi-
`zation we allowed the curvature of the imaging surface to vary,
`the individual lens optical power to vary, and the aperture stop
`location to also vary. We found that the optimum location of
`the stop was between the first and second lenses for a given
`system total length. Since no correction for field curvature
`was necessary, a faster lens, f ∕1.6, with excellent image quality
`was found. The layout of the evaluation lens is shown at the
`bottom of Fig. 4. The prescription is given in Table 3.
`
`5. LENS COMPARISON
`In this section we illustrate the advantages of the curved image
`surface by comparing the optical performance of the bench-
`mark design and the evaluation design.
`
`A. First-Order Properties
`The layouts of both benchmark and evaluation lenses are
`shown in Fig. 4. Both designs have similar configuration:
`the first three lenses provide most of the optical power while
`last two elements are weak correcting lenses.
`
`The power distribution in these two lenses is different.
`Consider the paraxial refraction equation:
`n0u0 ˆ nu − yϕ:
`(1)
`In this equation n0u0
`and nu are products of the index of re-
`fraction and paraxial marginal ray slope before and after refraction
`at the surface, y is the marginal ray height at the surface, and Φ is
`the surface optical power. Consequently, the weighted surface
`power is given by the difference between the marginal paraxial
`ray slope before and after refracting at the surface. We normalize
`the total weighted power to 1. For each optical element we plot
`the difference between the marginal paraxial ray slope before and
`after passing through the element. The plot in Fig. 5 provides an
`indication where the optical power originates within the system.
`
`B. Field Curvature
`The field curves of the benchmark design are typical for a flat
`field mobile lens and are shown in Fig. 6 on the left. The Petzval
`radius is −19.12 mm. The residual field curvature is corrected by
`balancing higher orders of the field curvature and astigmatism.
`The field curves are wavy with multiple crossing across the FOV.
`In contrast the field curves for the evaluation lens are much more
`smooth and are shown in Fig. 6 on the right. The Petzval radius
`is −8.74 mm (about 2 times the focal length); this clearly indi-
`cates that the field curvature is compensated by the curved image
`surface. The image surface radius of curvature is −10.86 mm.
`
`C. Image Quality
`The polychromatic modulation transfer function (MTF) was
`calculated in Zemax lens design software [15] and shown in
`Fig. 7. The pupil is sampled with a 128 × 128 ray grid. The
`MTF of the evaluation lens is calculated on the curved image
`surface. Both lenses show very good performance over the en-
`tire FOV with an average MTF of about 70% at 112 lp∕mm
`(gray-scale Ny/4 frequency for a 1.1 μm pixel) and over 45%
`MTF at 225 lp∕mm (gray-scale Ny/2 frequency for a 1.1 μm
`pixel). However, the lens designed for the curved image sensor
`is about one f -stop faster compared to the conventional design.
`We think it would be impossible to achieve similar aberration
`correction for this f ∕# for a flat sensor with five lens elements.
`In the benchmark lens the MTF at high frequencies varies sig-
`nificantly with the field angle. The MTF of the evaluation lens
`is more uniform over the field.
`The limiting aberrations for the evaluation lens are spherical
`aberration, oblique spherical aberration, and spherochromatism.
`Figure 8 shows the variation of spherical aberration and oblique
`spherical aberration across the field of view.
`
`Fig. 4. Layout: (top) benchmark design; (bottom) evaluation lens.
`
`Fig. 5. Normalized weighted power per optical element.
`
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`Fig. 6. Field curves: (left) benchmark design; (right) evaluation lens.
`
`Fig. 8. Residual spherical and oblique spherical aberration across
`the field of view.
`
`Spherical and oblique spherical aberrations are balanced at
`about the middle of the FOV, leaving a small amount of
`residual aberration elsewhere.
`As the field-dependent aberrations increase with the field of
`view, the usable depth of focus is reduced. The image quality of
`the evaluation lens is uniform over the entire FOV. Although
`the evaluation lens is one f -stop faster compared to the bench-
`mark design, it provides similar depth of focus. For example, if
`the MTF reduction from nominal value to 40% at 112 lp∕mm
`is chosen as the depth of focus criteria, both designs provide
`about 10 μm of focus range, as shown in Fig. 9.
`D. Distortion
`The meaning of distortion aberration on a curved image surface
`changes; the f -tan(theta) mapping is no longer valid. A reason-
`able assumption for distortion would be a mapping where equal
`distances on a flat object are mapped to equal distances along
`the curved image surface.
`
`Using this equal distances mapping we calculated distortion
`for the benchmark lens. We found that less than 1% of f -tan
`(theta) distortion corresponds to about 9% of equal distances
`mapping distortion. Thus we limited the distortion of the
`evaluation lens to no more than 9% of equal mapping distor-
`tion, as shown in Fig. 10.
`
`E. Chief Ray Incidence Angle on the Sensor
`Significant pupil spherical aberration helps control the CRA in
`the benchmark design. The maximum chief ray incidence angle
`on the sensor is 30 deg. In the evaluation lens the maximum
`chief ray incidence angle, which is calculated relative to the
`curved sensor normal, is only 18 deg. Figure 11 compares beam
`footprint at the exit pupil for different field positions of the
`benchmark and evaluation lens. The beam footprints at the exit
`pupil of the evaluation lens mostly overlap. The pupil spherical
`aberration is not as large as in the benchmark design.
`
`Fig. 7. Modulation transfer function (MTF): (top) benchmark de-
`sign; (bottom) evaluation lens.
`
`Fig. 9. Thru-focus MTF at 112 lp∕mm: (top) benchmark design;
`(bottom) evaluation lens.
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`Research Article
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`Fig. 10. Equal distances mapping distortion: (top) benchmark de-
`sign; (bottom) evaluation.
`
`Fig. 12. Refraction angles: (top) benchmark design; (bottom) evalu-
`ation lens.
`
`Fig. 11.
`Spot diagram at the exit pupil: (left) benchmark design;
`(right) evaluation lens.
`
`F. ANGLES OF REFRACTION
`It is well known in lens design that minimizing the ray angles of
`incidence and refraction in a system is important to reduce
`aberration. For example, large or small ray angles are indicative
`of stress or relaxation in a lens system [16]. When the refraction
`angles are minimized, the surfaces tend to contribute smaller
`amounts of aberration. However, the angle of incidence and
`angle of refraction are different in air and in the material and
`this can lead to confusion in assessing a given lens. As an
`insightful graph shown in Fig. 12 we instead plot surface by
`surface the ray invariant product n sin(I) where I is the ray in-
`cidence angle and n is the index of refraction of the media. The
`rays used are the real marginal and chief rays. As shown, the
`angles of refraction of the marginal ray for the evaluation lens
`are slightly larger indicating more refraction and lens stress,
`though the evaluation design is faster.
`
`G. Tolerance Analysis
`It is important to provide insight in the sensitivity to manufac-
`turing tolerances. We use RMS wavefront error, root summed
`squared over the field, as a criterion. Tilts and decenters have
`the largest effect on the as-build performance of the mobile
`lens. We analyze 21 field points: five field points in each
`
`Fig. 13.
`Sensitivity to manufacturing tolerances: (top) element de-
`center; (bottom) element tilt (the horizontal line indicates nominal
`criterion value).
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`‡X, −X, ‡Y, −Y directions and the on-axis field. The criterion
`is evaluated for a lens element decenter of 5 μm and tilt of
`0.1 deg. Tolerance values of the same order of magnitude have
`been reported in other papers. These values are considered
`being commercial tolerances for small molded plastic optical
`elements [17–19]; however, in practice much tighter tolerances
`are specified to increase lens manufacturing yield.
`Since the tolerance sensitivity strongly depends on the f ∕#
`of the lens, we compare four designs: the evaluation lens at
`f ∕1.6, the evaluation lens stopped down to f ∕2.2, the bench-
`mark lens at f ∕2.2 and the benchmark lens stopped down to
`f ∕2.8. The results are summarized in Fig. 13.
`As expected, the f ∕1.6 lens is the most tolerance sensitive:
`the manufacturability may be a limiting factor for this fast
`design. Fortunately, the fabrication technologies of molded
`optics are constantly improving, allowing tighter tolerances.
`Comparison of lenses at f ∕2.2 shows that the evaluation lens
`is performing better under manufacturing tolerances than the
`benchmark design. In fact, the performance is close to the
`benchmark lens at f ∕2.8.
`
`6. CONCLUSION
`In this paper, we discussed how allowing for a curved imaging
`surface can benefit miniature mobile camera lenses. Recently,
`significant progress toward developing curved sensors has been
`reported. Working prototypes have been demonstrated by a
`number of researchers and commercial organizations.
`We conclude the following.
`First, the curved image surface allows producing an equiva-
`lently performing design with faster f ∕# than the conventional
`design. We found that about one f -stop improvement in speed
`can be achieved while preserving similar to the benchmark lens
`image quality. The limiting aberrations in this design are spherical
`and oblique spherical aberrations, and spherochromatism, which
`can be further reduced by replacing material on one of the ele-
`ments near the aperture stop by a light crown molded glass.
`Second, in the conventional design, the aberration correc-
`tion over the large FOV is extremely challenging: often MTF
`for the full field is significantly lower comparing to the on-axis
`field. The curved sensor helps achieve uniform image quality
`over the entire FOV. Uniform optical performance provides
`an advantage in the image postprocessing. Moreover, the full
`field aberrations often limit the depth of focus of the lens.
`The curved sensor helps increase the effective depth of focus.
`Third, we found that the aperture stop location between
`first and second elements is optimal for aberration balancing
`and controlling the total length of the system.
`Fourth, the radius of curvature of the sensor in our design is
`about 11 mm for a 4.5 mm focal length lens. This gives an idea
`of the required radius of curvature of the sensor for a mobile
`camera lens.
`Fifth, the tolerances for the evaluation lens are not too strin-
`gent. They compare favorably with the benchmark lens for
`lenses of the same f ∕number.
`Finally, we believe that in practice a curved image surface
`will not allow substantial reduction in length of a mobile cam-
`era. A shorter design would require further optically stressing
`
`the lens and departing more from symmetry. As the aberrations
`substantially increase with lens stress, it would be nearly impos-
`sible, even assuming that field curvature can be perfectly com-
`pensated by a curved image surface, to control the remaining
`aberrations for the required FOV and f ∕#.
`Although we are unaware of commercially available curved
`sensors suitable for mobile applications, we believe that potential
`benefits will force further development of this technology allowing
`a new generation of faster compact mobile cameras. Overall,
`this paper contributes specific information about what to expect
`from a mobile lens with an ad hoc curved imaging surface.
`
`Acknowledgment. This research has been sponsored
`by Mr. Gary Sutton. The authors acknowledge Mr. Sutton’s
`insightful comments and generous support.
`
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