Apparatus and method for achieving a ring-shaped rainbow using a refractive total reflection fresnel prism

Abstract

This invention relates to an improved Fresnel prism based on the principle of refraction-total internal reflection-refraction. To address the high cost of existing circular rainbow generation devices and supplement deficiencies in color order, it provides a reusable device and method for enhancing ring rainbows. The invention utilizes a refraction-total internal reflection Fresnel prism to achieve ring rainbows, comprising a Fresnel prism and a projection screen. The rotation axis of the Fresnel prism is perpendicular to the projection screen. One side of the Fresnel prism is flat, and the other side is toothed. Its cross-section is a triangle with two sets of connected bases symmetrically distributed along the rotation axis. The Fresnel prism is made of a transparent material. Incident light, parallel to the rotation axis of the Fresnel prism, enters and penetrates the prism from the connected bases, and is then projected perpendicularly onto the projection screen. This invention is primarily applied to the design and manufacture of optical equipment with high optical performance.

Description

Technical Field

[0001] This invention relates to an improved Fresnel prism based on the principle of refraction-total internal reflection-refraction, specifically to the design of a Fresnel prism for focusing and collecting light and its application in achieving a brightly colored ring rainbow effect. Background Technology

[0002] Existing rainbow generating devices are limited to projecting a rainbow through refraction-reflection-refraction within water droplets. The angle between the emitted red and violet light is small, resulting in a less bright rainbow phenomenon. They are also less flexible in operation and make it more difficult to achieve a circular rainbow.

[0003] The submitted invention patent, "Method and Device for Enhancing Circular Rainbows Based on the Principle of Triangular Prisms," uses parallel light to illuminate a gyroscopic toothed prism, producing a vibrant full-circle rainbow based on the principle of refraction-refraction, enabling direct observation of the rainbow. However, the side closest to the light source has a non-collinear concave structure, making the manufacturing process difficult. Moreover, it produces a rainbow with inner red and outer violet light (neon rainbow), rather than a rainbow with inner violet light and outer red light (rainbow).

[0004] A classic Fresnel prism has one flat side and one serrated side. The flat surface is used for incident light, while the serrated surface disperses the incident light in different directions. Light rays from different serrations converge at the focal plane. This design allows Fresnel prisms to achieve high optical performance within a relatively small thickness. Summary of the Invention

[0005] To address the high cost of existing circular rainbow generation devices and supplement deficiencies in color sequence, this invention utilizes mature Fresnel prism manufacturing processes and modifies the shape and size of the prism's teeth to provide a reusable device and method for enhancing annular rainbows. Specifically, this invention employs a device for generating annular rainbows using a refractive total internal reflection Fresnel prism, comprising a Fresnel prism and a projection screen. The rotation axis of the Fresnel prism is perpendicular to the projection screen. One side of the Fresnel prism is flat, and the other is toothed. Its cross-section consists of two sets of triangles with their bases connected sequentially and symmetrically distributed along the rotation axis. The Fresnel prism is made of a transparent material. Incident light, parallel to the rotation axis of the Fresnel prism, enters and penetrates the prism from the connected bases and is then projected perpendicularly onto the projection screen.

[0006] The Fresnel prism has a triangular cross-section for each tooth. To the right of the rotation axis is the central tooth OA1B1, followed by the next adjacent tooth A1A2B2, the third adjacent tooth A2A3B3, the fourth adjacent tooth A3A4B4, the fifth adjacent tooth A4A5B5, and so on up to A. N-1 A N B NN is the number of teeth;

[0007] To the right of the rotation axis, the incident edges of the light rays are OA1, A1A2, A2A3, A3A4, A4A5...A N-1 A N The points A1, A2, A3, A4, A5, ..., A4 form a straight line perpendicular to the incident light, and the sides of the total internal reflection are OB1, A1B2, A2B3, A3B4, A4B5...A N-1 B N The light rays exit from the edges A1B1, A2B2, A3B3, A4B4, A5B5...A N B N Connecting angles: ∠A1OB1=α1, ∠A2A1B2=α2, ∠A3A2B3=α3, ∠A4A3B4=α4, ∠A5A4B5=α5……∠A N A N-1 B N =α N The tooth angles are ∠OB1A1=θ1, ∠A1B2A2=θ2, ∠A2B3A3=θ3, ∠A3B4A4=θ4, ∠A4B5A5=θ5……∠A N-1 B N A N =θ N ;

[0008] The incident side of each serrated triangle has the same length, i.e., OA1=A1A2=A2A3=A3A4=A4A5…… = A N- 1A N The size and position of the total internal reflection side and the exit side of each serrated triangle are determined by the angles of the serrations and the connecting angles of the triangle.

[0009] The angles of each serrated triangle, namely θ1, θ2, θ3, θ4, θ5...θ N The differences lie in the connecting angles of each serrated triangle, namely α1, α2, α3, α4, α5...α N They are also different and need to be calculated according to the toothed edge emission convergence method.

[0010] The toothed edge emission and convergence method is that all monochromatic light incident parallel to the incident edge of the toothed edge is emitted along the edge of the emitted edge of the toothed edge, that is, the emission angle is 90°. Monochromatic light emitted from different teeth is simultaneously converged at the same point on the projection screen, and colored light with wavelengths greater than the wavelength of violet light is dispersed on the projection screen in sequence.

[0011] Light rays are incident perpendicularly from the incident surface OA1 of the central tooth OA1B1 along a direction parallel to the y-axis. They reach the OB1 surface within the prism, undergo total internal reflection at OB1, reach the A1B1 surface, are refracted at A1B1, exit from the central tooth, and finally reach the projection screen. Due to the dispersion of light, the various colors of light, from violet to red, reach the screen at different positions, ultimately creating a circular rainbow with violet on the inside and red on the outside.

[0012] According to the requirement that violet light is emitted and converged along the edges of the teeth, all the violet light emanating from the A1B1 surface of the central tooth OA1B1 converges to point P1 on the projection screen. Similarly, the violet light emanating from the A2B2 surface of the second tooth A1A2B2 also converges to point P1. The violet light emanating from the third, fourth, and fifth teeth must also converge to point P1. Therefore, the kth tooth A1B1... k-1 A k B k A k B k The violet light emitted from the surface must also converge at point P1 on the projection screen, where k = 1, 2, ..., N, representing the order of the Fresnel prism teeth. k = 1 represents the first tooth A0A1B1, i.e., the central tooth OA1B1, and k = N represents the Nth tooth A0A1B1. N-1 A N B N ;

[0013] On the k-th tooth of the Fresnel prism, the connecting angle... With light at the incident interface A k-1 A k and total reflection interface A k-1 B k Ejection interface A k B k Refraction - Total Internal Reflection - 5 Angles Related to Refraction , , , , The relationship between them is:

[0014]

[0015] Where the subscripts k = 1, 2, ..., N, and n is the refractive index of light in the transparent medium, which will be involved in the calculations below. or , which is the refractive index of red or violet light in a transparent medium;

[0016] When k=1, that is, on the central tooth, the light ray is incident perpendicularly on the OA1 interface, then the angle of incidence is... and angle of refraction All are 0. It is the angle of incidence of the light at interface OB1. It is the angle of incidence of the light at the A1B1 interface. It is the angle of refraction or emission of light at the A1B1 interface. According to the toothed edge emission and converging method, its violet emission angle is... The angle is 90°. The incident angle of violet light at the OB1 interface is calculated using formula (1). The angle of incidence of red light on the OB1 interface At this point in the formula Substitution Calculate the red light emission angle ;

[0017] In the prism A k-1 A k B k Above, the light at A k-1 A k If the light is incident perpendicularly to the interface, then the angle of incidence is... and angle of refraction All are 0. Is the light at A k-1 B k The angle of incidence of the interface Is the light at A k B k The angle of incidence of the interface Is the light at A k B k The angle of refraction or the angle of emission at the interface, where n is taken in formula (1) At that time, its violet light emission angle In formula (1), n ​​takes the following values: At that time, the red light emission angle is calculated according to formula (1).

[0018] To construct a Fresnel prism with N teeth, the following steps are required: the central tooth OA1B1, the second adjacent tooth A1A2B2, the third tooth A2A3B3, ..., the (N-1)th tooth A N-2 A N-1 B N-1 The Nth tooth A N-1 A N B N The angle parameters and vertex coordinates are determined, and the specific process is as follows:

[0019] (1) First, O is given as the origin (0, 0), the y-axis is parallel to the incident ray, and the x-axis is parallel to the projection screen. The position of the projection screen is given, that is, the distance OP0 from point O to the intersection point P0 of the y-axis and the projection screen. The incident edge A is given. k-1 A k The length L is given by the subscript k = 1, 2, ..., N;

[0020] (2) Determine the angular parameters of the center tooth OA1B1 and The specific process for determining the coordinates of the two vertices A1 and B1 is as follows: (Central tooth edge, tooth angle) The range is from 10° to 80°, given Calculate the connection angle for one of the specific values. Then calculate the coordinates of point A1, which is the center tooth and the first edge tooth. The coordinates of point B1, the tip of the first edge tooth Finally, the coordinates of point P1, where the violet light converges on the projection screen, are calculated. ;

[0021] (3) Determine the kth edge tooth A k-1 A k B k Angular parameters and and its two vertices A k and B k The process of obtaining coordinates is as follows: combining the coordinates of point P1 on the projection screen with the k-th edge tooth. coordinates The angle of the kth edge tooth Determined by the following formula

[0022]

[0023] Where the subscripts k = 1, 2, ..., N. The k-th edge tooth A k-1 A k B k Angular parameters Determined by the following formula

[0024]

[0025] The tip B of the kth edge tooth is calculated. k coordinates of the point From formula (3), it can be seen that in the design... Greater than the total internal reflection angle of red light in a transparent medium Light on interface A k-1 B k Total internal reflection occurs on the surface, avoiding interface A. k-1 B k Light energy loss on the surface;

[0026] (4) Determine the kth edge tooth A k-1 A k B k upper red light emission angle The calculation formula is as follows:

[0027]

[0028] From formula (4), it can be seen that when the transparent medium material is selected, the red light emission angle is... The red light emission angle on each tooth is a fixed value. and the angle of violet light emission The difference is also a fixed value. ,exist The left and right sides created the conditions for a large, brightly colored rainbow;

[0029] (5) The range of the red light emitted from the kth prism on the projection screen is analyzed as follows: In A k-1 A k The effective light-gathering range on the interface is the entire A k-1 A k Part, in A k-1 Point and A k The violet light incident between points reaches A. k B k After that, all along A k B k The light emitted from the surface converges at the same point P1 on the projection screen;

[0030] In A k-1 Point and A k Red light incident between points, at A k B k The rays emitted in parallel on the surface cannot converge to a single point on the projection screen. A k-1 The red light that enters from point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 1k Near the outer edge of the rainbow circle, and from A k Red light incident at point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 2k , compared to R 1k Stay close to the center.

[0031] R 2k and R 1k The distance between them is much smaller than that between P1 and R. 1k The distance between them ensures the vibrancy of the rainbow; with OP0 fixed, the center tooth's... As the size increases, the rainbow halo will become smaller; when P1 is near P0, the rainbow halo can turn into a colored spot.

[0032] The method for creating a ring rainbow using a refractive total internal reflection Fresnel prism, achieved with the aforementioned device, involves the following steps:

[0033] The toothed edge emission and convergence method is that all monochromatic light incident parallel to the incident edge of the toothed edge is emitted along the edge of the emitted edge of the toothed edge, that is, the emission angle is 90°. Monochromatic light emitted from different teeth is simultaneously converged at the same point on the projection screen, and colored light with wavelengths greater than the wavelength of violet light is dispersed on the projection screen in sequence.

[0034] Light rays are incident perpendicularly from the incident surface OA1 of the central tooth OA1B1 along a direction parallel to the y-axis. They reach the OB1 surface within the prism, undergo total internal reflection at OB1, reach the A1B1 surface, are refracted at A1B1, exit from the central tooth, and finally reach the projection screen. Due to the dispersion of light, the various colors of light, from violet to red, reach the screen at different positions, ultimately creating a circular rainbow with violet on the inside and red on the outside.

[0035] According to the requirement that violet light is emitted and converged along the edges of the teeth, all the violet light emanating from the A1B1 surface of the central tooth OA1B1 converges to point P1 on the projection screen. Similarly, the violet light emanating from the A2B2 surface of the second tooth A1A2B2 also converges to point P1. The violet light emanating from the third, fourth, and fifth teeth must also converge to point P1. Therefore, the kth tooth A1B1... k-1 A k B k A k B k The violet light emitted from the surface must also converge at point P1 on the projection screen, where k = 1, 2, ..., N, representing the order of the Fresnel prism teeth. k = 1 represents the first tooth A0A1B1, i.e., the central tooth OA1B1, and k = N represents the Nth tooth A0A1B1. N-1 A N B N ;

[0036] On the k-th tooth of the Fresnel prism, the connecting angle... With light at the incident interface A k-1 A k and total reflection interface A k-1 B k Ejection interface A k B k Refraction - Total Internal Reflection - 5 Angles Related to Refraction , , , , The relationship between them is:

[0037]

[0038] Where the subscripts k = 1, 2, ..., N, and n is the refractive index of light in the transparent medium, which will be involved in the calculations below. or , which is the refractive index of red or violet light in a transparent medium;

[0039] When k=1, that is, on the central tooth, the light ray is incident perpendicularly on the OA1 interface, then the angle of incidence is... and angle of refraction All are 0. It is the angle of incidence of the light at interface OB1. It is the angle of incidence of the light at the A1B1 interface. It is the angle of refraction or emission of light at the A1B1 interface. According to the toothed edge emission and converging method, its violet emission angle is... The angle is 90°. The incident angle of the violet light at the OB1 interface can be calculated using formula (1). The angle of incidence of red light on the OB1 interface At this point in the formula Substitution The red light emission angle can be calculated. ;

[0040] In the prism A k-1 A k B k Above, the light at A k-1 A k If the light is incident perpendicularly to the interface, then the angle of incidence is... and angle of refraction All are 0. Is the light at A k-1 B k The angle of incidence of the interface Is the light at A k B k The angle of incidence of the interface Is the light at A k B k The angle of refraction or the angle of emission at the interface, where n is taken in formula (1) At that time, its violet light emission angle In formula (1), n ​​takes the following values: At that time, the red light emission angle is calculated according to formula (1).

[0041] To construct a Fresnel prism with N teeth, the following steps are required: the central tooth OA1B1, the second adjacent tooth A1A2B2, the third tooth A2A3B3, ..., the (N-1)th tooth A N-2 A N-1 B N-1 The Nth tooth A N-1 A N B N The angle parameters and vertex coordinates are determined, and the specific process is as follows:

[0042] (1) First, O is given as the origin (0, 0), the y-axis is parallel to the incident ray, and the x-axis is parallel to the projection screen. The position of the projection screen is given, that is, the distance OP0 from point O to the intersection point P0 of the y-axis and the projection screen. The incident edge A is given. k-1 A k The length L is given by the subscript k = 1, 2, ..., N;

[0043] (2) Determine the angular parameters of the center tooth OA1B1 and The specific process for determining the coordinates of the two vertices A1 and B1 is as follows: (Central tooth edge, tooth angle) The range is from 10° to 80°, given Calculate the connection angle for one of the specific values. Then calculate the coordinates of point A1, which is the center tooth and the first edge tooth. The coordinates of point B1, the tip of the first edge tooth Finally, the coordinates of point P1, where the violet light converges on the projection screen, are calculated. ;

[0044] (3) Determine the kth edge tooth A k-1 A k B k Angular parameters and and its two vertices A k and B k The process of obtaining coordinates is as follows: combining the coordinates of point P1 on the projection screen with the k-th edge tooth. coordinates The angle of the kth edge tooth Determined by the following formula

[0045]

[0046] Where the subscripts k = 1, 2, ..., N. The k-th edge tooth A k-1 A k B k Angular parameters Determined by the following formula

[0047]

[0048] The tip B of the kth edge tooth is calculated. k coordinates of the point From formula (3), it can be seen that in the design... Greater than the total internal reflection angle of red light in a transparent medium Light on interface A k-1 B k Total internal reflection occurs on the surface, avoiding interface A. k-1 B k Light energy loss on the surface;

[0049] (4) Determine the kth edge tooth A k-1 A k B k upper red light emission angle The calculation formula is as follows:

[0050]

[0051] From formula (4), it can be seen that when the transparent medium material is selected, the red light emission angle is... The red light emission angle on each tooth is a fixed value. and the angle of violet light emission The difference is also a fixed value. ,exist The left and right sides created the conditions for a large, brightly colored rainbow;

[0052] (5) The range of the red light emitted from the kth prism on the projection screen is analyzed as follows: In A k-1 A k The effective light-gathering range on the interface is the entire A k-1 A k Part, in A k-1 Point and A k The violet light incident between points reaches A. k B k After that, all along A k B k The light emitted from the surface converges at the same point P1 on the projection screen;

[0053] In A k-1 Point and A k Red light incident between points, at A k B k The rays emitted in parallel on the surface cannot converge to a single point on the projection screen. A k-1 The red light that enters from point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 1k Near the outer edge of the rainbow circle, and from A k Red light incident at point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 2k , compared to R 1k Stay close to the center.

[0054] The features and beneficial effects of this invention are:

[0055] This invention solves the problem of high manufacturing costs in existing circular rainbow generation devices and addresses shortcomings in color sequence. Based on the principles of refraction-total internal reflection-refraction, it microstructurally adjusts the shape and size of the Fresnel prism's teeth, using existing mature processes to fabricate a Fresnel prism with one flat side and the other toothed side. This provides reusable Fresnel prisms that can project circular rainbows into colored light spots without the need for additional optical elements. These rainbow projections are visually impactful and have significant value for children's education, architectural design, and science popularization. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the device of the present invention.

[0057] Figure 2 This is a cross-sectional schematic diagram of the Fresnel prism of the present invention.

[0058] Figure 3 This is a schematic diagram of the ultraviolet light path on the center tooth and adjacent teeth of the present invention.

[0059] Figure 4 This is a schematic diagram of the red light path on the center tooth and adjacent teeth of the present invention.

[0060] Figure 5 This is an optical schematic diagram of the main cross section of the present invention.

[0061] Figure 6 This is a diagram showing the rainbow halo effect on the projection screen of the present invention (from the inside out, from purple to red).

[0062] Figure 7 This is a diagram showing the rainbow light spot effect on the projection screen of the present invention (from the inside out, from purple to red). Detailed Implementation

[0063] This invention relates to a Fresnel prism based on the principle of refraction-total internal reflection-refraction. When sunlight illuminates a flat surface, it produces a ring-shaped rainbow (inner ultraviolet, outer red), enabling direct observation of the rainbow. By adjusting the angle of the prism teeth, the color can be varied from a colored ring to a colored spot. The Fresnel prism in this device is relatively thin and lightweight, easy to manufacture, adjustable in size, and has a variable number of teeth. It can be optimized and customized to meet the needs of different applications to achieve specific optical performance and effects.

[0064] This invention mainly addresses the problems of high process cost and limited color sequence in existing circular rainbow generation devices, and provides a Fresnel prism that can generate rainbow rings and colored light spots.

[0065] This invention provides an improved Fresnel prism design consisting of a series of microstructures of specific shapes and sizes, which are precisely calculated to achieve accurate focusing and dispersion control of light.

[0066] A device for creating a ring rainbow using a Fresnel prism with total internal reflection includes a Fresnel prism and a projection screen. The rotation axis of the Fresnel prism is perpendicular to the projection screen. One side of the Fresnel prism is flat, and the other side is toothed. Its cross-section is a triangle with two sets of connected bases symmetrically distributed along the rotation axis. The Fresnel prism is made of transparent material. Incident light is parallel to the rotation axis of the Fresnel prism, enters from the connected bases, penetrates the Fresnel prism, and is then projected perpendicularly onto the projection screen.

[0067] The Fresnel prism has a triangular cross-section for each tooth. To the right of the rotation axis is the central tooth OA1B1, followed by the next adjacent tooth A1A2B2, the third adjacent tooth A2A3B3, the fourth adjacent tooth A3A4B4, the fifth adjacent tooth A4A5B5, and so on up to A. N-1 A N B N N is the number of teeth;

[0068] To the right of the rotation axis, the incident edges of the light rays are OA1, A1A2, A2A3, A3A4, A4A5...A N-1 A N The points A1, A2, A3, A4, A5, ..., A4 form a straight line perpendicular to the incident light, and the sides of the total internal reflection are OB1, A1B2, A2B3, A3B4, A4B5...A N-1 B N The light rays exit from the edges A1B1, A2B2, A3B3, A4B4, A5B5...A N B N Connecting angles: ∠A1OB1=α1, ∠A2A1B2=α2, ∠A3A2B3=α3, ∠A4A3B4=α4, ∠A5A4B5=α5……∠A N A N-1 B N =α N The tooth angles are ∠OB1A1=θ1, ∠A1B2A2=θ2, ∠A2B3A3=θ3, ∠A3B4A4=θ4, ∠A4B5A5=θ5……∠A N-1 B N A N =θ N ;

[0069] The incident side of each serrated triangle has the same length, i.e., OA1=A1A2=A2A3=A3A4=A4A5…… = A N- 1A N The size and position of the total internal reflection side and the exit side of each serrated triangle are determined by the angles of the serrations and the connecting angles of the triangle.

[0070] The angles of each serrated triangle, namely θ1, θ2, θ3, θ4, θ5...θ N The differences lie in the connecting angles of each serrated triangle, namely α1, α2, α3, α4, α5...α N They are also different and need to be calculated according to the toothed edge emission convergence method;

[0071] The toothed edge emission and converging method involves all monochromatic lights incident parallel to the incident edge of the toothed edge emanating along the edge of the exit edge of the toothed edge, i.e., the exit angle is 90°. Monochromatic lights emitted from different teeth simultaneously converge at the same point on the projection screen, and colored lights with wavelengths greater than the wavelength of the monochromatic light disperse sequentially on the projection screen. In this embodiment, purple is selected as the monochromatic light to enhance the rainbow effect.

[0072] Light rays are incident perpendicularly from the incident surface OA1 of the central tooth OA1B1 along a direction parallel to the y-axis. They reach the OB1 surface within the prism, undergo total internal reflection at OB1, reach the A1B1 surface, are refracted at A1B1, exit from the central tooth, and finally reach the projection screen. Due to the dispersion of light, the various colors of light, from violet to red, reach the screen at different positions, ultimately creating a circular rainbow with violet on the inside and red on the outside.

[0073] According to the requirement that violet light is emitted and converged along the edges of the teeth, all the violet light emanating from the A1B1 surface of the central tooth OA1B1 converges to point P1 on the projection screen. Similarly, the violet light emanating from the A2B2 surface of the second tooth A1A2B2 also converges to point P1. The violet light emanating from the third, fourth, and fifth teeth must also converge to point P1. Therefore, the kth tooth A1B1... k-1 A k B k A k B k The violet light emitted from the surface must also converge at point P1 on the projection screen, where k = 1, 2, ..., N, representing the order of the Fresnel prism teeth. k = 1 represents the first tooth A0A1B1, i.e., the central tooth OA1B1, and k = N represents the Nth tooth A0A1B1. N-1 A N B N ;

[0074] On the k-th tooth of the Fresnel prism, the connecting angle... With light at the incident interface A k-1 A k and total reflection interface A k-1 B k Ejection interface A k B k Refraction - Total Internal Reflection - 5 Angles Related to Refraction , , , , The relationship between them is:

[0075]

[0076] Where the subscripts k = 1, 2, ..., N, and n is the refractive index of light in the transparent medium, which will be involved in the calculations below. or , which is the refractive index of red or violet light in a transparent medium;

[0077] When k=1, that is, on the central tooth, the light ray is incident perpendicularly on the OA1 interface, then the angle of incidence is... and angle of refraction All are 0. It is the angle of incidence of the light at interface OB1. It is the angle of incidence of the light at the A1B1 interface. It is the angle of refraction or emission of light at the A1B1 interface, specifically the violet emission angle. The red light emission angle is 90°. It can be calculated;

[0078] In the prism A k-1 A k B k Above, the light at A k-1 A k If the light is incident perpendicularly to the interface, then the angle of incidence is... and angle of refraction All are 0. Is the light at A k-1 B k The angle of incidence of the interface Is the light at A k B k The angle of incidence of the interface Is the light at A k B k The angle of refraction or the angle of emission at the interface, where n is taken in formula (1) At that time, its violet light emission angle In formula (1), n ​​takes the following values: At that time, its red light emission angle It can be calculated.

[0079] To construct a Fresnel prism with N teeth, the following steps are required: the central tooth OA1B1, the second adjacent tooth A1A2B2, the third tooth A2A3B3, ..., the (N-1)th tooth A N-2 A N-1 B N-1 The Nth tooth A N-1 A N B N The angle parameters and vertex coordinates are determined, and the specific process is as follows:

[0080] (1) First, O is given as the origin (0, 0), the y-axis is parallel to the incident ray, and the x-axis is parallel to the projection screen. The position of the projection screen is given, that is, the distance OP0 from point O to the intersection point P0 of the y-axis and the projection screen. The incident edge A is given. k-1 A k The length L is given by the subscript k = 1, 2, ..., N;

[0081] (2) Determine the angular parameters of the center tooth OA1B1 and The specific process for determining the coordinates of the two vertices A1 and B1 is as follows: (Central tooth edge, tooth angle) The range is from 10° to 80°, given Calculate the connection angle for one of the specific values. Then calculate the coordinates of point A1, which is the center tooth and the first edge tooth. The coordinates of point B1, the tip of the first edge tooth Finally, the coordinates of point P1, where the violet light converges on the projection screen, are calculated. ;

[0082] (3) Determine the kth edge tooth A k-1 A k B k Angular parameters and and its two vertices A k and B k The process of obtaining coordinates is as follows: combining the coordinates of point P1 on the projection screen with the k-th edge tooth. coordinates The angle of the kth edge tooth Determined by the following formula

[0083]

[0084] Where the subscripts k = 1, 2, ..., N. The k-th edge tooth A k-1 A k B k Angular parameters Determined by the following formula

[0085]

[0086] The tip B of the kth edge tooth is calculated. k coordinates of the point From formula (3), it can be seen that in the design... Greater than the total internal reflection angle of red light in a transparent medium Light on interface A k-1 B k Total internal reflection occurs on the surface, which can prevent interface A from being affected. k-1 B kLight energy loss on the surface;

[0087] (4) Determine the kth edge tooth A k-1 A k B k upper red light emission angle The calculation formula is as follows:

[0088]

[0089] From formula (4), it can be seen that when the transparent medium material is selected, the red light emission angle is... The red light emission angle on each tooth is a fixed value. and the angle of violet light emission The difference is also a fixed value. ,exist The left and right sides created the conditions for a large, brightly colored rainbow;

[0090] (5) The range of the red light emitted from the kth prism on the projection screen is analyzed as follows: In A k-1 A k The effective light-gathering range on the interface is the entire A k-1 A k Part, in A k-1 Point and A k The violet light incident between points reaches A. k B k After that, all along A k B k The light emitted from the surface converges at the same point P1 on the projection screen;

[0091] In A k-1 Point and A k Red light incident between points, at A k B k The rays emitted in parallel on the surface cannot converge to a single point on the projection screen. A k-1 The red light that enters from point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 1k Near the outer edge of the rainbow circle, and from A k Red light incident at point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 2k , compared to R 1k Get closer to the center;

[0092] Although red light emitted from the same tooth and red light emitted from different teeth cannot converge at a single point, R 2k and R 1k The distance between them is generally much smaller than that between P1 and R. 1kThe distance between them ensures the vibrancy of the rainbow. With OP0 fixed, the center tooth's... As the size increases, the rainbow halo decreases. When P1 is near P0, the rainbow halo can transform into a colored spot.

[0093] The circular rainbow device includes a Fresnel prism 1 and a projection screen 2. The rotation axis of the Fresnel prism 1 is perpendicular to the projection screen 2. The distance between the Fresnel prism and the projection screen on one side is determined by design; sunlight is parallel to the rotation axis of the Fresnel prism and perpendicular to the projection screen.

[0094] In the construction of the Fresnel prism 1, this invention proposes a method for violet light to be emitted and converged along the edge of the teeth, so that violet light from the same tooth and violet light from different teeth can be converged at the same position on the projection screen. Moreover, the angle between the emitted red light and the emitted violet light can reach about 10 degrees, which creates a premise for maximizing the expansion of rainbow colors, while ensuring the color saturation of the rainbow.

[0095] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0096] See Figures 1-7 This invention provides a Fresnel prism design and a method for achieving a vibrant circular rainbow effect:

[0097] It includes a Fresnel prism 1 and a projection screen 2. The rotation axis of the Fresnel prism 1 is perpendicular to the projection screen 2. The projection screen 2 is set at a certain distance away from one side of the Fresnel prism 1. The sunlight is parallel to the rotation axis y of the Fresnel prism and perpendicular to the projection screen.

[0098] Fresnel prism 1 is a prism group consisting of a series of prism teeth, the size of which and the number of teeth N can be adjusted.

[0099] For a cross-sectional diagram of Fresnel prism 1 with N=5 teeth, see [link / reference]. Figure 2 The y-axis is symmetrical to the right and left, and the cross-section of each tooth is triangular. To the right of the y-axis is the center tooth OA1B1, the second adjacent tooth A1A2B2, the third adjacent tooth A2A3B3, the fourth adjacent tooth A3A4B4, and the fifth adjacent tooth A4A5B5.

[0100] Rotating the five prism teeth OA1B1, A1A2B2, A2A3B3, A3A4B4, and A4A5B5 on the right side of the y-axis around the y-axis for one revolution yields a complete Fresnel prism. See [link to Fresnel prism]. Figure 2 The teeth OC1D1, C1C2D2, C2C3D3, C3C4D4, and C4C5D5 on the left side of the y-axis are exactly the same as those on the right side. Therefore, when constructing Fresnel prism 1, only the five teeth on the right side of the y-axis need to be constructed.

[0101] The five serrated triangles to the right of the y-axis have incident sides OA1, A1A2, A2A3, A3A4, and A4A5 that form a straight line perpendicular to the incident light. The sides of total internal reflection are OB1, A1B2, A2B3, A3B4, and A4B5, and the sides of outgoing light are A1B1, A2B2, A3B3, A4B4, and A5B5. The connecting angles are ∠A1OB1=α1, ∠A2A1B2=α2, ∠A3A2B3=α3, ∠A4A3B4=α4, and ∠A5A4B5=α5. The serrated angles are ∠OB1A1=θ1, ∠A1B2A2=θ2, ∠A2B3A3=θ3, ∠A3B4A4=θ4, and ∠A4B5A5=θ5.

[0102] The incident side of each serrated triangle has the same length, i.e., OA1=A1A2=A2A3=A3A4=A4A5. The size and position of the total internal reflection side and the exit side of each serrated triangle are determined by the serrated angle and the connecting angle of the triangle.

[0103] The tooth angles of each tooth triangle, namely θ1, θ2, θ3, θ4, and θ5, are different, and the connection angles of each tooth triangle, namely α1, α2, α3, α4, and α5, are also different. They need to be calculated according to the tooth edge emission convergence method.

[0104] The edge-emission converging method involves all monochromatic light rays incident parallel to the incident edge of the teeth, i.e., light rays of a specific wavelength, exiting along the edge of the exit edge of the teeth at an exit angle of 90°. These monochromatic lights exiting from different teeth simultaneously converge at the same point on the projection screen. Light rays with wavelengths greater than the monochromatic light's wavelength then disperse sequentially on the projection screen. In this embodiment, purple is chosen as the monochromatic light to enhance the rainbow effect.

[0105] See Figure 3 As shown, ray 1 is incident perpendicularly from the incident surface OA1 of the central tooth OA1B1 along a direction parallel to the y-axis. It enters the prism as ray 2, reaches the OB1 surface, undergoes total internal reflection at the OB1 surface, becomes ray 3, reaches the A1B1 surface, is refracted at the A1B1 surface, exits from the central tooth as ray 4, and finally reaches the projection screen. Due to the dispersion of light, the various colors of light, from violet to red, reach the projection screen at different positions, ultimately presenting a circular rainbow with violet on the inside and red on the outside.

[0106] According to the requirement that violet light is emitted and converged along the edges of the prism, all the violet light emanating from the A1B1 surface of the central prism OA1B1 converges to point P1 on the projection screen. Similarly, the violet light emanating from the A2B2 surface of the second prism A1A2B2 also converges to point P1. The violet light emanating from the third, fourth, and fifth prisms must also converge to point P1. The violet light paths of the central prism and the second adjacent prism of Fresnel prism 1 are shown in [reference needed]. Figure 3As shown, the position of point P1 varies depending on the position of the projection screen. Therefore, the position of point P1 directly affects the size, angle, and position of the second edge tooth A1A2B2, and similarly, the position of point P1 also affects the size, angle, and position of the third, fourth, and fifth edge teeth.

[0107] The kth tooth A of the Fresnel prism k-1 A k B k Above, connecting corner With light at the incident interface A k-1 A k and total reflection interface A k-1 B k Ejection interface A k B k Refraction - Total Internal Reflection - 5 Angles Related to Refraction , , , , The relationship between them is:

[0108]

[0109] Where the subscripts k = 1, 2, ..., N, k = 1 represents the first edge tooth A0A1B1, i.e., the center tooth OA1B1, and k = N represents the Nth edge tooth A0A1B1. N-1 A N B N ; n is the refractive index of light in a transparent medium, which is involved in the calculations below. or , which is the refractive index of red or violet light in a transparent medium such as glass;

[0110] At the kth edge A k-1 A k B k Above, the light at A k-1 A k If the light is incident perpendicularly to the interface, then the angle of incidence is... and angle of refraction All are 0. Is the light at A k-1 B k The angle of incidence of the interface Is the light at A k B k The angle of incidence of the interface Is the light at A k B k The angle of refraction or the angle of emission at the interface, where n is taken in formula (1) At that time, its violet light emission angle In formula (1), n ​​takes the following values: At that time, its red light emission angle It can be calculated.

[0111] See Figure 3 The light path diagram shows that the violet light from the central tooth and the adjacent second tooth exits along the exit edge: When k=1, that is, on the central tooth, the light ray is perpendicularly incident at the OA1 interface, then the incident angle is... and angle of refraction All are 0. It is the angle of incidence of the light at interface OB1. It is the angle of incidence of the light at the A1B1 interface. It is the angle of refraction or emission of light at the A1B1 interface, specifically the violet emission angle. The red light emission angle is 90°. It can be calculated;

[0112] When k=2, that is, on the second serration, the light ray is incident perpendicularly on the A1A2 interface, then the angle of incidence is... and angle of refraction All are 0. It is the angle of incidence of the light at the interface A1B2. It is the angle of incidence of the light at the A2B2 interface. It is the angle of refraction or emission of light at the A2B2 interface, specifically the violet emission angle. The red light emission angle is 90°. It can be calculated;

[0113] To construct a Fresnel prism with 5 teeth, the angular parameters and vertex coordinates of the central tooth OA1B1, the second adjacent tooth A1A2B2, the third tooth A2A3B3, the fourth tooth A3A4B4, and the fifth tooth A4A5B5 need to be determined sequentially. The specific process is as follows:

[0114] (1) First, O is given as the origin (0, 0), the y-axis is parallel to the incident ray, and the x-axis is parallel to the projection screen. The position of the projection screen is given, that is, the distance from point O to the intersection point P0 of the y-axis and the projection screen is OP0 = 10 cm. The incident side A is given. k-1 A k The length L of (OA1=A1A2=A2A3=A3A4=A4A5) is 1 cm;

[0115] (2) Determine the angular parameters of the center tooth OA1B1 and The specific process for determining the coordinates of the two vertices A1 and B1 is as follows: (Central tooth edge, tooth angle) The range is from 10° to 80°, given For one of the specific values, such as Calculate the connection angle ,in, =1.5308, take the refractive index of 400 nm violet light in the glass; then calculate the coordinates of point A1 of the central tooth, i.e., the first edge tooth. The coordinates of point B1, the tip of the first edge tooth Finally, the coordinates of point P1, where the violet light converges on the projection screen, are calculated. ;

[0116] (3) Determine the kth edge tooth A k-1 A k B k Angular parameters and and its two vertices A k and B k The process of obtaining coordinates is as follows: combining the coordinates of point P1 on the projection screen with the k-th edge tooth. coordinates The angle of the kth edge tooth Determined by the following formula

[0117]

[0118] Where the subscripts k = 1, 2, ..., N. The k-th edge tooth A k-1 A k B k Angular parameters Determined by the following formula

[0119]

[0120] The tip B of the kth edge tooth is calculated. k coordinates of the point From formula (3), it can be seen that in the design... Greater than the total internal reflection angle of red light in a transparent medium Light on interface A k-1 B k Total internal reflection occurs on the surface, which can prevent interface A from being affected. k-1 B k Light energy loss on the surface;

[0121] When k=2, that is, the angle parameter of the second adjacent edge tooth A1A2B2 And the coordinates of its two vertices A2 and B2 can be determined as follows: Combining the coordinates of point P1 and point A1, the following can be calculated using formulas (2) and (3): , The coordinates of point A2 of the second edge tooth , The coordinates of point B2 at the tip of the second tooth. , ;

[0122] When k=3 The coordinates of point A3 , The coordinates of point B3 , ;

[0123] When k=4 The coordinates of point A4 , The coordinates of point B4 cm, ;

[0124] When k=5 The coordinates of point A5 , The coordinates of point B5 , ;

[0125] (4) Determine the kth edge tooth A k-1 A k B k upper red light emission angle The calculation formula is as follows:

[0126]

[0127] Where the subscript k = 1, 2, ..., N; =1.5116, taking the refractive index of 760 nm red light in glass. From formula (4), it can be seen that when the transparent medium material is selected, the red light emission angle is a fixed value. The red light emission angle on each edge tooth and the angle of violet light emission The difference is also a fixed value. This creates the conditions for a large, brightly colored rainbow. (See also...) Figure 4 ;

[0128] (5) The range of the red light emitted from the kth prism on the projection screen is analyzed as follows: In A k-1 A k The effective light-gathering range on the interface is the entire A k-1 A k Part, in A k-1 Point and A k The violet light incident between points reaches A. k B k After that, all along A k B k The light emitted from the surface converges at the same point P1 on the projection screen;

[0129] In A k-1 Point and Ak Red light incident between points, at A k B k The rays emitted in parallel on the surface cannot converge to a single point on the projection screen. A k-1 The red light that enters from point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 1k Near the outer edge of the rainbow circle, and from A k Red light incident at point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 2k , compared to R 1k Get closer to the center;

[0130] Although red light emitted from the same tooth and red light emitted from different teeth cannot converge at a single point, R 2k and R 1k The distance between them is generally much smaller than that between P1 and R. 1k The distance between them ensures the vibrancy of the rainbow. With OP0 fixed, the center tooth's... As the size increases, the rainbow halo decreases. When P1 is near P0, the rainbow halo can transform into a colored spot.

[0131] when At 20.9°, the violet light on the central tooth converges at point P1, whose x-coordinate is... The red light entering at point O passes through the prism and then travels along... Figure 4 The red light emanating from the center points to the innermost edge (R) of the rainbow circle on the projection screen. 11 The x-coordinate of this point is The red light incident at point A1, after passing through the prism, travels along... Figure 4 The red light from the middle comes out onto the R on the projection screen. 21 The x-coordinate of this point is You need to be closer to the center. The distance between them is very small, only It is much smaller than the width of a rainbow from red to purple. This makes the colors of the rainbow relatively purer and more vibrant.

[0132] Calculations show that when k=2, the range of the red light emitted from the second adjacent tooth on the projection screen is... The width of the rainbow is , The superposition of this second tooth of the rainbow makes the colors of the rainbow more vibrant.

[0133] When k=3, that is, the range of the red light emitted from the third adjacent edge on the projection screen is: The width of the rainbow is When k=4, the range of red light on the projection screen is: The width of the rainbow is When k=5, the range of red light on the projection screen is: The width of the rainbow is .

[0134] Sunlight shines on the entire Fresnel prism 1, and light of different colors will be dispersed from the prism's serrated sides. See the schematic diagrams for the violet and red light paths. Figure 5 As shown.

[0135] Using ray tracing software, at an irradiance of 1000 W / m 2 See the example of a ring-shaped rainbow effect on a projection screen 10 cm away under sunlight. Figure 6 As shown.

[0136] In the design of Fresnel prisms, the angle of the prism teeth is changed. This can achieve a focusing effect. For example, keeping N=5, =1cm and OP0=10 cm, when At 24.7°, a colored light spot with a diameter of about 5 cm is formed on the projection screen. See [reference needed]. Figure 6 As shown. Further increase the tooth angle. This allows for a smaller focused light spot.

[0137] In Fresnel prism design, increasing the projection screen distance OP0 can increase the radius and radial width of the ring rainbow, achieving a better rainbow effect. For example, keeping N=5, =20.9°, =1 cm, increasing the projection screen distance to OP0=100 cm, the maximum radius P0R of the rainbow on the projection screen. 11 =30.49 cm, rainbow radial width R 11 P1 = 16.49 cm, both more than 10 times larger than when OP0 = 10 cm, while the range of red light on the projection screen remains the same. It is much smaller than the width of a rainbow from red to purple. This makes the colors of the rainbow still very vibrant.

[0138] Furthermore, given a fixed Fresnel prism size, the larger the number of teeth N, the more pronounced the projected ring-shaped rainbow effect.

[0139] In this invention, sunlight can also be replaced with a strong artificial white parallel light source, which can still project a ring rainbow.

[0140] The embodiments of the present invention have been described above, but the scope of the present invention is not limited thereto. Users can make various changes and implement them without departing from the spirit of the present invention, but all of them are included within the protection scope of the present invention.

Claims

1. A device for realizing a ring rainbow using a refracting total internal reflection Fresnel prism, characterized in that, The system includes a Fresnel prism and a projection screen. The rotation axis of the Fresnel prism is perpendicular to the projection screen. One side of the Fresnel prism is flat, and the other side is toothed. Its cross-section is a triangle with two sets of connected bases symmetrically distributed along the rotation axis. The Fresnel prism is made of transparent material. Incident light is parallel to the rotation axis of the Fresnel prism, enters from the connected bases, penetrates the Fresnel prism, and is then projected perpendicularly onto the projection screen. Each tooth of the Fresnel prism has a triangular cross-section. The center tooth OA1B1 is located to the right of the rotation axis, followed by the next adjacent tooth A1A2B2, the next adjacent tooth A2A3B3, the next adjacent tooth A3A4B4, the next adjacent tooth A4A5B5, and so on up to A. N-1 A N B N N is the number of teeth; To the right of the rotation axis, the incident edges of the light rays are OA1, A1A2, A2A3, A3A4, A4A5...A N-1 A N The points A1, A2, A3, A4, A5, ..., A4 form a straight line perpendicular to the incident light, and the sides of the total internal reflection are OB1, A1B2, A2B3, A3B4, A4B5...A N-1 B N The light rays exit from the edges A1B1, A2B, A3B3, A3B4, A4B5...A N B N Connecting angles: ∠A1OB1=α1, ∠A2A1B2=α2, ∠A3A2B3=α3, ∠A4A3B4=α4, ∠A5A4B5=α5……∠A N A N-1 B N =α N The tooth angles are ∠OB1A1=θ1, ∠A1B2A2=θ2, ∠A2B3A3=θ3, ∠A3B4A4=θ4, ∠A4B5A5=θ5……∠A N-1 B N A N =θ N ; The incident side of each serrated triangle has the same length, i.e., OA1=A1A2=A2A3=A3A4=A4A5…… = A N-1 A N The size and position of the total internal reflection side and the exit side of each serrated triangle are determined by the angles of the serrations and the connecting angles of the triangle. The angles of each serrated triangle, namely θ1, θ2, θ3, θ4, θ5...θ N The differences lie in the connecting angles of each serrated triangle, namely α1, α2, α3, α4, α5...α N They are also different and need to be calculated according to the toothed edge emission convergence method; The toothed edge emission and convergence method is that all monochromatic light incident parallel to the incident edge of the toothed edge is emitted along the edge of the emitted edge of the toothed edge, that is, the emission angle is 90°. Monochromatic light emitted from different teeth is simultaneously converged at the same point on the projection screen, and colored light with wavelengths greater than the wavelength of violet light is dispersed on the projection screen in sequence. Light rays are incident perpendicularly from the incident surface OA1 of the central tooth OA1B1 along a direction parallel to the y-axis. They reach the OB1 surface in the prism, undergo total internal reflection at the OB1 surface, reach the A1B1 surface, are refracted at the A1B1 surface, exit from the central tooth, and finally reach the projection screen. Due to the dispersion of light, the various colors of light, from violet to red, reach the screen at different positions, ultimately presenting a circular rainbow with violet inside and red outside.

2. The device for realizing a ring rainbow using a refractive total internal reflection Fresnel prism as described in claim 1, characterized in that, According to the requirement that violet light is emitted and converged along the edges of the teeth, all the violet light emanating from the A1B1 surface of the central tooth OA1B1 converges to point P1 on the projection screen. Similarly, the violet light emanating from the A2B2 surface of the second tooth A1A2B2 also converges to point P1. The violet light emanating from the third, fourth, and fifth teeth must also converge to point P1. Therefore, the kth tooth A1B1... k-1 A k B k A k B k The violet light emanating from the surface must also converge at point P1 on the projection screen, where k = 1, 2, ..., N, represents the order of the Fresnel prism teeth. k = 1 represents the first tooth A0A1B1, i.e., the central tooth OA1B1, and k = N represents the Nth tooth A0A1B1. N-1 A N B N ; On the k-th tooth of the Fresnel prism, the connecting angle... With light at the incident interface A k-1 A k and total reflection interface A k-1 B k Ejection interface A k B k Refraction - Total Internal Reflection - 5 Angles Related to Refraction , , , , The relationship between them is: Where the subscripts k = 1, 2, ..., N, and n is the refractive index of light in the transparent medium, which will be involved in the calculations below. or , which is the refractive index of red or violet light in a transparent medium; When k=1, that is, on the central tooth, the light ray is incident perpendicularly on the OA1 interface, then the angle of incidence is... and angle of refraction All are 0. It is the angle of incidence of the light at interface OB1. It is the angle of incidence of the light at the A1B1 interface. It is the angle of refraction or emission of light at the A1B1 interface. According to the toothed edge emission and converging method, its violet emission angle is... The angle is 90°. The incident angle of violet light at the OB1 interface is calculated using formula (1). The angle of incidence of red light on the OB1 interface At this point in the formula Substitution Calculate the red light emission angle ; In the prism A k-1 A k B k Above, the light at A k-1 A k If the light is incident perpendicularly to the interface, then the angle of incidence is... and angle of refraction All are 0. Is the light at A k-1 B k The angle of incidence of the interface, Is the light at A k B k The angle of incidence of the interface, Is the light at A k B k The angle of refraction or the angle of emission at the interface, where n is taken in formula (1) At that time, its violet light emission angle In formula (1), n ​​takes the following values: At that time, the red light emission angle is calculated according to formula (1). To construct a Fresnel prism with N teeth, the following steps are required: the central tooth OA1B1, the second adjacent tooth A1A2B2, the third tooth A2A3B3, ..., the (N-1)th tooth A N-2 A N-1 B N-1 The Nth tooth A N-1 A N B N The angle parameters and vertex coordinates are determined, and the specific process is as follows: (1) First, O is given as the origin (0, 0), the y-axis is parallel to the incident ray, and the x-axis is parallel to the projection screen. The position of the projection screen is given, that is, the distance OP0 from point O to the intersection point P0 of the y-axis and the projection screen. The incident edge A is given. k-1 A k The length L is given by the subscript k = 1, 2, ..., N; (2) Determine the angular parameters of the center tooth OA1B1 and The specific process for determining the coordinates of the two vertices A1 and B1 is as follows: (Central tooth edge, tooth angle) The range is from 10° to 80°, given Calculate the connection angle for one of the specific values. Then calculate the coordinates of point A1, which is the center tooth and the first edge tooth. The coordinates of point B1, the tip of the first tooth Finally, the coordinates of point P1, where the violet light converges on the projection screen, are calculated. ; (3) Determine the kth edge tooth A k-1 A k B k Angular parameters and and its two vertices A k and B k The process of obtaining coordinates is as follows: Combine the coordinates of point P1 on the projection screen with the coordinates of the k-th edge. coordinates The angle of the kth edge tooth Determined by the following formula Where the subscript k = 1, 2, ..., N, the k-th edge tooth A k-1 A k B k Angular parameters Determined by the following formula The tip B of the kth edge tooth is calculated. k coordinates of the point From formula (3), it can be seen that in the design... Greater than the total internal reflection angle of red light in a transparent medium Light on interface A k-1 B k Total internal reflection occurs on the surface, avoiding interface A. k-1 B k Light energy loss on the surface; (4) Determine the kth edge tooth A k-1 A k B k upper red light emission angle The calculation formula is as follows: From formula (4), it can be seen that when the transparent medium material is selected, the red light emission angle is... The red light emission angle on each tooth is a fixed value. and the angle of violet light emission The difference is also a fixed value. ,exist The left and right sides created the conditions for a large, brightly colored rainbow; (5) The range of the red light emitted from the kth prism on the projection screen is analyzed as follows: In A k-1 A k The effective light-gathering range on the interface is the entire A k-1 A k Part, in A k-1 Point and A k The violet light incident between points reaches A. k B k After that, all along A k B k The light emitted from the surface converges at the same point P1 on the projection screen; In A k-1 Point and A k Red light incident between points, at A k B k The rays emitted in parallel from the surface cannot converge to a single point on the projection screen. (A) k-1 The red light that enters from point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 1k Near the outer edge of the rainbow circle, and from A k Red light incident at point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 2k , compared to R 1k Stay close to the center.

3. The device for realizing a ring rainbow using a refractive total internal reflection Fresnel prism as described in claim 2, characterized in that, R 2k and R 1k The distance between them is much smaller than that between P1 and R. 1k The distance between them ensures the vibrancy of the rainbow; with OP0 fixed, the center tooth's... As the size increases, the rainbow halo will become smaller; when P1 is near P0, the rainbow halo can turn into a colored spot.

4. A method for realizing a ring rainbow using a refracting total internal reflection Fresnel prism, characterized in that, The following steps are achieved using the device described in claim 2: According to the requirement that violet light is emitted and converged along the tooth edges, all the violet light emanating from the A1B1 surface of the central tooth OA1B1 converges to point P1 on the projection screen. Similarly, the violet light emanating from the A2B2 surface of the second tooth A1A2B2 also converges to point P1 on the projection screen. The violet light emanating from the third, fourth, and fifth teeth must also converge to point P1 on the projection screen. Therefore, the kth tooth A1B1... k-1 A k B k A k B k The violet light emitted from the surface must also converge at point P1 on the projection screen, where k = 1, 2, ..., N, which are the order of the Fresnel prism; k = 1 represents the first prism A0A1B1, i.e., the central prism OA1B1, and k = N represents the Nth prism A0A1B1. N-1 A N B N ; On the k-th tooth of the Fresnel prism, the connecting angle... With light at the incident interface A k-1 A k and total reflection interface A k-1 B k Ejection interface A k B k Refraction - Total Internal Reflection - 5 Angles Related to Refraction , , , , The relationship between them is: Where the subscripts k = 1, 2, ..., N, and n is the refractive index of light in the transparent medium, which will be involved in the calculations below. or , which is the refractive index of red or violet light in a transparent medium; When k=1, that is, on the central tooth, the light ray is incident perpendicularly on the OA1 interface, then the angle of incidence is... and angle of refraction All are 0. It is the angle of incidence of the light at interface OB1. It is the angle of incidence of the light at the A1B1 interface. It is the angle of refraction or emission of light at the A1B1 interface. According to the toothed edge emission and converging method, its violet emission angle is... The angle is 90°. The incident angle of the violet light at the OB1 interface can be calculated using formula (1). The angle of incidence of red light on the OB1 interface At this point in the formula Substitution The red light emission angle can be calculated. ; In the prism A k-1 A k B k Above, the light at A k-1 A k If the light is incident perpendicularly to the interface, then the angle of incidence is... and angle of refraction All are 0. Is the light at A k-1 B k The angle of incidence of the interface, Is the light at A k B k The angle of incidence of the interface, Is the light at A k B k The angle of refraction or the angle of emission at the interface, where n is taken in formula (1) At that time, its violet light emission angle In formula (1), n ​​takes the following values: At that time, the red light emission angle is calculated according to formula (1). To construct a Fresnel prism with N teeth, the following steps are required: the central tooth OA1B1, the second adjacent tooth A1A2B2, the third tooth A2A3B3, ..., the (N-1)th tooth A N-2 A N-1 B N-1 The Nth tooth A N-1 A N B N The angle parameters and vertex coordinates are determined, and the specific process is as follows: (1) First, O is given as the origin (0, 0), the y-axis is parallel to the incident ray, and the x-axis is parallel to the projection screen. The position of the projection screen is given, that is, the distance OP0 from point O to the intersection point P0 of the y-axis and the projection screen. The incident edge A is given. k-1 A k The length L is given by the subscript k = 1, 2, ..., N; (2) Determine the angular parameters of the center tooth OA1B1 and The specific process for determining the coordinates of the two vertices A1 and B1 is as follows: (Central tooth edge, tooth angle) The range is from 10° to 80°, given Calculate the connection angle for one of the specific values. Then calculate the coordinates of point A1, which is the center tooth and the first edge tooth. The coordinates of point B1, the tip of the first tooth Finally, the coordinates of point P1, where the violet light converges on the projection screen, are calculated. ; (3) Determine the kth edge tooth A k-1 A k B k Angular parameters and and its two vertices A k and B k The process of obtaining coordinates is as follows: Combine the coordinates of point P1 on the projection screen with the coordinates of the k-th edge. coordinates The angle of the kth edge tooth Determined by the following formula Where the subscript k = 1, 2, ..., N; the k-th edge tooth A k-1 A k B k Angular parameters Determined by the following formula The tip B of the kth edge tooth is calculated. k coordinates of the point From formula (3), it can be seen that in the design... Greater than the total internal reflection angle of red light in a transparent medium Light on interface A k-1 B k Total internal reflection occurs on the surface, avoiding interface A. k-1 B k Light energy loss on the surface; (4) Determine the kth edge tooth A k-1 A k B k upper red light emission angle The calculation formula is as follows: From formula (4), it can be seen that when the transparent medium material is selected, the red light emission angle is... The red light emission angle on each tooth is a fixed value. and the angle of violet light emission The difference is also a fixed value. ,exist The left and right sides created the conditions for a large, brightly colored rainbow; (5) The range of the red light emitted from the kth prism on the projection screen is analyzed as follows: In A k-1 A k The effective light-gathering range on the interface is the entire A k-1 A k Part, in A k-1 Point and A k The violet light incident between points reaches A. k B k After that, all along A k B k The light emitted from the surface converges at the same point P1 on the projection screen; In A k-1 Point and A k Red light incident between points, at A k B k The rays emerge parallel to each other on the surface and cannot converge to a single point on the projection screen.

5. The method for achieving a ring rainbow using a refractive total internal reflection Fresnel prism as described in claim 4, characterized in that, A k-1 The red light that enters from point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 1k Near the outer edge of the rainbow circle, and from A k Red light incident at point A k B k After being emitted from the surface, the image reaches position R on the projection screen. 2k , compared to R 1k Stay close to the center.

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