CN104237992A - Method for analyzing eccentricity errors of multilayer diffractive optical elements - Google Patents

Abstract

The invention discloses a method for analyzing eccentricity errors of multilayer diffractive optical elements, and belongs to the technical field of optical designs. By the aid of the method, the problem of deficiency of a scientific and reliable method for analyzing influence of eccentricity errors on diffraction efficiency in the prior art can be solved. The method for analyzing the eccentricity errors of the multilayer diffractive optical elements includes steps of 1, solving the diffraction efficiency of the continuous face type or stepped face type multilayer diffractive optical elements according to a phase delay expression and a diffraction efficiency formula for the multilayer diffractive optical elements; 2, determining the average diffraction efficiency of polychromatic light integration according to a polychromatic light integration diffraction efficiency theory in integral working wave bands; 3, determining the maximum values of the diffraction efficiency of the polychromatic light integration and corresponding working wavelengths in the integral working wave bands, substituting design wavelengths into a microstructure height formula and determining height values of microstructures of diffractive elements of various layers; 4, deducing a diffraction efficiency expression of the multilayer diffractive optical elements with the eccentricity errors according to a wave optical model. The corresponding working wavelengths are the design wavelengths.

Description

Multilayer diffraction optical element eccentric error analytical approach

Technical field

The present invention relates to a kind of multilayer diffraction optical element eccentric error analytical approach, for the design of the folding containing multilayer diffraction optical element/diffraction hybrid optical system, the method can determine the allowed band of the eccentric error when obtaining high-diffraction efficiency, theoretical foundation can be provided for the control of the eccentric error scope of multilayer diffraction optical element, improve the image quality of the folding/hybrid optical system that spreads out containing multilayer diffraction optical element, belong to optical design techniques field.

Technical background

Along with the development of advanced Optical manufacture technology, diffraction optical element creates an independently branch in contemporary optics, brings revolutionary change to traditional optical design theory and manufacturing process.Diffraction optical element has the advantages such as high-diffraction efficiency, unique dispersion characteristics, flexibly material selectivity and special bit phase function, thus can realize special optical function.Rolling over/spreading out in hybrid optical system design, the diffraction efficiency due to individual layer diffraction optical element departs from centre wavelength along with operation wavelength and sharply declines, and image quality is affected.Therefore, individual layer diffraction optical element can only be used for the optical system of limited wavestrip width.In recent years, the multilayer diffraction optical element of appearance overcame this shortcoming, and the structure of multilayer diffraction optical element as shown in Figure 1, Figure 2, Figure 3 shows, achieves the raising of broadband diffraction efficiency.

The diffraction efficiency of the diffraction optical element in prior art application scalar diffraction theory analysis folding/diffraction hybrid optical system, this theory is thought in the assembling process of multilayer diffraction optical element, it is inevitable that eccentric error occurs, as shown in Figure 4, Figure 5, and, this eccentric error can bring adverse effect to diffraction efficiency, this just needs to need this eccentric error of look-ahead analysis on the impact of diffraction efficiency when optical design, controls multilayer diffraction optical element eccentric error scope to take measures.

But, also there is no a kind of science, reliably eccentric error in the prior art to the analytical approach of the impact of diffraction efficiency.

Summary of the invention

The object of the present invention is to provide a kind of method can analyzed multilayer diffraction optical element eccentric error and diffraction efficiency is affected, thering is provided technical foundation for controlling multilayer diffraction optical element eccentric error scope, we have invented a kind of multilayer diffraction optical element eccentric error analytical approach for this reason.

The method of the present invention comprises following four steps:

1, according to multilayer diffraction optical element bit phase delay expression formula and diffraction efficiency formula, the diffraction efficiency of continuous surface type or step surface type multilayer diffraction optical element is obtained;

2, in whole service band, theoretical according to polychromatic light Integral Diffraction Efficiency, determine polychromatic light integral mean diffraction efficiency;

3, in whole service band, polychromatic light Integral Diffraction Efficiency maximal value is determined, and the operation wavelength of correspondence, the operation wavelength of described correspondence is design wavelength, design wavelength is substituted into microstructure height formula, determines each layer diffraction element microstructure height value;

4, according to wave optics model, the expression of diffraction efficiency of the multilayer diffraction optical element containing eccentric error is derived.

Its technique effect of the method for the present invention is, according to multilayer diffraction optical element bit phase delay expression formula and diffraction efficiency formula, can calculate the diffraction efficiency of continuous surface type or step surface type multilayer diffraction optical element.In 400 ~ 700nm wave band, with polymethylmethacrylate and polycarbonate for base material, theoretical according to polychromatic light Integral Diffraction Efficiency, determine polychromatic light integral mean diffraction efficiency, and then obtain the distribution of polychromatic light Integral Diffraction Efficiency; When polychromatic light Integral Diffraction Efficiency reaches maximal value 99.25%, corresponding operation wavelength is to being 435nm and 598nm, and this operation wavelength is to being confirmed as design wavelength pair.By design wavelength to substitution microstructure height formula, determine that each layer diffraction element microstructure height value is respectively 16.7112 μm and 13.0242 μm.Finally determine the relation of diffraction efficiency and eccentric error, as shown in Fig. 6, Fig. 7, Fig. 8.Thus in optical design process, higher diffraction efficiency can be obtained according to eccentric error scope.

Accompanying drawing explanation

Fig. 1 is continuous surface type multilayer diffraction optical element structural representation.Fig. 2 is step surface type multilayer diffraction optical element structural representation.Fig. 3 is the index path of light oblique incidence step surface type multilayer diffraction optical element.Fig. 4 is multilayer diffraction optical element generation eccentric error schematic diagram.Fig. 5 is multilayer diffraction optical element generation eccentric error close-up schematic view.Fig. 6 is multilayer diffraction optical element diffraction efficiency and operation wavelength and eccentric error relation surface chart when service band is 400 ~ 700nm.Fig. 7 is multilayer diffraction optical element diffraction efficiency and eccentric error graph of relation when operation wavelength is respectively 435nm and 598nm, in figure, curve 1 is 435nm operation wavelength relation curve, curve 2 is 598nm operation wavelength relation curve, and this figure is simultaneously as Figure of abstract.Fig. 8 is multilayer diffraction optical element polychromatic light Integral Diffraction Efficiency and eccentric error graph of relation when service band is 400 ~ 700nm.

Embodiment

The method of the present invention furtherly below, the multilayer diffraction optical element number of plies is double-deck, as shown in Figure 1, Figure 2, Figure 3 shows.

The first step: according to multilayer diffraction optical element bit phase delay expression formula and diffraction efficiency formula, calculates the diffraction efficiency of continuous surface type multilayer diffraction optical element.According to scalar diffraction theory, the diffraction efficiency formula of continuous surface type multilayer diffraction optical element is:

${\mathrm{\η}}_m\left(\mathrm{\λ}\right)={\sin c}^2[m-\mathrm{\φ}\left(\mathrm{\λ}\right)]---\left(1\right)$

In formula: η _m(λ) be diffraction efficiency, m is the order of diffraction time, and λ is operation wavelength, the bit phase delay that φ (λ) is multilayer diffraction optical element.Formula (1) is a kind of sinc (x) function, i.e. sinc (x)=sin (π x)/(π x), now, and x=[m-φ (λ)].When light oblique incidence, as shown in Figure 3, the bit phase delay φ (λ) between adjacent periods is:

$\mathrm{\φ}\left(\mathrm{\λ}\right)=\frac{H_1}{\mathrm{\λ}}[\sqrt{(1-n_1^2\left(\mathrm{\λ}\right){\sin }^2{\mathrm{\θ}}_{i1})}-n_1\left(\mathrm{\λ}\right)\cos {\mathrm{\θ}}_{i1}]+\frac{H_2}{\mathrm{\λ}}[\sqrt{(n_2^2\left(\mathrm{\λ}\right)-{\sin }^2{\mathrm{\θ}}_{i2})}-\cos {\mathrm{\θ}}_{i2}]---\left(2\right)$

In formula, H ₁, H ₂represent the microstructure height of first, second layer of diffraction element, θ _i1, θ _i2for light enters the incident angle of ground floor, second layer diffraction element, n ₁(λ), n ₂(λ) be the refractive index of base material when operation wavelength is λ.

Second step: in whole service band, theoretical according to polychromatic light Integral Diffraction Efficiency, determine polychromatic light integral mean diffraction efficiency.Polychromatic light integral mean diffraction efficiency formula is:

${\stackrel{\‾}{\mathrm{\η}}}_m({\mathrm{\λ}}_1,{\mathrm{\λ}}_2)=\frac{1}{{\mathrm{\λ}}_{\max }-{\mathrm{\λ}}_{\min }}\underset{{\mathrm{\λ}}_{\min }}{\overset{{\mathrm{\λ}}_{\max }}{\∫}}{\mathrm{\η}}_m\left(\mathrm{\λ}\right)\mathrm{d\λ}---\left(3\right)$

In formula: for polychromatic light integral mean diffraction efficiency, λ _min, λ _maxfor minimum value and the maximal value of wavelength in service band, λ ₁, λ ₂for the operation wavelength pair in service band.

3rd step: in whole service band, according to formula (3), determine the operation wavelength of polychromatic light Integral Diffraction Efficiency maximal value and correspondence, corresponding operation wavelength is design wavelength, design wavelength is brought into microstructure height formula (4) and determines that each layer diffraction element microstructure height value formula is:

$\left\{\begin{array}{c}{H}_1=\frac{{\mathrm{\λ}}_1(n_2\left({\mathrm{\λ}}_2\right)-1)-{\mathrm{\λ}}_2(n_2\left({\mathrm{\λ}}_1\right)-1)}{(n_1\left({\mathrm{\λ}}_1\right)-1)(n_2\left({\mathrm{\λ}}_2\right)-1)-(n_1\left({\mathrm{\λ}}_2\right)-1)(n_2\left({\mathrm{\λ}}_1\right)-1)}\\ H_2=\frac{{\mathrm{\λ}}_2(n_1\left({\mathrm{\λ}}_1\right)-1)-{\mathrm{\λ}}_1(n_1\left({\mathrm{\λ}}_2\right)-1)}{(n_1\left({\mathrm{\λ}}_1\right)-1)(n_2\left({\mathrm{\λ}}_2\right)-1)-(n_1\left({\mathrm{\λ}}_2\right)-1)(n_2\left({\mathrm{\λ}}_1\right)-1)}\end{array}\right.---\left(4\right)$

In formula: λ ₁, λ ₂for design wavelength pair, n ₁(λ ₁), n ₁(λ ₂), n ₂(λ ₁) and n ₂(λ ₂) to be respectively base material at design wavelength be λ ₁, λ ₂time refractive index.

Bring formula (3) into formula (4), namely obtain polychromatic light Integral Diffraction Efficiency at service band λ _min~ λ _maxinterior distribution.

4th step: according to wave optics model, derives the expression of diffraction efficiency of the multilayer diffraction optical element containing eccentric error.As shown in Figure 4, the additional phase that eccentric error causes in one-period postpones:

$\mathrm{\φ}{\left(\mathrm{\λ}\right)}_{\mathrm{\Δ}}=\frac{1}{\mathrm{\λ}}[n_2\left(\mathrm{\λ}\right)L_1-L_2]---\left(5\right)$

Have according to refraction law:

$\left\{\begin{array}{c}{L}_1=\frac{\mathrm{\Δ}\sin {\mathrm{\β}}_2}{\sqrt{1-n_1^2\left(\mathrm{\λ}\right){\sin }^2{\mathrm{\θ}}_{i1}}}\\ L_2=L_1[\sqrt{1-n_1^2\left(\mathrm{\λ}\right){\sin }^2{\mathrm{\θ}}_{i1}}\sqrt{1-\frac{n_1^2\left(\mathrm{\λ}\right)\sin {\mathrm{\θ}}_{i1}}{n_2^2\left(\mathrm{\λ}\right)}}+\frac{n_1^2\left(\mathrm{\λ}\right){\sin }^2{\mathrm{\θ}}_{i1}}{n_2\left(\mathrm{\λ}\right)}]\end{array}\right.---\left(6\right)$

In formula: L ₁, L ₂light path when being respectively ideally eccentric with generation, Δ is eccentric error, and T is the cycle, and bit phase delay is now φ (λ)+φ (λ) _Δ.Formula (2), (4), (5) and (6) are substituted into formula (1), calculates and work as incidence angle θ _i1when being 0, the expression of diffraction efficiency of the multilayer diffraction optical element containing eccentric error:

${\mathrm{\η}}_m\left(\mathrm{\λ}\right)={\sin c}^2\{m-[\frac{H_1}{\mathrm{\λ}}(n_1\left(\mathrm{\λ}\right)-1)+\frac{H_2}{\mathrm{\λ}}(n_2\left(\mathrm{\λ}\right)-1)+\frac{{\mathrm{\ΔH}}_2}{\mathrm{\λ}\sqrt{T^2+H_2^2}}(n_2\left(\mathrm{\λ}\right)-1)\left]\right\}---\left(7\right)$

From formula (7), diffraction efficiency is the function of operation wavelength and eccentric error.In formula: η _m(λ) be diffraction efficiency, m is the order of diffraction time, and λ is operation wavelength, H ₁, H ₂represent the microstructure height of first, second layer of diffraction element, n ₁(λ), n ₂(λ) be the refractive index of base material when operation wavelength is λ, Δ is eccentric error, and T is the cycle.

We further illustrate the method for the present invention for the multilayer diffraction optical element that polymethylmethacrylate and polycarbonate are base material below.

Select optically conventional material polymethylmethacrylate and polycarbonate respectively as the base material of multilayer diffraction optical element ground floor and the second layer.When service band is 400 ~ 700nm, when polychromatic light integral mean diffraction efficiency reaches maximal value, corresponding operation wavelength is 435nm and 598nm, the design wavelength pair that Here it is determines.By design wavelength to substitution formula (4), obtain microstructure height H1=13.0242 μm, H2=16.7112 μm, cycle T=1000 μm, the order of diffraction time m=1.As shown in Figure 6, in 400 ~ 700nm service band, the relation of diffraction efficiency and eccentric error.Fig. 7 is operation wavelength when being 435nm and 598nm, the relation of diffraction efficiency and eccentric error.Operation wavelength is 435nm, and when eccentric error increases within 0.13%, diffraction efficiency is more than 99.8%; When eccentric error is increased to 0.65%, diffraction efficiency drops to 95.3%.When eccentric error increases to 0.93%, diffraction efficiency drops to 90.1%.Operation wavelength is 598nm, and when eccentric error increases in 0.18% scope, diffraction efficiency is more than 99.8%; When eccentric error increases to 0.9%, diffraction efficiency drops to 95.1%; When eccentric error increases to 1.2%, diffraction efficiency drops to 91.6%, and when eccentric error increases further, diffraction efficiency sharply declines.As shown in Figure 8, in 400 ~ 700nm service band, eccentric error is when 0.43% scope increases, and polychromatic light Integral Diffraction Efficiency can reach more than 99.4%; When eccentric error increases to 1.15%, polychromatic light Integral Diffraction Efficiency drops to 95%.

In sum, when diffraction efficiency is below 91%, in other words, when eccentric error is more than 0.93%, error should be taken into full account multilayer diffraction optical element performance impact, now should assess the impact of diffraction efficiency on whole system further.

Claims (5)

1. a multilayer diffraction optical element eccentric error analytical approach, is characterized in that, the method comprises following four steps:

(1) according to multilayer diffraction optical element bit phase delay expression formula and diffraction efficiency formula, the diffraction efficiency of continuous surface type or step surface type multilayer diffraction optical element is obtained;

(2) in whole service band, theoretical according to polychromatic light Integral Diffraction Efficiency, determine polychromatic light integral mean diffraction efficiency;

(3) in whole service band, determine polychromatic light Integral Diffraction Efficiency maximal value, and the operation wavelength of correspondence, the operation wavelength of described correspondence is design wavelength, design wavelength is substituted into microstructure height formula, determine each layer diffraction element microstructure height value;

(4) according to wave optics model, the expression of diffraction efficiency of the multilayer diffraction optical element containing eccentric error is derived.

2. multilayer diffraction optical element eccentric error analytical approach according to claim 1, is characterized in that, described diffraction efficiency formula is:

η _m(λ)＝sinc ²[m-φ(λ)]，

In formula: η _m(λ) be diffraction efficiency, m is the order of diffraction time, and λ is operation wavelength, the bit phase delay that φ (λ) is multilayer diffraction optical element;

This diffraction efficiency formula is a kind of sinc (x) function, i.e. sinc (x)=sin (π x)/(π x), now, and x=[m-φ (λ)];

When light oblique incidence, the bit phase delay φ (λ) between adjacent periods is:

$\mathrm{\φ}\left(\mathrm{\λ}\right)=\frac{H_1}{\mathrm{\λ}}[\sqrt{(1-n_1^2\left(\mathrm{\λ}\right)){\sin }^2{\mathrm{\θ}}_{i1}}-n_1\left(\mathrm{\λ}\right)\cos {\mathrm{\θ}}_{i1}]+\frac{H_2}{\mathrm{\λ}}[\sqrt{(n_2^2\left(\mathrm{\λ}\right)-{\sin }^2{\mathrm{\θ}}_{i2})}-{\cos \mathrm{\θ}}_{i2}],$

In formula, H ₁, H ₂represent the microstructure height of first, second layer of diffraction element, λ is operation wavelength, θ _i1, θ _i2for light enters the incident angle of ground floor, second layer diffraction element, n ₁(λ), n ₂(λ) be the refractive index of base material when operation wavelength is λ.

3. multilayer diffraction optical element eccentric error analytical approach according to claim 1, is characterized in that, described polychromatic light integral mean diffraction efficiency formula is:

${\stackrel{\‾}{\mathrm{\η}}}_m({\mathrm{\λ}}_1,{\mathrm{\λ}}_2)=\frac{1}{{\mathrm{\λ}}_{\max }-{\mathrm{\λ}}_{\min }}\underset{{\mathrm{\λ}}_{\min }}{\overset{{\mathrm{\λ}}_{\max }}{\∫}}{\mathrm{\η}}_m\left(\mathrm{\λ}\right)\mathrm{d\λ},$

In formula: for polychromatic light integral mean diffraction efficiency, λ _min, λ _maxfor minimum value and the maximal value of wavelength in service band, λ ₁, λ ₂for the operation wavelength pair in service band.

4. multilayer diffraction optical element eccentric error analytical approach according to claim 1, is characterized in that, described each layer diffraction element microstructure height value formula is:

$\left\{\begin{array}{c}{H}_1=\frac{{\mathrm{\λ}}_1(n_2\left({\mathrm{\λ}}_2\right)-1)-{\mathrm{\λ}}_2(n_2\left({\mathrm{\λ}}_1\right)-1)}{({\mathrm{\η}}_1\left({\mathrm{\λ}}_1\right)-1)(n_2\left({\mathrm{\λ}}_2\right)-1)-(n_1\left({\mathrm{\λ}}_2\right)-1)(n_2\left({\mathrm{\λ}}_1\right)-1)}\\ H_2=\frac{{\mathrm{\λ}}_2(n_1\left({\mathrm{\λ}}_1\right)-1)-{\mathrm{\λ}}_1(n_1\left({\mathrm{\λ}}_2\right)-1)}{(n_1\left({\mathrm{\λ}}_1\right)-1)(n_2\left({\mathrm{\λ}}_2\right)-1)-(n_1\left({\mathrm{\λ}}_2\right)-1)(n_2\left({\mathrm{\λ}}_1\right)-1)}\end{array}\right.,$

In formula: λ ₁, λ ₂for design wavelength pair, n ₁(λ ₁), n ₁(λ ₂), n ₂(λ ₁) and n ₂(λ ₂) to be respectively base material at design wavelength be λ ₁, λ ₂time refractive index.

5. multilayer diffraction optical element eccentric error analytical approach according to claim 1, is characterized in that, the expression of diffraction efficiency of the described multilayer diffraction optical element containing eccentric error is:

${\mathrm{\η}}_m\left(\mathrm{\λ}\right)={\sin c}^2\{m-[\frac{H_1}{\mathrm{\λ}}(n_1\left(\mathrm{\λ}\right)-1)+\frac{H_2}{\mathrm{\λ}}(n_2\left(\mathrm{\λ}\right)-1)+\frac{\mathrm{\Δ}{H}_2}{\mathrm{\λ}\sqrt{T^2+H_2^2}}(n_2\left(\mathrm{\λ}\right)-1)\left]\right\},$

In formula: η _m(λ) be diffraction efficiency, m is the order of diffraction time, and λ is operation wavelength, H ₁, H ₂represent the microstructure height of first, second layer of diffraction element, n ₁(λ), n ₂(λ) be the refractive index of base material when operation wavelength is λ, △ is eccentric error, and T is the cycle.