Modeling of diffractive optical elements for lens design.

Transcription

1 Juan L. Rayces and Lan Lebich OA Applied Optics, 7421 Orangewood Ave., Garden Grove, A Abstract. The use of a standard aspheric profile to describe conventional optical elements in lens design programs has advantages such as independent verification of lens system performance by different individuals, ease of interpretation by different manufacturers, etc. A model of diffractive optical element is proposed for consideration as a standard form to have similar benefits. Diffractive elements are characterized here by (1) the equation of the supporting surface (substrate) and (2) the phase function. The phase function itself consists of two terms: (a) the stigmatic phase term and (b) the astigmatic phase term The stigmatic phase term is a rotationally symmetric function and is related to the property of the diffraction element to image free from aberration one axial point onto another, as in holograms, at a specific (reference) wavelength. The astigmatic phase term is an arbitrary function of the space coordinates, it takes several forms, it can impart aspheric deformations (or corrections) to incident wave fronts or it can be used to model different types of gratings. The substrate can be any of the standard surfaces already in use in lens design. The parameters of the phase function are paraxial diffractive power (the counterpart of paraxial curvature), stigmatic foci position index (the counterpart of the conic constant), coefficients of the astigmatic phase function, reference wavelength and diffracted order number. 1 Introduction Modeling of diffractive optical elements for lens design. The well-known sag equation 1 used to describe rotationally symmetric aspheric conventional optical elements 2 ρ x = ( K+ 1) ρ (1) a ρ + a ρ + a ρ + a ρ, has become a standard In the notation used here is paraxial curvature, K conic constant and ρ 2 = y 2 + z 2. The axis of symmetry is the x-axis. The first term of the equation is the general equation of a surface of revolution generated by a conic section often called the base curve. It corresponds to a class of reflective surfaces that image one axial point onto another free from aberration at any aperture This first term may be called the stigmatic part of the equation.. A four-term polynomial, as used to describe Schmidt plates, is appended to the base curve equation probably on the grounds that the sag equation, thus written, reduces to the most commonly used aspherics thirty years ago. These four terms may be called.the astigmatic or polynomial part of the equation. Eq. (1) has some shortcomings. such. as ripples generated by the polynomial terms, that cannot be abated by increasing the number of terms,. when used to describe very steep curves. Yet it covers very many cases and has been successfully used in a large variety of aspheric systems. Other equations have been proposed, and perhaps used to manufacture optical elements, but the standard sag equation is still alive and well. In this paper we use a similar approach for describing rotationally symmetric diffractive optical elements that is, to combine in one equation a stigmatic part with an astigmatic part. This idea is not new: it is used in ODE V TM software as disclosed by Michael Hayford 2 in very general terms. It is probably used in other design codes, but details are not widespread. Here we shall derive the pertinent equations from basic principles, and present them in sufficient detail so that other individuals can use them or perhaps propose better solutions. 1.1 Ray-tracing through conventional surfaces The essential entity necessary for tracing rays and computing optical path differences through conventional elements is the equation of the surface: Eikonal 07/14/2007 1

2 S( x, y, z ) = 0. (2) It takes four steps per surface to do ray tracing through conventional optical surfaces: (1) The coordinates of the point of intersection of a selected ray with the surface, are found by eliminating x, y, z, from the parametric equations of the ray x = X + tcosα.. y = Y + tcosβ (3) z = Z + tcos γ, and the equation of the surface, Eq.(2), and solving for the parameter t, then substituting t into Eq.(3). In the latter, cosα, cos β, cosγ are components of the unit vector r in the direction of the incident ray. (2) The partial derivatives of Eq.(2) are evaluated at the point of intersection. From these the components of the unit vector n in the direction of the normal.are obtained. (3) The unit vector r' parallel to the refracted rays is solved with the vectorial Snell-Descartes equation: µ r n = µ r n, (4) where µ, µ' are refractive indices in the object and image spaces, respectively, of the OE. (4) Optical path differences are computed by adding the product of refractive indices and distances form point of intersection to point of intersection of the selected ray and subtracting from it a similar quantity previously computed for the principal ray. Alternately they may be computed with Hopkins' formula 3 (Appendix I) 1.2 Ray-tracing throught diffractive elements Mathematically a diffractive optical element is a family of curves (fringes, grooves) in space. Any curve in space must be defined as the intersection of two surfaces. For this reason the diffractive element needs to be defined by two equations: (a) the equation of the supporting surface or substrate, of the same form as Eq.(2), and (b) the phase equation that represents a family of surfaces in space: Φ = 1 ϕ ( xyz,, ), (5) RAY SUPPORTING SURFAE PHASE ISOPLETHS Figure 1. The intersection of a family of phase surfaces with the supporting surface define the locus of fringes or grooves o where is a reference wavelength and ϕ( xyz,, ) is the phase function discussed below. The intersection of surfaces in this family with the supporting surface defines the fringes or grooves as shown schematically in figure 1. Each integer value of Φ corresponds to one fringe. It takes five steps to do ray-tracing through diffractive optical elements: (1) The coordinates of the point of intersection of a selected ray with the supporting surface are found using ray-tracing methods for conventional optics, as discussed already. (2) The components of the normal n to the supporting surface are also computed with the same methods. Eikonal 07/14/2007 2

3 P o n grad φ t PAPERS AND PRESENTATIONS (3) The phase funtion ϕ and its gradient ϕ are evaluated are evaluated at the point of intersection. (4) The unit vector r' in the direction of the diffracted ray is Figure 2. Point solved with the vectorial grating equation 1 : of intersection r n = r n+ A t (6) O of ray with supporting where surface. Unit vector n is the A= m ϕ s (7) normal to the surface and ϕ and m is the integral order number, is the current wavelength, is the phase and is the reference wavelength. Solution of Eq.(6) is given in Appendix II gradient. Unit vector s lies on (5) The optical path difference of the diffractive element is the intersection made itself of two parts. One part is computed exactly the same as in conventional systems as of the plane of explained above. The other is the phase function contribution of the diffractive element. With n and ϕ, and Hopkins' notation this can be written: the supporting ( OPD) = W + ϕ (8) surface. Unit vector t is where W is Hopkin's OPD contribution term (Appendix I) and normal to n and to s. Fringes ϕ = ϕβxyz,, γ ϕβxyz,, γ, (9) are parallel to vector t. where ϕ is the phase function, xyz,,, are coordinates of the point of intersection of the selected ray and x, y, z, are coordinates of the point of intersection of the reference (principal or chief) ray. 2 The phase function The phase function is often mentioned in the discussion of diffraction and diffractive elements but hardly ever it is defined. In holography the phase function is clearly a phase term introduced by the hologram to satisfy Fermat's principle when the the wavefronts are reconstructed. However if the phase function is expressed by an arbitrary mathematical expression, such as a polynomial, its meaning remains obscure as there is no pair of points where the condition of stigmatism must be satisfied. The gradient of the phase function is needed, allright, to solve the vectorial grating equation (6), but what about optical path differences needed to compute wave aberrations?. Two postulates are needed to explain and define the phase function in diffraction optical elements: (1) In an infinitesimal area the diffraction optical element reduces to a plane diffraction grating. (2) The effect of a plane grating on an incident wavefront, regardless of angle, is to impart to the wavefront a phase shift (advance or retard) of one wave length per fringe width (first diffraction order), or two wave lengths per fringe width (second diffraction order), etc. The first postulate is not difficult to accept if we image ourselves observing a diffractive element through a microscope. We shall see in the field of view a pattern of apparently parallel straight lines with apparently uniform separation. When we move the microscope to a different point the separation and orientation of the line change. Eikonal 07/14/2007 3

4 (a) (b) (c) PAPERS AND PRESENTATIONS The second postulate is not so easy to accept. But if we believe Huyugens' principle and add periodicity to it as Fresnel did, and then we say that the elementary wavelets recombined are those with the same amplitude rather than the same phase, we may very well understand the second postulate with the help of figure 3. Then, we can define the phase function as the phase Figure 3. Diffraction grating effect on wavefronts: (a) zero change imparted on a wavefront at each fringe, regardless of order, (b) first order, (c) second order. Wavelets are shown whether stigmatic points are involved or not separated by one wave length steps. As an elementary proof of the validity of the proposed definition, it may be used to derive the scalar grating equation, as shown in figure 4. onsiderable difficulty in the formulation of the phase function arises if its definition is restricted to the supporting P surface because it is necessary to express it in terms of t curvilinear coordinates unless the surface is flat. O m -t It is best, then, to accept that the phase function is defined at all points in the three-dimensional space, as shown schematically in Figure 1, although it manifests itself over the d Q supporting surface only. α β The phase function may be thought of as made of d ( layers one wavelength thick, just like an onion. The grooves or fringes are the onion layers that show up when it is carved. As already explained we shall derive a phase equation that combines one stigmatic part ϕ S and one astigmatic or polynomial part ϕ A, thus Eq.(5) becomes 1 Φ = ( ϕs + ϕ A) (10) OPQ=m OP=t PQ=m - t t=d sinα m - t =d sin β sin α + sin β ) =m Figure 4. Derivation of the scalar grating equation. Eikonal 07/14/2007 4

5 ========================================== X Appendix 1: Hopkins' OPD formula. OPD = W + W k 1 δ j δ k, ( ) j= 1 ( x x)( cosα + cosα ) + ( y y)( cos β + cos β ) + ( z z)( cosγ + cosγ ) ( 1+ cosα cosα + cos β cos β + cosγ cosγ ) W j = n n ( x x)( cosα + cosα) + ( y y)( cos β + cos β) + ( z z)( cosγ + cosγ ) ( 1+ cosαcosα + cos βcos β + cosγ cosγ ) j. j, x, y, z, are coordinates of the point of incidence of the ray on the surface, and α, β, γ are direction cosines. Primed symbols denote quantities associated with the ray just after surface j and unprimed symbols are associated with the ray just after the same surface. Bars over symbols denote quantities associated with the principal ray while symbols without bars denote quantities associated with the typical ray. The term after the summation is a focal shift term derived by B. Tatian 6, equivalent to the last term in equation (52). 1. G. H. Spencer and M. V. R. K.Murty,"General ray-tracing procedure", J. Opt. Soc. Am., Vol. 52, No. 6, p , (1962). 2. Michael J. Hayford, "Holographic optical design using ODE V TM ", in Proceeding of SPIE, Vol. 883, January 1988, Los Angeles, alifornia. 3. H. H. Hopkins, "The wave aberration associated with skew rays" Proc. Phys. Soc. 65B, p , Eikonal 07/14/2007 5