# Flowline Optical Simulation to Refractive/Reflective 3D Systems: Optical Path Length Correction

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## Abstract

**:**

**J**, is one of these techniques. The main advantage of the flowline method is its capability to visualize and estimate how radiant energy is transferred by the optical systems using the concepts of vector field theory, such as field line or flux tube, which overcomes traditional raytrace methods. The main objective this paper is to extend the flowline method to analyze and design real 3D concentration and illumination systems by the development of new simulation techniques. In this paper, analyzed real 3D refractive and reflective systems using the flowline vector potential method. A new constant term of optical path length is introduced, similar and comparable to the gauge invariant, which produces a correction to enable the agreement between raytrace- and flowline-based computations. This new optical simulation methodology provides traditional raytrace results, such as irradiance maps, but opens new perspectives to obtaining higher precision with lower computation time. It can also provide new information for the vector field maps of 3D refractive/reflective systems.

## 1. Introduction

**J**, was first introduced as a photometrical theory by Gershun [1] and later developed by Moon [2]. Such a concept applies vector field theory to the study of irradiance transfer, which is based on the definition of the geometrical flux vector

**J**produced by a Lambertian source at any point in the space P. The modulus of

**J**is the maximum irradiance value at point P; the direction of the vector is the direction of the maximum irradiance at P. The projection of the

**J**vector upon a plane directly corresponds to the irradiance on that plane. Over the last decades, vector flowline formalism has been applied to several problems in optics. In the seventh decade of last century, Winthrop [3] applied it to the study of propagation of structural information; at the end of that decade, Winston [4,5] applied it successfully to the study of concentrators, showing that reflective concentrators with the geometry of the field lines achieve the theoretical limit of concentration. Later, Welford and Winston [6] developed this method among others to create a new topic called nonimaging optics. Recently, flowline method has been used to design so-called hyperparabolic concentrators (HPC) [7], which extends the compound parabolic concentrator (CPC) to multiple reflections and approaches the thermodynamic limit of concentration. Another interesting result from the application of field theory concepts to optical design is that refractive elements with the geometry of orthogonal surfaces to the field lines produce ideal concentrators [8].

## 2. Theoretical Background and Flowline Methodology for 3D Optical Systems Analysis

**J**, in the context of applying the etendue conservation law for a loss-free optical system. The Cartesian components of the geometrical flux vector

**J**are

_{x}, dp

_{y}, and dp

_{z}are the direction cosines or, for more general considerations, the generalized optical momenta in phase space. The physical meaning of geometrical flux vector is that the J

_{z}component at any point in the space P, for instance, is proportional to the total flux per unit area (irradiance/illuminance) entering to a plane parallel to the XY plane at point P. The most important property of flowline method is that it represents a new way to construct concentrators with maximum concentration ratio, which is to place mirrors with the geometry of flowlines [6]. In an analogous way, following Gershun [1], the concept of light vector was introduced, and the magnitude of light vector equals the net radiant power per unit area at point P, and the direction of the vector is the direction of the flow of the radiant energy at that point. To analyze 3D optical systems, we used the concept of vector potential of this light vector, in way that the geometrical flux vector can be computed from vector potential

**A**[1] as

**dl**is an infinitesimal vector along the direction of the contour of the light source. B is the luminance/radiance of the source—it is a constant for Lambertian radiators and therefore we can get it out of the integral.

**J**vector indirectly from vector potential computation for different real 3D optical systems. We will replace r by the optical path length, L, in Equation (3) for systems with refractive elements. One of the potential advantages of the proposed flowline method is the computation of a contour integral, Equation (3), instead of the surface computation needed for raytrace, as it can improve computation efficiency. Another potential advantage of the flowline method is the possibility of calculating field lines, which provides information about the energy flow related with ideal optical designs. We studied optical systems with an optical axis coincident to the OZ axis, and we analyzed irradiance distribution at a detector plane perpendicular to the optical axis, and then we computed the J

_{z}component at the detector plane. We also compared the results obtained from the computation of Equations (2) and (3) of J

_{z}with raytrace computations.

#### Irradiance Computation from J Vector for Simple Free Space 3D Non Symmetric Systems

## 3. Irradiance Computation from J Vector for Asymmetric 3D Systems with Refractive Elements

**A**at the position of the detector by replacing the distance r in Equation (3) with optical path length (OPL), L. This introduced a new challenge in the computation of the contour integral of Equation (3). Before calculating the integral value at any point in the detector, such as point P, it is necessary to compute the optical path length between point P and all the points in the closed contour of the source C. In this way, the irradiance E at point P on the detector for the system of Figure 5 with Lambertian source can be computed from the contour source integral to Equation (3) with the log of OPL and the derivatives of Equation (4). It is possible to do so using Fermat’s principle. We developed some routines in Matlab based on Nelder–Mead minimum algorithm to compute the OPL for this simple optical system. Another option was to use the routines incorporated in the Raytracer package software, which includes the computation of OPL. Once we computed the OPL between one point in the detector and the closed contour of the source, it was then possible to compute the vector potential

**A**at any point in the detector from the integral of Equation (3). Finally, we obtained the irradiance pattern E at detector from

**A**following Equation (4). Figure 5 shows a sketch of the analyzed optical system. A Lambertian square source of 10 mm side was located at the origin, and a plano-convex lens was located at z = 300 mm (plane size of the lens). The convex surface of the lens had a radius R = 150 mm, the thickness of the lens at the axis was 30 mm, the refraction index was n = 1.5, and the lens had a square perimeter of 120 mm per side. Finally, there was also a square detector of 80 mm per side, located at z = 950 mm.

## 4. Irradiance Computation from J Vector for Asymmetric 3D Systems with Reflective Elements

## 5. Optical Path Length Correction for Standard Vector Potential Method

_{i}is the refractive index, s

_{i}is the distance of the ray along the optical media, and λ is a constant. This transformation of the optical path length is fully compatible with Fermat’s principle. All classical consequences of Fermat’s principle remain unaltered. We also proposed that the vector potential can be computed by the integral of Equation (3) by replacing the distance r with the proposed corrected optical path length L′.

**A**in Equation (2) is not unique. The gradient of a scalar function can be always added to

**A**with Equation (2) remaining unaltered, which is in agreement with gauge transformation.

## 6. Conclusions

**A**of flowline vector

**J.**We applied this analysis to non-axisymmetric refractive and reflective optical systems. As prescribed by the theoretical method, we replaced the standard distance r by classical optical path length L. With this replacement, we obtained interesting results, such as the precise location of irradiance maximum in the distribution. However, this method seems to need corrections to obtain agreement between raytrace simulations and the vector potential of

**J**vector computations. We then proposed a corrected version of the optical path length L′ by introducing a constant parameter, which was similar to the gauge invariant. This corrected version of the optical path length is compatible with the Fermat principle, and provided better agreement between the raytrace simulations and the vector potential computations. Nevertheless, the study of the physical meaning of this corrected optical path length L′ remains an open question.

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Irradiance map and central irradiance profile of square source of 160 × 160 mm

^{2}; detector of 200 × 200 mm

^{2}; z detector distance 20 mm. (

**a**)Flowline computation (

**b**) Raytrace computation

**Figure 3.**Irradiance map and central irradiance profile of rectangular source of 20 × 180 mm

^{2}; detector of 200 × 200 mm

^{2}; z detector distance 5 mm. (

**a**) Flowline computation (

**b**) Raytrace computation

**Figure 4.**Irradiance map and central irradiance profile of rectangular source of 20 × 180 mm

^{2}; detector of 200 × 200 mm

^{2}; z detector 50 mm. (

**a**)Flowline computation (

**b**) Raytrace computation

**Figure 6.**Irradiance pattern and central irradiance profile at the detector: (

**a**) computed by flowline vector potential method; (

**b**) computed by raytrace.

**Figure 8.**Normalized irradiance pattern of reflective system, detector at z = 250 mm: (

**a**) flowline computation; (

**b**) raytrace computation.

**Figure 9.**Central irradiance profile of distributions of Figure 8: (

**a**) flowline computation; (

**b**) raytrace computation.

**Figure 10.**Normalized irradiance pattern of reflective system, detector located at z = 550 mm: (

**a**) flowline computation; (

**b**) raytrace computation.

**Figure 11.**Central irradiance profile of distributions of Figure 10: (

**a**) flowline computation; (

**b**) raytrace computation.

**Figure 12.**Irradiance pattern and central irradiance profile at the detector for refractive optical system of Section 3: (

**a**) computed by standard flowline vector potential method; (

**b**) computed by optical path length (OPL) correction in flowline vector potential method; (

**c**) computed by raytrace.

**Figure 13.**Irradiance pattern and central irradiance profile at the detector for the reflective system, with the detector located at z = 250 mm: (

**a**) computed by standard flowline vector potential method; (

**b**) computed by OPL correction in flowline vector potential method; (

**c**) computed by raytrace.

**Figure 14.**Irradiance pattern and central irradiance profile at the detector for the reflective system with the detector located at z = 550 mm: (

**a**) computed by flowline vector potential method; (

**b**) computed by OPL correction in flowline vector potential method; (

**c**) computed by raytrace.

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## Share and Cite

**MDPI and ACS Style**

García-Botella, A.; Jiang, L.; Winston, R.
Flowline Optical Simulation to Refractive/Reflective 3D Systems: Optical Path Length Correction. *Photonics* **2019**, *6*, 101.
https://doi.org/10.3390/photonics6040101

**AMA Style**

García-Botella A, Jiang L, Winston R.
Flowline Optical Simulation to Refractive/Reflective 3D Systems: Optical Path Length Correction. *Photonics*. 2019; 6(4):101.
https://doi.org/10.3390/photonics6040101

**Chicago/Turabian Style**

García-Botella, Angel, Lun Jiang, and Roland Winston.
2019. "Flowline Optical Simulation to Refractive/Reflective 3D Systems: Optical Path Length Correction" *Photonics* 6, no. 4: 101.
https://doi.org/10.3390/photonics6040101