Optical superlattice structure design method

Abstract

Disclosed in the present invention is an optical superlattice structure design method, aiming to use the design of a superlattice structure to compensate for phase mismatch in a nonlinear frequency conversion position. The method comprises: determining the magnitude of a reciprocal lattice vector required by a reciprocal space; on the basis of Fourier transform, generating an intensity modulation function of a real space; then using a transverse integral effect to transform the reciprocal lattice vector into the real space; and then using the change of a transverse pattern to realize the reciprocal lattice vector, and generating a second-order nonlinear coefficient for intensity modulation. Use of the method designed by the present invention generates a reciprocal lattice vector that can be randomly designed, thereby eliminating high-order and stray reciprocal lattice vectors, and breaking through the limitation of a manufacturing process.

Description

A method for designing optical superlattice structures Technical Field

[0001] The present invention relates to the field of optical nonlinear frequency conversion, and in particular to a method for designing an optical superlattice structure. Background Art

[0002] Optical nonlinear frequency conversion, such as frequency doubling and optical parametric amplification, is widely used in industry, national defense and other fields. In the process of nonlinear frequency conversion, phase matching technology is generally required to compensate for momentum mismatch, such as birefringence phase matching (BPM) and quasi-phase matching (QPM). Quasi-phase matching was first proposed by Bloembergen in 1962. The original form was to use a periodically arranged inverted domain structure to compensate for the three-wave phase mismatch. In the 1980s, the periodically arranged domain structure was expanded to the concept of dielectric superlattice, and the multiple quasi-phase matching theory of quasi-periodic superlattice emerged, and the quasi-periodic superlattice was used to realize the generation of red, green and blue lasers. Later, fan-shaped, arc-shaped, two-dimensional array and even three-dimensional superlattice structures were proposed and realized.

[0003] The basic principle of quasi-phase matching is to periodically arrange the optical axes of a nonlinear crystal in opposite directions, thereby creating positive and negative modulation of the nonlinear coefficient. Taking the frequency doubling process of a basic lithium niobate crystal as an example, the frequency of the 1064nm fundamental wave is doubled to 532nm, requiring a period of approximately Λ = 7μm. In other words, through room-temperature electric field poling, the z-axis of the lithium niobate crystal is reversed every approximately 3.5μm, creating a periodic change from d33 to -d33, thereby generating a spatial frequency component of 1 / Λ in the reciprocal lattice space.

[0004] According to the Fourier transform principle, this +1 and -1 modulation not only produces a reciprocal lattice vector at 1 / Λ, but also produces higher-order reciprocal lattice vectors, such as 3 / Λ and 5 / Λ. However, if the nonlinear coefficient can be arbitrarily modulated, for example, to form a sine or cosine curve X(x) = cos(2πx / Λ), only ±1 / Λ reciprocal lattice vectors will be generated. However, due to manufacturing process limitations, it is currently impossible to arbitrarily modulate the nonlinear coefficient X, so high-order and spurious reciprocal lattice vectors are inevitable.

[0005] Summary of the Invention

[0006] The purpose of the present invention is to provide an optical superlattice structure design method to solve the problem of size modulation of the nonlinear coefficient X, generate arbitrarily designed reciprocal lattice vectors, and eliminate high-order and stray reciprocal lattice vectors.

[0007] To achieve the above functions, the present invention designs an optical superlattice structure design method, which performs the following steps S1 to S3 to complete the design of the superlattice polarization pattern:

[0008] Step S1: Establish an XYZ space coordinate system and determine the size and number of reciprocal lattice vectors required for the reciprocal space;

[0009] Step S2: Based on the size and number of reciprocal lattice vectors required in the reciprocal space, the reciprocal lattice vectors are transformed into real space using the Fourier transform method, an expression for the target superlattice polarization pattern is determined, a curve corresponding to the expression is obtained, a graph of the target superlattice polarization pattern in the Y-axis direction is designed, and the corresponding nonlinear coefficient is obtained;

[0010] Step S3: The pattern obtained in step S2 is reversed and spliced ​​up and down to form a basic unit pattern within a polarization period of the target superlattice polarization pattern, and the basic unit pattern is repeatedly arranged to form the target superlattice polarization pattern.

[0011] As a preferred technical solution of the present invention, the specific steps of step S1 are as follows:

[0012] Step S1.1: Define the laser transmission direction as the positive X-axis direction, the direction perpendicular to the X-axis direction as the Y-axis direction, and form an XOY plane. The Z axis is perpendicular to the XOY plane. The superlattice polarization pattern consists of polarization-reversed regions and non-reversed regions. The polarization-reversed regions are the positive Z-axis direction, and the non-reversed regions are the negative Z-axis direction.

[0013] Step S1.2: The expression of the reciprocal lattice vector is as follows:

[0014] Where f(x,y)=±1 represents the spatial pattern, F(u,v) represents the reciprocal lattice vector corresponding to the spatial pattern, x is the coordinate in the X-axis direction, y is the coordinate in the Y-axis direction, u is the spatial frequency corresponding to the X-axis direction, v is the spatial frequency corresponding to the Y-axis direction, and i is the imaginary unit;

[0015] Step S1.3: Set v = 0 and change the expression of the reciprocal lattice vector to the following formula:

[0016] Where F(u) represents the reciprocal lattice vector.

[0017] As a preferred technical solution of the present invention, the specific steps of step S2 are as follows:

[0018] Step S2.1: Express the integral of the spatial pattern in the Y-axis direction as follows:

[0019] Where X(x) is the nonlinear coefficient, f(x, y) represents the spatial pattern, x is the X-axis coordinate, and y is the Y-axis coordinate;

[0020] Step S2.2: Express the reciprocal lattice vector as follows:

[0021] Where u is the spatial frequency corresponding to the X-axis direction, v is the spatial frequency corresponding to the Y-axis direction, i is the imaginary unit, and F(u) represents the reciprocal lattice vector.

[0022] As a preferred technical solution of the present invention: the target superlattice polarization pattern requires a reciprocal lattice vector, the size of which is 1 / Λ, and the superlattice polarization pattern is expressed as follows:

[0023] Where A is the period width of the superlattice polarization pattern in the Y-axis direction, x is the coordinate in the X-axis direction, and Λ is the polarization period;

[0024] By integrating the above equation with respect to y, we can obtain the nonlinear coefficient when the superlattice polarization pattern contains a reciprocal lattice vector as follows:

[0025] In the formula, the nonlinear coefficient in the polarization reversal region is recorded as -X0, and the nonlinear coefficient in the non-reversal region is recorded as +X0.

[0026] As a preferred technical solution of the present invention: the superlattice polarization pattern requires two reciprocal lattice vectors, the sizes of which are 1 / Λ1 and 1 / Λ2 respectively, and the nonlinear coefficient is as follows:

[0027] Wherein, the nonlinear coefficient of the polarization reversal region is recorded as -X0, the nonlinear coefficient of the non-reversal region is recorded as +X0, Λ1 and Λ2 are polarization periods, and x is the coordinate in the X-axis direction.

[0028] As a preferred technical solution of the present invention: the superlattice polarization pattern requires an arbitrary shape of the reciprocal lattice vector, and its nonlinear coefficient is as follows:

[0029] Where T is the reciprocal lattice width, Λ is the polarization period, and x is the X-axis coordinate. The superlattice polarization pattern is expressed as follows:

[0030] Where Δ is the difference between the maximum and minimum values ​​of the nonlinear coefficient X(x).

[0031] Beneficial effects: Compared with the prior art, the advantages of the present invention include:

[0032] This paper proposes a method for designing an optical superlattice structure, using it to compensate for phase mismatch in nonlinear frequency conversion. The basic approach involves using Fourier transforms to transform the reciprocal lattice vector into real space. This reciprocal lattice vector is then realized by varying the transverse pattern. This allows for arbitrary modulation of the nonlinear coefficient X, resulting in a freely designable reciprocal lattice vector and eliminating high-order and spurious reciprocal lattice vectors. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] FIG1 is a superlattice polarization pattern including a reciprocal lattice vector provided according to an embodiment of the present invention;

[0034] FIG2 is a curve of a superlattice polarization pattern including a reciprocal lattice vector within a polarization period according to an embodiment of the present invention;

[0035] FIG3 is a superlattice polarization pattern including two reciprocal lattice vectors provided according to an embodiment of the present invention;

[0036] FIG4 is a superlattice polarization pattern including an arbitrary-shaped reciprocal lattice vector according to an embodiment of the present invention;

[0037] FIG5 is a diagram showing a portion of a frequency-doubling polarization pattern provided according to an embodiment of the present invention;

[0038] FIG6( a ) is a reciprocal lattice vector of a superlattice polarization pattern designed by a method for designing an optical superlattice structure provided according to an embodiment of the present invention;

[0039] Figure 6(b) shows the inverse lattice vector of the polarization pattern of a normal superlattice. DETAILED DESCRIPTION

[0040] The present invention will be further described below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solutions of the present invention and are not intended to limit the scope of protection of the present invention.

[0041] An embodiment of the present invention provides a method for designing an optical superlattice structure, which performs the following steps S1 to S3 to complete the design of a superlattice polarization pattern:

[0042] Step S1: Establish an XYZ space coordinate system and determine the size and number of reciprocal lattice vectors required for the reciprocal space;

[0043] The specific steps of step S1 are as follows:

[0044] Step S1.1: With reference to Figures 1 and 2, define the laser transmission direction as the positive X-axis direction, the direction perpendicular to the X-axis direction as the Y-axis direction, and form an XOY plane. The Z axis is perpendicular to the XOY plane. The superlattice polarization pattern includes polarization reversal regions and non-reversal regions. The black portion in Figures 1 and 2 represents the polarization reversal region, representing the positive Z-axis direction, with a nonlinear coefficient of X = +1. The white portion represents the non-reversal region, representing the negative Z-axis direction, with a nonlinear coefficient of X = -1. According to the quasi-phase matching principle, the superlattice polarization pattern should be vertical black and white stripes, and the width of the black and white stripes should be the same, both Λ / 2.

[0045] Step S1.2: The expression of the reciprocal lattice vector is as follows:

[0046] Where f(x,y)=±1 represents the spatial pattern, F(u,v) represents the reciprocal lattice vector corresponding to the spatial pattern, x is the coordinate in the X-axis direction, y is the coordinate in the Y-axis direction, u is the spatial frequency corresponding to the X-axis direction, v is the spatial frequency corresponding to the Y-axis direction, and i is the imaginary unit;

[0047] Step S1.3: Set v = 0 and change the expression of the reciprocal lattice vector to the following formula:

[0048] Where F(u) represents the reciprocal lattice vector.

[0049] Step S2: Based on the size and number of reciprocal lattice vectors required in the reciprocal space, the reciprocal lattice vectors are transformed into real space using the Fourier transform method, an expression for the target superlattice polarization pattern is determined, a curve corresponding to the expression is obtained, a graph of the target superlattice polarization pattern in the Y-axis direction is designed, and the corresponding nonlinear coefficient is obtained;

[0050] The specific steps of step S2 are as follows:

[0051] Step S2.1: Express the integral of the spatial pattern in the Y-axis direction as follows:

[0052] Where X(x) is the nonlinear coefficient, f(x,y) represents the spatial pattern, x is the X-axis coordinate, and y is the Y-axis coordinate;

[0053] Step S2.2: Express the reciprocal lattice vector as follows:

[0054] Where u is the spatial frequency corresponding to the X-axis direction, v is the spatial frequency corresponding to the Y-axis direction, i is the imaginary unit, and F(u) represents the reciprocal lattice vector;

[0055] The above formula means that if the required reciprocal lattice vector F(u) is determined, the expression for f(x) can be obtained through inverse Fourier transform, where f(x) is the result of integrating y with respect to the spatial pattern f(x,y). Then, based on the X(x) expression (3), the Y-direction pattern can be designed to satisfy the X(x) expression (3).

[0056] Step S3: The pattern obtained in step S2 is reversed and spliced ​​up and down to form a basic unit pattern within a polarization period of the target superlattice polarization pattern, and the basic unit pattern is repeatedly arranged to form the target superlattice polarization pattern.

[0057] In one embodiment, referring to FIG2 , the target superlattice polarization pattern requires a reciprocal lattice vector with a size of 1 / Λ. Based on the inverse Fourier transform, X(x) = sin(2πx / Λ) or X(x) = cos(2πx / Λ). Furthermore, based on the expression for X(x), it is necessary to satisfy the requirement that at any point on the X-axis, the length of the black line segment on the Y-axis minus the length of the white line segment equals X(x). The boundary between the black and white areas is a cosine curve, expressed as follows:

[0058] Where A is the period width of the superlattice polarization pattern in the Y-axis direction, x is the coordinate in the X-axis direction, and Λ is the polarization period, that is, the period width of the superlattice polarization pattern in the X-axis direction. Only for y in the range of 0-A / 2, since the black part in Figure 2 represents +1 and the white part represents -1, at a certain x value, the length of the black line segment minus the length of the white line segment is:

[0059] In Figure 2, the nonlinear coefficient for the white portion is denoted as +X0, and the nonlinear coefficient for the black portion is denoted as -X0. If the area between 0 and A / 2 is completely white, the nonlinear coefficient is +X0; otherwise, it is -X0. In other words, the ratio of the length of the black portion corresponding to +1 to the length of the white portion corresponding to -1 to A / 2 determines the magnitude of the nonlinear coefficient, as expressed in the following formula:

[0060] In the formula, the nonlinear coefficient in the polarization reversal region is recorded as -X0, and the nonlinear coefficient in the non-reversal region is recorded as +X0.

[0061] Therefore, the pattern in FIG2 realizes modulation with a nonlinear coefficient of cosine. The pattern in FIG2 is reversed upside down and pieced together to form a basic unit within the dotted rectangular frame as shown in FIG1. ​​Then, this unit is repeatedly arranged in the X and Y directions to form the pattern as shown in FIG1.

[0062] As shown in Figure 1, the X-axis period Λ determines the size of the reciprocal lattice vector. The Y-axis period A needs to be sufficiently small, for example, comparable to the wavelength of the fundamental light. If A is too large, the incident laser will only sense a portion of the fundamental unit, failing to achieve an integration effect.

[0063] In one embodiment, the superlattice polarization pattern requires two reciprocal lattice vectors, the magnitudes of which are 1 / Λ1 and 1 / Λ2, respectively. The nonlinear coefficient is as follows:

[0064] Wherein, the nonlinear coefficient of the polarization reversal region is recorded as -X0, the nonlinear coefficient of the non-reversal region is recorded as +X0, Λ1 and Λ2 are polarization periods, and x is the coordinate in the X-axis direction.

[0065] 3 , a superlattice polarization pattern including two reciprocal lattice vectors has polarization periods Λ1 = 6 μm and Λ2 = 6 μm.

[0066] In one embodiment, the superlattice polarization pattern requires an arbitrary shape reciprocal lattice vector with the following nonlinear coefficient:

[0067] Where T is the reciprocal lattice width, Λ is the polarization period, and x is the X-axis coordinate;

[0068] The difference between the maximum and minimum values ​​of the nonlinear coefficient X(x) is recorded as Δ. Then, according to y-(Δ-y)=2y-Δ=X(x), the following formula can be obtained:

[0069] 4 , the reciprocal lattice vector required for the superlattice polarization pattern in the figure is a rectangular function with a center position of Λ and a width of T.

[0070] The following are application examples of the optical superlattice structure design method designed by the present invention:

[0071] Magnesium-doped lithium niobate (Mg:LiNbO3) is used to frequency-double a nanosecond laser with a wavelength of 1064 nm. The nonlinear coefficient d33 of lithium niobate is chosen, which means that the polarization reversal direction is the Z axis (c axis) of the crystal, and the pattern is in the XY plane. First, the reciprocal lattice vector of the superlattice is calculated according to the Sellmier equation of the material: We get Λ = 6.97 μm. According to formula (5), the size of A needs to be compared with the wavelength, and the selection range can be between 1 μm and 5 μm. Considering the difficulty of process processing, A = 3.85 μm is selected. A part of the polarization pattern designed according to formula (5) is shown in Figure 5, which is an enlarged view of the 48.79 μm × 50 μm (X × Y) square area. The room temperature electric field polarization method is used to polarize according to the pattern in Figure 5, where the black part is the polarization reversal area. When performing frequency doubling, the laser is focused and incident along the X axis, and the crystal temperature is fine-tuned to obtain the highest frequency doubling power.

[0072] As shown in Figure 6, Figure 6(a) shows the components of the Fourier transform of Figure 5 along the X-axis, and Figure 6(b) shows the Fourier transform of a conventional superlattice pattern. The pattern of the present invention only has ±1-order reciprocal lattice vectors, while conventional superlattice patterns have higher-order reciprocal lattice vectors. Therefore, the present invention has the advantage of low stray light during nonlinear frequency conversion.

[0073] The embodiments of the present invention are described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Various changes can be made within the scope of knowledge possessed by ordinary technicians in this field without departing from the spirit of the present invention.

Claims

1. A method for designing an optical superlattice structure, characterized in that: Perform the following steps S1 to S3 to complete the design of the superlattice polarization pattern: Step S1: Establish an XYZ space coordinate system and determine the size and number of reciprocal lattice vectors required for the reciprocal space; Step S2: according to the size and number of the reciprocal lattice vectors required in the reciprocal space, the reciprocal lattice vectors are transformed into the real space by the Fourier transform method, the expression of the target superlattice polarization pattern is determined, the curve corresponding to the expression is obtained, the graph of the target superlattice polarization pattern in the Y-axis direction is designed, and the corresponding nonlinear coefficient is obtained; Step S3: The pattern obtained in step S2 is reversed and spliced ​​up and down to form a basic unit pattern within a polarization period of the target superlattice polarization pattern, and the basic unit pattern is repeatedly arranged to form the target superlattice polarization pattern.

2. The optical superlattice structure design method according to claim 1, characterized in that: The specific steps of step S1 are as follows: Step S1.1: define the laser transmission direction as the positive direction of the X-axis, the direction perpendicular to the X-axis as the Y-axis, and form an XOY plane. The Z axis is perpendicular to the XOY plane. The superlattice polarization pattern is composed of a polarization reversal region and a non-reversal region. The polarization reversal region is the positive direction of the Z-axis, and the non-reversal region is the negative direction of the Z-axis. Step S1.2: The expression of the reciprocal lattice vector is as follows: Wherein, f(x,y)=±1 represents the spatial pattern, F(u,v) represents the reciprocal lattice vector corresponding to the spatial pattern, x is the coordinate in the X-axis direction, y is the coordinate in the Y-axis direction, u is the spatial frequency corresponding to the X-axis direction, v is the spatial frequency corresponding to the Y-axis direction, and i is the imaginary unit; Step S1.3: Set v = 0 and change the expression of the reciprocal lattice vector to the following: Where F(u) represents the reciprocal lattice vector.

3. The optical superlattice structure design method according to claim 1, characterized in that: The specific steps of step S2 are as follows: Step S2.1: The integral of the spatial pattern in the Y-axis direction is expressed as follows: Where X(x) is the nonlinear coefficient, f(x,y) represents the spatial pattern, x is the coordinate in the X-axis direction, and y is the coordinate in the Y-axis direction; Step S2.2: Express the reciprocal lattice vector as follows: Wherein, u is the spatial frequency corresponding to the X-axis direction, v is the spatial frequency corresponding to the Y-axis direction, i is the imaginary unit, and F(u) represents the reciprocal lattice vector.

4. The optical superlattice structure design method according to claim 1, characterized in that: The target superlattice polarization pattern requires a reciprocal lattice vector, and the reciprocal lattice vector size is 1 / Λ, so the superlattice polarization pattern is expressed as follows: Where A is the period width of the superlattice polarization pattern in the Y-axis direction, x is the coordinate in the X-axis direction, and Λ is the polarization period; By integrating y with respect to the above equation, the nonlinear coefficient when the superlattice polarization pattern contains a reciprocal lattice vector is obtained as follows: In the formula, the nonlinear coefficient of the polarization reversal region is recorded as -X0, and the nonlinear coefficient of the non-reversal region is recorded as +X0.

5. The optical superlattice structure design method according to claim 1, characterized in that: The superlattice polarization pattern requires two reciprocal lattice vectors, the sizes of which are 1 / Λ1 and 1 / Λ2 respectively. The nonlinear coefficient is as follows: Wherein, the nonlinear coefficient of the polarization reversal region is recorded as -X0, the nonlinear coefficient of the non-reversal region is recorded as +X0, Λ1 and Λ2 are polarization periods, and x is the coordinate in the X-axis direction.

6. The optical superlattice structure design method according to claim 1, characterized in that: The superlattice polarization pattern requires an arbitrary shape reciprocal lattice vector, and its nonlinear coefficient is as follows: Where T is the reciprocal lattice width, Λ is the polarization period, and x is the coordinate in the X-axis direction. The superlattice polarization pattern is expressed as The following formula: Where Δ is the difference between the maximum and minimum values ​​of the nonlinear coefficient x(x).

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