JP6784710B2 - Diffractive element design method - Google Patents

Description

The present invention relates to a diffraction element having a function of converting a light intensity pattern and a method for designing the diffraction element.

An optical diffraction element represented by a Fresnel lens is an optical component that converts a pattern of light intensity by utilizing the property as a wave of light, and is used in various industrial fields. A Fresnel lens utilizes the fact that light having a certain wavelength has periodicity at a wavelength pitch, and is generally a thin lens with a thick wall, and has a function of condensing light. In addition to Fresnel lenses, many diffractive elements that use wave optics to change the shape of the light beam have been developed and used.

Holography is a technology for converting the wave surface of an optical beam with a high degree of freedom. In holography, light containing a lot of information called object light and light called reference light are made to interfere with each other, and the interference fringes at this time are copied to a photosensitive medium. This copied interference fringe is called a hologram. When this hologram is irradiated with only the reference light used earlier, the intensity and phase are modulated, and light that reproduces the object light used at the time of fabrication is generated. That is, the light beam is converted from the reference light to the object light by this kind of diffraction element. Diffractive elements that use the principles of holography have very high fidelity and can reproduce the original object light.

However, as described above, in holography, the phase of the incident light (reference light) is modulated and the intensity is also modulated. Light that is not needed to faithfully reproduce object light is absorbed or scattered and removed. Therefore, it is inevitable that the power of the emitted light (reproduced object light) is attenuated with respect to the total power of the incident light, and the conversion efficiency is not always sufficiently high in terms of power.

On the other hand, unlike the hologram, which is a diffraction element that realizes holography described above, there is a diffraction element that only modulates the optical phase and does not change the light intensity, and this diffraction element is generally called a kinoform (). Non-Patent Document 1). Kinoform processes a curved surface pattern on the surface of a glass substrate and modulates the optical path length of light incident on this substrate approximately perpendicularly, thereby modulating the optical phase, but this does not cause intensity modulation. .. The Fresnel lens mentioned above can be said to be a special case of this quinoform. When high power light is incident, in a diffractive element that absorbs light such as a hologram, it is assumed that the element will be destroyed by heat generated by the absorbed power, and quinoform that does not perform intensity modulation is more advantageous. There is.

Yoshiki Ichioka, "Kinoform and its Applications," Optics Vol. 2, No. 3, pp. 133-152, 1973.

As described above, quinoforms that only perform phase modulation have advantages over holograms that perform intensity modulation, but instead, they are restricted only to phase modulation, making it difficult to convert the wave surface similar to holograms. And that limited the efficiency of power conversion.

The present invention has been made in view of such conventional problems, and an object of the present invention is to provide a diffraction element that realizes high power conversion efficiency in a quinoform that does not perform intensity modulation.

In order to solve the above-mentioned problems, the invention described in one embodiment causes incident light having a predetermined wavelength λ ₀ incident on the diffraction grating surface with a first electromagnetic field intensity distribution to have a desired light intensity on the imaging surface. It is a method of designing a diffraction element that phase-modulates so that it becomes an emitted light having a second electromagnetic field intensity distribution corresponding to the distribution, and is a second electromagnetic field on the imaging plane from a desired light intensity distribution on the imaging plane. The step of determining the intensity distribution, the step of determining the third electromagnetic field intensity distribution on the diffraction element surface based on the second electromagnetic field intensity distribution, and the diffraction element surface determined from the third electromagnetic field intensity distribution. The step of determining the phase modulation amount Φ _H (x, y) in the diffraction element from the phase after the phase conversion in the above and the phase of the incident light determined from the first electromagnetic field intensity distribution, and the determined phase modulation. A phase modulation amount Φ _HR (x, y) of 0 or more and less than 2π using the quantity Φ _H (x, y) and Φ _HR (x, y) = Φ _H (x, y) -2πN (N is an integer). A step of converting the converted phase modulation amount Φ _HR (x, y) to any one of the phase modulation amounts Φa (x, y) of 8 gradations or more between 0 and less than 2π. the thickness d (x, y) of the diffractive element on the basis the 8 phase modulation amount Φa approximating above gradation (x, y) to the wavelength lambda ₀ of the incident light to include the step of determining the This is a characteristic method for designing a diffraction element.

It is a figure explaining the relationship of the coordinates of a diffraction element and an image plane. It is a figure explaining the state of limitation of the range of phase modulation by a diffractive element. It is a figure explaining the difference between the curved surface for target phase modulation and the unevenness which can be actually processed. It is a figure of the gradation number dependence of the conversion efficiency of the light power from the incident light to the emitted light. It is a figure which shows the processing flow of the design method of a diffraction element. It is a figure of the light intensity pattern intended to generate in Example 1. FIG.

Hereinafter, embodiments of the present invention will be described in detail.

(About phase modulation to be performed on the diffraction element surface)
FIG. 1 is a diagram for explaining the relationship between the coordinates of the diffraction element and the image plane. In FIG. 1, when light is incident on a diffraction element arranged on the diffraction element surface 1, it proceeds to the right after phase modulation is applied by the diffraction element, and a case where a desired image is formed on the imaging surface 2 is considered. ..

As shown in FIG. 1, xy orthogonal coordinates are set in the plane of the diffraction element surface 1, and z coordinates orthogonal to the orthogonal coordinates are set. Further, in the same manner, the orthogonal coordinates of x'y'are set in the plane of the image plane 2, and the z'coordinates orthogonal to the orthogonal coordinates are set. In such an explanation, it is easy to understand that the axis of the xy coordinate and the axis of the x'y'coordinate are parallel and the origin is only deviated in the z direction, and it is often used, but the diffraction element emits light. In the case of the reflective type, the diffraction element surface 1 and the imaging surface 2 may not be parallel. Therefore, in the present embodiment, as shown in FIG. 1, the diffraction element surface 1 and the imaging surface 2 are not parallel. Represents a thing. Further, in the present specification, the "vector" is represented by bold (bold) or enclosed by <>.

Here, the coordinates of an arbitrary point P are <r> = (x, y, z) in the xyz coordinate system, and <r'> = (x', y', z') in the x'y'z'coordinate system. Then, it is assumed that the conversion between these coordinate systems is possible by rotation by the Cartesian matrix C and translation by the vector <b>. In other words

And. Here, assuming that the scalar electromagnetic field on the diffraction element surface 1 is U (x, y) with a complex amplitude, the scalar electromagnetic field U'(x', y') on the image plane 2 is

It is expressed as. The integration is performed over the entire in-plane area of the diffraction element. Also,

Is one point on the diffraction element surface _{1 <r p> = (x} , y, 0) and a point on the imaging plane _{2 <r p '> = (} x', y ', 0) the correlation between Representing function

Is. Here, j is an imaginary unit, λ ₀ is a wavelength, k is a wave number, and k = 2π / λ ₀ . As a simple example, if the x'axis and x-axis, the y'axis and y-axis are parallel, and the z'axis and z-axis overlap and there is a z ₀ deviation.

Because it becomes

Because it is

Can be written as. As another example, when the x'axis and the x-axis are parallel, but the y'axis and the z'axis have an inclination of 45 degrees with respect to the y-axis and the z-axis, and there is a z ₀ deviation. Is

Because it becomes

Can be written as.

The method of obtaining the electromagnetic field on the image plane 2 from the electromagnetic field on the diffraction element surface 1 has been described above, but conversely, the electromagnetic field on the diffraction element surface 1 can be obtained from the electromagnetic field on the image plane 2.

Here, * represents the complex conjugate and also

Is. Or

May be. The integral of Eq. (2) may be considered to express the so-called Huygens principle by Eq. That is, G (x, y: x', y') represents a spherical wave that propagates radially from one point to the surroundings, and represents a spherical wave generated from all points on the surface of the diffraction element surface 1. It is considered that what is overlapped is an electromagnetic field formed in the plane of the image plane 2. Alternatively, according to the equation (8), the in-plane electromagnetic field of the diffraction element surface 1 is calculated by superimposing the spherical waves generated from the point light source on the surface of the image plane 2. In order to convert the light of the scalar electromagnetic field U ₀ (x, y) incident on the diffraction element surface 1 from the left side into the light of U'(x', y') on the image plane surface 2, first of all, it is concluded. The U (x, y) of the diffraction element surface 1 is calculated from the U'(x', y') of the image plane 2. from here,

The diffraction element may be provided with a function of performing optical modulation H (x, y) such that. By the way, since U ₀ (x, y) and the like are complex numbers, they are displayed in polar coordinates using two real functions.

A and Φ are real functions. Then, from equation (11),

It can be seen that the diffraction element should be designed so as to be. The latter Φ _H of the equation (13) can be realized by processing the surface of a glass plate or the like with a curved surface as described above. For example, when the surface of a glass plate having a refractive index n has a curved surface and the thickness has a distribution of d (x, y), light is vertically incident on the glass plate and transmitted therethrough.

The phase is modulated as follows. In this way, the phase modulation Φ _H to be performed on the diffraction element surface can be determined.

(About the fabrication of a diffraction element that realizes the desired phase modulation)
In the description so far, the phase range is not limited and may be any wide range. However, in a general diffraction element, the phase modulation is usually set to 0 to 2π by utilizing the periodicity of light. That is, using the integer N

A phase modulation Φ _HR limited to 0 to 2π that satisfies is used instead of Φ _H. In this case, according to the equation (14), the thickness of the diffraction element is

It will be modulated in the range of. d ₀ is a constant. Since the refractive index n of glass is about 1.5, when the wavelength λ ₀ is 1 μm, λ ₀ / (n-1) is about 2 μm. That is, an example of an actual diffraction element has a structure in which a fine curved surface pattern having a height of about ₀ to 2 μm is formed on the surface of a glass plate having a thickness of 1 mm, where d ₀ is 1 mm.

FIG. 2 shows how the range of phase modulation in the diffraction element is limited. The phase modulation 3 is the phase modulation Φ _H before limiting the phase range, and the phase modulation 4 is the phase modulation Φ _HR after the limitation. The phase modulation 3 changes the phase beyond 2π, but the phase modulation 4 limited to the range of 0 to 2π is also equivalent due to the periodicity of light. In the phase modulation 4, 0 on the vertical axis corresponds to the thickness d ₀ of the glass plate of the diffraction element, and 2π corresponds to the thickness d ₀ + λ ₀ / (n-1).

Further, in the case of a reflection type diffraction element in which light is incident at an incident angle of 45 °, is also reflected at 45 °, and the light is bent by 90 ° to perform conversion, instead of equation (14).

Therefore, 0 on the vertical axis of the phase modulation 4 corresponds to the thickness d ₀ of the diffraction grating plate, and 2π is the thickness d ₀ + λ ₀ / √2.
Corresponds to. In the reflective diffraction element, a plate made of a material having a high reflectance such as metal is used rather than glass having a low reflectance.

FIG. 3 is a diagram showing the deviation between the curved surface corresponding to the phase modulation and the unevenness that can be actually processed on the diffraction element. As described above, in the diffraction element that can be manufactured to perform the target light conversion, there is a deviation between the curved surface 5 determined by the phase modulation Φ _HR and the unevenness 6 that can be actually processed, and this deviation is Affects light conversion efficiency.

FIG. 4 illustrates the effect of deviation on the unevenness that can be actually processed on the light conversion efficiency. The horizontal axis of FIG. 4 indicates the number of steps of the unevenness 6 that can be actually processed as the number of gradations. FIG. 3 shows an example of 5 gradations, and the unevenness 6 that can be actually processed has a configuration of 5 steps. The vertical axis of FIG. 4 shows the ratio of the total power of the light pattern that was planned to be generated on the imaging element surface 2 to the total power of the incident light as a percentage.

As shown in FIG. 4, when the number of gradations of the unevenness processed on the diffraction element exceeds 10, the ratio of the incident light to the total power (power conversion efficiency) is almost 100%, and in theory, the power is completely converted. It shows that it can be done. This power conversion efficiency decreases as the number of gradations decreases.

Power conversion efficiency is one of the most important performance indexes for diffraction elements. Especially in laser processing applications, the power of the laser is large, so if the efficiency is poor, energy loss will be large, and the target light will be obtained. The remaining power that could not be converted becomes stray light and scatters, which poses a major safety problem.

On the other hand, considering the manufacturing cost, the cost increases as the number of gradations increases. In order to perform step-like microfabrication as shown in FIG. 3, the etching process is repeated. At this time, it is common to prepare a structure having 2 ^N gradations by N times of etching steps. For example, in the case of 16 gradations, the order of powers of 2 is 4, and since it is manufactured by four etching steps, the cost is four times that of two gradations in one step. On the other hand, the machining tolerance associated with microfabrication is a factor that reduces the power conversion efficiency. If the number of gradations is increased for the purpose of obtaining high power conversion efficiency, the number of processing steps increases, so that the influence of processing tolerances also increases accordingly.

From this point of view, the present inventors can approximate the phase modulation amount with a phase modulation amount of 8 gradations or more between 0 and less than 2π, and perform uneven processing to a thickness that realizes this approximate phase modulation amount. I found a good thing. In particular, considering the balance between power conversion efficiency, cost and processing tolerance, it can be said that it is preferable to approximate with the phase modulation amount of 8 gradations. According to FIG. 4, it can be seen that the power conversion efficiency is 95% for the eight gradations produced in the three steps, and the power (loss) that could not be converted is 5%. On the other hand, if the number of steps is increased by one to 16 gradations, the loss can be reduced to 1.3%. However, this is a case where there is no processing tolerance, and when the influence of the processing tolerance is taken into consideration, a loss of 10% or more occurs in 16 gradations. On the other hand, in the case of 8 gradations, the loss including the processing tolerance is about 13%, which is not much different from 16 gradations, so that the same safety measures can be taken. Therefore, it is not necessary to increase the cost from 8 gradations to 16 gradations. However, on the contrary, if the number of steps is reduced once to 4 gradations, the loss becomes more than 23%. Since it exceeds twice the 16 gradations, stricter safety measures are required, which is not preferable. For this reason, the present inventors have found that phase modulation with 8 gradations is most preferable.

The height of the curved surface corresponding to the desired phase modulation differs depending on whether the phase modulation is from 0 to 2π, whether it is a transmission type or a reflection type, and the incident condition and the exit condition, but the unevenness processed in the diffraction element is this. The height of the curved surface is not limited, and 0 to 2π is divided into 8 gradations or more to create unevenness.

FIG. 5 is a flow chart when designing the diffraction element of the present embodiment. In this embodiment, an example in base thickness (the thickness of the thinnest portion) is that the machining (roughened 8 gradations) to increase or decrease the thickness in 8 steps the diffraction element d ₀ explain.

First, the desired light intensity distribution I'(x', y') to be obtained on the image plane is determined, converted to the amplitude distribution A'(x', y'), and then the electromagnetic field intensity on the image plane. The distribution U'(x', y') = A'(x', y') exp [jΦ'(x', y')] is determined (S1).

From the electromagnetic field intensity distribution U'(x', y') on the image plane, the electromagnetic field intensity distribution U (x, y) on the diffraction element surface is determined based on the equation (10) (S2).

From the phase portion Φ (x, y) of the electromagnetic field distribution U (x, y) on the diffraction element surface and the phase Φ ₀ (x, y) of the incident light, the phase modulation amount Φ _H (x, y) is used by the equation (13). y) is determined (S3).

The phase modulation amount Φ _H (x, y) is converted into a value Φ _HR (x, y) (0 ≦ Φ _HR (x, y) <2π) from 0 to 2π using Eq. (15) (S4). ..

Phase modulation amount Φ _HR (x, y) converted from 0 to 2π is converted into 8 gradations (0, π / 4, π / 2, 3π / 4, π, 5π / 4, 3π / 2, 7π / 4) = Approximate to any of Φa (x, y) (S5).

The value obtained by multiplying the approximate phase modulation amount Φa (x, y) by λ ₀ / 2π (n-1) is added to the base thickness d _0, and the value d ₀ + λ ₀ Φa / 2π (n−) 1) is obtained as the thickness of the unevenness to be processed on the diffraction element (S6).

In the above configuration, the diffraction element is described by taking as an example a transmissive type. If the diffractive element is a reflective type, the value multiplied by the phase modulation amount Φa (x, y) approximated in S6 is not λ ₀ / 2π (n-1) but λ ₀ / (2π × 2 ^1/2 ). Therefore, the thickness of the unevenness to be processed is d ₀ + Φa (x, y) (x, y) λ ₀ / (2π × 2 ^1/2 ).

It can be said that the diffraction element manufactured with the thickness designed as described above has a diffraction element power conversion efficiency of 95% and has substantially no loss. In addition, the desired light intensity pattern could be realized with high fidelity.

A diffraction element that generates a light intensity pattern I'(x', y') as shown in FIG. 6 was produced by the method described in the above embodiment. The upper row is the profile in the x'axis direction, and the lower row is the profile in the y'axis direction. The width of the light pattern portion in the upper row was 1 cm, and the width of the lower row was also 1 cm. This pattern is generated at a distance of 20 cm on the optical axis from the diffraction element. The incident light is a collimated Gaussian beam having a wavelength of 1.06 μm and a diameter of 1 / e ^{2 and a} diameter of 1.5 cm, which is U ₀ (x, y).

'From, I the light intensity pattern I to be generated' (x ', y)' (x ', y') = A '2 (x', y ') by A' (x ', y' ) to determine the .. From this A'(x', y'), U'(x', y') is determined using the equation (12). However, in reality, the phase distribution Φ'(x', y') cannot be determined unless the phase distribution Φ'(x', y') on the image plane 2 is determined. ') Will be tentatively determined to determine U'(x', y'). Further, U (x, y) was calculated from U'(x', y') using the equation (10). Here, the phase distribution Φ'(on the image plane 2) is tentatively determined so that this U (x, y) matches well with U ₀ (x, y) and the power conversion efficiency here is optimized. x', y') was adjusted. The adjustment method of the phase distribution Φ'(x', y') is to determine U'(x', y') by changing the phase distribution Φ'(x', y') to be tentatively determined several times. Then, the method of finding the optimum phase distribution Φ'(x', y') is adopted by repeating the calculation of U (x, y) and checking whether it matches well with U ₀ (x, y). However, it is not particularly limited.

From the finally obtained U (x, y), Φ _H (x, y) is determined using the equations (12) and (13), and the phase is further limited to 0 to 2π by the equation (15). The thickness d (x, y) of the transmission type diffraction element was determined by the equation (14). This thickness distribution was produced by microfabrication of the surface of the glass substrate. However, two types of gradations, one having four gradations and the other having eight gradations, were produced in this processing.

A plate-like diffraction element manufactured, wavelength 1.06 .mu.m, generated when a collimated Gaussian beam 1 / ^{e 2} diameter 1.5cm incident perpendicularly, the light pattern of Figure 6 for the purpose, away 20cm from the element Was done. The conversion efficiency of the optical power was 95% for the 8-gradation, but 81% for the 4-gradation, and the diffraction element produced by the method described in the embodiment has sufficient power conversion efficiency. It was confirmed that it was expensive.

In Example 1, a transmission type diffraction element was produced, but in this example, a type of diffraction element that reflects light to a plate-shaped element inclined by 45 ° with respect to the optical axis was produced. The conditions are the same as those in the first embodiment except for the configuration of the diffraction element. Therefore, it is the same from the finally obtained U (x, y) to the point where Φ _H (x, y) is determined using the equations (12) and (13), but the thickness distribution of the element. Equation (17) was used for the conversion to. Also in the case of this reflection type, one having four gradations and one having eight gradations were produced.

A plate-like diffraction element manufactured by the above structural, wavelength 1.06 .mu.m, when a collimated Gaussian beam 1 / ^{e 2} diameter 1.5cm at an incident angle of 45 °, and 90 ° reflection, away 20cm from the element However, the intended optical pattern of FIG. 6 was generated. The conversion efficiency of the optical power was 93% for the 8-gradation, but 79% for the 4-gradation, and the diffraction element produced by the method described in the embodiment has sufficient power conversion efficiency. It was confirmed that it was expensive.

1 Diffraction element surface 2 Imaging surface 3 Phase modulation 4 Phase modulation 5 Curved surface corresponding to phase modulation 6 Concavities and convexities that can be actually processed

Claims (4)

The phase of the incident light having a predetermined wavelength λ ₀ incident on the diffraction element surface with the first electromagnetic field intensity distribution becomes the emitted light having the second electromagnetic field intensity distribution corresponding to the desired light intensity distribution on the imaging surface. It is a design method of a diffractive element to be modulated .
A step of determining a second electromagnetic field intensity distribution on the image plane from a desired light intensity distribution on the image plane, and
A step of determining a third electromagnetic field intensity distribution on the diffraction element surface based on the second electromagnetic field intensity distribution, and
The phase modulation amount in the diffraction element Φ _H ( from the phase after phase conversion on the diffraction element surface determined from the third electromagnetic field intensity distribution and the phase of the incident light determined from the first electromagnetic field intensity distribution. The process of determining x, y) and
The determined phase modulation amount Φ _H (x, y) _{and, Φ HR (x, y)} = Φ H (x, y) -2πN phase modulation amount of 0 to less than 2π with (N integer) [Phi _HR The process of converting to (x, y) and
A step of approximating the converted phase modulation amount Φ _HR (x, y) to any one of the phase modulation amounts Φa (x, y) of 8 gradations or more between 0 and less than 2π.
The thickness d (x, y) of the diffractive element on the basis the 8 phase modulation amount Φa approximating above gradation (x, y) to the wavelength lambda ₀ of the incident light to include the step of determining the A method for designing a characteristic diffraction element. The diffraction element is a transmission type diffraction element having a base thickness of d ₀ and a refractive index of n.
In the step of determining the thickness d (x, y) of the diffractive element, d (x, y) = d 0 + λ 0 Φa (x, y) / 2π (n-1) the thickness of the diffractive element by d The method for designing a diffraction element according to claim 1, wherein (x, y) is determined. The diffraction element is a reflection type diffraction element having a base thickness of d ₀ .
In the step of determining the thickness d (x, y) of the diffractive element, d (x, y) = λ 0 Φa (x, y) / thickness of the diffraction element by (2π × 2 ^1/2) d The method for designing a diffraction element according to claim 1, wherein (x, y) is determined. The method for designing a diffraction element according to any one of claims 1 to 3 , wherein the approximate phase modulation amount Φa (x, y) has 8 gradations.

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