The typical layout of a panoramic annular lens (PAL) can be divided into two parts—a rotationally symmetrical catadioptric optical component, certain portions of which deposited with reflection coatings, and a standard collector lens that transforms those virtual images formed by the catadioptric component into real ones [1
P. Greguss, “Panoramic imaging block for three-dimensional space,” U.S. Patent 4566763 (January 28, 1986).
D. R. Matthys, J. A. Gilbert, and P. Greguss, “Endoscopic measurement using radial metrology with digital correlation,” Opt. Eng.
30(10), 1455–1460 (1991). [CrossRef]
]. The path of rays traveling inside a PAL is schematically shown in Fig. 1
. As a well-known flat-cylinder-perspective imaging model [3
M. A. Stedham and P. P. Banerjee, “The panoramic annular lens attitude determination system (PALADS),” Proc. SPIE
2466, 108–117 (1995). [CrossRef]
], a panoramic annular imaging system transforms from a 3-D omni-directional cylindrical field of view (FOV) between the minimum and the maximum acceptance angles into a 2-D annular image plane perpendicular to the optical axis. Since the central zone of the catadioptric component is blocked by a front mirrored surface, rays between the optical axis and the minimum acceptance angle are excluded from the FOV, and a corresponding blind zone will appear in the central area of the image plane. Usually in conventional central-convergence-perspective optical systems, referring to the human visual habit, the target object would be imaged on the central area of a sensor and the optical axis would be at the center of the FOV. Obviously, this conventional definition about the center of the FOV does not apply to PALs because of a central blind zone of images. Therefore the center of the FOV of a PAL should be redefined explicitly according to the human dynamic-sweeping-vision mode.
Fig. 1 Path of rays traveling inside a PAL.
In general, a specified circular plane corresponding to a certain acceptance angle can be regarded as the center of the FOV. But it raises a problem in the conventional zoom mechanism—the optical power of a lens system varies with the changes of element positions along the optical axis. The FOV of a PAL may deviate from the target scope during the zooming process. The conventional zoom mechanism is still based on the principle of the optical axis being the center, which is not suitable for a zoom PAL. Figure 2(a)
shows the schematic for a typical change of FOV in a PAL with the conventional zoom method. In fact, the expected change of FOV is illustrated in Fig. 2(b)
. Therefore, it is necessary to adopt a new zoom mechanism to fix the specified center of the FOV.
Fig. 2 Schematics of the changing FOV of conventional zoom mechanism and the new zoom mechanism with freeform lenses. (a) The conventional zoom mechanism. (b) The new zoom mechanism with freeform lenses.
As we know, the zoom effect of an optical system can also be achieved by tilted prisms, deformable surfaces, gradient refractive lenses, and so on [4
A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt.
9(7), 1669–1671 (1970). [CrossRef]
]. These components can be thought of as generalized freeform optical elements. The focal lengths of a freeform varifocal lens are changed through variation of the optical power distributions of its lenses rather than moving these lenses back and forth along the optical axis. With the advantages of flexible geometry and great control of rays, freeform optical elements can adjust the local surface shape actively according to the imaging requirements, and the imaging quality at other parts of the aperture may not be affected. The focal lengths of a freeform lens are even varied in different directions of the FOV. This will play an important part in the realization of a varifocal PAL with a specified annular center of the FOV.
In this paper, a varifocal PAL with a FOV from 80°–100° at 0° to 70°–110° at ± 180° was designed based on freeform lenses. The specified annular center of the FOV is a circle plane corresponding to the acceptance angle of 90°. The absolute FOV in the direction of the acceptance angles is relative to the specified annular center with the cosine deviation from ± 20° at 0° rotational angle to ± 10° at ± 180° rotational angle on both sides around the optical axis. The focal length in each direction of the half-rotation angle of the FOV is different due to the application of freeform elements. The conjugate relations between the object space and the image space are changed when the freeform PAL rotates around the optical axis. The zooming effect of each FOV can be realized by the rotation of the freeform PAL, as shown in Fig. 3
. The main purpose of this design is to explain the freeform zoom mechanism keeping the annular center of the FOV unchanged during the zooming operation. The freeform varifocal PAL operates at a single wavelength of 650 nm, and the chromatic aberration of this lens system will not be discussed.
Fig. 3 Zooming effect through the rotation of freeform lenses.
2. Astigmatism and distortion of freeform lens system
Consider a simplified ray-tracing model of a freeform imaging optical system, such as the thin lens system shown schematically in Fig. 4
, which consists of k
surfaces with refractive powers
, refractive indices
Fig. 4 Simplified ray-tracing model of a freeform imaging system.
The propagation of rays between the two adjacent surfaces is written as
are the position vectors of the intersection points of the ray with the two surfaces,
is the vector for the direction of the ray, and
is the distance between the two intersection points.
Based on the refractive law, the relationship of an incident ray and an emergent ray can be written as
. Equations (1)
are the vector forms of the ray-tracing equations for the simplified freeform lens system. By ray-tracing calculation, the refractive powers of the freeform surfaces acting on the rays can be obtained along the paths of these rays.
The refractive power
of a simplified model with two surfaces having powers of
separately can be written as
is the equivalent distance of the two surfaces,
and the refractive power
of a three-surface model is
where the equivalent distance
By iteration calculation, the refractive power
of the whole system is written as
where the equivalent distance
If the system layout of the freeform lens model is explicit, which means the parameters of refractive indices
are constant, the optical power of the model can be described as a function related to the powers
of all of the freeform surfaces,
As we know, the refractive power of each optical surface is
Thus, the refractive power of the model is a function of
At one point on a surface, the normal vector
and the two tangential vectors
of the two local main curvatures of the surface form two orthogonal curvature planes separately. Similarly, the direction vector
of a ray through the intersection point and the normal vector
determine an incident plane. If θ
is the angle between the incident plane and the first main curvature plane, the local curvatures for the ray are obtained by the projection of the two main curvatures on the incident plane and the plane perpendicular to it:
Therefore, the refractive power
can be divided into two orthogonal components,
The astigmatism is expressed by the absolute value of the refractive power difference between the two orthogonal components:
The astigmatism can be eliminated by configuring the curvature distributions of the optical surfaces rationally, but for a freeform lens system, the astigmatism is not only concerned with the rays on tangential and radial planes but also with all the rays in the aperture for the shape flexibility of freeform surface at local positions. It requires the same power of all rays from one objective point for the correction of astigmatism.
Since in a freeform optical system the unit normal vector of a freeform surface is not always in the tangential plane, which is defined by the incident principal ray and the optical axis, it can be resolved into a tangential component and a radial component:
The incident principal ray and the emergent principal ray refracted by the freeform surface resolve a component along the radial direction and a component along the tangential direction in the tangent plane of the normal vector, respectively:
For most imaging optical systems with freeform elements, the principal rays emitted from the objective points are set in the tangential plane, which means that their radial components are zero:
Although the principal rays will deviate from the tangential plane during propagation in the lens system, the radial components of these rays are expected to be zero when they reach the image surface:
determines the position distribution of the optical rays on the image surface and can be used to control the irregular distortion of the freeform optical system.
3. Freeform varifocal PAL design
Based on the principle of correction for astigmatism and distortion analyzed in Section 2, the design and optimization process of a freeform varifocal PAL will be discussed in this section.
3.1 Original configuration and optical specification
A simple and complete optical lens is a system composed of a FOV, lenses, aperture, and imaging surface. Therefore, these four basic factors should be included in the specification of the freeform varifocal PAL design.
A rotationally symmetric PAL with a FOV of 70°–110° × 360° is taken as the original configuration of the varifocal PAL being designed. The optical layout is shown in Fig. 5
. The maximum range of the acceptance angles is from 70° to 110°, and the specified annular center is a circle plane corresponding to the acceptance angle of 90°. If the optical axis of this lens is located in a vertical direction, the annular center of the FOV corresponds to a horizontal circular plane. The imaging result is like a person turning in a circle on the spot and looking around.
Fig. 5 Optical layout of the original configuration.
The variation of the FOV of the freeform varifocal PAL based on this original design is specified from 80°–100° at 0° to 70°–110° at ± 180° on both sides around the optical axis. The annular center of the FOV at the 90° acceptance angle will remain unchanged. Relative to the specified annular center, the absolute acceptance angle of the FOV is from ± 20° at a 0° rotational angle to ± 10° at a ± 180° rotational angle by the cosine transition on both sides around the optical axis. The overall optical system is set to be symmetrical about the YOZ plane.
Five acceptance angles of each set of FOVs at the same interval of a 15° rotational angle are sampled in the design. The acceptance angles vary with the cosine law along the rotational direction. The set of annular FOVs and the transformation from polar to rectilinear coordinates are shown in Fig. 6
. When the freeform varifocal PAL rotates on its optical axis, the mapping from the FOV to the image plane will change with its varying optical power.
Fig. 6 Transition process of FOV of the varifocal PAL. (a) Annular FOV. (b) The transformation of annular FOV from polar to rectilinear coordinate.
The aperture of a PAL system is usually set at the last surface of the catadioptric component, which is helpful in reducing the stray light brought in by reflection [5
S. Niu, J. Bai, X. Y. Hou, and G. G. Yang, “Design of a panoramic annular lens with a long focal length,” Appl. Opt.
46(32), 7850–7857 (2007). [CrossRef]
], but the position of the entrance pupil of the freeform varifocal PAL is easy to offset for the irregular aperture spherical aberration caused by freeform surfaces. It is complicated to make the radial positions of entrance pupils of each FOV in all rotational angle directions both consistent and variable during the optimization process. Therefore, we set an auxiliary aperture (the entrance pupil) as the virtual field stop in the varifocal PAL to constrain the positions and angles of the incident rays. The intersection points of the principal rays of each FOV and the optical axis were constrained at the last surface of the catadioptric component. Thus, the position of the real aperture is determined. The setting of the auxiliary aperture can be seen in the original configuration as shown in Fig. 5
. It should be noted that the auxiliary aperture is used only for the lens design and will not appear in the real freeform PAL.
A large-size CCD sensor with a high resolution of 4368 × 2912 pixels and an 8.2 um pixel size was selected for image acquisition. Its small side length of about 24 mm is regarded as the outer diameter of the annular imaging area. According to the original PAL system, the effective imaging area is an annular ring from 8 to 12 mm in radius. The points at the specified annular center of the FOV correspond to a circle with a 10 mm radius in the image surface. The requirement of f-theta distortion is less than 5%. The target of MTF evaluated at a spatial frequency of 50 lp/mm is larger than 0.1 with the consideration of freeform surface accuracy.
Freeform optical elements are the key and basic parts to the varifocal PAL system, the zoom effect of which is different from the conventional zooming method by moving lenses back and forth along the optical axis. An X–Y polynomial surface was chosen from the many freeform surface representations and used in the varifocal PAL design. It is a simple and effective surface type. The accuracy of freeform geometry can be improved with the increasing order of its coefficients, but the convergence rate in the optimization process would become weak with that. Therefore, it is very important to decide the order of the asymmetric coefficients of the X–Y polynomial surface to balance the accuracy and the optimization efficiency. All of the X–Y polynomial surfaces used in the design share the same Z axis in the mathematics expressions of the form,
is the vertex curvature, r
is the radial coordinate, k
is the conic constant, and
are the coefficients of the various
terms. According to the symmetric setting of the FOV about the
plane, only the even powers of x
terms in the X–Y polynomials were used to describe the freeform surfaces. The optical power changes depend mainly on the variation of the curvature distribution.
The overall specification of the freeform varifocal PAL is summarized in Table 1
Table 1 Specifications of the Freeform Varifocal PAL
|CCD size||35.8 mm × 25.9 mm|
|Resolution||4368 × 2912 pixels|
|Effective imaging area||Outer diameter of 25.9 mm|
|Variation of FOV||80°–100°@0° to 70°–110°@ ± 180°|
|Center of FOV||90° × 360°|
|Optical stop aperture||2 mm|
|Number of freeform surfaces||5|
|Image quality||MTF>10%@61 lp/mm|
|Zooming mode||Rotational freeform varifocal lens system|
3.2 Optimization methods
The freeform varifocal PAL system was optimized with ray-tracing in the optical software ZEMAX [6
ZEMAX Optical Design Program, User’s Guide Version 8.0, Focus Software, Inc. (1999).
]. The basic optical definitions, such as effective focal length, were not applied in the optimization due to the non-rotational symmetric layout, the hyper-semispherical FOV, and the center transition of the FOV. In this freeform varifocal PAL design, the entrance pupil plane is vertical to the optical aperture, which easily causes a calculation error for the effective focal length of the whole system due to the aperture spherical aberration. For example, the effective focal length (EFFL) of the freeform PAL given by ZEMAX is −0.002mm. Obviously, the unexplained value of the focal length does not conform to actual conditions. So, the focal length was not controlled in the optimizing process. The mapping of rays from objective points to the designated positions on the imaging surface is taken as the basic principle for the configuration optimization of the freeform varifocal PAL. The centroid coordinates of the imaging point formed by a grid of rays from a single objective point were set as operand targets to constrain large and irregular distortion. In this way, the distortion of the freeform varifocal PAL can be easily controlled, although the optimization operands about distortion in ZEMAX cannot be used directly in this design.
In addition, the incident angles of real rays striking on the image surface were set constraints to prevent local large deformation caused by the high-order polynomial representations of those freeform surfaces. This helps to improve the smoothness of the shape transformation of freeform surfaces. The surface decenter in the Y and/or Z directions and tilt about the X axis were not set as variables because of their mutual influence on the X–Y polynomial representations of freeform surfaces. This is good for simplifying the space structures of freeform surfaces and accelerating convergence rates of system optimization. In optimization, the surface coefficients up to the sixth order in both tangential and radial planes and the spacing between those elements were set as variables. Making a brief review of this freeform varifocal PAL design, the characteristics of its configuration are plane symmetry and co-axial freeform surfaces. Therefore, only half of the full FOV (the rotational angles from 0° to 180°) needs to be optimized.
3.3 Optimization results and image evaluation
The layout of the final freeform varifocal PAL design is shown in Fig. 7
. It consists of one freeform catadioptric component and two pieces of plane-freeform collector lenses. A total of five pieces of freeform surfaces were used in this PAL design. The first mirror and the last refractive surface of the catadioptric component share the same X–Y polynomial representation to keep the freeform surface uniform, although it will decrease the degrees of freedom of the lens design. The aperture was set to shift from the last surface of the catadioptric component, which is helpful for efficient use of the freeform surface. The shapes and sizes for different fields are not the same, which means that the focal lengths are different even within one set of acceptance angle of the FOV corresponding to the same rotational angle. This is related to the limitation of the degrees of freedom of the XYP surfaces and the cosine transition of the FOV. The minimum size of optical stop is adopted in a real production. The incident directions of principal rays were constrained perpendicular to the image surface. The total length of the freeform varifoal PAL system is 75 mm, the diameter of the lens is 40 mm and the back focal length (BFL) is 18 mm. The material of the whole system is PMMA.
Fig. 7 Layout of the freeform varifocal PAL. (a) 2-D layout. (b) 3-D layout.
The MTF of the freeform varifocal PAL was evaluated at the spatial frequency of 61 cycles per mm. Figure 8
shows the MTF sampled at each FOV from 0°–180° rotational angle at the same interval of 15°. As illustrated in Fig. 8(a)
to Fig. 8(m)
, the values of the diffractive limitation for each rotation angle of the FOV decrease with the variation of the rotation angle. The MTF values of both ends of the half-rotational angle of the FOV are a little better than those of the middle part. Frankly, the design result is not good enough for an 8.2 um CCD sensor since some MTF values are less than 0.1 at 61 lp/mm. However, it is worth our effort to find out the cause of the low image quality. The reasons are that the cosine variation of the acceptance angle FOV is not in an optimal transition way for the freeform varifocal PAL, and the image telecentric optical layout acts as a constraint on the path of rays. Although five freeform surfaces were used in the PAL design, the sixth order degrees-of-freedom of these surfaces cannot provide near-diffraction-limit imaging quality; but the XYP freeform surfaces and the FOV transition by cosine curve are able to satisfy the requirement of the varifocal PAL design. Through comparative analyses, the differences of resolution among all FOV are acceptable. During the rotation zooming process, the imaging quality of the freeform varifocal PAL remains unaffected.
Fig. 8 MTF of the FOV at different rotational angle.
One of the goals of this freeform varifocal PAL design is zero f-theta distortion from the specified annular center of the FOV to both ends in the acceptance angle directions. While in the rotational angle directions, the image points were not expected to deviate from their tangential planes. The target centroid coordinates of the image points focused by the rays from 0, ± 0.5, and ± 1 FOV were separately (0, 8), (0, 9), (0, 10), (0, 11) and (0, 12) in the tangential planes. The real positions of image points in the varifocal PAL design were listed in Table 2
, and the distortion of the PAL at the half-annular FOV was shown in Fig. 9
. As illustrated in Fig. 9
, the distortion did not present any regularity. In the optimization, the distortion distribution of the varifocal PAL changed with the weight of constraints on the positions of the image points.
Table 2 Coordinates of Image Points in the Tangential Planes
|Rotation Angles (Degree)||Coordinates of image points in the tangential and radial directions (mm)
|X (0.00)||Y (8.00)||X (0.00)||Y (9.00)||X (0.00)||Y (10.00)||X (0.00)||Y (11.00)||X (0.00)||Y (12.00)|
Fig. 9 F-theta distortion of the freeform varifocal PAL.
The requirement on a 5% distortion is claimed for constraint of the image deformation. It is still within the conventional definition of relative distortion, which is the change in the image height with respect to the paraxial one. In this sense, the distortion is nearly 1%, but for reasons of the center transformation of the FOV, the image center is changed from the intersection point of the optical axis and the image plane to a circle corresponding to the specified annular center of the FOV. The image points focused by the rays from the center of the FOV do not coincide with the image circle center due to the distortion. The calculation of the relative distortion cannot take place because of the lacking of a consistent reference position. Therefore, only the positions of these sampled points are listed in Table 2
In this paper, we have proposed and analyzed a new zoom mechanism for the realization of a freeform varifocal PAL system with a specified annular center of FOV. Unlike a conventional zooming method, the optical axis was not taken as the zooming center of the FOV. Instead, optical zooming is implemented by a rotation of the freeform varifocal PAL around the optical axis, which solves the problem of deviation of the FOV from the target scope during the zooming process. The design principle and optimization process of the freeform varifocal PAL was introduced and discussed. The variation of the FOV is from 80°–100° at 0° to 70°–110° at ± 180°, which changes with the cosine law on both sides around the axis. The annular center of the FOV was specified at acceptance angle of 90°. It consists of one freeform catadioptric component and two pieces of plane-freeform collector lenses. Through comparative analyses of MTF at each FOV from a 0°–180° rotational angle at the same interval of 15°, the imaging quality of the varifocal PAL is good. The irregular f-theta distortion changes with the weight of the constraints on positions of the image points. The relationship between the distortion and the shape of freeform surfaces will be studied in our next work.