CN107589480B - Diaphragm and method for inhibiting axial light intensity oscillation of diffraction-free light beam with limited size - Google Patents
Diaphragm and method for inhibiting axial light intensity oscillation of diffraction-free light beam with limited size Download PDFInfo
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- CN107589480B CN107589480B CN201711001169.9A CN201711001169A CN107589480B CN 107589480 B CN107589480 B CN 107589480B CN 201711001169 A CN201711001169 A CN 201711001169A CN 107589480 B CN107589480 B CN 107589480B
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Abstract
The invention discloses a diaphragm and a method for inhibiting axial light intensity oscillation of a diffraction-free light beam with a limited size; the duty ratio of the transmission area on the circumference corresponding to any radius r on the light hole of the diaphragm is equal to the amplitude transmission coefficient on the circumference of the radius r in the soft boundary apodization technology, and the amplitude transmission coefficient is expressed as:wherein T (r) is the amplitude transmission coefficient function in the soft boundary apodization technique,and the central angle is the central angle corresponding to the transmission area on the circumference corresponding to any radius r on the light transmission hole of the diaphragm. Compared with the existing soft boundary apodization technology, the invention is easier to implement on the premise of effectively inhibiting the axial light intensity oscillation of the diffraction-free light beam with limited size, obviously improves the working efficiency, and can also improve the precision and reduce the cost.
Description
Technical Field
The invention relates to the technical field of optics, in particular to a diaphragm and a method for inhibiting axial light intensity oscillation of a diffraction-free light beam with a limited size.
Background
The concept of a non-diffracted beam was first proposed by Durnin et al, which is a special type of beam whose intensity distribution remains unchanged over any cross-section perpendicular to the direction of propagation as it propagates in free space. Because the light intensity distribution of the non-diffracted light beam does not depend on the axial position, the requirement on the accurate alignment of an experimental system is greatly reduced, and the stability and the accuracy of the system are improved, the method has wide application prospects in the aspects of laser processing, interferometry, optical capturing and the like. From a mathematical point of view, an undiffracted beam can satisfy the wave equation of a light wave in free space, or an undiffracted beam is a solution to the wave equation in free space.
However, since an undiffracted light beam of an infinite size contains infinite energy, an undiffracted light beam of an infinite size cannot be generated in practical applications, and only an undiffracted light beam of a finite size can be obtained. Theoretically, an undiffracted beam of finite size can be obtained after passing through a circular hole of finite size.
After the undiffracted light beam with infinite size passes through the circular hole with finite size, the light intensity distribution has obvious oscillation in the direction of the optical axis passing through the center of the circular hole, which is caused by the strong diffraction effect generated by the sudden blocking of the incident light field by the boundary of the circular hole. In order to suppress the oscillation effect of the axial light intensity, the following methods are generally adopted in the prior art: a gradual amplitude modulation plate is arranged on the plane of the round hole so as to gradually reduce the amplitude transmission coefficient of incident light from 1 to 0 near the boundary of the round hole, thereby inhibiting the oscillation of axial light intensity. For example, when the amplitude transmission coefficient is gradually reduced from 1 to 0 by using a trigonometric function, perfect suppression of the axial light intensity oscillation can be achieved, and stable axial light intensity distribution can be recovered. In the above scheme, since the amplitude transmission coefficient gradually decreases from 1 to 0, it is generally called a soft boundary, and a technique for realizing the soft boundary is called a soft boundary apodization technique.
Although a stable axial light intensity distribution can be theoretically obtained by the above method, the following two difficulties are faced in practical applications: first, amplitude modulation plates are typically implemented using spatial light modulators, but the amplitude modulation obtained with spatial light modulators is difficult to achieve exactly the same value as theoretically designed, and the effect of such differences in amplitude transmission coefficients on the final diffracted light intensity is to be studied further; secondly, to achieve accurate control of amplitude modulation, both the spatial light modulator is required to be large in overall size and to be as high in spatial resolution as possible, which are difficult to achieve simultaneously.
Disclosure of Invention
In view of the above, the present invention is directed to an effective and easy-to-implement diaphragm and method for suppressing axial light intensity oscillation of a diffraction-free beam of limited size.
Based on the above object, the invention provides a diaphragm for suppressing axial light intensity oscillation of a diffraction-free light beam with a limited size, wherein the duty ratio of a transmission area on the circumference corresponding to any radius r on a light hole of the diaphragm is equal to an amplitude transmission coefficient on the circumference of the radius r in a soft boundary apodization technology, and the amplitude transmission coefficient is expressed as:
wherein T (r) is the amplitude transmission coefficient function in the soft boundary apodization technique,and the central angle is the central angle corresponding to the transmission area on the circumference corresponding to any radius r on the light transmission hole of the diaphragm.
In some embodiments, the amplitude transmission coefficient function in the soft boundary apodization technique is a trigonometric function expressed as:
wherein R is the outer circle radius of the light hole in the soft boundary apodization technology, εR is the inner circle radius of the light hole in the soft boundary apodization technology, ε is the smoothing parameter;
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in some embodiments, the amplitude transmission coefficient function in the soft boundary apodization technique is a linear function expressed as:
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in some embodiments, the amplitude transmission coefficient function in the soft boundary apodization technique is a gaussian function expressed as:
wherein ω is an optimization parameter, and ω is not equal to 0;
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in some embodiments, the amplitude transmission coefficient function in the soft boundary apodization technique is a butterworth function expressed as:
wherein, D and N are both optimization parameters, D is not equal to 0, N is a natural number;
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in some embodiments, the amplitude transmission coefficient function in the soft boundary apodization technique is a sine function, expressed as:
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in some embodiments, the amplitude transmission coefficient function in the soft boundary apodization technique is a bessel function expressed as:
wherein J is 1 Is a first order Bessel function;
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in some embodiments, the amplitude transmission coefficient function in the soft boundary apodization technique is a flat top gaussian function expressed as:
wherein N and omega are optimization parameters, omega is not equal to 0, and N is a natural number; l (L) n Is Laguerre polynomial, C n The expansion coefficient is given by the specific form:
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in another aspect, the present invention also provides a method for suppressing axial light intensity oscillation of a limited-size non-diffracted beam, including:
an undiffracted beam of infinite size is transmitted through the diaphragm as claimed in any one of the preceding claims.
From the above, it can be seen that the diaphragm and the method for suppressing the axial light intensity oscillation of the non-diffraction beam with limited size provided by the invention are easier to implement, remarkably improve the working efficiency, and can also improve the precision and reduce the cost compared with the existing soft boundary apodization technology on the premise of effectively suppressing the axial light intensity oscillation of the non-diffraction beam with limited size.
Drawings
In order to more clearly illustrate the embodiments of the invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, it being obvious that the drawings in the following description are only some embodiments of the invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a schematic view of a light hole in soft-boundary apodization;
FIG. 2 is a schematic diagram of duty cycle of a transmission region on a circumference corresponding to an arbitrary radius r on a light hole according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of the outline shape of the edge of the light hole according to an embodiment of the present invention;
FIG. 4 is an axial intensity oscillation curve after Bessel beam transmission through a circular aperture;
FIG. 5 is a graph of the axial intensity distribution of a Bessel beam after using a soft boundary apodization technique and a hard boundary apodization technique according to an embodiment of the invention, respectively;
FIG. 6 is a graph showing the axial intensity distribution of a Bessel beam when different ε values are selected in the hard boundary apodization technique according to an embodiment of the present invention.
Detailed Description
The present invention will be further described in detail below with reference to specific embodiments and with reference to the accompanying drawings, in order to make the objects, technical solutions and advantages of the present invention more apparent.
The embodiment of the invention provides a diaphragm and a method for inhibiting axial light intensity oscillation of a diffraction-free light beam with a limited size. The following describes the technical scheme of the invention by taking Bessel beams as examples. Obviously, the diaphragm and the method of the present invention can be applied to other finite-size non-diffracted beams, such as plane waves, based on the nature of the finite-size non-diffracted beams.
First, consider the diffraction case of a bessel beam after passing through a circular hole of a finite size. The incident plane is xy plane, positioned at z 0 At=0m. The incident Bessel beam propagates along the z-axis with an amplitude that satisfies J 0 = (βr) function, where J 0 Is a zero-order bessel function of the first class, beta is a propagation constant,is the distance between any point (x, y, 0) on the incident surface and the origin (0, 0).
Since the size of the circular hole is far greater than the wavelength of the incident light wave, the diffraction effect of the incident light at the boundary of the circular hole is negligible, and for the observation point (x ', y', z) behind the circular hole, the light field distribution is calculated by adopting a complete Rayleigh Li-Soxhlet method as follows:
wherein the method comprises the steps ofRepresenting the distance between the source point on the entrance face and the field point on the viewing face. />Representing the incident field at any point on the incident surface. In the above light field distribution formula, when x '=y' =0, the intensity distribution of the bessel beam in the axial direction after passing through the circular hole can be obtained.
According to the calculation result of the light field distribution formula, when the Bessel light beam passes through a circular hole with a limited size, the axial light intensity can generate strong oscillation, which is caused by the sudden blocking of the incident field on the boundary of the circular hole. In order to suppress axial light intensity oscillation, in the prior art, there is a soft boundary apodization technique based on trigonometric functions, and the amplitude transmission coefficient function inside a circular hole on an incident surface is:
where T (r) represents the amplitude transmission coefficient at the entrance face and ε [0,1] is a smoothing parameter. In the case of soft boundary apodization technique, the amplitude transmission coefficient of the incident field gradually decreases from 1 to 0 in the annular region R e [ epsilon R, R ], as shown in FIG. 1, where the white and black dashed lines represent two circles of radius R and epsilon R, respectively. In a circle with radius epsilon R, the amplitude transmission coefficient of the incident light is 1; in the annular region between the radius epsilon R and the radius R, the amplitude transmission coefficient of the incident light monotonically decreases from 1 to 0; outside the circle with radius R, the amplitude transmission coefficient of the incident light is 0. By this soft boundary apodization technique, stable transmission of the bessel beam can be achieved over a longer axial transmission distance range.
Corresponding to the soft boundary apodization technique, embodiments of the present inventionThe proposed diaphragm and method for suppressing the axial light intensity oscillation of the diffraction-free light beam with limited size can be called as hard boundary apodization technology. In the above trigonometric-function-based amplitude transmission coefficient function formula, the light field at any observation point (x ', y', z) in the transmission region is the diffraction superposition at that point of all sources on the incident plane. Since the incident field on the incident plane satisfies the Bessel function distribution, has circular symmetry, and the amplitude transmission coefficient on the incident plane also satisfies the circular symmetry according to the above-mentioned trigonometric function-based amplitude transmission coefficient function formula, the radial distance on the incident planeThe diffraction contributions at the on-axis observation points (x ', y', 0) are equal for the same point light source. Thus, we can do an equivalent exchange of converting the amplitude transmission coefficient into the duty cycle of the transmission region. For example, referring to FIG. 1, if a certain radius r 0 Amplitude transmission coefficient T (r) 0 ) =0.5, then we can let the incident light field have a radius r 0 The above mentioned transformation will not affect the light field distribution of the on-axis viewpoint, while the other half is not. When the amplitude transmission coefficient is other, the only change that is required is the duty cycle of the light transmitting region and the light non-transmitting region, as shown in fig. 2, where the hatched region is the light transmitting region.
In the embodiment of the invention, the shape (boundary outline) of the light hole in the hard boundary apodization technology and the amplitude transmission coefficient in the soft boundary apodization technology are in a corresponding relation. Specifically, referring to FIG. 2, the shaded portion represents a radius R (where R ε εR, R]) A light-transmitting part on the circumference of the circular ring, the opening angle of which to the center of the circle isTherefore, the proportion of the light-transmitting portion over the entire circumference is +.>Whereas in the trigonometric-based amplitude transmission coefficient function formula described above, at any radius r,the amplitude transmission coefficient of the incident light is T (r). From the idea of converting the amplitude transmission coefficient into the transmission area duty cycle, the following formula can be obtained:
the above is mathematically transformed to obtain the boundary equation (polar coordinates) of the transmission aperture:
in the plane of incidence, if all points are to be determined by the above equation of the boundary of the transmission apertureIt is drawn that the outline shape of the edge of the transmission hole as shown in fig. 3 is to be formed, i.e., the transmission hole is a heart-like hole (in fig. 3, a white dotted line and a black dotted line represent two circles with radii R and er, respectively). When the incident Bessel beam transmits the heart-like hole, the amplitude transmission coefficient inside the heart-like hole is 1, and the amplitude transmission coefficient outside the heart-like hole is 0, then the oscillation of the transmitted light intensity in the transmission axis direction is effectively suppressed.
To further illustrate the technical effects of the embodiments of the present invention, the inventors have selected a set of parameters and have performed numerical simulations. The selected numerical simulation parameters are as follows: the transverse transmission parameters of the bessel beam are: beta=10 4 m -1 The wavelength of the incident light is: λ=0.5 μm, and the radius of the outer circle of the light transmitting hole in the soft boundary apodization technique is: r=50 mm.
Referring to fig. 4, an axial light intensity oscillation curve after the bezier beam transmits through the circular hole is shown. It can be seen that the axial light intensity of the Bessel beam after being transmitted through the round hole has obvious oscillation, and the relative error of the light intensity is +/-7%.
Referring to fig. 5, the bezier beam axial intensity distribution curves after using the soft boundary apodization technique and the hard boundary apodization technique of the embodiments of the present invention, respectively, are shown.
When the soft boundary apodization technology is used, an amplitude modulation plate is added on an incident surface, and the amplitude distribution is given by the above-mentioned trigonometric function-based amplitude transmission coefficient function formula, at the moment, the light field distribution on the incident surface is as follows:wherein->When the parameter epsilon=0.5 is selected, the axial light intensity distribution of the transmission region is shown as a gray solid line in fig. 5. Obviously, by utilizing the soft boundary apodization technology, axial light intensity oscillation is effectively restrained, and the diffraction-free transmission characteristic of the incident light beam is obtained in a relatively long axial transmission range.
When the hard boundary apodization technique of the embodiment of the present invention is used, when the parameter epsilon=0.5 is selected, the heart-like light transmitting holes obtained as in the foregoing embodiment are used, and the axial light intensity distribution is calculated. In the hard boundary apodization technique of the embodiments of the present invention, two points need to be noted: firstly, the light holes are heart-like holes, so that the integral range is not a whole round hole, but is a heart-like hole; second, in the light-transmitting region, the light field on the incident surface isThe effect of the amplitude modulation T (r) no longer has to be taken into account. In the case of the hard boundary apodization technique, the resulting axial light intensity distribution is shown by the black dashed line in fig. 5. As can be seen from fig. 5, the black dotted line completely coincides with the gray solid line, which indicates that the same effect as that of the soft boundary apodization technique is obtained by using the hard boundary apodization technique, that is, the oscillation of the axial light intensity is effectively suppressed, and the diffraction-free transmission characteristic of the transmitted light is obtained. In contrast, by using the hard boundary apodization technology of the embodiment of the invention, only the light holes with special shapes are required to be processed, and an amplitude modulation plate with gradually changed amplitude transmission coefficients in the soft boundary apodization technology is not required to be manufactured, so that the implementation is easier, the working efficiency is obviously improved, and the cost can be reduced.
Further, the inventors have optimized the smoothing parameter epsilon in order to obtain a more excellent axially stable light intensity distribution performance. A series of values epsilon are selected every 0.05 between 0.7 and 0.9, and under the hard boundary apodization technical condition, the axial light intensity distribution of the transmission area is shown in fig. 6, and different lines correspond to different epsilon under the condition of incident Bessel light beams. As can be seen from fig. 6, as the smoothing parameter epsilon increases, the axial diffraction distance also increases, but the uniformity of the axial light intensity decreases, for the following reasons: when ε is larger, the speed at which the transmittance decreases from 1 to 0 is faster, and therefore, the diffraction effect of incident light at the edge of the circular hole is more remarkable, so that the oscillation of the axial light intensity gradually increases. When epsilon goes towards 1, it then goes back to the situation shown in fig. 4. And comprehensively considering the uniformity of the axial light intensity and the maximum transmission distance, and selecting epsilon=0.8 as a preferred value.
Obviously, the aperture provided by the invention is not particularly limited to the amplitude transmission coefficient function under the soft boundary apodization technology condition, and if other forms of amplitude transmission coefficient functions are selected, the transmission hole boundary equation under the corresponding condition can be obtained respectively. Several other forms of amplitude transmission coefficient distribution are given below by way of several examples.
In an alternative embodiment, the amplitude transmission coefficient function in the soft boundary apodization technique is a linear function expressed as:
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in an alternative embodiment, the amplitude transmission coefficient function in the soft boundary apodization technique is a gaussian function expressed as:
wherein ω is an optimization parameter, and ω is not equal to 0;
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in an alternative embodiment, the amplitude transmission coefficient function in the soft boundary apodization technique is a butterworth function expressed as:
wherein, D and N are both optimization parameters, D is not equal to 0, N is a natural number;
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in an alternative embodiment, the amplitude transmission coefficient function in the soft boundary apodization technique is a sine function expressed as:
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in an alternative embodiment, the amplitude transmission coefficient function in the soft boundary apodization technique is a Bessel function expressed as:
wherein J is 1 Is a first order Bessel function;
thenAnd carrying out mathematical transformation to obtain a boundary equation of the transmission hole:
in an alternative embodiment, the amplitude transmission coefficient function in the soft boundary apodization technique is a flat-top gaussian function expressed as:
wherein N and omega are optimization parameters, omega is not equal to 0, and N is a natural number; l (L) n Is Laguerre polynomial, C n The expansion coefficient is given by the specific form:
thenThrough the process ofMathematical transformation can obtain the boundary equation of the transmission aperture:
based on the same inventive concept, the embodiment of the invention also provides a method for inhibiting the axial light intensity oscillation of the diffraction-free light beam with a limited size, which comprises the following steps: an undiffracted beam of infinite size is transmitted through the diaphragm as in the previous embodiment.
As can be seen from the above examples, the diaphragm and the method for suppressing the axial light intensity oscillation of the diffraction-free light beam with limited size convert the amplitude transmission coefficient into the duty ratio of the transmission area, compared with the existing soft boundary apodization technology, the hard boundary apodization technology provided by the invention is easier to realize in practical application, and the actual error only comes from the boundary, thereby greatly improving the accuracy of the experimental result and providing a more effective implementation scheme for practical application.
Those of ordinary skill in the art will appreciate that: the discussion of any of the embodiments above is merely exemplary and is not intended to suggest that the scope of the disclosure, including the claims, is limited to these examples; the technical features of the above embodiments or in the different embodiments may also be combined within the idea of the invention, the steps may be implemented in any order and there are many other variations of the different aspects of the invention as described above, which are not provided in detail for the sake of brevity.
The embodiments of the invention are intended to embrace all such alternatives, modifications and variances which fall within the broad scope of the appended claims. Therefore, any omission, modification, equivalent replacement, improvement, etc. of the present invention should be included in the scope of the present invention.
Claims (9)
1. A diaphragm for suppressing axial light intensity oscillation of a diffraction-free beam of finite size, characterized in that a duty ratio of a transmission area on a circumference corresponding to an arbitrary radius r on a light hole of the diaphragm is equal to an amplitude transmission coefficient on the circumference of the radius r in a soft boundary apodization technique, expressed as:
wherein T (r) is the amplitude transmission coefficient function in the soft boundary apodization technique,and the central angle is the central angle corresponding to the transmission area on the circumference corresponding to any radius r on the light transmission hole of the diaphragm.
2. The diaphragm for suppressing axial intensity oscillation of a finite-size diffraction-free beam of light according to claim 1, wherein the amplitude transmission coefficient function in the soft-boundary apodization technique is a trigonometric function expressed as:
wherein R is the outer circle radius of the light hole in the soft boundary apodization technology, εR is the inner circle radius of the light hole in the soft boundary apodization technology, ε is the smoothing parameter;
thenAnd the boundary equation of the light hole can be obtained through mathematical transformation:
3. the diaphragm for suppressing axial intensity oscillation of a finite-size non-diffracted beam of light according to claim 1, wherein the amplitude transmission coefficient function in the soft-boundary apodization technique is a linear function expressed as:
thenAnd the boundary equation of the light hole can be obtained through mathematical transformation:
4. the diaphragm for suppressing axial intensity oscillation of a finite size diffraction-free beam of light of claim 1, wherein the amplitude transmission coefficient function in the soft-boundary apodization technique is a gaussian function expressed as:
wherein ω is an optimization parameter, and ω is not equal to 0;
thenAnd the boundary equation of the light hole can be obtained through mathematical transformation:
5. the diaphragm for suppressing axial intensity oscillation of a finite-size undiffracted light beam according to claim 1, wherein the amplitude transmission coefficient function in the soft-boundary apodization technique is a butterworth function expressed as:
wherein, D and N are both optimization parameters, D is not equal to 0, N is a natural number;
thenAnd the boundary equation of the light hole can be obtained through mathematical transformation:
6. the diaphragm for suppressing axial intensity oscillation of a finite-size diffraction-free beam of light according to claim 1, wherein the amplitude transmission coefficient function in the soft-boundary apodization technique is a sine function expressed by:
thenAnd the boundary equation of the light hole can be obtained through mathematical transformation:
7. the diaphragm for suppressing axial intensity oscillation of a finite-size undiffracted light beam according to claim 1, wherein the amplitude transmission coefficient function in the soft-boundary apodization technique is a bessel function expressed as:
wherein J is 1 Is a first order Bessel function;
thenAnd the boundary equation of the light hole can be obtained through mathematical transformation:
8. the diaphragm for suppressing axial intensity oscillation of a finite-size diffraction-free beam of light according to claim 1, wherein the amplitude transmission coefficient function in the soft-boundary apodization technique is a flat-top gaussian function expressed as:
wherein N and omega are optimization parameters, omega is not equal to 0, and N is a natural number; l (L) n Is Laguerre polynomial, C n The expansion coefficient is given by the specific form:
thenAnd the boundary equation of the light hole can be obtained through mathematical transformation:
9. a method of suppressing axial intensity oscillations of a finite-sized non-diffracted beam, comprising:
an undiffracted beam of infinite size is transmitted through the diaphragm of any one of claims 1 to 8.
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