## Abstract

The simultaneous multiple surface (SMS) design method is extended to include design of diffractive optical surfaces besides refractive and reflective ones. This method involves the simultaneous and direct (no optimization) calculation of diffractive and refractive/reflective surfaces. Using the phase-shift properties of diffractive elements as an extra degree of freedom, two rays for each point on each diffractive surface are controlled. Representative diffractive systems designed by the SMS method are shown.

© 2016 Optical Society of America

## 1. Introduction

The simultaneous multiple surface (SMS) method was initially developed as a method to design nonimaging optics. Later, it was extended to Imaging Optics [1]. The SMS method involves the simultaneous calculation of J optical surfaces (refractive or reflective) using J one-parameter wavefronts, for which specific conditions are imposed. The relationship between degrees of freedom (J surfaces) and the number of wavefronts is not exactly one to one [2]. An optical system can control more than J wavefronts with J surfaces, provided the footprint of the rays of the wavefronts on the optical surfaces is not covering 100% of those optical surfaces [3]. This footprint can be controlled with the pupil definition. Here we only consider 100% footprint coverage. In typical SMS imaging applications, the wavefronts to couple in the design are spheres centered at J points of the object plane. The J design conditions are such that the wavefronts are perfectly focused at J image points [4].

Hybrid lenses, i.e., optical systems that combine both reflective and refractive with diffractive optical elements (DOE), bring attractive solutions and have been applied in IR and broad-band applications [5]. Recently, their application to non-imaging systems has been studied [6,7]. These hybrid lenses are effective for controlling chromatic and field curvature aberrations. A kinoform is a lens with discrete jumps in its profile such that the position and depth of each jump causes 2π phase shifts on the design wavefront, so it results in positive interference due to the discrete structure. Each jump on the profile defines the edges of a diffractive zone. If the zones shape and height are properly defined, a 100% diffraction efficiency is obtained for a specific wavefront and wavelength.

This paper shows an extension of the SMS method to design DOEs. Kinoform surfaces are calculated together with the remaining surfaces of the optical system and through a direct method, i. e. with no multi-parametric optimization. The phase-shift properties of DOEs provide us an extra degree of freedom, that enables the control of two rays per each point of the surface (instead of one ray as is the case of non-diffractive surfaces). For a single diffractive surface (as in section 2.1), in which the controlled rays footprint coverage of the surfaces is 100%, the set of extra rays form, in general, a wavefront, which means two wavefronts can be controlled with a single surface. Wavefronts of different wavelengths can also be coupled (with obvious applications in achromatic designs).

## 2. Diffractive SMS method

A kinoform lens can be designed in two steps: first, the diffractive macro surface and the phase function on this surface are calculated. This step only considers the order of diffraction *m* for which the system is designed, typically *m* = ± 1. The phase function on the macro surface is used to calculate the loci where the 2π phase shifts happen, i.e., the transition points between each diffractive zone. In the second step, the depth and shape of each diffractive zone is designed to optimize the diffraction efficiency, i.e., to ensure that the highest amount of light is diffracted into the desired order of diffraction. A 100% diffraction cannot be obtained for multiple wavefronts, which is the interest of the method described hereafter. Several methods and proposals to design the diffractive zone shapes to achieve a high diffraction efficiency are described in the literature [8,9]. This topic is not covered in this work.

The phase function *ϕ* can be defined with respect the eikonal functions of the incident *S _{i}* and diffracted wavefronts

*S*, as [10]

_{d}**r**

_{i,}

**r**

_{d}and ${n}_{i}$, ${n}_{d}$ are the incident, diffracted ray vectors and the refractive index of the incident and diffracted side, respectively.

**N**is the normal vector to the surface, and

**∇**

*ϕ*is the gradient of the phase function [11]. Comparing Eq. (2) with Snell's law, the difference is the term dependent on

*ϕ*, which provides a variable to control an extra wavefront.

#### 2.1 Diffractive oval

Let *Wa _{i}* and

*Wb*be two wavefronts impinging on a diffractive surface and, let the diffracted wavefronts, be respectively,

_{i}*Wa*and

_{d}*Wb*

_{d}_{.}The phase function

*ϕ*jumps are determined by the surface discontinuities so we will consider that

*ϕ*as well as

*m*are equal for both couples of wavefronts. Then we have two equations like Eq. (1),

Equation (4) is the implicit equation of the macro diffractive surface. When *λ _{a}* =

*λ*, this equation is the same obtained for the macro surface profile on the design of microstructured surfaces [13,14], which are not necessarily diffractive. The phase shift in the discontinuities of these microstructured surfaces is not necessarily a multiple of 2π as it is in the diffractive surfaces. We shall call a surface like the one given by Eq. (4) a Generalized Diffractive Oval (GDO) for wavefronts

_{b}*a*and

*b*. Once the macro surface is calculated,

*ϕ*can be calculated using any of the expressions in Eq. (3). The discontinuity curves on the macro surface separating diffractive zones are a family of curves

*ϕ*= 2π

*l*+

*A*, where

*A*is a family constant that can be freely chosen and the integer number

*l*is the parameter of the family.

As an example, Fig. 1 shows the design of a diffractive surface for imaging two points, **a** and **b** into points **a’** and **b'**. For a generic point **p** at the Diffractive Oval, ${S}_{{a}_{i}}={n}_{i}\left|p-a\right|+{A}_{i}$ and${S}_{{a}_{d}}=-{n}_{d}\left|a\text{'}-p\right|+{A}_{d}$, where *A _{i}* and

*A*are constants. The same equivalent expressions are valid for points

_{d}**b**and

**b’**. If we consider the optical path length

*L*

_{a}

_{-}

_{p}

_{-}

_{a}*of the path*

_{’}**a**-

**p**-

**a**’ as ${L}_{a-p-a\text{'}}={n}_{i}\left|p-a\right|+{n}_{d}\left|p-a\text{'}\right|$, and the equivalent case for

*L*

_{b}

_{-}

_{p}

_{-}

_{b}*, we can rewrite Eq. (4), as*

_{’}*L*

_{a}

_{-}

_{p}

_{-}

_{a}*/*

_{’}*λ*) and second (2π

_{a}*L*

_{b}

_{-}

_{p}

_{-}**/**

_{b}*λ*) term are, respectively, the phase along the path

_{b}**a**-

**p**-

**b**’ and

**b**-

**p**-

**b**’. Equation (5) shows that the diffractive macrosurface is such that the difference between the phases is a constant. This calculation is similar to that of a Cartesian Oval [14], in which a refractive/reflective surface focuses the rays from a point into another point . For this reason we will call the surface of Eq. (5) a Diffractive Oval (which is the same as the GDO of Eq. (4) but restricted to spherical wavefronts). In summary, a Diffractive Oval focuses 2 points into another 2 points.

#### 2.2 Diffractive SMS method for a hybrid system

The theory described is applicable to 2D and 3D geometries. In this section, we restrict the analysis to 2D. The diffractive SMS method is applied to a hybrid system (one refractive and one diffractive surface). The DOE’s phase shift allows a perfect coupling of three wavefronts.

The SMS method calculates different portions of the surfaces by a sequential application of the Generalized Cartesian Oval (GCO) procedure. The diffractive SMS method uses a similar strategy, except that the GCO is replaced by the GDO procedure of Section 2.1 when portions of the diffractive surface are calculated.

Consider Fig. 2. Let **a**, **b** and **c**, be the origin points of the spherical wavefronts (with wavelengths *λ _{a}*,

*λ*and

_{b}*λ*) to be coupled into wavefronts with end points in

_{c}**a**

*’*,

**b**

*’*and

**c**

*’*, respectively. Assume that

**d**

_{0}, an initial point on the diffractive surface

*d*, the normal of the macro surface

**N**

_{0}and the gradient (

**∇**

*ϕ*)

_{0}at this point are given. Let’s take

**d**

_{0}as the origin of the phase function

*ϕ*(

**d**

_{0}) = 0. The refractive index relative to the outside medium is

*n*. Let L

_{a}, L

_{b}and L

_{c}be optical path lengths from

**a**to

**a**’,

**b**to

**b**’ and

**c**to

**c**’, respectively. The optical path length along a ray is calculated as in a non-diffractive configuration plus the term

*mλ ϕ*(

**p**)/(2π) of the

**p**where the ray crosses the diffractive surface. The procedure is as follows.

- 1. Since
**d**_{0},**N**_{0}and**∇***ϕ*(**d**_{0}) are known we can trace back 3 rays from**a***’*,**b***’*and**c**’ passing through**d**_{0}and diffract them using Eq. (2) [Fig. 2(a)]. Along the ray from**a**’ we can calculate the point**r**_{a0}where the ray must refract to get an optical path length**a**-**r**_{a0}-**d**_{0}-**a**’ equal to L_{a}. In a similar way we can calculate the points**r**_{b0}and**r**_{c0}on the trajectories of the rays coming from**b**’ and**c**’. As we know the deflection of the rays at**r**_{a0},**r**_{b0}and**r**_{c0}, we can calculate the normal vectors to the refractive surface*r*at these 3 points. Now we can choose as first portion of the surface*r*to any surface passing through the points**r**_{a0},**r**_{b0}and**r**_{c0}, whose normals fit the preceding calculation. As in the conventional SMS method it is advisable to choose this surface as well as L_{a}, L_{b}and L_{c}so this portion of*r*is smooth. The design procedure works also with more “wild” surfaces, but the final design, although mathematically correct, could be physically undoable. - 2. The rays with origin in
**b**and**c**are refracted on the previously calculated portion of*r*, and the GDO that focuses these rays into**b**’ and**c**’ is calculated. The constant of the GDO calculation [Eq. (5)] has to be selected such that the GDO passes through**d**_{0}. Once the GDO is calculated we can calculate the phase function*ϕ*as well as the tangential component of**∇***ϕ*(i.e.**N**×**∇***ϕ*) along this portion of diffractive surface*d*(using any of the equations of Eq. (3)).The first section of*d*can be calculated until**d**_{1}[Fig. 2(b)]. - 3. Now we can diffract back rays from
**a**’ through this recently calculated portion of surface*d*and calculate a new portion of*r*as the GCO that passes through**r**_{a0}and focuses these rays into the point**a**. The new portion can be calculated until point**r**_{a1}[Fig. 2(c)]. - 4. Steps 2 and 3 are repeated iteratively to calculate the remaining extensions of
*d*and*r.*

## 3. Design examples

This section describes three examples of systems designed using the diffractive SMS method. Since the method is based on perfect coupling of wavefronts, our performance evaluation is focused on the RMS spot size over a range of fields and for *m* = 1. The systems are designed for acrylic (PMMA) refractive index. The systems were analyzed using Code V^{®}.

#### 3.1 Freeform monochromatic Diffractive Oval coupling 2 wavefronts

This example shows a freeform monochromatic diffractive surface coupling 2 spherical wavefronts (a diffractive Oval) [Fig. 3]. Both origin points in the object plane **a** and **b** are embebbed in PMMA (*n*_{i} = 1.4914, *n*_{d} = 1 at λ = 587.6nm). **a** and **b**, **a**’ and **b**’ are 8 and 22 mm apart, respectively. The lens vertex is at 49.9 and 50 mm from the object and image plane, respectively, with a lens aperture of 20mm diameter. RMS spot size field map is displayed in Fig. 3. The two design wavefronts are perfectly coupled (RMS = 0) and the RMS spot size curve over the *y* axis [Fig. 3 right] has the typical shape of a two surface non-diffractive SMS design [4], the difference being only one surface is required in the diffractive case.

#### 3.2 Freeform bichromatic diffractive oval

DOE's are typically used for correcting chromatic aberration, so it is interesting to design a system for two spherical wavefronts originated in the same point, but with two different wavelengths, 670.4 and 403.2nm [Fig. 4]. The lens vertex is at 72 and 28 mm from the object and image plane, respectively, with a lens aperture of 35mm. RMS spot size map is also used but, unlike Fig. 3 center, now one of the axis represents the wavelength, to evaluate chromatic behavior. The *V-shaped* profile of the RMS spot size over the object height [Fig. 4 right] is typical from devices that perfectly image only one wavefront. It is noticeable that the RMS line is exactly the same for both wavelengths. The systems perfectly images the wavefront on the design wavelengths. Although only two wavelengths are being controlled, RMS spot size values are very low on the wavelengths between the two design ones, which emphasizes the reason why diffractive devices are so successfully used for chromatic corrections.

#### 3.3 One refractive + one diffractive surface coupling three wavefronts

The third example is a hybrid system, with a refractive and a diffractive surface, and is designed using the diffractive SMS method in section 2.2. This is a 2D design and only rays contained in the lens plane are considered in the ray tracing. The system images 3 points, 2 of them at one wavelength and the third point at another wavelength [Fig. 5].

The central design object point (0 mm) has λ = 403.2nm and the two outer design points (4 and −4mm) have λ = 670.4nm. The vertex of the first surface is at 56.7mm from the object plane, while the second surface vertex is at 49mm from the image plane. The lens aperture is 23mm. The results are, again, represented by a RMS spot size map with one axis representing the object height, and the other axis representing the wavelength. We can see, for both wavelengths, that the design points are perfectly imaged. For λ = 403.2nm, where only one field is being controlled, we can observe that the RMS curve presents a much smoother behavior than the previous example [Fig. 4]. The shape of the RMS curve is typical of an aplanatic system [15]. The most interesting result is the *ring-shaped* RMS spot size map, which shows a set of combination of wavefronts with specific wavelengths that, although not perfect solutions as in the case of the design points, still have a very low RMS spot size value.

## 4. Conclusions

The diffractive SMS method uses the extra degree of freedom provided by the phase shifting properties of DOEs to control two rays on each point of each diffractive surface. The concept of diffractive oval (in 3D geometry) is introduced, and the diffractive SMS method (in 2D) is presented as a sequential application of the diffractive oval design procedure. The method can decrease the number of elements of a system, thus decreasing size and weight. Design examples of a 3D freeform diffractive surface coupling two wavefronts and a polychromatic hybrid 2D lens controlling three wavefronts with different wavelengths have been successfully implemented.

## Acknowledgments

UPM authors thank the European Commission (ADOPSYS: FP7-PELE-2013-ITN Grant Agreement No. 608082, NGCPV: FP7-ENERGY.2011.1.1 Grant Agreement No. 283798), the Spanish Ministries (OPTIVAR: TEC2014-56867-R, GUAKS: RTC-2014-2091-7, SUPERRESOLUCION: TEC2011-24019, PMEL: IPT-2011-1212-920000), UPM (Q090935C59) and the academic license for CodeV from Synopsys for the support given to the research activity of the UPM-Optics Engineering Group. This work was partly supported by Fundação para a Ciência e Tecnologia, Scholarship SFRH / BD / 80892 / 2011 and by Consejo Social de la Universidad Politécnica de Madrid.

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**15. **B. Narasimhan, P. Benítez, and Juan C. Miñano, Cedint, Polytechnical University of Madrid, Spain are preparing a manuscript to be called as “Aplanatic systems as a limiting case of SMS – a new look”.