When designing an optical system, the combination of optical material and element power has a great effect on aberrations, including color aberration and Petzval field curvature. Accordingly, the proper selection of optical material and element power distribution is crucial to obtaining an optical system that is capable of good performance.

To correct for such aberrations that are sensitive to material properties and the element’s power, many graphical methods have been reported [1-3]. However, they can be used only to design a system composed of contact lenses or fewer than three elements. To solve these shortcomings, other graphical methods for proper selection of materials and element powers have been proposed. In a multilens system with many elements, these methods could correct the color and thermal aberrations using the linear relationship between the thermal and optical properties [4-6].

To additionally correct the field curvature along with these two aberrations, we reported a graphical method on a three-dimensional glass chart consisting of axes for inverse refractive index, chromatic power, and thermal power [7]. However, this graphical method requires cumbersome processes to solve multiple simultaneous equations. Also, the glass chart is composed of complex and processed parameters (inverse refractive index, chromatic power, and thermal power), and thus it is difficult to intuitively obtain the desired materials and element powers to simultaneously correct the aberrations. In addition, since this design approach handles the thermal behavior, it is not the best method to obtain a system corrected for color aberration and Petzval curvature, rather than thermal aberration.

To solve these problems, in this study we suggest a new graphical method, called the symmetric method, to simultaneously correct the color aberration and Petzval curvature, by introducing a new glass chart. Unlike in previous studies, we can simply correct these two aberrations by locating both lenses on lines that are symmetric to each other on a new glass chart, through changing the lens parameters effectively, without solving the several simultaneous equations. To locate a system on this glass chart, an optical system with an arbitrary number of elements is reduced to a simpler doublet system of the specific lens and an equivalent single lens. Since the chart also consists of basic optical parameters, such as Abbe number, refractive index, and optical power, we can easily obtain the materials and element powers to realize a system that both is achromatic and has a flat Petzval curvature.

As a design example using this proposed method, the catadioptric system for a forward-looking camera operating in a wide field of view is presented. The patented refractive lens system is taken, and then two mirrors are added to it. This configuration yields a wide-field system, ranging from ±30° to ±110°, so that the Petzval curvature should be corrected along with the color aberration. Using the proposed symmetric method, we correct these aberrations and then find a good solution that has a small color aberration and a very small Petzval sum.

Figure 1 shows a doublet system composed of two thin lenses separated by an air spacing of d . The positions of the second principal points (, ) depend on wavelength, as shown in Fig. 1. Therefore, for the system to be achromatic, the doublet system should have same back focal lengths at both wavelength extremes, i.e. In that figure, and are the focal points at the C and F lines, and and the focal lengths at each wavelength. Finally, h_iλ is the paraxial ray height of the i^th element at wavelength λ.

In a doublet system composed of two thin lenses separated by a spacer, the total power ϕ_T and the achromatic condition are given by [8, 9]:

Achromatic condition :

where h_i is the paraxial ray height, ϕ_i is the lens power, and v_i is the Abbe number of the i^th element. Here the primed parameters are weighted by the ratio of the paraxial ray heights, i.e. the weighted element power and weighted Abbe number are and respectively.

The optical system generally has a basic field curvature, called the Petzval curvature, which is a function of the optical power and refractive index of each element [10]. In the separated doublet of Fig. 2, the zero Petzval sum condition is given by

Zero Petzval sum :

where n_i is the refractive index. Like in Eqs. (1) and (2), the weighted refractive index is . Therefore, for this doublet system to be achromatic and have a flat Petzval surface at a given optical power, the system must satisfy Eqs. (1)~(3) simultaneously.

An equivalent single lens is used to effectively simplify an optical system with an arbitrary number of elements to a doublet system [5-7]. That is, an optical system with k elements can be recomposed into a doublet system comprising the specific j^th element L_j and an equivalent single lens L_e . This equivalent single lens consists of the remaining k −1 elements (i.e. all individual elements except the j^th element L_j). Thus, in this separated doublet system composed of L_j and L_e, the total power ϕ_T , the achromatic, and the zero Petzval sum conditions are respectively given by

where , , and .

To graphically obtain a pairing of optical material and power that satisfies Eqs. (4)~(6) for a doublet system, we introduce the two glass charts of Figs. 3 and 4. In the glass chart of Fig. 3, the vertical axis corresponds to the weighted optical power ϕ′, while the horizontal axis corresponds to the weighted Abbe number v′. To choose a pairing of optical material and power that correct the color aberration, two elements of a doublet are located as and in the glass chart of optical power versus Abbe number.

From Eq. (5), we know that any two elements L_j and L_e will provide an achromatic solution if the two lines connecting them to the origin have slopes of opposite sign and are symmetric to each other with respect to the v′ axis. Thus, to obtain the combination of material and power, two methods can be considered graphically, as in Fig. 3. In the first method (A), the equivalent lens is fixed and its symmetric point with respect to the horizontal axis is . In this case the material M_A, being located on a crossing point between the line of and the extended line passing through the origin and , is selected as the new optical glass for the specific j^th element L_J(M_A).

In the second method (B), the material M_B is given as an optical glass for the specific j^th element, , for which the symmetric point with respect to the horizontal axis is . The new equivalent single lens , being on the crossing point between the line of and the line connecting the origin and L_{j_sym}(M_B) , is obtained by redistributing the power of each element [6, 7].

Here, since an equivalent Abbe number is dependent on the ray height and power of each element, a starting equivalent single lens L_e can be altered to L_E through a redistribution of the element powers in the equivalent single lens. Hence, even if a particular material combination is unavailable, we can continuously change the element powers to have the required equivalent Abbe number .

Using the two symmetric methods outlined above, an achromatic system can be obtained from the combination of L_J(M_A) and L_e , or the combination of L_J(M_B) and L_E .

To satisfy the zero Petzval sum condition given in Eq. (6), a similar rule applies to the other glass chart of weighted optical power ϕ′ versus weighted refractive index n′. To choose a pairing of material and power that corrects the Petzval curvature, two elements of a doublet are located as and on the glass chart, as shown in Fig. 4. Using the symmetric methods outlined above, a system corrected for Petzval curvature can be obtained from the combination of L_J(M_A) and L_e , or the combination of L_J(M_B) and L_E .

This study suggests an alternate graphical method to obtain an achromatic and flat Petzval curvature system by merging the two glass charts of Figs. 3 and 4 into a three-dimensional glass chart.

Thus, to graphically obtain a pairing of optical material and power that satisfies Eqs. (4)~(6) simultaneously, we plot the three-dimensional glass chart composed of axes for the weighted Abbe number v′, weighted refractive index n′, and weighted optical power ϕ′, as shown in Fig. 5. This chart is designated as a glass chart of optical power versus Abbe diagram. This figure includes the coordinates of the specific j^th element and the equivalent single lens . Assuming for convenience that the powers of L_j and L_e are respectively positive and negative, the design approaches to correct color aberration and Petzval curvature will be presented using a symmetric method.

When the equivalent lens is fixed, as in method (A) of Fig. 5, its symmetric point with respect to the horizontal (v′ − n′) plane is . In this case the material M_A , being located on a crossing point between the plane of and the extended line passing through both the origin and , is selected as the new optical glass for the specific j^th element L_J(M_A).

When the material M_B is given as the optical glass for the specific j^th element , as in method (B) of Fig. 5, its symmetric point with respect to the horizontal (v′ − n′) plane is . A starting equivalent single lens L_e is changed to a new lens L_E that exists at the crossing point between the plane of and the line connecting the origin and L_{j_sym}(M_B), by redistributing the power of each element in an equivalent single lens. Thus, using two methods alternately, an achromatic system with flat Petzval curvature can be easily obtained from either the combination of L_J(M_A) and L_e or the combination of L_J(M_B) and L_E.

As a design example using this proposed symmetric method, a forward-looking camera operating in a wide field range is presented. The initial refractive lens with field angle of ±36° is taken from an existing U.S. Patent [11], and two mirrors are added to it. This configuration yields the catadioptric system for a forward-looking camera shown in Fig. 6 [12, 13]. The convex primary mirror M₁ is used to enlarge the field, and the flat secondary mirror M₂ relays the reflected rays to the refractive lenses.

In the primary mirror M₁ of Fig. 7, to realize a field angle from ±30° to ±110°, the upper chief ray angle β_u and lower chief ray angle β_l on the primary mirror should be −60° and 20° respectively. When the distance between the center of curvature of M₁ and the entrance pupil is S = 700mm, and if the angles of incidence of these chief rays into the entrance pupil are set to be α_u = −9.8° and α_l = −27.8° at the two extremal fields, then the radius of curvature R₁ for the primary mirror is given by [14]:

Accordingly, adding a convex mirror of R = 350mm and a flat secondary mirror to the initial refractive lens system, and then scaling this system down to yield an image size of 1/6 inch at a field angle of ±110°, we obtain an initial catadioptric system for a forward-looking camera, serving a wide field ranging from ±30° to ±110°, as illustrated in Fig. 6. Tables 1 and 2 list the specifications and optical properties of the elements used in this initial catadioptric system. Table 3 lists the third-order axial color aberration (AX) and the Petzval curvature (PTZ).

In this system, the shift of back focal length between the C and F lines is evaluated to be 34.3µm , which is greater than the depth of focus. Additionally, the Petzval sum is still quite large at −0.228mm⁻¹, which is increased by adding mirror M₁. For this catadioptric system to be used for a forward-looking system, therefore, the color aberration and Petzval curvature should be corrected.

Figure 8 illustrates the catadioptric system obtained from the thin-lens approximation for Fig. 6. When this catadioptric system is recomposed into a doublet system comprising the specific j^th element L_j and an equivalent single lens L_e, the condition to correct the Petzval curvature is given by

where , .

Meanwhile, there are two mirrors in this catadioptric system, unlike the doublet outlined in Section 2. A mirror has an effect on the total power and Petzval curvature. However, because the mirror in an optical system has no dispersion, the mirrors M₁ and M₂ are excluded from the achromatic condition, as in Eq. (10). Thus the power ϕ_L (generated by refractive lenses only) and the achromatic condition for this system are respectively given by

Accordingly, if the refractive lenses L₃ ~ L₈ that affect the color aberration are recomposed into a doublet system of lenses L_j and L_e, the achromatic condition for this refractive lens system can be rewritten as follows:

where , .

In this system, and of the specific lens L_j are defined on the plane of a glass chart of optical power versus the Abbe diagram. However, and of an equivalent single lens exist on different planes: on the plane of , and on the plane of .

To utilize the symmetric method with the glass chart of Fig. 5, we replace on the plane of with on the plane of , which results in a new achromatic condition, rewritten from Eq. (11), as

where .

There are two mirrors and six refractive lenses in the initial catadioptric system, but only six refractive lenses can be selected as the specific j^th lens, because of their finite indices and Abbe numbers. To correct the color aberration and Petzval curvarture, the six doublet systems can be created using each refractive element as a specific j^th lens in the initial catadioptric system, as listed in Table 4. The first column lists the six possible choices for the specific j^th lens L_j(j = 3, 4, ⋯ , 8). The second column shows the material properties, the index n_j and Abbe number v_j, of the lens L_j in the initial catadioptric system. Two quantities (n_j, v_j), required to correct the two aberrations of Eqs. (8) and (12), are listed in the final column of Table 4. These two quantities depend on the element selected as an L_j .

From the irrational index n listed in the final column, obtained by solving Eq. (8), we estimate that there is no material to correct the Petzval curvature in all six cases. Meanwhile, available Abbe numbers in the cases of L_j = L₄ and L_j = L₇ yield the opportunity to correct color aberration. Among them, the color aberration can be easily corrected by slightly changing the Abbe number of the specific lens L_j = L₇ (Abbe number : 36.4309 → 31.4787), rather than the case of L_j = L₄ (Abbe number : 60.3236 → 37.0851), as listed in Table 4.

Moreover, the lateral color aberration in this initial system is sufficiently small, less than 1.4µm. Since L₇ is located near the stop, the chief ray height at L₇ is small enough that the new Abbe number required to correct the axial color aberration hardly changes the lateral color aberration. Therefore, we select the lens L₇ as a specific lens rather than L₄, and then determine a pairing of optical material to correct the color aberration and power to realize flat Petzval curvature respectively.

Using Tables 2 and 4, the lenses L_j and L_e are first located on a glass chart of optical power versus Abbe diagram, as illustrated in Fig. 9. Next, the solution material M_solution for L₇ is determined to satisfy the achromatic and zero Petzval sum conditions, using the symmetric method.

To select a material that can correct the color aberration and Petzval curvature, observing the Abbe diagram at of Fig. 10, it is evident that the initial glass NF2 (620.364) of L₇ should be switched, if possible, to a new glass having the same refractive index and Abbe number as those of the solution marked by [15]. In that figure, since no glass having this refractive index actually exists, the Petzval curvature cannot be corrected.

Meanwhile, if we select a glass having the same Abbe number as that of the solution, an achromatic system can be obtained. Because the lens L₇ is negatively powered, glass with a small refractive index is preferred, to prevent increase of the Petzval curvature. Accordingly, NSF8 (689.313), having a small refractive index and Abbe number similar to that of the solution marked by , is selected as the new glass for L₇. By replacing the glass of L₇ with NSF8, the color aberration is significantly reduced without changing the optical power or paraxial ray height. Subsequently, moving L_e to L_E through a redistribution of element powers in the equivalent single lens yields the new system, corrected for Petzval curvature and color aberration, satisfying the symmetric conditions as shown in Fig. 11.

Next, this thin catadioptric system is redesigned into a thick-lens system of suitable thickness. In this process, the symmetric conditions are kept to satisfy the achromatic and flat Petzval conditions. Since this initial catadioptric system is designed to fulfill Eqs. (1)~(3), the aberrations are evaluated to be very small in the limited aperture and field domain. To meet the overall performance required for a modern forward-looking camera, the aperture and field size should be increased. The f-number was extended to F/4, and the maximum field angle was set to be ±110° for a 1/6-inch image sensor. To improve the overall performance in an extended-aperture and -field system, we balance the residual aberrations of the starting data using the optimization method. While designing the system, the material and power of each element should be reconfigured to satisfy the symmetric conditions.

A sketch of the achromatic catadioptric system with corrected Petzval curvature, finally designed following the above processes, is shown in Fig. 12. The glass of L₇ has been changed to NSF8 through the material-selection process. Also, the optical powers and paraxial ray heights of the elements have been changed by the power-redistribution method. The optical properties of the elements finally designed through the optimization process are listed in Table 5. Table 6 illustrates the third-order axial color and Petzval curvature of the final catadioptric system. Comparing this table to Table 3 for the initial system, it is evident that the third-order axial color aberration is reduced from −0.004mm to 0.001mm . Moreover, the Petzval sum also undergoes a significant reduction, from −0.228mm⁻¹ to −0.001mm⁻¹.

In this system, the shift of back focal length between the C and F lines is evaluated to be −7.53µm, which is less than the depth of focus. Even though it is a wide-field system, our design shows low distortion of −5% at a margin field of ±110°.

Figure 13 shows the modulation transfer functions (MTFs) of the final catadioptric system at several fields. The MTFs at the maximum frequency of 130 cycles/mm are greater than 34.1% for all fields. Thus the final catadioptric system satisfies the overall performance required for a forward-looking camera. In conclusion, the system designed using the symmetric method is achromatic for visible light and has greatly reduced Petzval curvature over its field of view, verifying the effectiveness of this design method.

To correct color aberration and Petzval curvature, which are significantly affected by the optical material and the element power respectively, this study suggests a graphically symmetric method based on a glass chart of optical power versus Abbe diagram, through changing the lens parameters effectively. To simultaneously correct these two aberrations, the glass chart is created to contain the Abbe number, the refractive index, and the optical power, which are closely related to these aberrations. Therefore, an optical designer can intuitively select the optimal combination of optical material and element power to correct these aberrations, using the symmetric method with this enhanced glass chart.

Utilizing this method to design a catadioptric system for a forward-looking camera, we have obtained an optical system that has corrected Petzval curvature and very small color aberration. In conclusion, the graphically symmetric method outlined here is expected to serve as a useful way to find an achromatic catadioptric system with corrected Petzval curvature on this new glass chart.