When considering the design of an optical system that is capable of steady performance in various environmental conditions, the proper selection of materials and element power distribution is crucial. Such care is necessary since all optical systems generally suffer from chromatic and thermal defocuses, due to wide changes in wavelength and temperature.

For the visible and infrared wavebands, many graphical methods used to reduce such defocuses have been reported [1-7]. Although these methods provide achromatic and athermal solutions, they do not treat the refractive index of the material, which directly affects the field curvature. In particular, since a wide-angle system such as a fisheye lens has a large field curvature compared to normal or telephoto systems, the Petzval curvature of such a system should be reduced to some extent.

This study suggests a new graphical method to correct the Petzval curvature, in addition to the color and thermal aberrations, by introducing a three-dimensional glass chart to select an appropriate material and power combination. The chart consists of the inverse refractive index, the chromatic power, and the thermal power, which are the parameters that most significantly affect the Petzval curvature, color aberration, and thermal defocus, respectively.

In order to locate a system on this glass chart, an optical system with an arbitrary number of elements is reduced to a simpler doublet system that uses an equivalent single lens [7]. Thus, an optical system can be recomposed as a doublet of the specific lens and an equivalent single lens. First, by finding the proper element and material as a specific lens, and then by redistributing the element’s powers of an equivalent single lens, we can identify a pair of element’s material and power that satisfy the achromatic, athermal, and flat Petzval curvature conditions reasonably well.

Using this method to design a fisheye lens, we have found a good solution that has small color aberration and thermal defocus, along with a very slight Petzval sum. The Petzval sum and color aberration are reduced to −0.00064 mm⁻¹ and −0.4 μm between the C- and F-lines, respectively. Moreover, the thermal defocus under a temperature change from −40°C to +80°C is found to be less than the depth of focus.

As outlined above, the selection of the material and power distribution used for a lens element are very important in the design of optical systems, since these parameters significantly affect the aberrations. Among them, the Petzval curvature, chromatic aberration, and thermal aberration are representative aberrations that depend on the material’s properties and the power of a lens element. In an optical system composed of k thin lenses separated by spacers, the total power, the zero Petzval sum, the achromatic, and athermal conditions are given by [4, 8-10]:

where h_i is the paraxial ray height, ϕ_i is the lens power, ε_i is the inverse refractive index (1/n_i) at the central wavelength, ω_i is the chromatic power, and γ_i is the thermal power of the i-th element, and α_h is the coefficient of thermal expansion (CTE) of the housing material. In Eqs. (1)-(4), the primed parameters denote that they are weighted by the ratio of the paraxial ray heights, i.e., the weighted element power is .

The axial color aberration can be expressed as the difference between the back focal lengths of both wavelength extremes. Therefore, for the system to be achromatic, the system must satisfy Eq. (3) [4, 8-10]. The thermal aberration, as given by Eq. (4), is obtained by differentiating the total power ϕ_T of Eq. (1) with respect to the temperature T. Here, we assume that the first summation is negligible when compared to the second summation, since the change of lens power with temperature has a greater influence on the total power fluctuation than that of ray height in a refractive optical system [7].

An equivalent single lens is used to effectively simplify the optical system with an arbitrary number of elements into a doublet system. 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. For the equivalent single lens L_e consisting of the remaining k −1 elements (i.e., all individual elements except the j-th element L_j), the power ( ), inverse refractive index ( ), chromatic power (), and the thermal power (γ_e) of the equivalent single lens may be determined from Eqs. (1) to (4), and are given by [7]:

In this doublet system, which has total power ϕ_T, the conditions to correct the above three aberrations are given by

For a doublet system to be considered athermal, each element must have the power given in the following Eq. (13), which is obtained by solving Eqs. (9) and (12):

Inserting Eq. (13) into Eqs. (10) and (11) results in expressions for the zero Petzval sum and achromatic condition in a doublet system, which simultaneously satisfy the athermal condition:

In order to graphically obtain a pair of materials satisfying Eqs. (14) and (15) for the doublet system, we plot two glass charts as shown in Fig. 1. In the first glass chart, the vertical axis corresponds to the thermal power γ , while the horizontal axis corresponds to the inverse refractive index ε . From Eq. (14), we know that any two elements, L_j and L_e, will provide a zero Petzval sum and an athermal solution, if the γ -intercept of the line connecting them is the negative of the housing’s CTE [see Fig. 1(a)]. In the second glass chart of the thermal power γ versus the chromatic power ω, a similar rule applies. We know from Eq. (15) that any two elements, L_j and L_e, will provide an achromatic and athermal solution if the γ -intercept of the line connecting them is the negative of the housing’s CTE [see Fig. 1(b)].

Many graphical methods which simultaneously correct the color and thermal aberrations exist [1-7]. However, although these methods provide achromatic and athermal solutions, they do not consider the correction of the field curvature, which is an extremely important aberration in a wide-angle system. To solve this problem, this study suggests a new graphical method to obtain an achromatic and athermal system with flat Petzval curvature by introducing a three-dimensional glass chart. This is one of the key points of this study.

Thus, to graphically obtain a pair of materials which satisfy Eqs. (14) and (15) simultaneously, we plot the three-dimensional glass chart composed of the ε , ω, and γ axes, as shown in Fig. 2. This figure includes the coordinates of the specific j-th element L_j(, , γ_j), the equivalent single lens L_e(, , γ_e), and the housing material H(ε_h, ω_h, γ_h). In the chart, the line LjLe that connects the two elements, L_j(, , γ_j) and L_e(, , γ_e), can be expressed as

If this system is mounted in a housing material with a CTE of α_h, as given in Eq. (12), then, its thermal aberration is zero and the thermal power γ_h of a housing H(ε_h, ω_h, γ_h) should be −α_h. Accordingly, in an optical system corrected for thermal aberrations, the inverse refractive index ε_h and chromatic power ω_h of a housing are given by

The housing coordinate H(ε_h, ω_h, γ_h) indicates the state to which aberrations are corrected. Thus, the housing H(0, 0, −α_h) defines the optical system satisfying the achromatic and athermal conditions, along with a zero Petzval sum, provided that α_h is the CTE of an available housing material (α_H), i.e., α_h = α_H.

As illustrated in Fig. 2(a), if any two elements, L_j(, , γ_j) and L_e(, , γ_e), can be connected by a line passing through the available housing H(0, 0, −α_h), then this combination simultaneously satisfies the necessary conditions to correct the three basic aberrations given in Eqs. (10)-(12).

On the other hand, although the coordinates of elements L_j, L_e, and H may lie on the same line, if such a housing material is not available [as for H_non(ε_non, ω_non, γ_non) of Fig. 2(b)], then this combination does not simultaneously satisfy the required conditions to correct the three aberrations. To replace this H_non(ε_non, ω_non, γ_non) with an available housing H(0, 0, −α_H), two graphical methods can be visualized. The first is by changing the material of L_j directly [see Method (A) of Fig. 2(b)]. The second is by altering the equivalent single lens L_e through a redistribution of the element powers [see Method (B) of Fig. 2(b)].

Since the available optical and housing materials are limited, the proper material selection for the specific j-th lens L_j does not frequently occur in a glass chart. Additionally, design approaches that only redistribute the element powers can generate great changes in the power of each element. This approach often leads to bad lens shapes and additional aberrations. To solve these difficulties, an iterative execution for both procedures is desirable.

First, the specific lens L_j(, , γ_j) is replaced by L_j(, , γ_j) in order to minimize the magnitude of the element power changes in L_e, since large power changes may cause unstable configurations and poor optical performance. Thus, the choice of which element and what glass are used for the specific lens L_j is very important in order to correct aberrations and to reduce the cost.

Next, moving L_e(, , γ_e) to L_E(, , γ_E) by redistributing the element powers of the equivalent single lens places L_J, L_E, and H(0, 0, −α_H) on a line, as shown in Fig. 2(b). In Method (B) of Fig. 2(b), the material properties (, , γ_J) of the new specific lens L_J are always fixed, while keeping the total power and the element materials constituting the equivalent single lens unchanged, the power of each element is redistributed to move L_e to L_E. Here, when the housing material is determined to be H(0, 0, −α_H), the material properties (, , γ_E) of the new equivalent single lens L_E are chosen to satisfy the constraints derived from Eqs. (9)-(12). Thus, in order to have a proper material and power combination that can simultaneously correct three aberrations, the power (i ≠ J) of each element should be altered to satisfy the following three constraints:

Thus, by iterative application of this design approach, a multilens system with many elements can be achromatized, passively athermalized, and corrected for Petzval curvature.

As a design example using this proposed method, the redesign of an F/2 fisheye lens operating in the visible range from −40°C to +80°C is presented, as shown in Fig. 3. The initial fisheye lens is taken from an existing U.S. Patent [11], and redesigned to have nearly diffraction-limited performance, while keeping the first-orders intact. Tables 1 and 2 list the specifications and optical properties of the elements used in the initial fisheye lens. In Table 2, the labels G and P denote glass and plastic materials. In Fig. 3, the elements marked L₂ and L₇ are aspherized plastic lenses.

In this system, the color aberration between the C- and F-lines is evaluated to be +8.0 μm. Additionally, the Petzval sum is still quite large at a value of +0.0232 mm⁻¹. Among the available housing materials, AL7075 with a CTE of α_H = 23.6×10⁻⁶°C was chosen to mount the lenses and evaluate the optical performance, because it is a cost-effective material and commonly used to build the refractive system.

The thermal properties of this system are shown in Fig. 4. As seen in Fig. 4(a), the effective focal length of this system is thermally unstable from −40°C to +80°C. The label HML that appears in the figure corresponds to the housing material length (i.e., the length of the housing), while the label EFL corresponds to the effective focal length. The thermal defocus, which is expressed as Δ = HML − EFL, ranges from −10.8 μm to +11.4 μm at the two extremal temperatures, which is greater than the depth of focus. This large thermal defocus leads to an unstable modulation transfer functions (MTFs) at frequencies greater than 100 cycles/mm, as shown in Figs. 4(b)-4(d).

A number of doublet systems can be created from the elements of the initial optical system. In Table 3, the first column lists the seven possible choices of the specific j-th lens L_j(j = 1, 2, ..., 7), used to recompose the optical system as a doublet system with a corresponding equivalent single lens. The second column shows the material properties of the index (n_j), Abbe number (v_j), and thermal power (γ_j) of the lens L_j in the initial fisheye lens. The three quantities (n_J, v_J, γ_J) required in order to correct the three aberrations of Eqs. (10), (11), and (12) are listed in the third column of Table 3. These three quantities are dependent on the element selected as L_j. Finally, the last column denotes the differences (Δn, Δv, Δγ) between the initial values and their solutions for each material property.

From the small values of Δn and Δv listed in the final column, we estimate that the Petzval curvature and color aberration may be corrected without difficulty. However, in addition to the large Δγ values outlined in Table 3, since dn/dT and the CTE of plastic materials are orders of magnitude larger than those of glass materials, the thermal aberration is too great to correct by only changing the glasses. Thus, in order to design a fisheye lens corrected for the three aberrations, two steps can be considered graphically, as in Fig. 5. In step (1), we correct the Petzval curvature by selecting an appropriate glass for L_j, and then correct the color aberration by redistributing the element powers in L_e. In step (2), the thermal aberration is eliminated by only redistributing the element powers, without changing the material used for each element.

In a wide-angle lens, the front lens L₁ is generally much larger than the rear lenses, as outlined in Fig. 3. In addition, the material used for L₁ is almost twice as expensive as those used for all rear lenses, as listed in Table 2. Therefore, in order to reduce the cost of the lens system and correct the Petzval curvature from the appropriate index n_J value listed in Table 3, the glass of L₁ should be changed to a cheaper one with an appropriate index. In this lens system, it is preferable that the first lens be selected to act as the specific j-th lens (L_j=1) of the doublet system, with the equivalent single lens L_e comprising the remaining elements L₂, L₃, ... , L₇.

In the two glass charts of Fig. 6, the current glass NLAF21(788.475) used for L₁ is not on the line connecting the equivalent single lens L_e and the origin. Thus, the glass used for the first lens should be substituted for by a material with a larger chromatic power, i.e., a smaller Abbe number. However, in order to satisfy the zero Petzval sum condition of Eq. (10), we should also select a glass that has its inverse refractive index located on the violet dashed line. Following these requirements, the glass used in the first lens is changed to NF2(620.364), as marked by .

By replacing the glass used in the first lens with NF2, the Petzval curvature is corrected, and subsequently, moving L_e to through a redistribution of the element powers in the equivalent single lens yields the correction required to account for the color aberration. Consequently, the initial housing material H_initial(1.25×10⁻², 22.64×10⁻³, 51.58×10⁻⁶/°C) is changed to H_Step(1)(0, 0, 51.58×10⁻⁶/°C) , which is the γ-intercept of the line connecting and in the three-dimensional glass chart of Fig. 6(a). Since the new housing H_Step(1) has vanishing values for ε_h and ω_h, this temporary system is corrected for Petzval curvature and color aberration. Thus, all elements have the refractive index and Abbe number required to remove both aberrations in a temporary fisheye lens, as shown in Table 4.

Since the thermal aberration has not yet been corrected, each Δγ value listed in the last column of Table 4 is non-zero. Thus, a secondary design process is required. From the glass charts of Fig. 6, we know that the CTE of the housing material should be α_h = −51.58×10⁻⁶/°C, for the athermal condition to be satisfied. However, no housing material having this CTE actually exists. Moreover, there is a large gap between this housing CTE value and available housing CTE values (α_AL7075 = 23.6×10⁻⁶/°C) . In order to overcome these difficulties, we may move the equivalent single lens , to the coordinate L_E by redistributing the element powers, to place , L_E, and H_AL7075(0, 0, −23.6×10⁻⁶/°C) on the same line, as shown in step (2) of Fig. 7.

Here, while is moved to L_E, the total power and material combination of the equivalent single lens are not changed. This relocation of the equivalent single lens can be realized by redistributing the element’s powers to satisfy Eqs. (19)-(21), using the optimization design method of Code-V. In order to have a stable optical system, the magnitude of the power change of each element is constrained to be small in this process. We also note that the Petzval curvature and color aberration here have been corrected in step (1) already. When this system is mounted in a housing of AL7075, and the values of ε and ω of the lens are kept fixed, the element powers of an equivalent single lens are redistributed in order to correct the thermal aberration by forcing its thermal power to be γ = −19.93×10⁻⁶/°C. Thus, in order to obtain a suitable material and power combination for an achromatic and athermal design with zero Petzval sum, the power () of each element should be reconfigured to simultaneously satisfy the following four constraints:

In order to have an athermalized system, the CTE of a housing is required to be α_h = 23.6×10⁻⁶/°C. Such a value can be realized by AL7075. A sketch of the achromatic and athermal fisheye lens with corrected Petzval curvature, designed following the above processes, is shown in Fig. 8. The optical properties of the elements of our final system design are listed in Table 5. Comparing this table and Table 2 of the initial system, it is evident that the glass used in the first lens is changed into a glass with appropriate optical properties to correct the Petzval curvature and reduce the cost. In fact, the new NF2 glass used for the first lens is a quarter of the cost cheaper compared to the original NLAF21 glass [12]. Since the design of the final system simultaneously satisfies the achromatic and athermal conditions, along with the zero Petzval sum, the refractive index, Abbe number, and thermal power of all elements coincide with the values required to correct the three aberrations, as shown in Table 6.

From the new approach demonstrated in this example, we see how the three-dimensional glass chart may be utilized efficiently in order to yield achromatic and athermal solutions that have corrected Petzval curvature. This new design concept is a key aspect of this study.

When the redesigned lens elements are housed in an AL7075 material, the thermal defocus (Δ= HML − EFL) from −40°C to +80°C is significantly reduced to less than 3.7 μm, as illustrated in Fig. 9(a). We note that with this design the sign of ΔEFL is the same as that of ΔHML, which effectively reduces the thermal defocus across the entire temperature range. Thus, the thermal defocus due to a change in temperature is less than the depth of focus of ±4.7 μm. In addition, the color aberration between the C- and F-lines is reduced from +8.0 μm to −0.4 μm, which is much less than the depth of focus. Moreover, the Petzval sum also undergoes a significant reduction, decreasing from +0.0232 mm⁻¹ to −0.00064 mm⁻¹. Here, since the initial fisheye lens was approximated as a thin lens system, and the three aberrations were subsequently corrected by changing the glass and redistributing the element powers, the final redesigned lens has a slight, but non-zero aberration.

Figure 9 shows the modulation transfer functions (MTFs) at various temperatures. In comparison to those shown in Fig. 4, the MTFs of a fisheye lens corrected for three aberrations are much more stable than those of the initial lens over the specified waveband and temperature ranges. In addition, the MTFs at the maximum frequency of 180 cycles/mm are greater than 40% for all fields. Even though it is an extremely wide-angle system, our redesigned system reduces the distortion of the initial lens from −10% to −8%, when evaluated using the equi-angle mapping (f−θ) described in the aforementioned patent [11]. In conclusion, the designed fisheye lens is achromatic in visible light, passively athermalized across the temperature range of −40°C to +80°C, and has flat Petzval curvature.

In order to correct the three aberrations that are most significantly affected by the material properties and the element power distribution in an optical system, this study suggests a new graphical design method based on a three-dimensional glass chart to iteratively obtain the optimal material and power combinations. In summary, in order to correct the Petzval curvature in an achromatic and athermal lens, a two-dimensional glass chart is expanded into a three-dimensional glass chart by the introduction of the inverse refractive index axis. In comparison to previous glass charts, the iterative design method that results from the use of this expanded chart graphically provides the desired materials and element powers to simultaneously correct the three basic aberrations.

Although a particular material combination may not exist, the proposed method effectively provides athermal and achromatic solutions with corrected Petzval curvature. By utilizing this method to design a fisheye lens, we have obtained an optical system that has a flat Petzval curvature, small color aberration and thermal defocus. The proposed iterative design method outlined here is expected to serve as a useful way to find optical system solutions with a three-dimensional glass chart.