CN101794024A - Device and method for generating column vector beams - Google Patents

Abstract

The invention relates to a device and a method for generating column vector beams. TEM00 model beams outputted by a laser is changed into linearly polarized light through a polarizer after the beams are reshaped and expanded by a beam expander and a diaphragm; the beams enter a spatial light modulator or a complex filter, and form an array through an analyzer; the beams pass through a 4f system consisting of a pair of Fourier transform lenses and aperture diaphragms; the beams are added into a phase retarder for phase modulation so as to generate TEM10 (or TEM01) model beams, and are synthesized to the column vector beams by using an interference system. The device and the method have the advantages of low requirement on incident beam, simple device, easy operation, high polarization degree of the output column vector beams and the like.

Description

A kind of device and method that produces column vector beam

Technical field

The present invention relates to a kind of optical technology, particularly a kind of device and method that produces column vector beam.

Background technology

Radial polarisation light (Radial polarized beams) and position angle polarized light (azimuthalpolarized beams) are called column vector beam (Cylindrical polarized beams), and they have the symmetry of height.Since some characteristics of column vector beam may be used on the imprison of particle speed technology, optical tweezers, materials processing, particulate and control, fields such as large-numerical aperture lens and some special measuring methods.For example, column vector beam can be applicable to realize the super-resolution effect in the optical microscope system, because certain column vector beam is littler than the horizontal hot spot of the focus of linearly polarized light beam focusing in the same focusing system through focusing on back focus.The generation system of column vector beam is research and the indispensable precondition of utilizing circulation polarized light beam, has caused extensive concern.Produce column vector beam a variety of methods are arranged, such as, adjust the crystal of a branch of linearly polarized light by the nematic phase distortion of its directive aligned twisted, synthetic with the linearly polarized light that two bundle polarization directions are mutually orthogonal by interferometer, interfering after the phase place of modulated beam of light diverse location and synthesize, is exactly to utilize the method for interferometer and diffraction that the laser-based transverse mode is changed or the like simultaneously in addition.Every kind of method has superiority, and still, still exists some essence not enough, mainly is incident light requirement height and the column vector beam that can't produce high-polarization.

For the selection of zlasing mode, what common laser instrument was exported is that the fundamental transverse mode light beam (is designated as TEM ₀₀Mould).But, synthesize column vector beam, must use TEM ₁₀(or TEM ₀₁) the mould light beam.Like this, will change the output mode of laser instrument.By Principles of Laser as can be known, it promptly is to select fundamental transverse mode in a large amount of transverse modes that participate in vibration that common transverse mode is selected, greater than fundamental transverse mode, suppress its vibration according to the occupied space of high-order transverse mode thereby therefore in resonator cavity, add the diaphragm of certain size restriction and the chamber type of appropriate design resonator cavity and the diffraction loss that parameter will increase the high-order transverse mode.The method of selecting can be divided two kinds, and the first realizes the fundamental transverse mode running by the geometry and the chamber parameter of design resonator cavity, and it two is to add the modeling element in the chamber.Yet, keep TEM ₁₀(or TEM ₀₁) mould light beam and to suppress the mould on other rank will be more complicated, and the resonator cavity that changes laser also is cumbersome, such as, can design an annulus that has a line by the center of circle, can realize exporting TEM preferably in theory ₁₀(or TEM ₀₁) the mould light beam, yet very high for the accuracy requirement of intermediate filtered line, so be difficult in the reality realize.Along with developing rapidly of information society and kownledge economy, each rank pattern of laser is in various experiments and use more and more widely in manufacturing and designing, and seems more important in national defense construction and modern science and technology.The zlasing mode of therefore, can be simply, obtaining institute's palpus quickly is very crucial.

Summary of the invention

The problem that The present invention be directed to has proposed a kind of device and method that produces column vector beam, utilizes spatial light modulator to produce column vector beam, the method novelty, and have the advantage that is easy to handle.

Technical scheme of the present invention is: a kind of device that produces column vector beam, comprise laser instrument successively, beam expanding lens, diaphragm, the polarizer, spatial light modulator or complex filter, analyzer, Fourier transform lens, aperture, Fourier transform lens, phase delay device, and interference system, Laser Output Beam becomes linearly polarized light with expansion Shu Houzai through the polarizer through beam expanding lens and diaphragm shaping, light beam enters spatial light modulator or complex filter, light beam is through analyzer, form array, through forming the 4f system by a pair of Fourier transform lens and aperture, light beam adds phase delay device and carries out phase modulation (PM), obtains light beam through the synthetic column vector beam of interference system.

Described interference system comprises at least: semi-transparent semi-reflecting lens, plane mirror, relay system, phase-plate, 1/2nd wave plates, polarization close light microscopic.Described semi-transparent semi-reflecting lens is flat board or prism.It is prism or flat board that described polarization closes light microscopic.

Described laser instrument is a single mode fundamental transverse mode laser instrument.

Described beam expanding lens is Kepler's beam expanding lens or Galileo beam expanding lens.

Described spatial light modulator is the transmission-type spatial light modulator, is positioned on the front focal plane of first Fourier transform lens.

Described Fourier transform lens is at least two, and two lens place with optical axis, and the front focal plane of second lens overlaps with the back focal plane of first lens.

A kind of method that produces column vector beam comprises the device that produces column vector beam, it is characterized in that, comprises following concrete steps:

1) TEM ₀₀The mould light beam is an input beam: suppose input function be Gaussian function: g (x, y)=exp[-(x ²+ y ²)/w ²] TEM then ₀₀The light beam expression formula of mould is:

2) utilize spatial light modulator and optical element that it is sampled: to an input g (x y) samples, and then sampling function is:

Wherein

Be the Modulation and Amplitude Modulation effect of spatial light modulator or the effect of complex filter; Then the frequency spectrum of sampling function is

$G_s(f_x,f_y)=F\left\{\frac{x}{w}\mathrm{comb}(x/X)\mathrm{comb}(y/Y)g(x,y)\right\}$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right){\mathrm{\δ}}^{\left(\mathrm{1,0}\right)}(f_x,f_y)*F\left[\mathrm{comb}(x/X)\mathrm{comb}(y/Y)g(x,y)\right]$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right){\mathrm{\δ}}^{\left(\mathrm{1,0}\right)}(f_x,f_y)*\left[\underset{n=-\∞}{\overset{\∞}{\mathrm{\Σ}}}\underset{m=-\∞}{\overset{\∞}{\mathrm{\Σ}}}G(f_x-\frac{n}{X},f_y-\frac{n}{Y})\right]$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right)\underset{n=-\∞}{\overset{\∞}{\mathrm{\Σ}}}\underset{m=-\∞}{\overset{\∞}{\mathrm{\Σ}}}{G}^{\left(\mathrm{1,0}\right)}(f_x-\frac{n}{X},f_y-\frac{n}{Y})$

3) carry out Fourier transform: an input g (x, frequency spectrum y):

Carry out inversefouriertransform so and get TEM ₁₀The spectrum light beam of mould is:

In like manner can obtain TEM ₀₁The spectrum light beam of mould;

4) through interferometer system, synthetic at last needed column vector beam: establishing two input light is same-phase,

$E_{\left(x\right)}(r,\mathrm{\θ})=E_0\sqrt{\mathrm{\ρ}}\exp (-\mathrm{\ρ}/2)\cos \left(\mathrm{\θ}\right),$

$E_{\left(y\right)}(r,\mathrm{\θ})=E_0\sqrt{\mathrm{\ρ}}\exp (-\mathrm{\ρ}/2)\sin \left(\mathrm{\θ}\right),$

In the formula, r and θ represent polar coordinate system, E ₀Be amplitude, w is a waist of Gaussian beam spot radius, ρ=2r ²/ w ², carry out simple vector addition and subtract:

The position angle polarized light:

Radial polarisation light:

Beneficial effect of the present invention is: the present invention produces the device and method of column vector beam, and the manipulation of laser is to carry out under stable environment, and it is convenient to implement; System is unrestricted to the polarization state of incident beam, and low for the incident beam requirement, the system applies scope is wide; Can regulate the position of the polarizer and 1/2nd wave plates, produce the different an amount of light beams of post, need not change entire system structure, simple to operate.

Description of drawings

Fig. 1 produces the device example one optical system synoptic diagram of column vector beam for the present invention;

Fig. 2 produces the gray-scale map of the device example one spatial light modulator write signal of column vector beam for the present invention;

Fig. 3 produces the device example two optical system synoptic diagram of column vector beam for the present invention;

Fig. 4 produces the gray-scale map of the device example two spatial light modulator write signals of column vector beam for the present invention;

Fig. 5 produces the device example three optical system synoptic diagram of column vector beam for the present invention.

Embodiment

Basic design of the present invention is: with TEM ₀₀The mould light beam is an input beam, utilizes spatial light modulator and relevant optical that it is carried out the sampling of certain way then, or utilizes complex filter, and the Fourier transform that makes output beam is TEM ₁₀(or TEM ₀₁) frequency spectrum of mould light beam, on the frequency plane of above-mentioned frequency spectrum, carry out filtering, carry out again Fourier transform is obtained TEM ₁₀(or TEM ₀₁) the mould light beam; Make TEM ₁₀(or TEM ₀₁) mould light beam process interferometer system, synthetic at last needed column vector beam.Its theoretical foundation is:

Suppose that input function is a Gaussian function

g(x，y)＝exp[-(x ²+y ²)/w ²] (1)

Know that easily following formula is TEM ₀₀The light beam expression formula of mould;

Adopt function

$g_c(x,y)=\frac{x}{w}\mathrm{comb}(x/X)\mathrm{comb}(y/Y)---\left(2\right)$

(x y) samples, and then sampling function is to an input g

$g_s(x,y)=\frac{x}{w}\mathrm{comb}(x/X)\mathrm{comb}(y/Y)g(x,y)---\left(3\right)$

Wherein

Be the Modulation and Amplitude Modulation effect of spatial light modulator or the effect of complex filter; Then the frequency spectrum of sampling function is

$G_s(f_x,f_y)=F\left\{\frac{x}{w}\mathrm{comb}(x/X)\mathrm{comb}(y/Y)g(x,y)\right\}$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right){\mathrm{\δ}}^{\left(\mathrm{1,0}\right)}(f_x,f_y)*F\left[\mathrm{comb}(x/X)\mathrm{comb}(y/Y)g(x,y)\right]$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right){\mathrm{\δ}}^{\left(\mathrm{1,0}\right)}(f_x,f_y)*\left[\underset{n=-\∞}{\overset{\∞}{\mathrm{\Σ}}}\underset{m=-\∞}{\overset{\∞}{\mathrm{\Σ}}}G(f_x-\frac{n}{X},f_y-\frac{n}{Y})\right]---\left(4\right)$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right)\underset{n=-\∞}{\overset{\∞}{\mathrm{\Σ}}}\underset{m=-\∞}{\overset{\∞}{\mathrm{\Σ}}}{G}^{\left(\mathrm{1,0}\right)}(f_x-\frac{n}{X},f_y-\frac{n}{Y})$

Input g (x, frequency spectrum y)

$G(f_x,f_y)=\mathrm{\π}{w}^2\exp [-{\mathrm{\π}}^2{w}^2(f_x^2+f_y^2)]---\left(5\right)$

So

$G^{\left(\mathrm{1,0}\right)}(f_x,f_y)=-2{\mathrm{\π}}^3{w}^4{f}_x\exp [-{\mathrm{\π}}^2{w}^2(f_x^2+f_y^2)]---\left(5\right)$

(5) formula is carried out inversefouriertransform get TEM ₁₀The light beam of mould in like manner can obtain TEM ₀₁The light beam of mould.The spatial light modulator that is utilized is the spatial light modulator of transmission-type electrical addressing, and its light of reading is TEM ₀₀The mould light beam, writing () signal is straight line two dimensional gray figure.

Utilize TEM ₀₁And TEM ₁₀The mould light beam can synthesize column vector beam, and establishing two input light is same-phase,

$E_{\left(x\right)}(r,\mathrm{\θ})=E_0\sqrt{\mathrm{\ρ}}\exp (-\mathrm{\ρ}/2)\cos \left(\mathrm{\θ}\right),$

$E_{\left(y\right)}(r,\mathrm{\θ})=E_0\sqrt{\mathrm{\ρ}}\exp (-\mathrm{\ρ}/2)\cos \left(\mathrm{\θ}\right)$

In the formula, r and θ represent polar coordinate system, E ₀Be amplitude, w is a waist of Gaussian beam spot radius, ρ=2r ²/ w ²Carrying out simple vector addition subtracts:

$E_{\left(\mathrm{\θ}\right)}(r,\mathrm{\θ})=\hat{y}{E}_{\left(x\right)}(r,\mathrm{\θ})-\hat{x}{E}_{\left(y\right)}(r,\mathrm{\θ})=\widehat{\mathrm{\θ}}{E}_0\sqrt{\mathrm{\ρ}}\exp (-\mathrm{\ρ}/2),---\left(6\right)$

$E_{\left(r\right)}(r,\mathrm{\θ})=\hat{x}{E}_{\left(x\right)}(r,\mathrm{\θ})+\hat{y}{E}_{\left(y\right)}(r,\mathrm{\θ})=\hat{r}{E}_0\sqrt{\mathrm{\ρ}}\exp (-\mathrm{\ρ}/2),---\left(7\right)$

In the formula

With Be position angle and unit vector radially, its Chinese style (6) is represented position angle polarized light and radial polarisation light respectively with formula (7).

By being equipped with beam expanding lens, diaphragm, the polarizer, spatial light modulator or complex filter, analyzer, Fourier transform lens, aperture, Fourier transform lens, phase delay device successively with symmetrical optical axis on the incident beam direction of propagation, and interference system.

Laser instrument output TEM ₀₀The mould light beam; This light beam becomes linearly polarized light with expansion Shu Houzai through the polarizer through beam expanding lens and diaphragm shaping, if laser instrument is output as the then above-mentioned polarizer of linearly polarized light and can omits; Light beam enters spatial light modulator or complex filter; Light beam forms array through analyzer; Light beam is through forming the 4f system by a pair of Fourier transform lens and aperture; Lobe adding phase delay device to light beam carries out phase modulation (PM) again, and making two lobe phase differential is (2n+1) π, and n is the natural number more than or equal to 0, obtains TEM ₁₀(or TEM ₀₁) the mould light beam; TEM ₁₀(or TEM ₀₁) the mould light beam is through the synthetic column vector beam of interference system; The above-mentioned interference system comprises at least: semi-transparent semi-reflecting lens, plane mirror, relay system, phase-plate, 1/2nd wave plates, polarization close light microscopic.

Produce the device example one optical system synoptic diagram of column vector beam as shown in Figure 1, single-mode laser 10 output fundamental transverse mode light beams; Through beam expanding lens 11; Beam center is through the center of diaphragm 21, obtains expanding the TEM after the bundle shaping ₀₀The mould light beam; Light beam becomes linearly polarized light through the polarizer 31; Light beam enters spatial light modulator 35; Light beam passes through analyzer 32 again, and get a beam array this moment, utilize the 4f system to carry out spatial filtering; Fourier transform lens 41 is coaxial with the front optical system, and its front focal plane is positioned at spatial light modulator 35 places; Aperture 22 is positioned at the back focal plane place of Fourier transform lens 41, and its center is on optical axis; The front focal plane of Fourier transform lens 42 overlaps with the back focal plane of Fourier transform lens 41, and two lens are coaxial; Light beam carries out being divided into two lobes behind the spatial filtering through the 4f system, need carry out phase modulation (PM) to a lobe wherein, a lobe that adds light beam in Fourier transform lens 42 backs adds phase delay device 33 and carries out phase modulation (PM), and making two lobe phase differential is (2n+1) π, and n is the natural number more than or equal to 0; Can obtain TEM this moment ₁₀(or TEM ₀₁) the mould light beam;

Gained TEM ₁₀(or TEM ₀₁) the mould light beam; Through becoming the aplanatic light beam of two bundles behind the semi-transparent semi-reflecting lens 51; A branch of light becomes TEM through relay system 52 ₀₁(or TEM ₁₀) mould light beam and change optical axis direction, select one of plus thirty wave plate 54 according to required post vector light beam; Another Shu Guang passes through the plane mirror of placing into 45 degree 55 and changes optical axis direction through variable phase plate 53; The effect of variable phase plate 53 is to regulate and makes the preceding light path of two light beam coherence stack identical; Two-beam closes light microscopic 56 coherence stack at polarization, from other end output column vector beam.

Fig. 2 is the gray-scale map of example one spatial light modulator write signal, and this gray-scale map can make TEM when being used in the described optical system of Fig. 1 ₀₀The mould light beam becomes TEM ₁₀The mould light beam.Its gray scale function is: T=|x|; Wherein be with the horizontal direction x axle its represent normalized distance value, with the vertical direction be the T axle its represent normalized rate value, be initial point with the picture centre.During Gaussian beam incident, the center of this gray level image on the center that makes Gaussian beam and the spatial light modulator overlaps.

Produce the device example two optical system synoptic diagram of column vector beam as shown in Figure 3, single-mode laser 10 output fundamental transverse mode light beams; Through beam expanding lens 11; Beam center is through the center of diaphragm 21, obtains expanding the TEM after the bundle shaping ₀₀The mould light beam; Be divided into aplanatic two-beam behind this light beam process becomes miter angle to place with optical axis the semi-transparent semi-reflecting lens 71, the first bundle light is propagated along former optical axis direction, the second bundle light through with catoptron 72 reflection of the parallel placement of semi-transparent semi-reflecting lens after to restraint the direction of propagation of light parallel in its direction of propagation and first, and close together; Two light beams becomes linearly polarized light through the polarizer 31; Light beam enters spatial light modulator 35; Light beam passes through analyzer 32 again, and get two beam arrays this moment, and two arrays all will utilize the 4f system to carry out spatial filtering; Fourier transform lens 41 is coaxial with the front optical system, and its front focal plane is positioned at spatial light modulator 35 places; Aperture 22 is positioned at the back focal plane place of Fourier transform lens 41, and its center is on optical axis; The front focal plane of Fourier transform lens 42 overlaps with the back focal plane of Fourier transform lens 41, and two lens are coaxial; Every light beam carries out being divided into two lobes behind the spatial filtering through the 4f system, need carry out phase modulation (PM) to a lobe wherein, the lobe adding phase delay device 33 and the phase delay device 34 that add two light beams in Fourier transform lens 42 backs carry out phase modulation (PM), making two lobe phase differential of every light beams is (2n+1) π, n is the natural number more than or equal to 0, and above-mentioned phase delay device only carries out phase modulation (PM) to a wherein lobe of each light beam; Can obtain TEM this moment ₁₀And TEM ₀₁The mould light beam;

Gained TEM ₁₀And TEM ₀₁The mould light beam; Two-beam makes optical propagation direction change 90 degree after through the plane mirror 58 that becomes miter angle with optical axis and place and plane mirror 59; Wherein a branch ofly can select one of plus thirty wave plate 54 according to required post vector light beam; Another Shu Guang passes through the plane mirror of placing into 45 degree 55 and changes optical axis direction through variable phase plate 53; The effect of variable phase plate 53 is to regulate and makes the preceding light path of two light beam coherence stack identical; Two-beam closes light microscopic 56 coherence stack at polarization, from other end output column vector beam.

Fig. 4 is the another kind of gray-scale map of example two spatial light modulator write signals, and this gray-scale map can make TEM when being used in the described optical system of Fig. 3 ₀₀The mould light beam becomes TEM ₁₀And TEM ₀₁The mould light beam.Its left-half gray scale function is: T=|x|; Wherein be with the horizontal direction x axle its represent normalized distance value, be the T axle with the vertical direction, it represents normalized rate value, is initial point with this half parts of images center.During Gaussian beam incident, the center of this gray level image left-half on the center that makes Gaussian beam and the spatial light modulator overlaps.Its right half part gray scale function is: T=|y|; Wherein be with the vertical direction y axle its represent normalized distance value, be the T axle with the horizontal direction, it represents normalized rate value, is initial point with this half parts of images center.During Gaussian beam incident, the center of this gray level image right half part on the center that makes Gaussian beam and the spatial light modulator overlaps.

Similar to the principle of embodiment 1 or 2, produce the device example three optical system synoptic diagram of column vector beam as shown in Figure 5, utilize complex filter or transparent display screen etc. to replace the Modulation and Amplitude Modulation effect of spatial light modulator and the phase modulation (PM) effect of 1/2nd wave plates, also can produce column vector beam.

The transmitance of complex filter 39 is: T=|x|, wherein be with the horizontal direction x axle its represent normalized distance value, with the vertical direction be the T axle its represent normalized rate value, be initial point with the complex filter center; Complex filter plays the effect of phase modulation simultaneously, promptly has on the direction of incident light polarization, and the left side of complex filter and right-hand part optical path difference are (2n+1) π, and n is the natural number more than or equal to 0.During Gaussian beam incident, the center of Gaussian beam is overlapped with the center of complex filter.

Claims (9)

1. device that produces column vector beam, it is characterized in that, comprise laser instrument successively, beam expanding lens, diaphragm, the polarizer, spatial light modulator or complex filter, analyzer, Fourier transform lens, aperture, Fourier transform lens, phase delay device, and interference system, Laser Output Beam becomes linearly polarized light with expansion Shu Houzai through the polarizer through beam expanding lens and diaphragm shaping, light beam enters spatial light modulator or complex filter, light beam is through analyzer, form array, through forming the 4f system by a pair of Fourier transform lens and aperture, light beam adds phase delay device and carries out phase modulation (PM), obtains light beam through the synthetic column vector beam of interference system.

2. according to the device of the described generation column vector beam of claim 1, it is characterized in that described interference system comprises at least: semi-transparent semi-reflecting lens, plane mirror, relay system, phase-plate, 1/2nd wave plates, polarization close light microscopic.

3. according to the device of the described generation column vector beam of claim 2, it is characterized in that described semi-transparent semi-reflecting lens is flat board or prism.

4. according to the device of the described generation column vector beam of claim 2, it is characterized in that it is prism or flat board that described polarization closes light microscopic.

5. according to the device of the described generation column vector beam of claim 1, it is characterized in that described laser instrument is a single mode fundamental transverse mode laser instrument.

6. according to the device of the described generation column vector beam of claim 1, it is characterized in that described beam expanding lens is Kepler's beam expanding lens or Galileo beam expanding lens.

7. according to the device of the described generation column vector beam of claim 1, it is characterized in that described spatial light modulator is the transmission-type spatial light modulator, be positioned on the front focal plane of first Fourier transform lens.

8. according to the device of the described generation column vector beam of claim 1, it is characterized in that described Fourier transform lens is at least two, and two lens place with optical axis, the front focal plane of second lens overlaps with the back focal plane of first lens.

9. a method that produces column vector beam comprises the device that produces column vector beam, it is characterized in that, comprises following concrete steps:

1) TEM ₀₀The mould light beam is an input beam: suppose input function be Gaussian function: g (x, y)=exp[-(x ²+ y ²)/w ²] TEM then ₀₀The light beam expression formula of mould is:

2) utilize spatial light modulator and optical element that it is sampled: to an input g (x y) samples, and then sampling function is:

Wherein

Be the Modulation and Amplitude Modulation effect of spatial light modulator or the effect of complex filter; Then the frequency spectrum of sampling function is

$G_s(f_x,f_y)=F\left\{\frac{x}{w}\mathrm{comb}(x/X)\mathrm{comb}(y/Y)g(x,y)\right\}$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right){\mathrm{\δ}}^{\left(\mathrm{1,0}\right)}(f_x,f_y)*F\left[\mathrm{comb}(x/X)\mathrm{comb}(y/Y)g(x,y)\right]$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right){\mathrm{\δ}}^{\left(\mathrm{1,0}\right)}(f_x,f_y)*\left[\underset{n=-\∞}{\overset{\∞}{\mathrm{\Σ}}}\underset{m=-\∞}{\overset{\∞}{\mathrm{\Σ}}}G(f_x-\frac{n}{X},f_y-\frac{n}{Y})\right]$

$=\frac{1}{w}\left(\frac{1}{-j2\mathrm{\π}}\right)\underset{n=-\∞}{\overset{\∞}{\mathrm{\Σ}}}\underset{m=-\∞}{\overset{\∞}{\mathrm{\Σ}}}{G}^{\left(\mathrm{1,0}\right)}(f_x-\frac{n}{X},f_y-\frac{n}{Y})$

3) carry out Fourier transform: an input g (x, frequency spectrum y):

Carry out inversefouriertransform so and get TEM ₁₀The spectrum light beam of mould is:

In like manner can obtain TEM ₀₁The spectrum light beam of mould;

4) through interferometer system, synthetic at last needed column vector beam: establishing two input light is same-phase,

$E_{\left(x\right)}(r,\mathrm{\θ})=E_0\sqrt{\mathrm{\ρ}}\exp (-\mathrm{\ρ}/2)\cos \left(\mathrm{\θ}\right),$

$E_{\left(y\right)}(r,\mathrm{\θ})=E_0\sqrt{\mathrm{\ρ}}\exp (-\mathrm{\ρ}/2)\sin \left(\mathrm{\θ}\right),$

In the formula, r and θ represent polar coordinate system, E ₀Be amplitude, w is a waist of Gaussian beam spot radius, ρ=2r ²/ w ², carry out simple vector addition and subtract:

The position angle polarized light:

Radial polarisation light:

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