CN102043177A - Radial logarithm spiral phase plate (SPP) - Google Patents

Abstract

The invention provides a radial logarithm spiral phase plate (SPP) which is applied to generating a confluent hypergeometric beam. The radial logarithm SPP is the SPP that the phase modulation changes spirally along the azimuth angle direction, and along the radial logarithm change direction. The SPP is formed by the following steps: drawing a high-resolution grayscale image based on a phase expression program; copying the grayscale image to a reversal film by a photoetching method and then copying to a holographic plate by an exposure method; and developing, fixing and bleaching to obtain the SPP with refractive index and thickness modulation. The confluent hypergeometric light beam can be generated by incidence of a Laguerre-Gaussian beam on the radial logarithm SPP.

Description

Logarithm spiral phase board radially

Technical field

Patent of the present invention relates to a generation position phase singular point, and the optics position of optical vortex and hypergeometry interflow hollow beam is modulation panel mutually.

Background technology

Empty in the dark light beam with spiral position phase wavefront and phase place singular point has a very wide range of applications.It not only can be used for increasing the model volume of laser cavity, the photoconduction of light, and frequency shifts, the change of angular momentum, but also can be used as dark orphan in the self-focusing medium.The orbital angular momentum that the vortex light beam is had more can be used for the information decoding of free space optical communication.Particularly, not only, also can catch high refractive index medium, and damage less catching of biological living cells because empty light beam can be caught low refractive index dielectric in the dark in optical control field very advantageous.The more important thing is that in addition the entrained orbital angular momentum of optical vortex can rotate captive microscopic particle.Generally the formation of vortex light beam can realize by the modulation to laser beam.Modulated laser beam just can be produced the empty in the dark light beam with spiral position phase wavefront and phase place singular point by the spiral phase board with following position phase modulating function.

Common empty in the dark light beam has Laguerre-Gaussian beam and Bessel-Gauss beams at present.But also do not produce interflow the hypergeometry device or the method for empty light beam in the dark.

Summary of the invention

Produce the interflow hypergeometry problem of empty light beam in the dark in order to solve, the present invention's design provides a kind of radially logarithm spiral phase board.Its position phase modulating function is:

(1)

r ₀, θ ₀Be position phase modulation panel plane polar coordinates.N is a topological charge.w ₀For the radial coordinate amplification factor or dwindle the factor.γ is for logarithm amplification factor radially or dwindle the factor.

The technical scheme that patent of the present invention adopts is: to the programming of (1) formula, use different parameters w with Matlab under high voxel model ₀And γ, the phase grating pattern of having made several different topology exponent number n.For example accompanying drawing 1, n=4, w ₀=1, γ=1; Accompanying drawing 2, n=4, w ₀=1, γ=4.These grating modes figure is a gray-scale map, we convert gray-scale map to the bmp formatted file, by film recorder (Agfa plus CD writer, the corresponding 35mm of 4096 pixels * 2732 pixels * 23mm model film) directly imprinting on taking a picture (Kodak Ektachrome Professional 100 reversal films) with film.After the film imprinting finishes, need xerox and make holographic grating on the holographic dry plate.With the electronics exposure box holographic dry plate (Tianjin I type dry plate) is covered on the film operations such as it expose, develop then, stop showing, photographic fixing, flushing, processing such as bleaching obtain almost completely transparent phase grating at last.This grating has index modulation and thickness modulation concurrently, and based on index modulation, thereby light path and the Wave-front phase of having modulated incident light distribute.

The invention has the beneficial effects as follows: compare with existing spiral phase board, radially logarithm spiral phase board can be used for producing interflow hypergeometry light beam.With Laguerre-Gaussian beam incident logarithm spiral phase board radially, through radially promptly producing the interflow hypergeometry light beam that following (2) formula is represented behind the logarithm spiral phase board diffraction

$\×\underset{s=0}{\overset{l}{\mathrm{\Σ}}}{(-1)}^s{2}^s\frac{(l+m)!}{(m+s)!(l-s)!s!}{(1-\frac{{\mathrm{ikw}}_0^2}{2z})}^{-(2m-n+\mathrm{i\γ}+2s+2)/2}\mathrm{\Γ}\left(\frac{2m-n+\mathrm{i\γ}+2s+2}{2}\right)$

${\&times;}_1{F}_1(\frac{2m-n+\mathrm{i\&gamma;}+2s+2}{2},m-n+1,-\frac{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009100193408E00013.png"\; state="\{\{state\}\}">k</figure-callout>}}^2{\mathrm{\&rho;}}^2{w}_0^2}{2(2z-{\mathrm{ikw}}_0^2)z})---\left(2\right)$

Wherein Γ () is a gamma function, ₁F ₁() is confluent hypergeometric function.

Description of drawings

Fig. 1 is n=4, w ₀=1, the gray scale simulation that the radially logarithm spiral phase board position of γ=1 o'clock distributes mutually

Fig. 2 is n=4, w ₀=1, the gray scale simulation that the radially logarithm spiral phase board position of γ=10 o'clock distributes mutually

Fig. 3 uses the radially enforcement figure of logarithm spiral phase board generation interflow hypergeometry light beam

1. incident Laguerre-Gaussian beams among the figure, 2. outgoing interflow hypergeometry light beam, 3. logarithm spiral phase board radially

Embodiment

In Fig. 3, incident Laguerre Gaussian beam (1) is represented with following (4) formula:

Utilize beam-expanding system with incident beam change over diameter for 3mm (millimeter) about, radially logarithm spiral phase board position phase modulating part diameter is controlled at 4mm between the 6mm. and developer solution adopts Tianjin I type holographic plate D-19 developer solution commonly used. and stop bath is that aquae destillata adds a small amount of glacial acetic acid preparation stop bath employing F-5 stop bath. bleaching liquid prescription: 1.8g potassium dichromate, the 1ml concentrated sulphuric acid, the 4g potassium bromide.They are dissolved in temperature successively is not higher than in 40 ℃ the distilled water and adds to 1 L. flushing and all use aquae destillata. handling procedure and time: developed 4 minutes; Stop showing 1 minute; Washed 1 minute; Photographic fixing 5 minutes; Washed 1 minute; Bleached 5 minutes.

At Fig. 3 logarithm spiral phase board (2) rear surface light field complex amplitude radially:

(5)

Utilize near field fresnel diffraction integration can obtain from logarithm spiral phase board z place planar lightfield complex amplitude radially as (2) formula with shown in (3) formula. can see that the diffracted wave that (2) formula and (3) formula are represented is that topological charge is the empty in the dark light beam of m-n. position singular point is mutually being arranged on the optical axis. during as l=0, (4) formula is reduced to:

And (2) formula is reduced to:

$\&times;{(1-\frac{{\mathrm{ikw}}_0^2}{2z})}^{-(2m-n+\mathrm{i\&gamma;}+2)/2}\mathrm{\&Gamma;}\left(\frac{2m-n+\mathrm{i\&gamma;}+2}{2}\right)---\left(7\right)$

(7) formula is represented a typical interflow hypergeometry light beam.During l=1, (2) formula is reduced to:

${\&times;}_1{F}_1(\frac{2m-n+\mathrm{i\&gamma;}+2}{2},m-n+1,-\frac{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009100193408E00013.png"\; state="\{\{state\}\}">k</figure-callout>}}^2{\mathrm{\&rho;}}^2{w}_0^2}{2(2z-{\mathrm{ikw}}_0^2)z})$

$-2A{(1-\frac{{\mathrm{ikw}}_0^2}{2z})}^{-(2m-n+\mathrm{i\&gamma;}+4)/2}\mathrm{\&Gamma;}\left(\frac{2m-n+\mathrm{i\&gamma;}+4}{2}\right)$

${\&times;}_1{F}_1(\frac{2m-n+\mathrm{i\&gamma;}+4}{2},m-n+1,-\frac{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009100193408E00013.png"\; state="\{\{state\}\}">k</figure-callout>}}^2{\mathrm{\&rho;}}^2{w}_0^2}{2(2z-{\mathrm{ikw}}_0^2)z})---\left(8\right)$

(8) formula is represented the stack of two interflow hypergeometry light beams.

Claims (4)

1. logarithm spiral phase board radially is along azimuth direction θ ₀Phase change increases twist, it is characterized in that being logarithm simultaneously along the radial direction phase change increases.

。

2. according to the described radially logarithm of claim 1 spiral phase board, it is characterized in that radius parameter can be added one as required amplifies or dwindles factor w ₀

3. according to the described radially logarithm of claim 1 spiral phase board, it is characterized in that logarithm itself can add one as required amplifies or dwindles factor gamma.

4. according to the described radially logarithm of claim 1 spiral phase board, it is characterized in that to produce the empty in the dark light beam of interflow hypergeometry with Laguerre-Gaussian beam incident.

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