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US-12617020-B2 - Precision freeform structuring for the fabrication of coded lenses

US12617020B2US 12617020 B2US12617020 B2US 12617020B2US-12617020-B2

Abstract

A system for fabricating coded lenses includes a cutting tool configured to controllably cut a workpiece at a specified position-dependent depth while traversing a surface of the workpiece along a specified two-dimensional path. A signal generator is operative to generate a signal for controlling fabrication of a coded lens from the workpiece. A vibration tool is operative to ultrasonically vibrate the cutting tool for cutting of gratings on the workpiece.

Inventors

  • Ping Guo
  • Yaoke Wang

Assignees

  • NORTHWESTERN UNIVERSITY

Dates

Publication Date
20260505
Application Date
20220426

Claims (17)

  1. 1 . A system for fabricating coded lenses comprising: a cutting tool configured to controllably cut a workpiece at a specified position-dependent depth while traversing a surface of the workpiece along a specified two-dimensional path; a signal generator operative to generate a signal for controlling fabrication of a coded lens from the workpiece; and a vibration tool operative to ultrasonically vibrate the cutting tool for cutting of gratings on the workpiece, wherein the cutting tool is configured to machine a freeform surface and fabricate a variable grating on the machined freeform surface; and a processor configured to control the cutting tool and the vibration tool for machining a freeform lens from the workpiece, wherein computing the three-dimensional freeform surface is based on a Chebyshev polynomial Z = f z ( X , Y ) = - ( c ⁡ ( X 2 + Y 2 ) 1 + 1 + c 2 ( X 2 + Y 2 ) + ∑ i = 0 I ∑ j = 0 J C ij ⁢ T i ( X L x ) ⁢ T j ( X L y ) + Z 0 T i ( x ) = cos ⁡ ( i ⁢ arccos ⁢ x ) , where x, y, z are global coordinates, X, Y, Z are workpiece coordinates, ρ, θ are workpiece polar coordinates, Z=f z (X,Y) is a freeform surface function, Z 0 is a reference depth, T i (x) is a i th order Chebyshev polynomials at x, C ij is a 2-D Chebyshev polynomials coefficient, i, j are a order of Chebyshev polynomials, I, J are a maximum order of Chebyshev polynomials, c is a curvature of toric term.
  2. 2 . The system of claim 1 , wherein the cutting tool is configured to rotate to cut the workpiece in directions that are perpendicular to local gradients on the workpiece.
  3. 3 . The system of claim 1 , further comprising a vacuum to remove cut material from the cutting tool while the cutting tool is cutting the workpiece.
  4. 4 . The system of claim 1 , further comprising a mechanism to feed the workpiece linearly along an axis relative to the cutting tool while the cutting tool cuts the workpiece.
  5. 5 . The system of claim 1 , further comprising a spindle operative to hold the workpiece and turn the workpiece about a central axis to facilitate the cutting tool to cut the workpiece while the workpiece is turning.
  6. 6 . The system of claim 1 , wherein the vibration tool includes an elliptical vibration tool.
  7. 7 . A method for fabricating coded lenses comprising: generating a signal for controlling fabrication of a coded lens from a workpiece; causing a cutting tool to traverse a surface of the workpiece along a specified two-dimensional path, based on the signal; controllably cutting a workpiece at a specified position-dependent depth while traversing the surface of the workpiece along the specified two-dimensional path, based on the signal; and ultrasonically vibrating the cutting tool for cutting gratings on the workpiece; and computing a three-dimensional freeform surface for machining a freeform lens from the workpiece, wherein computing the three-dimensional freeform surface is based on a Chebyshev polynomial Z = f z ( X , Y ) = - ( c ⁡ ( X 2 + Y 2 ) 1 + 1 - c 2 ( X 2 + Y 2 ) + ∑ i = 0 I ∑ j = 0 J C ij ⁢ T i ( X L x ) ⁢ T j ( X L y ) + Z 0 T i ( x ) = cos ⁡ ( i ⁢ arccos ⁢ x ) where x, y, z are global coordinates, X, Y, Z are workpiece coordinates, ρ, θ are workpiece polar coordinates, Z=f z (X,Y) is a freeform surface function, Z 0 is a reference depth, T i (x) is a i th order Chebyshev polynomials at x, C ij is a 2-D Chebyshev polynomials coefficient, i, j are a order of Chebyshev polynomials, I, J are a maximum order of Chebyshev polynomials, c is a curvature of toric term.
  8. 8 . The method of claim 7 , further comprising causing the cutting tool to traverse the surface of the workpiece based on the Chebyshev polynomial to machine a freeform surface.
  9. 9 . The method of claim 7 , wherein causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path comprises feeding the workpiece along a linear direction at a variable rate.
  10. 10 . The method of claim 7 , wherein causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path comprises causing the cutting tool to traverse the surface of the workpiece along quasi-spiral tool paths computed according to M n ≈ h / Δ ⁢ r min k = 1 , … , K  ∇ f ⁡ ( X k , n , Y k , n )  , where at a revolution m from 1 to M n , the quasi-spiral tool path is computed by an interpolation, where a k th point on a m th revolution is labelled as (X k,m , Y k,m ), where X k , m = m ⁡ ( k - 1 ) M n ⁢ K ⁢ ( X k , n + 1 - X k , n ) + X k , n , Y k , m = m ⁡ ( k - 1 ) M n ⁢ K ⁢ ( Y k , n + 1 - Y k , n ) + Y k , n , Z k , m = f z ( X k . m , Y k , m ) - ( n - 1 ) ⁢ h + e k ( X k , m , Y k , m ) , where e h (X k,m , Y k,m ) is a tool compensation term.
  11. 11 . The method of claim 7 , wherein the specified two-dimensional path is computed according to [ X k , m Y k , m ∇ x f ⁡ ( X k , m , Y k , m ) ∇ y f ⁢ ( X k , m , Y k , m ) ] = [ cos ⁡ ( φ k , m ) sin ⁡ ( φ k , m ) 0 0 - sin ⁡ ( φ k , m ) cos ⁢ ( φ k , m ) 0 0 0 0 cos ⁢ ( φ k , m ) sin ⁢ ( φ k , m ) 0 0 - sin ⁡ ( φ k , m ) cos ⁢ ( φ k , m ) ] [ x k , m y k , m - 1 0 ] , ∂ f ⁡ ( X , Y ) ∂ X = cX 1 - c 2 ( X 2 + Y 2 ) + ∑ i = 0 i ∑ j = 0 j C ij ⁢ i ⁢ sin ⁢ ( i ⁢ arc ⁢ cos ⁡ ( X L x ) ) 1 - ( X L x ) 2 ⁢ cos ⁡ ( j ⁢ arc ⁢ cos ⁡ ( Y L y ) ) ∂ f ⁡ ( X , Y ) ∂ Y = cX 1 - c 2 ( X 2 + Y 2 ) + ∑ i = 0 i ∑ j = 0 j C ij ⁢ cos ⁡ ( i ⁢ arc ⁢ cos ⁡ ( X L x ) ) ⁢ j ⁢ sin ⁡ ( j ⁢ arc ⁢ cos ⁡ ( Y L y ) ) 1 - ( Y L y ) 2 such that a cutting tool's trajectory in a workpiece coordinate system (X, Y, Z) is on a desired quasi-spiral, and a cutting direction is perpendicular to a local gradient.
  12. 12 . The method of claim 7 , further comprising compensating for a deviation of a cutting tool's actual cutting point from the desired cutting point on the workpiece by computing e h according to e h = R ⁢ tan ⁡ ( β ) ⁢ tan ⁡ ( β 2 ) β = arc ⁢ tan ⁡ (  ∇ f z ( X , Y )  ) and adjusting the cutting tool's actual cutting point based on e h .
  13. 13 . A non-transitory machine-readable storage medium having instructions stored thereon for causing a processor to execute the method: generating a signal for controlling fabrication of a coded lens from a workpiece; causing a cutting tool to traverse a surface of the workpiece along a specified two-dimensional path, based on the signal; controllably cutting a workpiece at a specified position-dependent depth while traversing the surface of the workpiece along the specified two-dimensional path, based on the signal; ultrasonically vibrating the cutting tool for cutting gratings on the workpiece; and computing a three-dimensional freeform surface for machining a freeform lens from the workpiece, wherein computing the three-dimensional freeform surface is based on a Chebyshev polynomial Z = f z ( X , Y ) = - ( c ⁡ ( X 2 + Y 2 ) 1 + 1 - c 2 ( X 2 + Y 2 ) + ∑ i = 0 I ∑ j = 0 J ⁢ C ij ⁢ T i ( X L x ) ⁢ T j ( Y L y ) ) + Z 0 , T i ( x ) = cos ⁡ ( i ⁢ arc ⁢ cos ⁢ x ) where x, y, z are global coordinates, X, Y, Z are workpiece coordinates, ρ, θ are workpiece polar coordinates, Z=f z (X,Y) is a freeform surface function, Z 0 is a reference depth, T i (x) is a i th order Chebyshev polynomials at x, C ij is a 2-D Chebyshev polynomials coefficient, i, j are a order of Chebyshev polynomials, I, J are a maximum order of Chebyshev polynomials, c is a curvature of toric term.
  14. 14 . The medium of claim 13 , wherein the instructions for causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path further comprise instructions for feeding the workpiece along a linear direction at a variable rate.
  15. 15 . The medium of claim 13 , wherein the instructions cause the processor to further execute the method operations: causing the cutting tool to traverse the surface of the workpiece along quasi-spiral tool paths computed according to M n ≈ h / Δ ⁢ r min k = 1 , … , K  ∇ f ⁡ ( X k , n , Y k , n )  , where at a revolution m from 1 to M n , the quasi-spiral tool path is computed by a interpolation, where a k th point on a m th revolution is labelled as (X k,m , Y k,m ), where X k , m = m ⁡ ( k - 1 ) M n ⁢ K ⁢ ( X k , n + 1 - X k , n ) + X k , n , Y k , m = m ⁡ ( k - 1 ) M n ⁢ K ⁢ ( Y k , n + 1 - Y k , n ) + Y k , n , Z k , m = f z ( X k , m , Y k , m ) - ( n - 1 ) ⁢ h + e k ( X k , m , Y k , m ) , where e h (X k,m , Y k,m ) is a tool compensation term.
  16. 16 . The medium of claim 13 , wherein the instructions cause the processor to further execute the method operations: causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path computed according to [ X k , m Y k , m ∇ x f ⁡ ( X k , m , Y k , m ) ∇ y f ⁢ ( X k , m , Y k , m ) ] = [ cos ⁡ ( φ k , m ) sin ⁡ ( φ k , m ) 0 0 - sin ⁡ ( φ k , m ) cos ⁢ ( φ k , m ) 0 0 0 0 cos ⁢ ( φ k , m ) sin ⁢ ( φ k , m ) 0 0 - sin ⁡ ( φ k , m ) cos ⁢ ( φ k , m ) ] [ x k , m y k , m - 1 0 ] , ∂ f ⁡ ( X , Y ) ∂ X = cX 1 - c 2 ( X 2 + Y 2 ) + ∑ i = 0 i ∑ j = 0 j C ij ⁢ i ⁢ sin ⁢ ( i ⁢ arc ⁢ cos ⁡ ( X L x ) ) 1 - ( X L x ) 2 ⁢ cos ⁡ ( j ⁢ arc ⁢ cos ⁡ ( Y L y ) ) ∂ f ⁡ ( X , Y ) ∂ Y = cX 1 - c 2 ( X 2 + Y 2 ) + ∑ i = 0 i ∑ j = 0 j C ij ⁢ cos ⁡ ( i ⁢ arc ⁢ cos ⁡ ( X L x ) ) ⁢ j ⁢ sin ⁡ ( j ⁢ arc ⁢ cos ⁡ ( Y L y ) ) 1 - ( Y L y ) 2 such that a cutting tool's trajectory in a workpiece coordinate system (X, Y, Z) is on a desired quasi-spiral, and a cutting direction is perpendicular to a local gradient.
  17. 17 . The medium of claim 13 , wherein the instructions cause the processor to further execute the method operations: compensating for a deviation of a cutting tool's actual cutting point from the desired cutting point on the workpiece by computing e h according to e h = R ⁢ tan ⁡ ( β ) ⁢ tan ⁡ ( β 2 ) β = arc ⁢ tan ⁡ (  ∇ f z ( X , Y )  ) and adjusting the cutting tool's actual cutting point based on e h .

Description

CROSS REFERENCE TO RELATED APPLICATIONS The present application claims the benefit of priority under 35 U.S.C. § 119 from U.S. Provisional Patent Application Ser. No. 63/180,388 entitled “Precision Freeform Structuring for the Fabrication of Coded Lenses,” filed on Apr. 27, 2021, the disclosure of which is hereby incorporated by reference in its entirety for all purposes. TECHNICAL FIELD The present disclosure generally relates to optical lenses, and more specifically relates to precision freeform structuring for the fabrication of coded lenses. BACKGROUND Integrated optical systems may include hybrid optical devices that integrate one or more of lenses, mirrors, and gratings. Mirrors provide the functionality of reflection, lenses provide the functionality of refraction, and gratings provide the functionality of diffraction. Examples of integrated optical systems include spectrometers, holographic projection systems, and integrated waveguides and gratings on semiconductor substrates such as silicon (Si). Fabrication processes for integrated optical systems include multi-step ruling and electron-beam (E-beam) processes. The description provided in the background section should not be assumed to be prior art merely because it is mentioned in or associated with the background section. The background section may include information that describes one or more aspects of the subject technology. SUMMARY According to certain aspects of the present disclosure, a system for fabricating coded lenses includes a cutting tool configured to controllably cut a workpiece at a specified position-dependent depth while traversing a surface of the workpiece along a specified two-dimensional path; a signal generator operative to generate a signal for controlling fabrication of a coded lens from the workpiece; and a vibration tool operative to ultrasonically vibrate the cutting tool for cutting of gratings on the workpiece. The cutting tool may be configured to machine a freeform surface and fabricate a variable grating on the machined freeform surface. The cutting tool may be configured to rotate to cut the workpiece in directions that are perpendicular to local gradients on the workpiece. The system may include a vacuum to remove cut material from the cutting tool while the cutting tool is cutting the workpiece. The system may include a mechanism to feed the workpiece linearly along an axis relative to the cutting tool while the cutting tool cuts the workpiece. The system may include a spindle operative to hold the workpiece and turn the workpiece about a central axis to facilitate the cutting tool to cut the workpiece while the workpiece is turning. The vibration tool may include an elliptical vibration tool. According to certain aspects of the present disclosure, a method for fabricating coded lenses includes generating a signal for controlling fabrication of a coded lens from a workpiece; causing a cutting tool to traverse a surface of the workpiece along a specified two-dimensional path, based on the signal; controllably cutting a workpiece at a specified position-dependent depth while traversing the surface of the workpiece along the specified two-dimensional path, based on the signal; and ultrasonically vibrating the cutting tool for cutting gratings on the workpiece. The method may further include computing a three-dimensional freeform surface for machining a freeform lens from the workpiece, wherein computing the three-dimensional freeform surface is based on a Chebyshev polynomial: Z=fz(X,Y)=-(c⁡(X2+Y2)1+1-c2(X2+Y2)+∑i=0I ∑j=0J Cij⁢Ti(XLx)⁢Tj(XLy))+Zo Ti(x)=cos⁡(i⁢arc⁢cos⁢x) where x, y, z are global coordinates; X, Y, Z are workpiece coordinates; ρ, θ are workpiece polar coordinates; φ is spindle angular position; Z=fz(X,Y) is the freeform surface function; Z0 is the reference depth; Ti(x) is the ith order Chebyshev polynomials at x; Cij is the 2-D Chebyshev polynomials coefficient; i, j are the order of Chebyshev polynomials; I, J are the maximum order of Chebyshev polynomials; c is the curvature of toric term; hn is the nth thread (facet) depth; Nis the maximum index of thread (facet); fc (X, Y)=0 is the contour profile of the nth thread; k, K are the index and maximum index of discretized θ; Mn is the number of revolutions in the nth thread; and m is the index of revolution. The method may further include causing the cutting tool to traverse the surface of the workpiece based on the Chebyshev polynomial to machine a freeform surface. Causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path may comprise feeding the workpiece along a linear direction at a variable rate. Causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path may comprise causing the cutting tool to traverse the surface of the workpiece along quasi-spiral tool paths computed according to Mn≈h/Δrmin1,…,K▽f⁡(Xk,n,Yk,n), where at a revo