Abstract

A monocentric lens combined with a multi-scale form can achieve a large field of view while maintaining the resolution. This report describes an analytical model that is suitable for both Galilean- and Keplerian-type monocentric multi-scale (MMS) systems; this model also analyzes the correlation between the two types of systems. Moreover, the off-axis aberration associated with the analytical model was derived, on this basis, the Galilean- and Keplerian-type MMS systems were compared. It was concluded that the Galilean-type MMS system performs better with respect to aberration performance. This report provides a useful reference for further applications and developments of MC systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Attaining a large field of view (FOV) while retaining small system dimensions is a longstanding goal pursued by researchers developing large FOV optical systems. A large FOV optical system is bound to undertake a certain FOV angle, often utilizing numerous lenses to converge the light to the image surface. The compact design of such systems is not easy to realize [1].

Compared with classic panoramic optical systems, the monocentric (MC) design facilitates the correction of system aberration. In general, MC lenses are constructed with multiple concentric surfaces, and such a concentric system only needs to correct the spherical aberration. Off-axis aberration is minimal for MC lenses because their optical form remains unchanged to ensure that each FOV is observed. The problem that needs to be addressed by MC lenses is the curved surface of the image. To deal with a curved surface, the MC system requires the use of curved fiber bundles to receive images or an image processing method [24].

The multi-scale form splits the system into different design levels, with each level smaller than the former one [5]. This design provides improved resolution without expanding the system dimensions dramatically and further enables the target information to be detected by a normal flat detector [6]. Combining an MC system with the multi-scale form exploits the advantage of both systems. Compared with a multi-camera array, the monocentric multi-scale (MMS) form can react with a larger entrance pupil diameter and smaller dimensions of multi-scale form under the same parameter requirements, providing an advantage because of its light weight [7]. Moreover, the MMS form offers greater flexibility when selecting the plane detector.

Optical systems generally follow the relationship $\mathrm{I}=\mathrm{I}_{0} \cdot \cos ^{4}(\mathrm{\theta})$ [8]. As the FOV increases, the system suffers a certain amount of light energy loss. If the aperture stop lies in the center of the MC ball, the highest peak illumination value of each subimager decreases significantly. To maintain a constant peak illumination value for each subimager, the aperture stop should lie within each subimager [9]. Generally, the MMS form can be classified as either Galilean-type or Keplerian-type depending on whether the system adopts an intermediate image plane between the MC lens and subsequent subimager.

As shown in Fig. 1, for the central FOV, the light passes through the head unit and then exits to the image plane through the relay system. For a central FOV of the MMS system, the ray path is symmetrical with respect to the field direction. However, when it comes to the marginal FOV, owing to the position of the aperture stop, the ray path is not in an axial symmetrical form but is instead replaced by an approximate symmetrical form. Thus, it no longer satisfies the MC lens principle that all field ray paths are identical. This problem occurs for both Galilean- and Keplerian-type MMS forms.

 figure: Fig. 1.

Fig. 1. Paths of different FOV rays transmitted by an MMS system.

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Axial asymmetric ray paths create off-axis aberration within the MC lens, which affects the subimagers and subsequently, the entire MMS system. No previous report has discussed the off-axis aberration of an MMS system in detail. Therefore, the purposes of this study were to build a simplified model of an MMS system and to evaluate the off-axis asymmetric effect within this MMS system. Our objective was to provide an analytic model for building MMS systems that not only models Galilean- and Keplerian-type MMS systems but also reveals their correlation and aberration performance. This information will facilitate the design of suitable initial structures, which will be convenient in the subsequent optimization process. In the following discussion, the Galilean-type system is called a GMMS and the Keplerian-type system is called a KMMS, and parameters with subscripts “G” and “K” are used to illustrate parameters associated with the Galilean- and Keplerian-type systems, respectively.

The remainder of this paper is organized as follows. In Section 2, the development of a simplified model of the MC lens is discussed. This model simplifies the MC lens while preserving its generality. In addition, a model of the entire MMS system, including the subimager, is provided alongside descriptions of the GMMS and KMMS systems. The relationship between the parameters of these two types of systems is considered. In Section 3, a generalized description of the geometric aberrations in the MMS system based on the results presented in Section 2 is provided. This description is accompanied by a detailed discussion of the MMS parameters that influence the performance, followed by examples for KMMS and GMMS systems. Section 4 presents our conclusions.

2. Analytical model of an MMS system

A simplified analytical model can facilitate the design and analysis of MMS systems. Firstly, we discuss the simplified model of an MC lens contained in an MMS system.

2.1 Simplified model of an MC lens

Many reports have provided discussions and analyses on the construction of MC lens [8,10]. An MC lens is composed of multiple concentric lenses. The cemented form is adopted to avoid chromatic aberration. By introducing additional degrees of freedom into the design, the air gap between the adjacent concentric surfaces can also be used to improve the system performance [10].

Figure 2 depicts a classic concentric system, which uses cemented surfaces to correct the chromatic aberration, spherical aberration, and spherochromatism. As a rough estimate of the system, a paraxial model is typically used, as shown in Fig. 2. Although this method is very practical, the simplification can be excessive. The height and axial location at which rays enter or exit the MC cannot be identical, whereas in the paraxial model, the entry and exit points are identical. Thus, the paraxial model is not completely authentic and produces imprecise analyses of MC lenses. Owing to the numerous elements comprising an MC system, it is often difficult to perform the preliminary estimation and evaluation of the MC lens directly. In addition, the variety of MC forms introduces uncertainty into the evaluation of MC lenses.

 figure: Fig. 2.

Fig. 2. Diagram illustrating the design of a classical MC lens.

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Typically, the glass comprising an MC lens is crown or flint material, where the crown material is located at the center of the MC lens. Typically, the refractive index of crown glass varies from approximately 1.5 to 1.6, with the flint material used more commonly as the edge of the MC lens (refractive index ∼1.7–1.8). The variation in the refractive index within the MC lens is much smaller than the index contrast between the air and lens material. Thus, the propagating ray path in the MC lens undergoes minor deviations. In our treatment of the system, the MC lens was simplified into a single medium sphere composed of two faces, facilitating the analysis of the MC system. The focal length and total length of the MC lens were used as the fundamental properties of the simplified model, designing the simplified MC model according to the actual desired parameters.

Figure 3 depicts the simplified MC lens, with point O representing the central point of the lens. We denote the focal length of the MC lens as fmc and the total length as LT. Next, we discuss the relationship between the design parameters of the MC lens and these two prerequisite parameters.

 figure: Fig. 3.

Fig. 3. Diagram of a simplified MC lens.

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Following the Gaussian equation [11], we obtain

$$\frac{{{n_{mc}}}}{{S_1^{\prime}}} - \frac{1}{{{S_1}}} = \frac{{{n_{mc}} - 1}}{{{R_1}}}$$
and
$$\frac{1}{{S_2^{\prime}}} - \frac{{{n_{mc}}}}{{{S_2}}} = \frac{{1 - {n_{mc}}}}{{{R_2}}},$$
where ${S_1}$ and $S_1^{\prime}$ represent the objective and image distances of the first surface, respectively, and ${S_2}$ and $S_2^{\prime}$ represent the objective and image distances of the second surface, respectively. ${R_1}$ and ${R_2}$ represent the radii of the front and rear surfaces of the MC lens, respectively. nmc denotes the average refractive index of the MC sphere material. Owing to its spherical shape, ${R_1}$ and ${R_2}$ can both be defined as ${R_{mc}}$.

As shown in Fig. 3, ${S_1}$ tends to infinity, ${S_2} = S_1^{\prime} - 2{R_{mc}}$, and $S_2^{\prime} = {L_T} - 2{R_{mc}}$. Then, we can derive the following equation:

$${S_2}^{\prime} = \frac{{2 - {n_{mc}}}}{{2({{n_{mc}} - 1} )}} \cdot {R_{mc}} $$
Combing Eq. (3) with Fig. 3, we can derive Eq. (4):
$${L_T} - 2{R_{mc}} = \frac{{2 - {n_{mc}}}}{{2({{n_{mc}} - 1} )}} \cdot {R_{mc}}$$
For a classical singlet lens, the focal length obeys the following equation:
$$\frac{1}{f} = ({n - 1} )\cdot \left( {\frac{1}{{{R_1}}} - \frac{1}{{{R_2}}}} \right) + \frac{{t{{({n - 1} )}^2}}}{{n{R_1}{R_2}}}$$

As shown in Fig. 3, we treated the MC system as a single lens, ${R_1} ={-} {R_2} = {R_{mc}}$ and $t = 2{R_{mc}}$. Substituting the above conditions into Eq. (5) leads to

$${f_{mc}} = \frac{{{n_{mc}} \cdot {R_{mc}}}}{{2({{n_{mc}} - 1} )}}$$

Then, by solving Eqs. (4) and (6) together, we can obtain the average refractive index and surface radius of the simplified model:

$${n_{mc}} = \frac{{2{f_{mc}}}}{{3{f_{mc}} - {L_T}}}$$
and
$${R_{mc}} = {L_T} - {f_{mc}}$$

The advantage of the simplified model is that only two surfaces need to be dealt with, which is very close to the arrangement of an MC lens and thus provides authenticity. Using the key parameters of the MC system, such as the total length and focal length, we can establish a reasonable initial structural model. This simplified model does not lose generality, and it will be used in the following sections.

As the above equations apply to both the GMMS and KMMS systems, the subscripts G and K are not used.

From Eq. (8), it can be observed that the focal length of the MC lens is equivalent to the distance from the center point O of the MC lens to the image plane. Point O is a crucial parameter that is used as the key factor in the following analysis. This article focuses mainly on monochromatic aberration without considering chromatic issues. Correction of the chromatic aberration of the system can be achieved by adding a cemented surface to the MC lens [12]. This procedure is not discussed in this article.

2.2 Establishment of GMMS model

The GMMS and KMMS systems are identified according to the location of the subimager relative to the image plane formed by the MC lens. In this section, we focus principally on the discussion and analysis of the GMMS system, followed by the discussion on KMMS system in Section 2.3.

As shown in Fig. 4, the GMMS-type system is composed of an MC lens and a subimager. To clarify the functionality of the components contained in the subimager, it is divided into a relay lens and field flattener. The distance between the center point O of the MC objective and the relay lens is denoted as L1G, and the distance between the relay lens and the image plane is L2G, whereas LTG represents the total length of the MC lens along its optical axis.

 figure: Fig. 4.

Fig. 4. Analytical model of the GMMS system.

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In the MMS subimager [13], the feasibility of the aperture stop location is relatively limited; thus, to simplify the GMMS analytical model, the position of the aperture stop coincides with the relay lens.

Because the image surface of the MC lens is a curved spherical surface, there is a certain Petzval field curvature associated with the subsequent system in addition to the field curvature problem introduced by the subimager itself. As the subimager predominantly uses a flat detector, the most specific method of correcting the Petzval field curvature involves placing a flat field lens in front of the image plane. Therefore, in the GMMS analysis, we added a field flattener in front of the image plane to complete the overall system model construction.

The rear surface of the field flattener resembles a flat surface to avoid collisions with the image plane. Thus, its lens power is defined by its front surface. However, the field flattener does not contribute to the lens bending power because it is too close to the image plane. The lens bending power of the entire subimager is mainly determined by the relay lens.

We define the focal lengths of the entire GMMS system and the MC lens as fTG and fmcG respectively. The correlation between fTG and fmcG can be inferred from Fig. 4, leading to

$${f_{TG}} = {f_{mcG}} \cdot \frac{{{L_{2G}}}}{{{f_{mcG}} - {L_{1G}}}}$$

In addition, for the relay system, the object and image distances are ${f_{mcG}} - {L_{1G}}$ and ${L_{2G}}$, respectively. According to the Gaussian lens formula, the power of the relay lens ${f_{rG}}$ can be derived as

$${f_{rG}} = \frac{{({{f_{mcG}} - {L_{1G}}} ){L_{2G}}}}{{{f_{mcG}} - {L_{1G}} - {L_{2G}}}}$$

To unify the variables, we define ${L_{1G}} = {f_{mcG}} \cdot {m_{1G}}$ and ${L_{2G}} = {f_{mcG}} \cdot {m_{2G}}$. Incorporating these relationships into Eqs. (9) and (10), we obtain

$${f_{TG}} = {f_{mcG}} \cdot \frac{{{m_{2G}}}}{{1 - {m_{1G}}}}$$
and
$${f_{rG}} = \frac{{{f_{mcG}}({1 - {m_{1G}}} ){m_{2G}}}}{{1 - {m_{1G}} - {m_{2G}}}}$$

For the MMS system, the image is best received by a flat image plane. Therefore, the Petzval field curvature must be considered. The Petzval field curvature formula of the system is determined by the following equation [11]:

$$\frac{1}{{{n_n}{r_{Pn}}}} = \mathop \sum \limits_{i = 1}^n \frac{1}{{{R_i}}} \cdot \left( {\frac{1}{{{n_i}}} - \frac{1}{{{n_{i - 1}}}}} \right)$$

In Eq. (13), ${R_{i1}}$ represents the ith surface radius and ${n_i}$ and ${n_{i - 1}}$ represent the refractive indices of media separated by the surface. The Petzval curvature is determined by the surface radius included in the system. In the actual design process, the index fluctuation is relatively small, and the material refractive index used by the subimager is unified as n for convenience. Assuming that the relay lens is composed of N lenses, the front and rear radii of ith lenses are ${R_{i1}}$ and ${R_{i2}}$, respectively. Applying these factors to Eq. (13), we obtain

$$\frac{1}{{{r_{Pn}}}} = \frac{2}{{{R_{mc}}}}\left( {\frac{1}{{{n_{mc}}}} - 1} \right) + \mathop \sum \limits_{i = 1}^N \left( {\frac{1}{{{R_{i1}}}}\left( {\frac{1}{n} - 1} \right) + \frac{1}{{{R_{i2}}}}\left( {1 - \frac{1}{n}} \right)} \right) + \frac{1}{{{R_{Flatten}}}}\left( {\frac{1}{n} - 1} \right)$$

From Eq. (14), the parameters of the subimager can be collated to

$$\mathop \sum \limits_{i = 1}^N \left( {\frac{1}{{{R_{i1}}}}\left( {\frac{1}{n} - 1} \right) + \frac{1}{{{R_{i2}}}}\left( {1 - \frac{1}{n}} \right)} \right) ={-} \frac{1}{n} \cdot \mathop \sum \limits_{i = 1}^N \frac{1}{{{f_{ri}}}},$$
where ${f_{ri}}$ represents the focal length of each relay lens. Owing to the dense arrangement of the relay lenses, the summations on the right-hand side of Eq. (15) can be condensed as the reciprocal of ${f_{rG}}$.

Considering the flat plane representation of the image plane, i.e., ${r_{Pn}} = \infty $ for Eqs. (14) and (15), the curvature radius of the flattener can be written as

$${R_{Flatten}} = \frac{{({1 - n} ){f_{rG}}{n_{mc}}{R_{mc}}}}{{ - 2{f_{rG}}n + 2{f_{rG}}n \cdot {n_{mc}} + {n_{mc}}{R_{mc}}}}.$$

Substituting Eqs. (7), (8), and (12) into Eq. (16) provides the final expression for the front curvature of the field flattener:

$${R_{FlatG}} ={-} \frac{{{f_{mcG}}({{m_{1G}} - 1} ){m_{2G}}({n - 1} )}}{{ - 1 + {m_{1G}} + {m_{2G}} - {m_{2G}}n + {m_{1G}}{m_{2G}}n}}$$

2.3 Establishment of the KMMS model

The KMMS system utilizes an intermediate image during the imaging process. One advantage of the KMMS system is that the imaging quality of each component can be checked prior to assembly.

Figure 5 provides a schematic of the KMMS. It comprises an MC, a relay lens, and a field flattener. The distance between the center point O of the MC objective and the relay lens is denoted as L1K, and the distance between the relay lens and field flattener is denoted as ${L_{2K}}$.

 figure: Fig. 5.

Fig. 5. Analytical model of the KMMS system.

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Similar to the analytical model of the GMMS system, the relay lens contained in the model of the KMMS system coincides with the stop lens. A field flattener is also included in the analytical model.

Assuming that the focal length of the whole system is ${f_{TK}}$, the correlation between ${f_{TK}}$ and ${f_{mcK}}$ can be inferred from Fig. 5, leading to

$${f_{TK}} = {f_{mcK}} \cdot \frac{{{L_{2K}}}}{{{L_{1K}} - {f_{mcK}}}}$$

In addition, for the relay system, the object and image distances are ${f_{mcK}} - {L_{1K}}$ and ${L_{2K}}$, respectively. Similar to the GMMS system, the power of the relay lens ${f_{rK}}$ can be expressed as

$${f_{rK}} = \frac{{({{f_{mcK}} - {L_{1K}}} ){L_{2K}}}}{{{f_{mcK}} - {L_{1K}} - {L_{2K}}}}$$

To unify the variables, we define ${L_{1K}} = {f_{mcK}} \cdot {m_{1K}}$ and ${L_{2K}} = {f_{mcK}} \cdot {m_{2K}}$. Substituting these relationships into Eqs. (18) and (19) leads to

$${f_{TK}} = {f_{mcK}} \cdot \frac{{{m_{2K}}}}{{{m_{1K}} - 1}}$$
$${f_{rK}} = \frac{{{f_{mcK}}({1 - {m_{1K}}} ){m_{2K}}}}{{1 - {m_{1K}} - {m_{2K}}}}$$

Following the approach demonstrated for the front radius of the flattener deduced in Section 2.2, the front radius of the field flattener of the KMMS can be derived as

$${R_{FlatK}} ={-} \frac{{{f_{mcK}}({{m_{1K}} - 1} ){m_{2K}}({n - 1} )}}{{ - 1 + {m_{1K}} + {m_{2K}} - {m_{2K}}n + {m_{1K}}{m_{2K}}n}}$$

It can be seen that the expressions for the parameters of the GMMS and KMMS systems tend to be consistent, which is convenient for subsequent analysis. Despite their similarities, however, the two systems cannot be considered as identical; their differences and relationship are discussed in the next section.

2.4. Correlation between the GMMS and KMMS systems

This section discusses the relationship between the two MMS models via the establishment of two system models, which considers the relationship between the MMS system parameters when they are equipped with the same optical characteristics, such as the F number and focal length. It can be seen from Sections 2.1–2.3 that the system construction parameters are mainly ${L_T}$, ${f_{mc}}$, ${m_1}$, and ${m_2}$, along with the aperture diameter r and half-image height h.

To obtain the same optical performance, the same focal length and F number are generally used, which can be expressed as

$${f_{mcK}} \cdot \frac{{{m_{2K}}}}{{{m_{1K}} - 1}} = {f_{mcG}} \cdot \frac{{{m_{2G}}}}{{1 - {m_{1G}}}}$$
and
$$\; \frac{{{m_{2K}} \cdot {f_{mcK}}}}{{2{r_K}}} = \frac{{{m_{2G}} \cdot {f_{mcG}}}}{{2{r_G}}}$$

In addition, to make a suitable comparison, the dimensions of the MC lens should be maintained. According to Eq. (8), we obtain

$${L_{TK}} - {f_{mcK}} = {L_{TG}} - {f_{mcG}}$$
The subimager should also try to maintain an identical aperture, which can be expressed as
$${r_K} = {r_G}$$

Nevertheless, it is difficult to maintain identical subimager focal lengths in both MMS systems. For the GMMS system, the object and image points of the subimager are on the same side, whereas for the KMMS system, the object and image points are located on opposite sides of the subimager. Thus, the bending power of the GMMS subimager cannot exceed the lens power of the KMMS subimager. In this study, the focal lengths of the subimagers of two types were not required to be identical.

The positions of the subimagers should follow the arrangement restriction. To illustrate the position of the relay lens more effectively, we define ${m_G} = {m_{1G}} \cdot {f_{mcG}}/({{L_{TG}} - {f_{mcG}}} )$ and ${m_K} = {m_{2K}}/({{m_{1K}} - 1} )$. The former expression represents the ratio of ${L_{1G}}$ to the MC radius, where a value larger than 1 guarantees that the relay lens cannot collide with the MC lens. Comparatively, when ${m_{1K}}$ is larger than 1, we can place the relay lens further behind the intermediate image plane. ${m_K}$ represents the magnification of the intermediate image by the subimager.

Based on Eqs. (23)–(26) and the expressions for mK and mG, we can determine one MMS parameter using another predefined MMS system. For example, the GMMS parameters can be derived from the KMMS system:

$${f_{mcG}} = \frac{{({{f_{mcK}} - {L_{TK}}} ){m_G}}}{{{m_{1K}} - 2}},$$
$${L_{TG}} = \frac{{({{L_{TK}} - {f_{mcK}}} )({{m_{1K}} - 2 - {m_G}} )}}{{{m_{1K}} - 2}},$$
$${m_{1G}} = 2 - {m_{1K}},$$
$${m_{2G}} = \frac{{{f_{mcK}}({{m_{1K}} - 2} ){m_{2K}}}}{{({{f_{mcK}} - {L_{TK}}} ){m_G}}}$$

There are two cases where the FOV is equivalent for MMS systems. In the first case, both types of subimagers have the same image height. Here, we consider that the KMMS subimager works at a negative magnification, ${h_k} ={-} {h_G}$.

The second case considers the entire MMS system, as depicted in Fig. 6, for which the arrangement of the array lens is limited by the feasible spatial dimensions. In extreme cases, the subimager is located adjacently. At this point, the angle $\alpha $ of the lens edge with respect to the center point O corresponds to the individual half-FOV (HFOV) of each subimager. The individual HFOVs for the respective systems can be expressed as

$$HFO{V_G} = \frac{{{r_G}}}{{{L_{1G}}}}$$
and
$$\; HFO{V_K} = \frac{{{r_K}}}{{{L_{1K}}}}$$

 figure: Fig. 6.

Fig. 6. Schematic of the (a) GMMS and (b) KMMS systems and the field domains of their respective subimagers.

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The image height h of the MMS system is related to the FOV of each subimager through $h = {f_T}HFOV$ when the focal lengths of both MMS systems are equal. Using Eqs. (27), (29), (31), and (32), we can obtain the height relationship between the GMMS and KMMS systems when the array lenses are placed adjacent to each other.

$$\frac{{{h_K}}}{{{h_G}}} = \frac{{({{f_{mcK}} - {L_{TK}}} ){m_G}}}{{{f_{mcK}}{m_{1K}}}}$$
Equation (33) shows the correlation between $\; {h_k}$ and $\; {h_G}$ when considering the arrangement factor of the entire MMS system. Other than the equivalent focal length, F number, and MC lens dimensions, the corresponding subimager image heights of the GMMS and KMMS systems are not identical and obey a certain ratio relationship.

3. Aberration analysis

In the previous discussion, we provided an analysis of the construction of the MMS system using the MC lens, subimager, and field flattener. That is, by knowing the specific parameters of the system, an analytical model of the MMS system could be built. For convenience, the number of variables within the system was restricted. Based on the approaches outlined in the previous sections, we analyzed the aberrations associated with the KMMS and GMMS systems. The aberrations associated with the different system components can be expressed as follows:

$$\begin{aligned}W\left( {H,\rho ,\theta } \right) = &\underbrace{{{W_{040}}{\rho ^4}}}_{{SPHA}} + \underbrace{{{W_{^131}}H{\rho ^3}cos\theta }}_{{COMA}} + \mathop {\mathop {\underbrace{{{W_{222}}{H^2}{\rho ^2}co{s^2}\theta }}_{{ASTI}}}\limits_{} }\limits_{} \\ &+ \mathop {\mathop {\underbrace{{{W_{220}}{H^2}{\rho ^2}}}_{{FCUR}}}\limits_{} }\limits_{} + \mathop {\mathop {\underbrace{{{W_{311}}{H^3}\rho cos\theta }}_{{DIST}}}\limits_{} }\limits_{} , \end{aligned}$$
where SPHA, COMA, ASTI, FCUR, and DIST represent the spherical aberration, coma, astigmatism, field curvature, and distortion, respectively. In the following section, we provide an analysis of the fourth-order aberration of the MMS system based on the analytical model constructed.

The aberrations can be classified into monochromatic and chromatic aberrations. The spherical aberration of the MMS system has been discussed previously [12], where the authors refined the structure of the system specifically to control its spherical aberration and spherochromatism. This article mainly focuses on the monochromatic off-axis aberrations of the MMS system. Typically, the off-axis aberration characteristics are better explained by core variables that can represent the system relatively accurately. Meanwhile, the Petzval field curvature is determined via the combined influences of astigmatism and field curvature [11]. As the correction of the Petzval field curvature was addressed in Section 2, there is no need to discuss the field curvature. Consequently, our discussion focuses primarily on the coma, astigmatism, and distortion associated with the MMS system. The coma, astigmatism, and distortion of each lens face can be expressed respectively as

$${W_{131}} = {a_s} \cdot {{({S^{\prime}/{L_{\textrm{exp}}}} )}^4} \cdot 4b{h_{\textrm{max}}}r_{\textrm{max}}^3,$$
$${W_{222}} = {a_s} \cdot {{({S^{\prime}/{L_{\textrm{exp}}}} )}^4} \cdot 4{b^2}h_{\textrm{max}}^2r_{\textrm{max}}^2,$$
and
$${W_{311}} = {a_s} \cdot {{({S^{\prime}/{L_{\textrm{exp}}}} )}^4} \cdot 4{b^3}h_{\textrm{max}}^3{r_{\textrm{max}}} + \frac{{n^{\prime}b({n - n^{\prime}} )}}{{2nR \cdot L_{\textrm{exp}}^2}}.$$

In Eqs. (35)–(37), $S^{\prime}$ represents the image distance, ${L_{\textrm{exp}}}$ represents the distance from the exit pupil to the image plane, and b denotes the ratio of the distance between the exit pupil and surface radius center point to the distance between the surface radius center point and image plane. n and $n^{\prime}$ represent the refractive indices before and after the interface surface, respectively, and R is the radius of the surface. The quantity ${a_S}$ can be expressed as

$${a_S} ={-} \frac{{n^{\prime}({n^{\prime} - n} )}}{{8{n^2}}}\left[ {{{\left( {\frac{1}{R} - \frac{1}{{S^{\prime}}}} \right)}^2} \cdot \left( {\frac{{n^{\prime}}}{R} - \frac{{n^{\prime} + n}}{{S^{\prime}}}} \right)} \right]$$

The image quality of a system is affected mainly by the aberration balance. The aberrations generated by one lens group are compensated by other lens groups. If the aberrations of each part of the system are not large, the overall aberration is relatively easy to balance, which ultimately benefits the implementation of each lens group in the system. The principal components of the MMS system are the MC, relay lens, and final field flattener. According to the discussion in Section 2, the parameters of the MC lens and field flattener are predetermined, whereas the design of the relay lens offers far greater flexibility. If the aberrations associated with lenses other than those in the relay lens group are small, then the pressure of the relay lens needs to compensate for the aberration, contributing to a concise relay system assembly [14]. As the number of relay lens groups contained in the MMS system is large [15], a concise relay lens greatly reduces the difficulty of mounting and adjusting the entire system.

3.1 Building of MMS aberration model

To evaluate the aberration rigorously, the aberration of the MMS needs to be deduced. The MC aberration parameter can be inferred from Fig. 7, where ${S_i}$ and $S_i^{\prime}$ denotes the objective and image distance of surface i, ${L_{\textrm{expi}}}$ represents the distance between the exit pupil position and image plane according to surface i, and ${n_i}$ and $n_i^{\prime}$ represent the refractive indices before and after surface i, respectively. The aperture stop is located just outside the MC lens. Then, the exit pupil position of surface 2, which is labeled as ExP2 in Fig. 7, can be firstly ascertained, followed by the position of ExP1 by substituting ExP2 into Eq. (2). In contrast, $S_1^{\prime}$ can be solved from the first surface because the object distance is infinity. ${h_2}\; $ denotes the image height of the second surface. Irrespective of whether the system is GMMS- or KMMS-type, ${h_2}\; $ can be expressed as h2 = (fmc-L1)/L2 according to the discussion in Section 2. Furthermore, h1 can be derived using h2 based on the object–image relationship on surface 2, as it can be inferred from Fig. 7 that ${n_1}$ and $n_2^{\prime}$ are both 1. $n_1^{\prime}$ and ${n_2}$ denote the refractive indices of the MC lens materials, which can be derived from Eq. (7). After all of the parameters have been determined, the aberration of the MC lens can be derived by substituting these expressions into Eqs. (35)–(38).

 figure: Fig. 7.

Fig. 7. Schematic of the simplified MC lens.

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The expression for the field flattener aberration can be inferred from Fig. 8. ${L_{\textrm{exp}0}}$ represents the distance between the aperture stop and image plane. Following the same method as for the MC lens, ${L_{\textrm{exp}1}}$ and ${L_{\textrm{exp}2}}$ can be obtained. In particular, as the field flattener is located extremely close to the imaging plane, the distance of each surface of the flat field lens are all close to 0; therefore, the image height h remains almost unchanged relative to each surface, besides, the field flattener only generate field curvature and distortion [11]. Consequently, we derived the aberration corresponding to the field flattener model. The deduction and simplification processes are omitted here.

 figure: Fig. 8.

Fig. 8. Schematic of the field-flattening lens placed near the image plane.

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Interestingly, after simplification, both the GMMS and KMMS system aberrations can be expressed via the general formulae in Table 1. Therefore, the subscripts that distinguish KMMS and GMMS can be omitted.

Tables Icon

Table 1. Aberrations within the MMS system

The full forms of coefficients A, B, C, and D appearing in the expressions provided in Table 1 are listed in Table 2.

Tables Icon

Table 2. Full expressions of coefficients A, B, C, and D.

Based on the aberration formula, the MMS systems can be compared. The parameter ${R_{Flat}}$ in Table 1 can be obtained using Eqs. (17) and (22), respectively.

3.2. Analysis and comparison of the two MMS-type systems

In the calculation process, the value of n was defined as 1.5. Based on the discussion in Section 2, the parameters of any KMMS or GMMS system can be determined equally using the other system parameters that are known in advance. Based on the aberration formula, we firstly analyzed the effects of mK and mG on the corresponding MMS systems. Next, the responses of the two MMS systems to each off-axis aberration were compared.

Figures 9 and 10 present the variations in the coma, astigmatism, and distortion for the GMMS and KMMS systems according to mG and mK, respectively. In the following discussion, all aberration values are counted as multiples of the central wavelength (0.586 µm). The purple line represents the distortion by considering the field flattener and MC objective, whereas the green line indicates the distortion created by the MC objective itself. As discussed in Section 2, if mG exceeds 1 then the relay lens is kept from colliding with the MC objective. As shown in Fig. 9, a relatively low value of mG corresponds to a decrease in the aberration of the GMMS system. However, this situation will reduce the distance between the MC and relay lenses. In practice, the thickness of the relay lens must be considered as it occupies the axial space. In addition, an extremely short dimension is unsuitable for the mechanical structure. Therefore, compromises are required when considering the value of mG.

 figure: Fig. 9.

Fig. 9. Aberration in the GMMS system as a function of mG.

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 figure: Fig. 10.

Fig. 10. Aberration in the KMMS system as a function of mK.

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Regarding the KMMS system, an ${m_{1K}}$ value exceeding 1 ensures the proper position of the relay lens because the relay lens should always lay behind the intermediate image plane. ${m_K}$ represents the image magnification ratio of the entire system relative to the MC objective. The curves in Fig. 10 show that both small and large values of ${m_K}$ benefit the aberration correction for coma and astigmatism. However, a larger value of ${m_K}$ results in the magnification of the spot imaged by the MC objective lens. Further, the entire system may suffer severe distortion in this case. In contrast, a small value of ${m_K}$ can alleviate all the off-axis aberration to a certain extent. A side effect is that the relay lens requires strong ray bending capability, which may introduce aberration by the relay lens itself. Therefore, a relatively low value of ${m_K}$ may be preferable. A larger value of ${m_K}$ can be chosen only if the MC objective has been designed perfectly and there is no distortion correction requirement. After the comparison, the rational basic parameters of the MMS systems are listed in Table 3, and their relationships are provided in Eqs. (27)–(33).

Tables Icon

Table 3. Parameters of the GMMS and KMMS systems.

In the following comparison, ${m_G}$ and ${m_K}$ are set equal to 1.5 and 0.75, respectively. The relationship between the parameters of the two types of MMS system follows the discussion in Section 2. We used the variations in the image height h and the aperture radius r to observe the aberration induced by the MC objective and field flattener. According to Eq. (33), ${h_K}$ and ${h_G}$ satisfy specific ratio relationships and are not identical. After choosing ${h_G}$ as the variation axis, the aberration of the KMMS system shown in Figs. 1113 varies according to ${h_G}$ (following the hK:hG ratio).

 figure: Fig. 11.

Fig. 11. Comparison of the W131 aberration for the GMMS and KMMS systems.

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 figure: Fig. 12.

Fig. 12. Comparison of the W222 aberration for the GMMS and KMMS systems.

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 figure: Fig. 13.

Fig. 13. Comparison of the W311 aberration for the GMMS and KMMS systems.

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Figures 11 and 12 depict the coma and astigmatism, respectively, of the two MMS systems. It is observed that the coma increases dramatically as r increases, whereas maximum astigmatism occurs when both r and h are large. The GMMS system shows marginally better performance than the KMMS system in terms of both coma and astigmatism.

Figure 13 illustrates the distortion of the MMS systems. It is observed that the distortion increases dramatically with increasing h. Conversely, in this case, the KMMS system outperforms the GMMS system.

From Figs. 1113, it can be observed that, in the KMMS system, the relay lens requires greater corrective modifications to alleviate the effects of coma and astigmatism, whereas in the GMMS system, distortion is the most serious issue affecting the relay lens. However, if the distortion in a system is not considered and this issue is left to be addressed during image processing, the GMMS form is the best choice. Meanwhile, the practical requirements and judgments of the designers are also significant factors. In contrast, if the aberration contributions from the MC objective and field flattener are low, then there are few compensation requirements for the relay lens, facilitating system simplification.

In addition, we can infer from Figs. 1113 that large values of both h and r may increase the aberration. Therefore, the subimager parameters must be chosen with care. The aberration model of the MMS systems can also be used to evaluate each system type separately during the design process.

One last factor that must be considered is that, in practice, KMMS system design may offer more MC form choices because, in this case, there are no restrictions on the space between the objective lens and MC objective image plane. Therefore, concentric lenses can be positioned far behind the center point to correct the aberrations [10]. This situation is unsuitable for a GMMS system.

In summary, Fig. 14 presents a flowchart of the modeling process and evaluation of the MMS systems.

 figure: Fig. 14.

Fig. 14. Flowchart of the modeling process and evaluation of the MMS systems.

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The following section presents design examples of the two MMS systems. The parameters of the two systems are listed in Table 4, with initial parameters provided in Table 3. The materials used are labeled in Fig. 15.

 figure: Fig. 15.

Fig. 15. Layouts of the two MMS systems and their corresponding subimager views.

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Tables Icon

Table 4. Parameters of example GMMS and KMMS systems.

With respect to the final design of the subimager, the form of a cemented lens and field flattener was adopted in both the GMMS and KMMS systems. The aperture and HFOV were chosen considering the aberration issue to alleviate the subimager difficulties, and the detectors of the GMMS and KMMS systems were accordingly chosen to be Sony IMX274 (8.51 megapixel) and Onsemi ARO543 (5.03 megapixel), respectively.

For aberration correction, conic surfaces were adopted on both the cemented lens and subimager of the KMMS system. During the design process, the distance and focal length operands were used to control the system, guaranteeing a stable form of MMS from start to end. The imaging quality of each MMS system is close to the diffraction limit. The imaging quality of the two type systems are shown in Fig. 16 [16]. The MTFs of both system are greater than 0.3 @ 150 cycles/mm, the distortions of both systems are within 1%, and the average RMS radius of the GMMS and KMMS systems are 1.732 µm and 2.377 µm, respectively. The radial diameters of the GMMS and KMMS systems are 92 mm and 132 mm, respectively, the performance of the GMMS system is slightly better than that of the KMMS system. It is observed that, when equipped with equivalent parameters, the GMMS system achieves the same performance while requiring fewer variables and smaller dimensions than the KMMS system. In contrast, the MC lens in the KMMS system affords greater flexibility as there is no need to consider the space reserved behind the last surface.

 figure: Fig. 16.

Fig. 16. Modulation transfer function of the (a) GMMS and (b) KMMS systems.

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4. Conclusion

This report proposes a novel analytical model that is suitable for Galilean-type and Keplerian-type MMS. This model is more accurate than the paraxial form and maintains the key point of the MMS system, avoiding the laborious analysis of different MC lens forms. The correlation between the design parameters of GMMS and KMMS systems was also discussed under the same working requirements.

An MMS system is the combination of an MC unit and array lens. When a single MMS subimager works in a certain FOV, the absolute symmetry of each FOV working at the MC lens is no longer particularly effective. In this event, off-axis aberration originates from the MC lens and subsequently, affects the subimager.

Based on the analytical model, the off-axis monochromatic aberration expressions for the MMS systems were derived, providing a convenient qualitative and quantitative aberration analysis of the MMS systems. The simplified aberration expression describes the MMS system efficiently, ensuring that appropriate system parameters can be chosen during the design process. Moreover, the GMMS and KMMS models were compared based on the above work, leading to the conclusion that the GMMS-type system provides better off-axis aberration performance than the KMMS-type system.

Further refinement of the MMS analysis model is possible, including incorporation of the combined effects of machining/ assembly tolerances. With the continuous demand for large FOV optics, MMS system designs will become increasingly achievable with a compact scale. The results presented in this study will serve as a useful platform to guide future research on MMS system design.

Funding

National Natural Science Foundation of China (61705018, 61805025); Jilin Scientific and Technological Development Program (20200401055GX).

Disclosures

The authors declare no conflicts of interest.

References

1. J. J. Kumler and L. M. Bauer, “Fish-eye lens designs and their relative performance,” Proc. SPIE 4093, 360–369 (2000). [CrossRef]  

2. S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015). [CrossRef]  

3. G. M. Schuster, D. G. Dansereau, G. Wetzstein, and J. E. Ford, “Panoramic single-aperture multi-sensor light field camera,” Opt. Express 27(26), 37257–37273 (2019). [CrossRef]  

4. I. Stamenov, A. Arianpour, S. J. Olivas, I. P. Agurok, A. R. Johnson, R. A. Stack, R. L. Morrison, and J. E. Ford, “Panoramic monocentric imaging using fiber-coupled focal planes,” Opt. Express 22(26), 31708–31721 (2014). [CrossRef]  

5. D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17(13), 10659–10674 (2009). [CrossRef]  

6. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. 28(23), 4996–4998 (1989). [CrossRef]  

7. D. L. Marks, E. J. Tremblay, J. E. Ford, and D. J. Brady, “Microcamera aperture scale in monocentric gigapixel cameras,” Appl. Opt. 50(30), 5824–5833 (2011). [CrossRef]  

8. I. P. Agurok and J. E. Ford, “Angle-invariant imaging using a total internal reflection virtual aperture,” Appl. Opt. 55(20), 5345–5352 (2016). [CrossRef]  

9. E. J. Tremblay, D. L. Marks, D. J. Brady, and J. E. Ford, “Design and scaling of monocentric multiscale imagers,” Appl. Opt. 51(20), 4691–4702 (2012). [CrossRef]  

10. I. Stamenov, I. Agurok, and J. E. Ford, “Optimization of high-performance monocentric lenses,” Appl. Opt. 52(34), 8287–8304 (2013). [CrossRef]  

11. V. N. Mahajan, Optical Imaging and Aberrations: Ray Geometrical Optics (SPIE Press, 1998).

12. I. Stamenov, I. P. Agurok, and J. E. Ford, “Optimization of two-glass monocentric lenses for compact panoramic imagers: general aberration analysis and specific designs,” Appl. Opt. 51(31), 7648–7661 (2012). [CrossRef]  

13. W. Pang and D. J. Brady, “Galilean monocentric multiscale optical systems,” Opt. Express 25(17), 20332–20339 (2017). [CrossRef]  

14. W. Smith, Modern Lens Design (McGraw-Hill, 1992).

15. H. S. Son, D. L. Marks, J. Hahn, J. Kim, and D. J. Brady, “Design of a spherical focal surface using close-packed relay optics,” Opt. Express 19(17), 16132–16138 (2011). [CrossRef]  

16. ZEMAX lens design program manual, www.zemax.com.

References

  • View by:
  • |
  • |
  • |

  1. J. J. Kumler and L. M. Bauer, “Fish-eye lens designs and their relative performance,” Proc. SPIE 4093, 360–369 (2000).
    [Crossref]
  2. S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
    [Crossref]
  3. G. M. Schuster, D. G. Dansereau, G. Wetzstein, and J. E. Ford, “Panoramic single-aperture multi-sensor light field camera,” Opt. Express 27(26), 37257–37273 (2019).
    [Crossref]
  4. I. Stamenov, A. Arianpour, S. J. Olivas, I. P. Agurok, A. R. Johnson, R. A. Stack, R. L. Morrison, and J. E. Ford, “Panoramic monocentric imaging using fiber-coupled focal planes,” Opt. Express 22(26), 31708–31721 (2014).
    [Crossref]
  5. D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17(13), 10659–10674 (2009).
    [Crossref]
  6. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. 28(23), 4996–4998 (1989).
    [Crossref]
  7. D. L. Marks, E. J. Tremblay, J. E. Ford, and D. J. Brady, “Microcamera aperture scale in monocentric gigapixel cameras,” Appl. Opt. 50(30), 5824–5833 (2011).
    [Crossref]
  8. I. P. Agurok and J. E. Ford, “Angle-invariant imaging using a total internal reflection virtual aperture,” Appl. Opt. 55(20), 5345–5352 (2016).
    [Crossref]
  9. E. J. Tremblay, D. L. Marks, D. J. Brady, and J. E. Ford, “Design and scaling of monocentric multiscale imagers,” Appl. Opt. 51(20), 4691–4702 (2012).
    [Crossref]
  10. I. Stamenov, I. Agurok, and J. E. Ford, “Optimization of high-performance monocentric lenses,” Appl. Opt. 52(34), 8287–8304 (2013).
    [Crossref]
  11. V. N. Mahajan, Optical Imaging and Aberrations: Ray Geometrical Optics (SPIE Press, 1998).
  12. I. Stamenov, I. P. Agurok, and J. E. Ford, “Optimization of two-glass monocentric lenses for compact panoramic imagers: general aberration analysis and specific designs,” Appl. Opt. 51(31), 7648–7661 (2012).
    [Crossref]
  13. W. Pang and D. J. Brady, “Galilean monocentric multiscale optical systems,” Opt. Express 25(17), 20332–20339 (2017).
    [Crossref]
  14. W. Smith, Modern Lens Design (McGraw-Hill, 1992).
  15. H. S. Son, D. L. Marks, J. Hahn, J. Kim, and D. J. Brady, “Design of a spherical focal surface using close-packed relay optics,” Opt. Express 19(17), 16132–16138 (2011).
    [Crossref]
  16. ZEMAX lens design program manual, www.zemax.com .

2019 (1)

2017 (1)

2016 (1)

2015 (1)

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

2014 (1)

2013 (1)

2012 (2)

2011 (2)

2009 (1)

2000 (1)

J. J. Kumler and L. M. Bauer, “Fish-eye lens designs and their relative performance,” Proc. SPIE 4093, 360–369 (2000).
[Crossref]

1989 (1)

Agurok, I.

Agurok, I. P.

Arianpour, A.

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

I. Stamenov, A. Arianpour, S. J. Olivas, I. P. Agurok, A. R. Johnson, R. A. Stack, R. L. Morrison, and J. E. Ford, “Panoramic monocentric imaging using fiber-coupled focal planes,” Opt. Express 22(26), 31708–31721 (2014).
[Crossref]

Bauer, L. M.

J. J. Kumler and L. M. Bauer, “Fish-eye lens designs and their relative performance,” Proc. SPIE 4093, 360–369 (2000).
[Crossref]

Brady, D. J.

Dansereau, D. G.

Ford, J. E.

G. M. Schuster, D. G. Dansereau, G. Wetzstein, and J. E. Ford, “Panoramic single-aperture multi-sensor light field camera,” Opt. Express 27(26), 37257–37273 (2019).
[Crossref]

I. P. Agurok and J. E. Ford, “Angle-invariant imaging using a total internal reflection virtual aperture,” Appl. Opt. 55(20), 5345–5352 (2016).
[Crossref]

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

I. Stamenov, A. Arianpour, S. J. Olivas, I. P. Agurok, A. R. Johnson, R. A. Stack, R. L. Morrison, and J. E. Ford, “Panoramic monocentric imaging using fiber-coupled focal planes,” Opt. Express 22(26), 31708–31721 (2014).
[Crossref]

I. Stamenov, I. Agurok, and J. E. Ford, “Optimization of high-performance monocentric lenses,” Appl. Opt. 52(34), 8287–8304 (2013).
[Crossref]

I. Stamenov, I. P. Agurok, and J. E. Ford, “Optimization of two-glass monocentric lenses for compact panoramic imagers: general aberration analysis and specific designs,” Appl. Opt. 51(31), 7648–7661 (2012).
[Crossref]

E. J. Tremblay, D. L. Marks, D. J. Brady, and J. E. Ford, “Design and scaling of monocentric multiscale imagers,” Appl. Opt. 51(20), 4691–4702 (2012).
[Crossref]

D. L. Marks, E. J. Tremblay, J. E. Ford, and D. J. Brady, “Microcamera aperture scale in monocentric gigapixel cameras,” Appl. Opt. 50(30), 5824–5833 (2011).
[Crossref]

Hagen, N.

Hahn, J.

Johnson, A. R.

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

I. Stamenov, A. Arianpour, S. J. Olivas, I. P. Agurok, A. R. Johnson, R. A. Stack, R. L. Morrison, and J. E. Ford, “Panoramic monocentric imaging using fiber-coupled focal planes,” Opt. Express 22(26), 31708–31721 (2014).
[Crossref]

Karbasi, S.

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

Kim, J.

Kumler, J. J.

J. J. Kumler and L. M. Bauer, “Fish-eye lens designs and their relative performance,” Proc. SPIE 4093, 360–369 (2000).
[Crossref]

Lareau, C.

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

Lohmann, A. W.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations: Ray Geometrical Optics (SPIE Press, 1998).

Marks, D. L.

Morrison, R.

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

Morrison, R. L.

Motamedi, N.

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

Olivas, S. J.

Pang, W.

Schuster, G. M.

Smith, W.

W. Smith, Modern Lens Design (McGraw-Hill, 1992).

Son, H. S.

Stack, R. A.

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

I. Stamenov, A. Arianpour, S. J. Olivas, I. P. Agurok, A. R. Johnson, R. A. Stack, R. L. Morrison, and J. E. Ford, “Panoramic monocentric imaging using fiber-coupled focal planes,” Opt. Express 22(26), 31708–31721 (2014).
[Crossref]

Stamenov, I.

Tennill, R.

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

Tremblay, E. J.

Wetzstein, G.

Appl. Opt. (6)

Opt. Express (5)

Proc. SPIE (2)

J. J. Kumler and L. M. Bauer, “Fish-eye lens designs and their relative performance,” Proc. SPIE 4093, 360–369 (2000).
[Crossref]

S. Karbasi, I. Stamenov, N. Motamedi, A. Arianpour, A. R. Johnson, R. A. Stack, C. Lareau, R. Tennill, R. Morrison, I. P. Agurok, and J. E. Ford, “Curved fiber bundles for monocentric lens imaging,” Proc. SPIE 9579, 95790G (2015).
[Crossref]

Other (3)

V. N. Mahajan, Optical Imaging and Aberrations: Ray Geometrical Optics (SPIE Press, 1998).

ZEMAX lens design program manual, www.zemax.com .

W. Smith, Modern Lens Design (McGraw-Hill, 1992).

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Figures (16)

Fig. 1.
Fig. 1. Paths of different FOV rays transmitted by an MMS system.
Fig. 2.
Fig. 2. Diagram illustrating the design of a classical MC lens.
Fig. 3.
Fig. 3. Diagram of a simplified MC lens.
Fig. 4.
Fig. 4. Analytical model of the GMMS system.
Fig. 5.
Fig. 5. Analytical model of the KMMS system.
Fig. 6.
Fig. 6. Schematic of the (a) GMMS and (b) KMMS systems and the field domains of their respective subimagers.
Fig. 7.
Fig. 7. Schematic of the simplified MC lens.
Fig. 8.
Fig. 8. Schematic of the field-flattening lens placed near the image plane.
Fig. 9.
Fig. 9. Aberration in the GMMS system as a function of mG.
Fig. 10.
Fig. 10. Aberration in the KMMS system as a function of mK.
Fig. 11.
Fig. 11. Comparison of the W131 aberration for the GMMS and KMMS systems.
Fig. 12.
Fig. 12. Comparison of the W222 aberration for the GMMS and KMMS systems.
Fig. 13.
Fig. 13. Comparison of the W311 aberration for the GMMS and KMMS systems.
Fig. 14.
Fig. 14. Flowchart of the modeling process and evaluation of the MMS systems.
Fig. 15.
Fig. 15. Layouts of the two MMS systems and their corresponding subimager views.
Fig. 16.
Fig. 16. Modulation transfer function of the (a) GMMS and (b) KMMS systems.

Tables (4)

Tables Icon

Table 1. Aberrations within the MMS system

Tables Icon

Table 2. Full expressions of coefficients A, B, C, and D.

Tables Icon

Table 3. Parameters of the GMMS and KMMS systems.

Tables Icon

Table 4. Parameters of example GMMS and KMMS systems.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

n m c S 1 1 S 1 = n m c 1 R 1
1 S 2 n m c S 2 = 1 n m c R 2 ,
S 2 = 2 n m c 2 ( n m c 1 ) R m c
L T 2 R m c = 2 n m c 2 ( n m c 1 ) R m c
1 f = ( n 1 ) ( 1 R 1 1 R 2 ) + t ( n 1 ) 2 n R 1 R 2
f m c = n m c R m c 2 ( n m c 1 )
n m c = 2 f m c 3 f m c L T
R m c = L T f m c
f T G = f m c G L 2 G f m c G L 1 G
f r G = ( f m c G L 1 G ) L 2 G f m c G L 1 G L 2 G
f T G = f m c G m 2 G 1 m 1 G
f r G = f m c G ( 1 m 1 G ) m 2 G 1 m 1 G m 2 G
1 n n r P n = i = 1 n 1 R i ( 1 n i 1 n i 1 )
1 r P n = 2 R m c ( 1 n m c 1 ) + i = 1 N ( 1 R i 1 ( 1 n 1 ) + 1 R i 2 ( 1 1 n ) ) + 1 R F l a t t e n ( 1 n 1 )
i = 1 N ( 1 R i 1 ( 1 n 1 ) + 1 R i 2 ( 1 1 n ) ) = 1 n i = 1 N 1 f r i ,
R F l a t t e n = ( 1 n ) f r G n m c R m c 2 f r G n + 2 f r G n n m c + n m c R m c .
R F l a t G = f m c G ( m 1 G 1 ) m 2 G ( n 1 ) 1 + m 1 G + m 2 G m 2 G n + m 1 G m 2 G n
f T K = f m c K L 2 K L 1 K f m c K
f r K = ( f m c K L 1 K ) L 2 K f m c K L 1 K L 2 K
f T K = f m c K m 2 K m 1 K 1
f r K = f m c K ( 1 m 1 K ) m 2 K 1 m 1 K m 2 K
R F l a t K = f m c K ( m 1 K 1 ) m 2 K ( n 1 ) 1 + m 1 K + m 2 K m 2 K n + m 1 K m 2 K n
f m c K m 2 K m 1 K 1 = f m c G m 2 G 1 m 1 G
m 2 K f m c K 2 r K = m 2 G f m c G 2 r G
L T K f m c K = L T G f m c G
r K = r G
f m c G = ( f m c K L T K ) m G m 1 K 2 ,
L T G = ( L T K f m c K ) ( m 1 K 2 m G ) m 1 K 2 ,
m 1 G = 2 m 1 K ,
m 2 G = f m c K ( m 1 K 2 ) m 2 K ( f m c K L T K ) m G
H F O V G = r G L 1 G
H F O V K = r K L 1 K
h K h G = ( f m c K L T K ) m G f m c K m 1 K
W ( H , ρ , θ ) = W 040 ρ 4 S P H A + W 1 31 H ρ 3 c o s θ C O M A + W 222 H 2 ρ 2 c o s 2 θ A S T I + W 220 H 2 ρ 2 F C U R + W 311 H 3 ρ c o s θ D I S T ,
W 131 = a s ( S / L exp ) 4 4 b h max r max 3 ,
W 222 = a s ( S / L exp ) 4 4 b 2 h max 2 r max 2 ,
W 311 = a s ( S / L exp ) 4 4 b 3 h max 3 r max + n b ( n n ) 2 n R L exp 2 .
a S = n ( n n ) 8 n 2 [ ( 1 R 1 S ) 2 ( n R n + n S ) ]

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