We propose an alternative method to design diffractive lenses free of spherical aberration for monochromatic light. Our method allows us to design diffractive lenses with the diffraction structure recorded on the last surface; this surface can be flat or curved with rotation symmetry. The equations that we propose calculate the diffraction profiles for any substratum, for any f-number, and for any position of the object. We use the lens phase coefficients to compensate the spherical aberration. To calculate these coefficients, we use an analytic-numerical method. The calculations are exact, and the optimization process is not required.

Spherical aberration is, in many cases, the most important of all primary aberrations, because it affects the whole field of the lens, including the vicinity of the optical axis. It is due to different focus positions for a marginal ray, meridional ray, and paraxial rays. An alternative to minimize the spherical aberration is to use diffractive optical elements (DOE). Diffractive lenses are essentially gratings with a variable spacing groove which introduces a chromatic aberration that is worse than conventional refractive/reflective optical elements. In some applications, an optical component may require a diffractive surface combined with a classic lens element. By using the diffractive properties, it is possible to design hybrid elements to obtain an achromatically corrected element [

We use lens phase coefficients to compensate spherical aberration. To calculate these coefficients, we use an analytic-numerical method. The calculations are exact, simple and quick. A process of optimization is not required.

The manufacturing problem of diffractive lenses is not considered here; to solve this problem you can read Castro-Ramos et al. [

First, we describe the diffractive lenses theory. Also, we give a brief derivation of the general grating equation to trace a couple of light rays through a rotationally symmetrical surface. Then, we establish the analytic-numerical method to minimize spherical aberration. We propose some heights to correct the spherical aberration. Finally, we conclude by providing a design example.

Diffractive lenses can be described by a polynomial phase function [

We will consider that the diffractive lens is rotationally symmetrical, so (

Designers usually use some commercial optical design programs to obtain the lens phase coefficients by using an optimization process. We will describe an analytical method to obtain these coefficients.

To trace a pencil of rays through the diffractive optical surface, we use the grating equation. For a flat surface, the grating equation is given by

To analyze the light propagation through a diffractive curved surface, we have to change the form of the last equation. After some algebra, we obtain the general grating equation

Lens parameter. The diffractive surface is on the second surface.

The grating frequency of (

Then, if we want to find

The spherical aberration of the ray in any optical system can be expressed as

It is possible to correct the residual spherical aberration by using the third term of the expansion (

Transversal spherical aberration of the example one.

If we consider f/numbers to be small, we should correct the spherical aberration residual, and its peaks fall at values

The points for Kingslake analysis are

The number of

We have proposed a general expression to compute the phase coefficients. Now, we will show how theses coefficients minimize the spherical aberration with some numerical examples. All examples considered in this section have the diffracted order

In this example, we consider that the diffractive surface is on a spherical surface (the last surface of the system) with 50 mm of diameter aperture, numerical aperture 0.375, object distance 200 mm, and

In Table

Example

Surfaces | Radius (mm) | Thickness (mm) | Radius aperture (mm) | Glass | |
---|---|---|---|---|---|

1 | ∞ | 200 | Air | ||

2 | 101.954 | 8.138 | 25 | BK7 | |

3 | DOE | −101.954 | 66.059 | 25 | Air |

4 | ∞ | 0 | Air |

We must trace light rays until the last surface, and then we can calculate all constants of (

The number of rays traced depends on the number of coefficients. In this example, we use two coefficients, and we get the next equations system

We have solved (

Coefficients value for the example

Coefficients | Aperture height (mm) | Value |
---|---|---|

Paraxial | −0.005 mm^{-1} | |

1.180409 × 10^{-6} mm^{-3} | ||

−1.270732 × 10^{-10} mm^{-5} |

Figure

We can see in the graphic a maximum transversal spherical aberration of about 0.004 mm, having zeros on two pupil positions. This is because we had computed two coefficients for the system. The corresponding Strehl Ratio is of about 0.151.

In the Figure

We consider the same optical system but now using four phase coefficients. Solving the next equations system

Coefficients value for the example

Coefficients | Aperture height (mm) | Value |
---|---|---|

Paraxial | −0.005 mm^{-1} | |

1.194430 × 10^{-6} mm^{-3} | ||

−1.684803 × 10^{-10} mm^{-5} | ||

3.602457 × 10^{-14} mm^{-7} | ||

−5.856954 × 10^{-14} mm^{-9} |

In Figure

Transversal spherical aberration of the example

Now we consider the same optical system but the diffractive surface on a hyperbolic surface (last surface) with conic constant, diameter aperture

Coefficients value for the example 3.

Coefficients | Aperture height (mm) | Value |
---|---|---|

Paraxial | −0.005 mm^{-1} | |

9.106091 × 10^{-7} mm^{-3} | ||

−1.941142 × 10^{-10} mm^{-5} | ||

5.218183 × 10^{-14} mm^{-7} | ||

−9.916744 × 10^{-18} mm^{-9} |

Figure

Transversal spherical aberration of the example 3.

We can see again a very small spherical aberration, and its maximum value is of around 2 × 10^{-5} mm. It has 4 zeros because we have used 4 phase coefficients. The irradiance distribution corresponding to this system is shown in Figure

The point spread function of the example 3.

Our proposed method is also for flat surfaces. We only use zero for the angle between the normal to the surface and optical axis in (

If the designer wants to use the first surface, the conjugates must be changed, and then the method proposed can be applied.

We have established a new exact method to correct the spherical aberration for any optical system using diffractive lenses; this method makes use of the general grating equation and exact ray trace. With our method, we can decide how many zeros the spherical aberration should have and fix its position in the exit pupil. The method can only be applied to the first and last surface of the optical system.

We also have proposed some heights to correct the spherical aberration and how many rays must be traced depending on the f/number.

In the first and second examples, we have shown that we can have a high control of spherical aberration, minimized at points on the surface where we have wanted. Also, we have shown that our method is valid for any rotationally symmetrical surface.

In general, spherical aberration will have as many zeros as the coefficients we calculate. It is very important to see that in order to minimize spherical aberration, we use only as many coefficients as necessary.

Finally, to calculate the coefficients, we only use the analytic-numerical method. The calculations are exact, simple, and quick. A process of optimization is not required.