Two-dimensional photonic crystal laser

Abstract

A two-dimensional photonic crystal laser (10) comprises: a pair of electrodes (171, 172); an active layer (11) that is provided between the pair of electrodes and that generates light having a prescribed wavelength in response to injection of current from the electrodes; and a two-dimensional photonic crystal layer (12) that is provided between the active layer and either one of the pair of electrodes, and that has a platelike base material (121) and a plurality of different refractive index regions (122) arranged on said base material and having refractive indices different from that of the base material. The plurality of different refractive index regions are arranged in a manner as to be deviated, by respectively different positional deviation amounts, from respective lattice points in a two-dimensional lattice arranged cyclically on the base material at an in-plane cycle corresponding to the prescribed wavelength, and / or are arranged at the respective lattice points or at positions deviated, by the positional deviation amounts, from said lattice points in areas having respectively different areal deviation amounts, which are differences from a prescribed reference area. The positional deviation amounts and / or the areal deviation amounts of the plurality of different refractive index regions are each represented by: a modulated value, which is defined by a composite cycle obtained by superposing mutually different plurality of in-plane cycles, in a different refractive index region in which the modulated value at a lattice point where the different refractive index region is arranged falls between a prescribed upper limit value and a prescribed lower limit value; said upper limit value in a different refractive index region in which the modulated value is more than the upper limit value; and said lower limit value in a different refractive index region in which the modulated value is less than the lower limit value.

Description

Two-dimensional photonic crystal laser

[0001] The present invention relates to a two-dimensional photonic crystal laser (also called a "two-dimensional photonic crystal surface-emitting laser") that amplifies light using a two-dimensional photonic crystal.

[0002] A two-dimensional photonic crystal laser includes an active layer, a two-dimensional photonic crystal layer, and a pair of electrodes sandwiching them. The active layer emits light in a specific emission wavelength band when carriers (holes and electrons) are injected into the active layer from the pair of electrodes. The two-dimensional photonic crystal layer has a structure in which modified refractive index regions, each having a different refractive index from that of a plate-shaped base material, are periodically arranged in a two-dimensional pattern. The modified refractive index regions are made of voids (air) formed in the base material or a material different from the base material. In such a two-dimensional photonic crystal laser, only light of a predetermined wavelength corresponding to the periodic length of the arrangement of the modified refractive index regions is amplified in the two-dimensional photonic crystal layer among the light generated in the active layer, causing laser oscillation, and a laser beam is emitted from the surface of the two-dimensional photonic crystal layer.

[0003] In a typical two-dimensional photonic crystal laser, modified refractive index regions have the same planar shape and are arranged at the lattice points of a two-dimensional lattice such as a square lattice, rectangular lattice, or triangular lattice, and the laser beam is emitted in a direction perpendicular to the two-dimensional photonic crystal layer.

[0004] In contrast, Patent Documents 1 and 2 describe a two-dimensional photonic crystal laser including a two-dimensional photonic crystal layer in which modified refractive index areas are arranged at each lattice point of a two-dimensional lattice with different offset amounts from the lattice point and / or in which the modified refractive index areas are arranged with different areas. Here, the offset amount is a vector quantity (i.e., a quantity with magnitude and direction) in a plane parallel to the two-dimensional photonic crystal layer, and "different offset amounts" also include cases in which the magnitude of the offset is the same but the direction of the offset is different. These offset amounts and / or areas change at a predetermined period (longer than the period of the lattice point arrangement) along a predetermined direction parallel to the two-dimensional photonic crystal layer. The period of change in this offset amount and / or area is called the "modulation period," and the offset amount and / or area of ​​each modified refractive index area modulated by this modulation period is called the "modulation phase." Such a two-dimensional photonic crystal laser emits two laser beams that are tilted from a direction perpendicular to the two-dimensional photonic crystal layer by an angle (tilt angle) that corresponds to the modulation period, and whose azimuth angles are 180° apart in the specified direction.

[0005] Furthermore, Patent Document 3 describes a two-dimensional photonic crystal laser having a two-dimensional photonic crystal layer in which modified refractive index regions are arranged at each lattice point of the two-dimensional lattice or at positions shifted from each lattice point, with a shift amount and / or area modulated by a composite period obtained by superimposing N different modulation periods (N is an integer of 2 or greater). Such a two-dimensional photonic crystal laser emits 2N (four or more) laser beams from the two-dimensional photonic crystal layer, each having an inclination angle and an azimuth angle corresponding to each of the superimposed modulation periods in the composite period. In fields such as LIDAR (Laser Imaging Detection And Ranging), a sensor used in remote sensing, a laser light source that emits multiple laser beams is required, and the two-dimensional photonic crystal laser described in Patent Document 3 can be suitably used for such applications.

[0006] International Publication No. WO2014 / 136607 U.S. Patent Publication No. US2016 / 0248224 International Publication No. WO2022 / 181723

[0007] The inventors of the present application have investigated the two-dimensional photonic crystal laser described in Patent Document 3 and found that in addition to 2N laser beams corresponding to the composite period (these will be referred to as "original laser beams"), many unnecessary laser beams with weaker intensities are emitted. When the original laser beams have an intensity similar to that shown in the examples of Patent Document 3, the intensities of these unnecessary laser beams are negligibly small, but when the intensity of the original laser beams is increased by increasing the current injected into the active layer, the intensity of the unnecessary laser beams also increases and becomes impossible to ignore.

[0008] The problem to be solved by the present invention is to provide a two-dimensional photonic crystal laser that emits multiple (four or more) laser beams while suppressing the emission of unnecessary laser beams.

[0009] The two-dimensional photonic crystal laser according to the present invention, which has been made to solve the above problems, comprises: a) a pair of electrodes; b) an active layer provided between the pair of electrodes and which generates light of a predetermined wavelength when a current is injected from the electrodes; and c) a two-dimensional photonic crystal layer provided between one of the pair of electrodes and the active layer, the two-dimensional photonic crystal layer having a plate-shaped base material and a plurality of modified refractive index areas arranged on the base material and having a refractive index different from that of the base material, wherein the plurality of modified refractive index areas are arranged at positions shifted by different amounts of positional deviation from each lattice point of a two-dimensional lattice which is periodically arranged on the base material with an in-plane period corresponding to the predetermined wavelength, and / or are arranged at each lattice point with an area shift amount which is a difference from a predetermined reference area, or at positions shifted from the lattice point by the position shift amount, The positional deviation amount and / or area deviation amount of each of the multiple modified refractive index areas is a modulation value determined by a composite period obtained by superimposing multiple different in-plane periods, and is a value between a predetermined upper limit value and a predetermined lower limit value at the lattice point where the modified refractive index area is located; the modulation value is the upper limit value in modified refractive index areas where the modulation value exceeds the upper limit value; and the modulation value is the lower limit value in modified refractive index areas where the modulation value is less than the lower limit value.

[0010] Here, the "composite period formed by superimposing multiple in-plane periods that are different from one another" may be a superimposition of modulations formed with multiple different period lengths along one direction (the same direction) parallel to the two-dimensional photonic crystal layer, a superimposition of modulations formed with one period length each along multiple different directions that are parallel to the two-dimensional photonic crystal layer (which may be the same period length in each direction or may be different period lengths), or a combination thereof (a superimposition of modulations formed with multiple different period lengths each along the multiple directions). The "amount of positional deviation" includes the distance of deviation from the lattice point, the magnitude of the azimuth angle of deviation, and a combination of these distances and the magnitude of the azimuth angle.

[0011] As a premise for explaining the two-dimensional photonic crystal laser according to the present invention, we will first further explain the two-dimensional photonic crystal laser described in Patent Document 3. In the two-dimensional photonic crystal laser described in Patent Document 3, the positional deviation amount and / or area deviation amount of the modified refractive index region at each lattice point of the two-dimensional lattice is modulated by a composite period in which a plurality of different periods are superimposed, and the modulation phase Ψ(r↑) using a vector r↑ indicating the position of each lattice point is where A n and exp(iα n ) are the amplitude and phase determined for each n. Vector k n ↑ indicates different inclination angles θ n and / or azimuth angle φ n is a wave vector determined by the combination of the tilt angle and azimuth angle of each of N laser beams (N is an integer of 2 or more, n is any value between 1 and N) having the following characteristics. In reality, 2N laser beams are emitted at each value of n, with azimuth angles differing by 180° from each other. For example, if the two-dimensional lattice is a square lattice with a periodic length a, then the vector k n ↑ is Using the modulation phase Ψ(r↑), the positional deviation Δd(r↑) and area deviation ΔS(r↑) at each lattice point are expressed as follows: It is expressed as:

[0012] Here, Δd and ΔS are constants, and Δd(r↑) and ΔS(r↑) are functions of the variable r↑. The values ​​of Δd and ΔS can basically be set arbitrarily. However, if Δd is too small, the positional deviation Δd(r↑) becomes negligibly small. Therefore, the constant Δd is set appropriately so that the value of Δd(r↑) at a position where Δd(r↑)≠0 is a value that cannot be ignored (for example, 0.1 nm or more). Similarly, ΔS is set so that the value of ΔS(r↑) at a position where ΔS(r↑)≠0 is a value that cannot be ignored (for example, 0.1 nm or more). 2 The above will be determined as appropriate.

[0013] Hereinafter, the positional deviation amount Δd(r↑) will be described, but the same description can be similarly applied to the area deviation amount ΔS(r↑).

[0014] The displacement Δd(r↑) expressed by equation (3) can be expressed as sin[arg C]=Im(C) / |C| …(5) (where Im(C) is the imaginary part of C), which holds for a general complex number C. The second term in the denominator of equation (6) is is a vector (k m ↑-k m' ↑) is the vector k m This is thought to be the cause of the emission of unnecessary laser beams other than the 2N laser beams corresponding to ↑ (m is any value between 1 and N).

[0015] Therefore, in the two-dimensional photonic crystal laser according to the present invention, the second term in the denominator of equation (6) is ignored, and the modulation value Δd'(r↑) determined by the composite period is expressed as It is stipulated that:

[0016] However, in equation (3), the maximum value that Δd(r↑) can take is the constant Δd, whereas the modulation value Δd'(r↑) in equation (8) may become too large due to ignoring the second term in the denominator of equation (6). Therefore, in the two-dimensional photonic crystal laser according to the present invention, the modulation value Δd'(r↑) calculated by equation (8) is set to a predetermined upper limit value Δd max and a predetermined lower limit Δd minAt the grid point having a value between Δd'(r↑) and Δd'(r↑), the modulation value Δd'(r↑) is set as the positional deviation amount Δd(r↑), and the modulation value Δd'(r↑) is set as the upper limit value Δd max At grid points exceeding the upper limit Δd max is the positional deviation Δd(r↑), and the modulation value Δd'(r↑) is the lower limit value Δd min At grid points where the lower limit Δd is less than min is the positional deviation Δd(r↑). max is typically set to Δd, but it is allowed to change between 0.25Δd and 4.0Δd. min is typically set to -Δd, but is allowed to vary between -4.0Δd and -0.25Δd.

[0017] As mentioned above, the above explanation for the positional deviation Δd(r↑) can be similarly applied to the area deviation ΔS(r↑), so the modulation value ΔS'(r↑) determined by the composite period for the area deviation is It is defined as follows.

[0018] The modulation value ΔS'(r↑) calculated by equation (9) is the predetermined upper limit value ΔS max and a predetermined lower limit ΔS min At the grid point where the value is between, the modulation value ΔS'(r↑) is set as the area deviation amount ΔS(r↑), and the modulation value ΔS'(r↑) is set as the upper limit value ΔS max At grid points exceeding the upper limit ΔS max is the area deviation amount ΔS(r↑), and the modulation value ΔS'(r↑) is the lower limit value ΔS min At grid points where the lower limit ΔS is less than min is the area deviation amount ΔS(r↑). Upper limit value ΔS max is typically set to ΔS, but it can be changed between 0.25ΔS and 4.0ΔS. min is typically set to -ΔS, but it can be changed between -4.0ΔS and -0.25ΔS. min And / or the reference area is the reference area and the lower limit value ΔS min The sum of these is set to be non-negative.

[0019] By defining the positional deviation amount Δd(r↑) and the area deviation amount ΔS(r↑) using these modulation values, upper and lower limits, the two-dimensional photonic crystal laser according to the present invention can eliminate the vector (k m ↑-k m' Without including the unwanted laser beam due to the term with n It is possible to emit 2N laser beams corresponding to the number ↑ (n is any number from 1 to N).

[0020] The two-dimensional photonic crystal laser according to the present invention may have the following configuration: the two-dimensional lattice is a first square lattice, and the plurality of modified refractive index areas are arranged without deviation from first lattice points that are lattice points of the first square lattice, and each have an area determined by the area deviation; and the laser further comprises a plurality of second modified refractive index areas having a refractive index different from that of the base material, and the plurality of second modified refractive index areas are arranged without deviation from second lattice points that are lattice points of a second square lattice that is a square lattice having the same periodic length as the first square lattice and arranged at a different position from the first square lattice; and the area of ​​the second modified refractive index area at each second lattice point is a value deviated from a predetermined second reference area by a second area deviation that is the modulation value between a predetermined second upper limit value and a predetermined second lower limit value at the second lattice point, the second upper limit value in modified refractive index areas where the modulation value exceeds the second upper limit value, and the second lower limit value in modified refractive index areas where the modulation value is less than the second lower limit value.

[0021] When a modified refractive index area is provided only at a first lattice point, if the modified refractive index area is positioned without deviation from the first lattice point (i.e., the amount of deviation is 0) and only the area deviation is modulated, the position of the node of the standing wave of the electric field formed in the two-dimensional photonic crystal layer coincides with the position of the modified refractive index area, and the intensity of the laser beam emitted from the two-dimensional photonic crystal layer is weakened. Therefore, by further positioning a second modified refractive index area having an area determined by the second reference area and the second area deviation at a second lattice point of a second square lattice positioned at a different position from the first square lattice without deviation from the second lattice point, the second modified refractive index area is present at a position other than the node of the standing wave of the electric field, thereby increasing the intensity of the laser beam.

[0022] In order to increase the intensity of the laser beam through the above-mentioned action, it is most desirable to arrange the modified refractive index areas in both the first square lattice and the second square lattice without shifting them from the lattice points, but it is also permissible to arrange the modified refractive index areas in one of the first square lattice and the second square lattice away from the lattice points and impart modulation limited by upper and lower limits to the amount of positional shift. Furthermore, other modulations may be additionally imparted as long as the modulation amount is small enough to have little effect on the characteristics of the laser beam.

[0023] Alternatively, the two-dimensional photonic crystal laser according to the present invention can be configured as follows: the two-dimensional lattice is a first square lattice, and the plurality of modified refractive index areas are arranged so as to be shifted by the positional shift amount from first lattice points that are lattice points of the first square lattice, and each have an area determined by the area shift amount; and further, the laser comprises a plurality of second modified refractive index areas having a refractive index different from that of the base material, and the plurality of second modified refractive index areas are arranged without shifting from second lattice points that are lattice points of a second square lattice that is a square lattice having the same periodic length as the first square lattice and arranged at a different position from the first square lattice, and each have the same area.

[0024] In a structure in which both the position and area of ​​the modified refractive index region are modulated at the first lattice point, increasing the area of ​​the region in which the laser oscillates can degrade the quality of the laser beam due to oscillation in higher-order modes, etc. Therefore, by further arranging a second modified refractive index region whose position and area are not modulated at the second lattice point of a second square lattice located at a position different from the first square lattice, the quality of the laser beam can be maintained. Note that, as long as the modulation amount is small enough to have little effect on the characteristics of the laser beam, modulation of the position and / or area may be additionally imparted to the second modified refractive index region.

[0025] The second reference area, second upper limit value, and second lower limit value may be the same as the reference area, upper limit value, and lower limit value in the modified refractive index area arranged in the first square lattice, or some or all of them may be different values ​​from those in the modified refractive index area.

[0026] In the configuration including the second modified refractive index areas, the second square lattice is preferably arranged in the same direction as one of the two primitive translation vectors of the first square lattice, shifted by a period within a range of 0.4 to 0.6 periods (more preferably 0.5 periods). This allows the second modified refractive index areas to be arranged near the loops of the standing wave formed by the periodicity of the modified refractive index areas arranged at the first lattice points, thereby increasing the intensity of the laser beam emitted from the two-dimensional photonic crystal layer.

[0027] In the two-dimensional photonic crystal laser according to the present invention, the positional deviation amount and / or area deviation amount of each of the plurality of modified refractive index areas can be expressed by a function that uses the position of the lattice point as a variable and saturates at the upper limit value and the lower limit value, respectively.

[0028] For example, the following σ is expressed using the vector r↑ that indicates the position of each lattice point. (k n ↑ and α n As described above, the function Δd(σ) of the amount of positional deviation and the function ΔS(σ) of the amount of area deviation can be expressed as follows: Here, t is an arbitrary constant. Δd and ΔS are constants as above. Note that Δd(σ) and ΔS(σ) are functions of the variable σ, but because σ is expressed as r↑, they are also functions of the variable r↑. These functions Δd(σ) and ΔS(σ) saturate at lower limit values ​​-Δd and -ΔS, respectively, as σ approaches negative infinity, and saturate at upper limit values ​​Δd and ΔS, respectively, as σ approaches positive infinity.

[0029] According to the present invention, a two-dimensional photonic crystal laser is obtained that emits multiple (four or more) laser beams while suppressing the emission of unnecessary laser beams.

[0030] 1A is a perspective view showing a first embodiment of a two-dimensional photonic crystal laser according to the present invention, and FIG. 1B is a partial plan view of a two-dimensional photonic crystal layer included in the two-dimensional photonic crystal laser. FIG. 1B is a partial plan view showing a modified example of the two-dimensional photonic crystal layer in the two-dimensional photonic crystal laser of the first embodiment, where the directions of deviation of the positions of the modified refractive index regions from the lattice points are different. FIG. 1C is a partial plan view showing a modified example of the two-dimensional photonic crystal layer in the two-dimensional photonic crystal laser of the first embodiment, where only the positions of the modified refractive index regions are modulated. FIG. 1D is a partial plan view showing a modified example of the two-dimensional photonic crystal layer in the two-dimensional photonic crystal laser of the first embodiment, where only the areas of the planar shapes of the modified refractive index regions are modulated. FIG. 1E is a diagram for explaining the inclination angle and azimuth angle of a laser beam emitted in an example of the two-dimensional photonic crystal laser of the first embodiment. FIG. 1F is a diagram showing the results of calculation of the spot intensity of a laser beam emitted in an example of the two-dimensional photonic crystal laser of the first embodiment. FIG. 1G is a diagram showing the results of calculation of the spot intensity of a laser beam emitted in a two-dimensional photonic crystal laser of a comparative example. 13A is a perspective view showing a second embodiment of a two-dimensional photonic crystal laser according to the present invention, and FIG. 13B is a partial plan view of a two-dimensional photonic crystal layer included in the two-dimensional photonic crystal laser. 13B is a partial plan view of a two-dimensional photonic crystal layer included in the two-dimensional photonic crystal laser. 13C is a diagram showing a calculation result of the intensity of the spot of a laser beam emitted in an example of the two-dimensional photonic crystal laser of the second embodiment. 13D is a diagram showing a calculation result of the electric field distribution in the two-dimensional photonic crystal layer in an example of the two-dimensional photonic crystal laser of the second embodiment (a), and an example of the two-dimensional photonic crystal laser of the first embodiment in which only the area is modulated (b). 13D is a partial plan view showing a modification of the two-dimensional photonic crystal layer in the two-dimensional photonic crystal laser of the second embodiment. 13E is a photograph showing a far-field pattern formed by a laser beam using the two-dimensional photonic crystal laser of the third embodiment. 13F is a graph showing the experimental results of the relationship between peak optical output and current magnitude for the third embodiment and a conventional two-dimensional photonic crystal laser that formed the far-field pattern shown in FIG. 13G.16 is a graph showing an example of a function of the area shift amount ΔS(σ) in an example of the two-dimensional photonic crystal laser of the fourth embodiment. 17 is a graph showing an example of a change in the area shift amount ΔS(σ) depending on σ in an example of the two-dimensional photonic crystal laser of the first embodiment, to which the function of Fig. 15 is approximated. 18 is a graph showing the results of calculating the intensity of the spot of the laser beam emitted in an example of the two-dimensional photonic crystal laser of the fourth embodiment.

[0031] 1 to 17, an embodiment of a two-dimensional photonic crystal laser according to the present invention will be described.

[0032] (1) First Embodiment As shown in FIG. 1( a), the two-dimensional photonic crystal laser 10 of the first embodiment has a configuration in which a first electrode 171, a first cladding layer 141, a two-dimensional photonic crystal layer 12, a spacer layer 13, an active layer 11, a second cladding layer 142, a substrate 16, and a second electrode 172 are stacked in this order. However, the order of the active layer 11 and the two-dimensional photonic crystal layer 12 may be reversed. For convenience, FIG. 1( a) shows the first electrode 171 on the bottom and the second electrode 172 on the top, but the orientation of the two-dimensional photonic crystal laser 10 during use is not limited to that shown in this figure. The configuration of each layer and electrode will be described below.

[0033] The active layer 11 emits light having a predetermined wavelength band when electric charges are injected from the first electrode 171 and the second electrode 172. The material of the active layer 11 may be, for example, an InGaAs / AlGaAs multiple quantum well (emission wavelength band: 935 to 945 nm).

[0034] As shown in FIG. 1B, the two-dimensional photonic crystal layer 12 is formed by arranging modified refractive index areas 122, each having a refractive index different from that of a plate-shaped base material 121, at each lattice point of the two-dimensional lattice. FIG. 1B shows an enlarged view of only a portion of the two-dimensional photonic crystal layer 12; in the actual two-dimensional photonic crystal layer 12, multiple modified refractive index areas 122 are arranged over an area roughly equivalent to the outer edge of the planar shape of the second electrode 172. The base material 121 can be made of, for example, p-type GaAs (a p-type semiconductor). The modified refractive index areas 122 are typically formed of air holes; however, instead of air holes, a material having a refractive index different from that of the base material 121 may be used. The positions at which the modified refractive index areas 122 are arranged and the planar shape of the modified refractive index areas 122 will be described in detail later.

[0035] The first cladding layer 141 and the second cladding layer 142 have the role of injecting charges from the first electrode 171 and the second electrode 172, and also the role of preventing in-plane guided light, which is guided parallel to the layers in the two-dimensional photonic crystal layer 12, from leaking from the layers. To fulfill the former role, the first cladding layer 141 is made of a p-type semiconductor (e.g., p-type Al 0.37 Ga 0.63 As) in the second cladding layer 142, and an n-type semiconductor (e.g., n-type Al 0.37 Ga 0.63 As) are used (the reason why the material of the base material 121 of the two-dimensional photonic crystal layer 12 is a p-type semiconductor is the same as this).

[0036] The spacer layer 13 is provided to allow holes injected from the first electrode 171 to pass through and be introduced into the active layer 11, while preventing electrons injected from the second electrode 172 from passing through the active layer 11 (and thereby combining with holes on the first electrode 171 side of the active layer 11). The spacer layer 13 is made of a material such as p-type Al 0.45 Ga 0.55 As can be used.

[0037] The substrate 16 is made sufficiently thicker than the other layers in order to maintain the mechanical strength of the entire two-dimensional photonic crystal laser 10. For the same reason as for the second cladding layer 142, an n-type semiconductor is used as the material for the substrate 16.

[0038] In the first embodiment, the first electrode 171 is made of a square metal plate-like material. In the first embodiment, the second electrode 172 is made of a square metal plate-like material with sides longer than the first electrode 171 and a square hole cut out in the center. The hole cut out portion of the plate-like material of the second electrode 172 is called a window portion 1722, and the remaining portion of the plate-like material is called a frame portion 1721. The window portion 1722 is provided to allow a laser beam emitted from the two-dimensional photonic crystal layer 12 to pass through, as will be described later. Note that in FIG. 1( a ), the first electrode 171 and the first cladding layer 141 are shown separated from each other to show the shape of the first electrode 171, but in reality, the first electrode 171 and the first cladding layer 141 are in contact with each other.

[0039] The materials for each layer described above are merely examples, and other materials may be used. Furthermore, the shapes of the first electrode 171 and the second electrode 172 are not limited to those described above, and other shapes (for example, a first electrode made of a circular metal plate member and a second electrode made of a circular metal plate member with a diameter larger than that of the first electrode and a circular hole cut out in the center) may be used.

[0040] The positions at which the modified refractive index areas 122 are disposed in the two-dimensional photonic crystal layer 12 and the planar shapes of the modified refractive index areas 122 will be described in detail below.

[0041] In the example shown in FIG. 1( b), the modified refractive index areas 122 are arranged so that the center of gravity G of the planar shape of each modified refractive index area 122 is located at a position shifted by a predetermined amount in the y direction from a lattice point, which is the intersection of a dot-dash line extending in the x direction (the horizontal direction in the figure) and a dot-dash line extending in the y direction (the vertical direction in the figure) in the square lattice indicated by the dot-dash line in the figure. The amount of shift varies for each modified refractive index area 122 (lattice point), as described below. The planar shape of each modified refractive index area 122 is an ellipse with its major axis in the x direction, and its area (the area shift amount, described below) also varies for each modified refractive index area 122. The lattice point period (length) a of the square lattice is determined appropriately depending on the material of the base material 121 and the emission wavelength band of the active layer 11. Instead of a square lattice, other two-dimensional lattices, such as a rectangular lattice or a triangular lattice, may be used. The planar shape of the modified refractive index area 122, including the amount of shift and area, can also be modified in various ways, as described below.

[0042] Here, the following value σ is introduced to determine the amount of positional deviation at each grid point, which is expressed by the position vector r↑ of each grid point. σ is where k n As mentioned above, the inclination angles θ n and / or azimuth angle φ n is a wave vector determined by a combination of the tilt angle and azimuth angle of each of N laser beams (N is an integer of 2 or more, n is any value between 1 and N), and since a square lattice is used in the first embodiment, it is expressed by equation (2). In the first embodiment, the positional deviation amount Δd(r↑) at each lattice point is expressed by It was decided.

[0043] The right side of the first line of equation (12) shows that the amplitude A n is defined as 1. In this way, regardless of the value of n, the amplitude A n Setting r to 1 is not an essential requirement of the present invention, but merely an example of the present invention. The second and third lines of equation (12) define the upper limit of the positional deviation amount Δd(r↑) as a constant +Δd and the lower limit as a constant −Δd. Note that, on the right side of the first line of equation (12), σ is set to N, which is the upper limit to which the first line of equation is applied. 1 / 2 and the lower limit -N 1 / 2 Then, the values ​​of Δd(r↑) become +Δd and -Δd, which correspond to the values ​​in the second and third rows, respectively.

[0044] In this way, in the first embodiment, the upper limit value Δd max = Δd and lower limit Δd inx =-Δd, so that no positional deviation outside the range between this upper and lower limit value occurs. Note that in the first embodiment, since the modified refractive index area 122 is shifted in the y direction, the positive and negative values ​​of Δd(r↑) represent the positive and negative values ​​in the y direction.

[0045] The area S(r↑) of the planar shape of the modified refractive index area 122 is the sum of a predetermined reference area S0 that does not depend on the position of the lattice point and an area deviation amount ΔS(r↑) that represents a deviation from the reference area S0, and is expressed as follows: S(r↑)=S0+ΔS(r↑) ... (13) The area deviation amount ΔS(r↑) is expressed as follows, similar to the position deviation amount Δd(r↑): In this way, the area deviation amount ΔS(r↑) is also determined by the modulation amount determined by the compound period of the linear sum of the sine functions of σ in the right-hand side of the first line of Equation (14), the upper limit value +ΔS, and the lower limit value −ΔS, so that an area deviation exceeding this upper limit value does not occur. Note that the reference area S0 is set to a value greater than ΔS so that no modified refractive index areas 122 with a negative S(r↑) value occur. On the other hand, if the reference area S0 is too large, adjacent modified refractive index areas 122 will overlap, so the reference area S0 is set to prevent such overlap from occurring.

[0046] The positional deviation amount Δd(r↑) shown in equation (12) and the area deviation amount ΔS(r↑) shown in equation (14) are defined by the same σ, and therefore have the same compound period. Therefore, as shown in Figure 1 (b), the modified refractive index area 1221 located at the lattice point (Δd(r↑) = 0) also has an area deviation amount ΔS(r↑) of 0 (hence, the area S is S0), the modified refractive index area 1222 arranged shifted in the positive direction in the y direction from the lattice point (Δd(r↑) > 0) has an area deviation amount ΔS(r↑) positive (S > S0), and the modified refractive index area 1223 arranged shifted in the negative direction in the y direction from the lattice point (Δd(r↑) < 0) has an area deviation amount ΔS(r↑) negative (S <S0)となっている。

[0047] In the examples shown so far, the position of the modified refractive index area 122 is modulated so as to be shifted in the y direction from the lattice point, but it may also be modulated so as to be shifted in other directions. In the example shown in FIG. 2, the major axis of the ellipse of the modified refractive index area 122 is oriented in the x direction, as in FIG. 1(b), and the direction of positional shift is the x direction. Also, as shown in FIG. 3, it is possible to modulate only the amount of positional shift without modulating the area of ​​the planar shape (using modified refractive index areas 122 of the same area). Alternatively, as shown in FIG. 4, it is possible to modulate only the area of ​​the planar shape without modulating the amount of positional shift (positioning the modified refractive index area 122 at a lattice point).

[0048] Next, the results of calculating the intensity of the laser beam emitted from the two-dimensional photonic crystal laser 10 of the first embodiment are shown. In this calculation example, the intensity of the spot of the laser beam incident on a plane parallel to the two-dimensional photonic crystal layer 12 and a predetermined distance away from the second electrode 172 in a direction perpendicular to the layer was calculated. In this calculation example, θ n and φ n Combination of (θ n , φ n) are (5°, 0°), (5°, 45°), (5°, 90°), (5°, 135°), (10°, 0°), (10°, 22.5°), (10°, 45°), (10°, 67.5°), (10°, 90°), (10°, 112.5°), (10°,135°), (10°, 157.5°), (15°, 9°), (15°, 27°), (15°, 45°), (15°, 63°), (15°, 81°), (15°, 99°), (15°, 117°), (15°, 135°), and (15°, 153°), (15°, 171°), (20°, 6.4°), (20°, 19.3°), (20°, 32.1°), (20°, 45°), (20°, 57.9°), (20°, 70.7°), (20°, 83.6°), (20°, 96.4°), (20°, 109.3°), (20°, 122.1°), (20°, 135°), (20°, 147.9°), (20°, 160.7°), (20°, 173.6°), (25°, 6.4°), (25°, 19.3°), (25°, 32.1°), (25°, 50 types (N=50) of vector k are: (25°, 45°), (25°, 57.9°), (25°, 70.7°), (25°, 83.6°), (25°, 96.4°), (25°, 109.3°), (25°, 122.1°), (25°, 135°), (25°, 147.9°), (25°, 160.7°), (25°, 173.6°). n These vectors k n ↑ is the inclination angle θ as shown in FIG. n A laser beam with an azimuth angle φ = 5° n = 8 lines in total, every 45°, θ n φ = 10° laser beam n = 16 lines in total, every 22.5°, θ n φ = 15° laser beam n = 20 lines in total, every 18°, θ n φ = 20° laser beam n = (180 / 14)°, total of 28 lines, and θ n φ = 25° laser beam n= (180 / 14)°, a total of 28 laser beams are emitted. The total number of emitted laser beams is 100 (= 2N).

[0049] These vectors k n Applying ↑ to equation (11), σ is expressed as a linear sum of sine functions with N=50 terms. In this calculation example, equations (12) and (14) are Δd=0.08a (i.e., the upper limit of the displacement is 8% of the lattice constant a of a square lattice), and S0=0.095a 2 (That is, S0 is the area a of the unit cell of a square lattice. 2 9.5%), ΔS=0.03a 2 (That is, the upper limit of the area deviation is the area a of the unit cell of the square lattice. 2 The calculation was made assuming 3% of the total.

[0050] As a comparison example, the same 50 types of vectors k as in the above calculation example are used. n Set ↑ and S0=0.095a 2 Then, the intensity of the laser beam spot was calculated for a laser with composite modulation applied without setting the upper limit +Δd and lower limit −Δd of the positional deviation amount, and the upper limit +ΔS and lower limit −ΔS of the area deviation amount. This comparative example corresponds to the two-dimensional photonic crystal laser of Patent Document 3. However, in this calculation, the injected current was larger than that in the calculation performed in Patent Document 3, and the laser beam intensity was also increased. In the calculation of the first embodiment, the same current magnitude as in the comparative example was set.

[0051] Figure 6 shows the calculation results for the first embodiment, and Figure 7 shows the calculation results for the comparative example. In the comparative example, in addition to the position of the original laser beam spot shown in Figure 5, many laser beam spots weaker than the original spot are observed. These weak laser beams are the result of increasing the injected current, making the unwanted laser beams that were negligibly weak in Patent Document 3 more noticeable. In contrast, in Figure 6, almost no unwanted laser beams are observed, and it can be seen that the emission of such unwanted laser beams is suppressed by the first embodiment.

[0052] In the first embodiment, too, weak unwanted laser beams are emitted, which can be ignored in Fig. 6. When the sum of the intensities of the unwanted laser beams is expressed as a ratio to the sum of the intensities of all laser beams (the original laser beams and the unwanted laser beams combined), it is 23.9% in the comparative example, while it is 8.5% in the first embodiment, and it was confirmed that the emission of unwanted laser beams is more suppressed in the first embodiment, even in terms of numerical calculations.

[0053] The two-dimensional photonic crystal laser of the first embodiment can be modified in various ways in addition to the variations in which the periodic structure of the two-dimensional photonic crystal layer is changed as shown in Figures 2 to 4. For example, in the above embodiment, the planar shape of the modified refractive index region is elliptical, but it may be various shapes such as an equilateral triangle, other triangles, polygons with more than one side, circles, or irregular shapes. Furthermore, one modified refractive index region may be a combination of multiple holes or a material different from the base material.

[0054] The positional deviation Δd(r↑) shown in equation (12) is A in equation (8) n The value of was calculated as 1 for all n, but A n The value of (r↑) may be set to any value (including values ​​other than 1). The same applies to the area deviation amount ΔS(r↑) shown in equation (14).

[0055] In the examples shown so far, the amount of misalignment was set to one of the two directions (x direction and y direction) in which the lattice points of the square lattice are arranged, but the lattice points may be arranged to be misaligned in a direction other than these. Also, in the examples shown so far, the distance from the lattice point was set as the amount of misalignment, but instead of or in addition to that, the angle ξ (see FIG. 8 , where the x direction is the reference direction) formed by the direction of misalignment from the lattice point with a predetermined reference direction (e.g., the x direction or the y direction) may be set as the amount of misalignment (i.e., the angle ξ is different for each lattice point).

[0056] In the example shown in Figure 1(b), the same vector k n ↑ (n is any value between 1 and N) to determine the positional deviation Δ(r↑) and the area deviation ΔS(r↑), and 2N laser beams are emitted. The positional deviation Δ(r↑) is determined by the vector kn ↑ (n is any value between 1 and N), and the area deviation ΔS(r↑) is k n ↑Different vector k m ↑ (m is any value between 1 and M, and M may be the same as or different from N).

[0057] (2) Second Embodiment As shown in Fig. 9, a two-dimensional photonic crystal laser 20 of the second embodiment has a similar configuration to the two-dimensional photonic crystal laser 10 of the first embodiment, except that the configuration of the two-dimensional photonic crystal layer 22 differs from that of the two-dimensional photonic crystal layer 12 in the two-dimensional photonic crystal laser 10 of the first embodiment. Therefore, only the configuration of the two-dimensional photonic crystal layer 22 will be described below.

[0058] The two-dimensional photonic crystal layer 22 is formed by periodically arranging first modified refractive index areas 2221 and second modified refractive index areas 2222, each having a refractive index different from that of the base material 221, in a plate-shaped base material 221. In this embodiment, the first modified refractive index areas 2221 and the second modified refractive index areas 2222 are both made of air (holes), but they may also be made of a material (other than air) different from that of the base material 221. When using such a material, the first modified refractive index areas 2221 and the second modified refractive index areas 2222 may be made of the same material or different materials.

[0059] The first modified refractive index area 2221 is arranged so that the center of gravity of its planar shape coincides with a first lattice point 2241, which is a lattice point of a first square lattice 2231, which is a square lattice with a period a, shown in Fig. 9(b) by a dashed line extending in the x direction and a dashed line extending in the y direction. The second modified refractive index area 2222 is arranged so that the center of gravity of its planar shape coincides with a second lattice point 2242, which is a lattice point of a second square lattice 2232, which is a square lattice with a period a, shown in Fig. 9(b) by a dashed line extending in the x direction and a dashed line extending in the y direction (the latter is common to the first square lattice 2231). That is, the first modified refractive index area 2221 and the second modified refractive index area 2222 are arranged at lattice points of square lattices that are shifted from each other by a / 2 in the y direction.

[0060] Each of the first modified refractive index areas 2221 has an elliptical planar shape, and its area S1(r↑) is equal to the first reference area S 01 and the sum of the area deviation amount ΔS1(r↑) that depends on the position vector r↑ of the first modified refractive index area 2221 (lattice point), S1(r↑)=S 01 +ΔS1(r↑) (15) ΔS1(r↑) is expressed by using σ shown in formula (11) and the first upper limit value ΔS1, which is the upper limit value of the area of ​​the first modified refractive index area 2221. Therefore, the first lower limit is −ΔS1.

[0061] Furthermore, each second modified refractive index area 2222 has an elliptical planar shape, and its area S2(r↑) is a second reference area S 02 and the sum of the area deviation amount ΔS2(r↑) that depends on the position vector r↑ of the second modified refractive index area 2222 (lattice point), S2(r↑)=S 02 +ΔS2(r↑) (17) ΔS2(r↑) is expressed by using σ shown in formula (11) and the second upper limit value ΔS2, which is the upper limit value of the area of ​​the first modified refractive index area 2221. Therefore, the second lower limit is −ΔS2. 01 and the second reference area S 02 The first upper limit value ΔS1 and the second upper limit value ΔS2 may be the same value or different values.

[0062] Next, the results of calculating the intensity of the laser beam emitted from the two-dimensional photonic crystal laser 20 of the second embodiment are shown. In this calculation example, the vector k n ↑ is θ in equation (2) n and φ n Combination of (θ n , φ n) were set to 12 types (N=12), with the angles (11°, 0°), (11°, 90°), (23°, 0°), (23°, 90°), (36°, 0°), (36°, 90°), (11×2½°, 45°), (11×2½°, 135°), (23×2½°, 45°), (23×2½°, 135°), (36×2½°, 45°), and (36×2½°, 135°). n ↑ is the inclination angle θ n The laser beams are oriented at azimuth angles φ = 11°, 23° and 36°, respectively. n = 0°, 90°, 180° and 270°, and the inclination angle θ n =11×2 1 / 2 °, 23 × 2 1 / 2 ° and 36 × 2 1 / 2 The laser beam is oriented at an azimuth angle of φ n The parameters in equations (16) and (18) are the first reference area S 01 is 0.075a 2 , second reference area S 02 is 0.055a 2 , the first upper limit value ΔS1 is 0.03a 2 , the second upper limit value ΔS2 is 0.025a 2 It was decided.

[0063] The calculation results are shown in Figure 10. 24 laser beam spots were obtained, and no other spots were observed. However, since unnecessary laser beams that were weak enough to be ignored were actually emitted in Figure 10, the ratio of the total intensity of the unnecessary laser beams to the total intensity of all the laser beams was calculated to be 22%.

[0064] As a comparative example, the first modified refractive index area 2221 modulates only the area based on equation (4) without setting upper and lower limits, and the second modified refractive index area 2222 modulates both the position and area and modulates the second reference area S 02 =0.055a 2Here, in the first modified refractive index area 2221, in order to correspond to the example of the second embodiment, A n = 1, and the constant ΔS is the first upper limit value ΔS1 = 0.03a 2 In this comparative example, the ratio of the sum of the intensities of the unnecessary laser beams to the sum of the intensities of all the laser beams was calculated to be 4.7%. Thus, it was confirmed that the emission of unnecessary laser beams was suppressed more effectively in the second embodiment than in the comparative example.

[0065] 11 shows the calculated electric field distribution in the two-dimensional photonic crystal layer for an example (a) of the two-dimensional photonic crystal laser of the second embodiment and an example (b) of the two-dimensional photonic crystal laser of the first embodiment in which only the area is modulated. (b) corresponds to a configuration in which only the first modified refractive index area 2221 is provided in the two-dimensional photonic crystal layer 22 of the two-dimensional photonic crystal laser of (a) (excluding the second modified refractive index area 2222). The direction and length of the numerous arrows shown in these figures indicate the direction and magnitude of the electric field at that position. In both (a) and (b), it can be seen that the electric field is smaller near the first modified refractive index area 2221. On the other hand, as shown in (a), the electric field is stronger near the second modified refractive index area 2222 than near the first modified refractive index area 2221. This makes it easier for the laser beam to exit the two-dimensional photonic crystal layer in (a) than in (b).

[0066] Various modifications are possible in the second embodiment. For example, the shapes of the first modified refractive index area 2221 and the second modified refractive index area 2222 are not limited to elliptical shapes, and various shapes can be used, as in the example given in the first embodiment. Furthermore, the direction and magnitude of the deviation between the first square lattice and the second square lattice are not limited to the above example (1 / 2a in the y direction), and various directions and magnitudes are possible.

[0067] 12, the first modified refractive index area 2221A may be arranged at each first lattice point of a first square lattice (a square lattice indicated by vertical and horizontal dashed lines in the figure) so as to have a predetermined positional deviation and area deviation, and the second modified refractive index area 2222A may be arranged at each second lattice point of a second square lattice (a square lattice indicated by horizontal dashed two-dot lines and vertical dashed one-dot lines in the figure) so as to have no positional deviation or area deviation. This prevents oscillation in higher-order modes, etc., even if the area of ​​the laser oscillation region is increased, and the quality of the laser beam can be maintained. Note that FIG. 12 shows an example in which the second square lattice is shifted from the first square lattice by 1 / 2 period in the vertical direction in the figure, but the magnitude and direction of this deviation are arbitrary.

[0068] (3) Third Embodiment Up to this point, we have described an embodiment of a two-dimensional photonic crystal laser that emits a large number of laser beams while suppressing the emission of unnecessary laser beams, but as described in Patent Document 3, if the number of laser beams is made sufficiently large, it is possible to form a distribution of laser spots that exhibit various patterns, such as a pattern that uniformly irradiates a wide angular range, or a pattern that represents characters or figures. Even in such cases, a method can be used that sets upper and lower limits on the positional deviation amount and area deviation amount of the modified refractive index areas.

[0069] The electric field distribution in wavenumber space corresponding to the distribution of the laser spot (far-field pattern) to be formed at a position distant from the two-dimensional photonic crystal laser is called the target far-field electric field distribution E far_iFFT (K↑), and the target far electric field distribution E far_iFFT The electric field distribution (near-field pattern) in real space at a cross section parallel to the two-dimensional photonic crystal layer of the laser beam emitted from the two-dimensional photonic crystal layer of the two-dimensional photonic crystal laser to form (K↑) is defined as the target radiated electric field distribution E rad_iFFT (r↑). Target radiation electric field distribution E rad_iFFT (r↑) is the target far electric field distribution E using the complex correction coefficient A(K↑). far_iFFT By performing an inverse Fourier transform on (K↑), the following equation (21) is obtained. It can be calculated as follows.

[0070] Therefore, the radiated electric field distribution E obtained from the modulation phase Ψ(r↑) of the modified refractive index area for each position r↑=(x, y) is rad (r↑) is the target radiated electric field distribution E obtained by equation (21). rad_iFFT The modulation phase Ψ(r↑) for each position r↑=(x, y) of the lattice point where the modified refractive index area is arranged is adjusted so as to approach (r↑). For example, the following equation (22) is obtained using equation (21): By applying the above to the modulation phase Ψ(r↑), the modulation phase Ψ(r↑) after adjustment can be determined (see Patent Document 3).

[0071] The positional deviation Δd(r↑) without considering the upper and lower limit values ​​is expressed by the following equation (23): Δd(r↑)=B·Δd·sin(Ψ(r↑)) ... (23) where B·Δd is a positive constant, but is expressed as the product of B, a constant greater than 1, and the value Δd set as the upper and lower limit values ​​in the following equation (24).

[0072] Applying the upper limit value Δd and the lower limit value -Δd to equation (23), the positional deviation Δd(r↑) is expressed by equation (24): Δd(r↑)=B·Δd·sin(Ψ(r↑)) (when |B·sin(Ψ(r↑))|≦1) Δd (when B·sin(Ψ(r↑)>1) −Δd (when B·sin(Ψ(r↑)<−1) ...(24)

[0073] Similarly, the area deviation ΔS(r↑) is expressed by the following equation (25) using C, a constant greater than 1, and upper and lower limits ΔS and -ΔS: ΔS(r↑)=C·ΔS·sin(Ψ(r↑)) (when |C·sin(Ψ(r↑))|≦1) ΔS (when C·sin(Ψ(r↑)>1) −ΔS (when C·sin(Ψ(r↑)<−1) ... (25) The area S(r↑) of the modified refractive index area is expressed as S(r↑)=S0+ΔS(r↑) using the reference area S0, as in the first embodiment.

[0074] Next, examples of the two-dimensional photonic crystal laser according to the third embodiment, which is fabricated to form characters as far-field patterns, will be described together with comparative examples. In both examples and comparative examples, the laser is fabricated to form a far-field pattern in which the horizontally written character string "Kyoto" and the vertically written character string "Univ." are arranged. far_iFFT (K↑) was determined. The modified refractive index areas were arranged with the positional deviation and area deviation (both of which are different between the example and the comparative example) described below so that when there was no positional deviation, the positions would be in the shape of a square lattice with a periodic length of a (=195 nm) in both the example and the comparative example. In the example, B in equation (24) was set to 2.5, Δd to 0.08a, and C in equation (25) was set to 2.5 and ΔS to 0.03a. 2 and S0 is 0.10a 2 In the comparative example, no upper or lower limit values ​​were set, and in all modified refractive index areas, the positional deviation amount Δd(r↑) was set to B·Δd·sin(Ψ(r↑)), the area deviation amount ΔS(r↑) was set to C·ΔS·sin(Ψ(r↑)), and the same values ​​as in the example were used for B, Δd, C, ΔS, and S0.

[0075] Figure 13 shows a photograph of the far-field pattern obtained in this example. A far-field pattern consisting of character strings was obtained as designed. This shows that the two-dimensional photonic crystal laser of the third embodiment can also suppress adverse effects such as loss of laser light due to higher-order diffraction, and form clear patterns such as characters.

[0076] Figure 14 shows the experimental results of the relationship between the peak optical output, which is the optical output of the laser beam while a pulse current is injected into the two-dimensional photonic crystal laser, and the magnitude of the current for each of the example and comparative example. Here, the pulse width of the pulse current was set to 100 nanoseconds. When compared at the same current value, the example has a higher peak optical output than the comparative example. The slope efficiency values ​​obtained from these experimental results are 0.70 W / A for the example and 0.31 W / A for the comparative example. In this way, the two-dimensional photonic crystal laser of the third embodiment can achieve high slope efficiency while suppressing high-order diffraction, as described above.

[0077] (4) Fourth Embodiment In the fourth embodiment, the positional deviation amount and / or area deviation amount of each of the multiple modified refractive index areas is defined by a function that uses the position of the lattice point as a variable and saturates at the upper and lower limit values. An example of such a function is a sigmoid function. There are several types of sigmoid functions, but for example, an error function is generally defined as follows: Here, the variable x is replaced with a variable σ including the position vector r↑ expressed by equation (11), the maximum positional deviation amount is Δd, and the maximum area deviation amount is ΔS, and then the positional deviation amount and area deviation amount are defined as functions Δd(σ) and ΔS(σ) of the variable σ, as shown in equation (27) below. Here, t is an arbitrary constant. Note that since the variable σ includes the position vector r↑, Δd(σ) and ΔS(σ) are also functions that use the position as a variable.

[0078] The function ΔS(σ) in equation (27) (hereinafter, this also applies to Δd(σ) in this paragraph) saturates at the lower limit value -ΔS as σ approaches negative infinity, and saturates at the upper limit value ΔS as σ approaches positive infinity. It increases monotonically with increasing σ near σ=0, and the slope is approximately (0.8 / t) at σ=0. ​​For example, when t=0.09 in equation (27), ΔS(σ) saturates at the lower limit value -ΔS and the upper limit value ΔS, respectively, and the slope is approximately 8.9 at σ=0, as shown in the graph in FIG. 15. In this way, the change in σ with the function ΔS(σ) in equation (27) at t=0.09 corresponds to the change in σ with the 50 types (N=50) of vectors k in the calculation example (FIG. 6) shown in the first embodiment. n The change is similar to that shown in the graph of FIG. 16, in which the area deviation amount determined using ↑ is expressed as σ.

[0079] In this way, when t=0.09 in equation (27), Δd=0.008a and S0=0.095a, as in the calculation example of FIG. 2 , ΔS=0.03a 2 The intensity of the laser beam spot was calculated under the conditions above. As a result, as shown in Fig. 17, the unwanted laser beam spot shown in the comparative example of Fig. 7 was hardly generated, and a spot close to the calculated example of Fig. 6 was obtained.

[0080] The above shows an example of determining the positional shift amount and area shift amount of the modified refractive index area using an error function, which is one of the sigmoid functions. However, a sigmoid function other than the error function or any other function may be used as long as it is a function that uses the position of the lattice point as a variable and saturates at the upper and lower limit values, respectively.

[0081] The method of determining the positional deviation amount and area deviation amount of the modified refractive index area using a sigmoid function can also be applied to the case of forming a distribution of laser spots showing various patterns described in the third embodiment. Specifically, using the modulation phase Ψ(r↑) expressed by equation (22) obtained from equation (21), the positional deviation amount is determined by using Δd(r↑)=B·Δd·sin(Ψ(r↑)), which is a function of r↑ expressed by equation (23), as a variable σ d The function Δd(σ d ), and for the area deviation, ΔS(r↑)=C・ΔS・sin(Ψ(r↑)) is used as the variable σ S Let function ΔS(σ S ) is the amount of positional deviation and the amount of area deviation. These functions Δd(σ d ) and ΔS(σ S ) is expressed as follows:

[0082] Aspects It will be apparent to those skilled in the art that the exemplary embodiments described above are examples of the following aspects.

[0083] (Item 1) A two-dimensional photonic crystal laser according to one aspect of the present invention comprises: a) a pair of electrodes; b) an active layer provided between the pair of electrodes, which generates light of a predetermined wavelength when a current is injected from the electrodes; and c) a two-dimensional photonic crystal layer provided between one of the pair of electrodes and the active layer, which has a plate-shaped base material and a plurality of modified refractive index areas arranged on the base material and having a refractive index different from that of the base material, wherein the plurality of modified refractive index areas are arranged at positions shifted by different amounts from each lattice point of a two-dimensional lattice that is periodically arranged on the base material with an in-plane period corresponding to the predetermined wavelength, and / or are arranged at the lattice points or at positions shifted from the lattice points by the same amount, with areas each having a different area shift amount, which is the difference from a predetermined reference area, The positional deviation amount and / or area deviation amount of each of the multiple modified refractive index areas is a modulation value determined by a composite period obtained by superimposing multiple different in-plane periods, and is a value between a predetermined upper limit value and a predetermined lower limit value at the lattice point where the modified refractive index area is located; the modulation value is the upper limit value in modified refractive index areas where the modulation value exceeds the upper limit value; and the modulation value is the lower limit value in modified refractive index areas where the modulation value is less than the lower limit value.

[0084] (2) The two-dimensional photonic crystal laser according to the second paragraph is the two-dimensional photonic crystal laser according to the first paragraph, further comprising: a vector r↑ indicating the position of each lattice point of the two-dimensional lattice; and a vector k (n is an integer of 2 or more) indicating a combination of the tilt angles and / or azimuth angles of each of n laser beams (n is an integer of 2 or more) whose tilt angles and / or azimuth angles are different from each other. n ↑, amplitude A determined for each n n and phase exp(iα n ), and a constant Δd, the modulation value Δ'd(r↑) of the positional deviation amount at each lattice point is The present invention is characterized in that it is represented by the following formula:

[0085] (Item 3) The two-dimensional photonic crystal laser according to item 3 is the two-dimensional photonic crystal laser according to item 1, further comprising: a vector r↑ indicating the position of each lattice point of the two-dimensional lattice; and a vector k (n is an integer of 2 or more) indicating a combination of the tilt angles and / or azimuth angles of each of n laser beams (n is an integer of 2 or more) whose tilt angles and / or azimuth angles are different from each other. n ↑, amplitude A determined for each n n and phase exp(iα n ), and a constant ΔS, the modulation value Δ′S(r↑) of the area deviation amount at each lattice point is The present invention is characterized in that it is represented by the following formula:

[0086] (Item 4) The two-dimensional photonic crystal laser according to item 4 is the two-dimensional photonic crystal laser according to item 1 or 3, wherein the two-dimensional lattice is a first square lattice, and the plurality of modified refractive index areas are arranged without deviation from first lattice points that are lattice points of the first square lattice, and each has an area determined by the amount of area deviation; and further comprises a plurality of second modified refractive index areas having a refractive index different from that of the base material, and the plurality of second modified refractive index areas are arranged without deviation from second lattice points that are lattice points of a second square lattice that is a square lattice having the same periodic length as the first square lattice and arranged at a position different from the first square lattice; The area of ​​the second modified refractive index region at each second lattice point is a value deviated from a predetermined second reference area by a second area deviation amount, which is the modulation value in modified refractive index regions where the modulation value is a value between a predetermined second upper limit value and a predetermined second lower limit value at the second lattice point, the second upper limit value in modified refractive index regions where the modulation value exceeds the second upper limit value, and the second lower limit value in modified refractive index regions where the modulation value is less than the second lower limit value.

[0087] (Item 5) The two-dimensional photonic crystal laser according to item 5 is the two-dimensional photonic crystal laser according to any one of items 1 to 3, wherein the two-dimensional lattice is a first square lattice, and the plurality of modified refractive index areas are arranged so as to be shifted by the positional shift amount from first lattice points that are lattice points of the first square lattice, and each have an area determined by the area shift amount; and further comprises a plurality of second modified refractive index areas having a refractive index different from that of the base material, and the plurality of second modified refractive index areas are arranged without shifting from second lattice points that are lattice points of a second square lattice that is a square lattice having the same periodic length as the first square lattice and arranged at a different position from the first square lattice, and each have the same area.

[0088] (Item 6) The two-dimensional photonic crystal laser according to item 6 is characterized in that, in the two-dimensional photonic crystal laser according to item 4 or 5, the second square lattice is arranged in the same direction as one of the two primitive translation vectors of the first square lattice, shifted by a period within a range of 0.4 to 0.6 periods (more preferably 0.5 periods).

[0089] (Item 7) The two-dimensional photonic crystal laser according to item 7 is the two-dimensional photonic crystal laser according to item 1, wherein an electric field distribution E in wavenumber space corresponding to a far-field pattern to be formed by a laser beam emitted from the two-dimensional photonic crystal laser is far_iFFT (K↑) is an electric field distribution in real space in a cross section of the two-dimensional photonic crystal layer parallel to the layer, obtained by inverse Fourier transform of the electric field distribution (K↑) using the position vector r↑ and wave number vector K↑ of each lattice point. The target radiation electric field distribution E rad_iFFT Using (r↑) With the modulation phase Ψ(r↑) expressed by the following formula, Δd(r↑) which is the positional shift amount of each of the plurality of modified refractive index areas and / or ΔS(r↑) which is the area shift amount of each of the plurality of modified refractive index areas is calculated using Δd which is the upper limit value of the positional shift amount and / or ΔS which is the upper limit value of the area shift amount, as follows: Δd(r↑)=B·Δd·sin(Ψ(r↑)) (when |B·sin(Ψ(r↑))|≦1, B is a constant) Δd (when B·sin(Ψ(r↑)>1) −Δd (when B·sin(Ψ(r↑)<−1) or / and ΔS(r↑)=C·ΔS·sin(Ψ(r↑)) (when |C·sin(Ψ(r↑))|≦1) ΔS (when C·sin(Ψ(r↑)>1) −ΔS (When C・sin(Ψ(r↑)<-1))

[0090] (Item 8) The two-dimensional photonic crystal laser according to item 8 is the two-dimensional photonic crystal laser according to item 1, in which the positional deviation amount and / or area deviation amount of each of the plurality of modified refractive index areas is expressed by a function that uses the position of the lattice point as a variable and saturates at the upper limit value and the lower limit value, respectively.

[0091] (Item 9) The two-dimensional photonic crystal laser according to Item 9 is the two-dimensional photonic crystal laser according to Item 8, wherein the position vector r↑ of each lattice point, and a vector k (n is an integer of 2 or more) indicating a combination of the tilt angles and / or azimuth angles of each of n laser beams (n is an integer of 2 or more) whose tilt angles and / or azimuth angles are different from each other are used. n ↑, and the phase exp(iα n ) a function having variables, Δd(σ) being the positional deviation amount of each of the plurality of modified refractive index areas and / or ΔS(σ) being the area deviation amount of each of the plurality of modified refractive index areas, and a constant t, It is expressed as:

[0092] (Item 10) The two-dimensional photonic crystal laser according to Item 10 is the two-dimensional photonic crystal laser according to Item 8, wherein an electric field distribution E in wavenumber space corresponding to a far-field pattern to be formed by a laser beam emitted from the two-dimensional photonic crystal laser is far_iFFT (K↑) is an electric field distribution in real space in a cross section of the two-dimensional photonic crystal layer parallel to the layer, obtained by inverse Fourier transform of the electric field distribution (K↑) using the position vector r↑ and wave number vector K↑ of each lattice point. The target radiation electric field distribution E rad_iFFT Using (r↑) and the variable σ, which is expressed by the modulation phase Ψ(r↑) and the constant B or C. d or σ S σ d =B・Δd・sin(Ψ(r↑)) σ S =C·ΔS·sin(Ψ(r↑)) and Δd(σ d ) or / and the area deviation amount ΔS(σ S ) is calculated by using Δd, which is the upper limit of the positional deviation amount, and / or ΔS, which is the upper limit of the area deviation amount, The present invention is characterized in that it is represented by the following formula:

[0093] 10, 20... Two-dimensional photonic crystal laser 11... Active layer 12, 22... Two-dimensional photonic crystal layer 121, 221... Base material 122... Modified refractive index area 1221... Modified refractive index area at lattice point 1222... Modified refractive index area arranged shifted in the positive direction in the y direction from the lattice point 1223... Modified refractive index area arranged shifted in the negative direction in the y direction from the lattice point 13... Spacer layer 141... First cladding layer 142... Second cladding layer 16... Substrate 171... First electrode 172... Second electrode 1721... Frame portion 1722... Window portion 2221, 2221A... First modified refractive index area 2222, 2222A... Second modified refractive index area 2231... First square lattice 2232... Second square lattice 2241... First lattice point 2242... Second lattice point

Claims

1. A photonic crystal layer comprising: a) a pair of electrodes; b) an active layer provided between the pair of electrodes and generating light of a predetermined wavelength when a current is injected from the electrodes; and c) a two-dimensional photonic crystal layer provided between one of the pair of electrodes and the active layer, the photonic crystal layer having a plate-shaped base material and a plurality of modified refractive index areas arranged on the base material and having a refractive index different from that of the base material, wherein the plurality of modified refractive index areas are arranged at different positional offsets from each lattice point of a two-dimensional lattice that is periodically arranged on the base material with an in-plane period corresponding to the predetermined wavelength, and / or are arranged at each lattice point with an area that is different from a predetermined reference area, or at a position offset from the lattice point by the positional offset, a positional deviation amount and / or area deviation amount of each of the multiple modified refractive index areas is a modulation value determined by a composite period obtained by superimposing multiple different in-plane periods, the modulation value being a value between a predetermined upper limit value and a predetermined lower limit value at a lattice point where the modified refractive index area is arranged, is the upper limit value in modified refractive index areas where the modulation value exceeds the upper limit value, and is the lower limit value in modified refractive index areas where the modulation value is less than the lower limit value.

2. A vector r↑ indicating the position of each lattice point of the two-dimensional lattice, and a vector k indicating a combination of the inclination angles and / or azimuth angles of each of n (n is an integer of 2 or more) laser beams whose inclination angles and / or azimuth angles are different from each other. n ↑, Amplitude A determined for every n n and phase exp(iα n ), and a constant Δd, the modulation value Δ'd(r↑) of the position shift amount at each lattice point is given by 2. The two-dimensional photonic crystal laser according to claim 1, characterized in that it is expressed by:

3. A vector r↑ indicating the position of each lattice point of the two-dimensional lattice, and a vector k indicating a combination of the inclination angles and / or azimuth angles of each of n (n is an integer of 2 or more) laser beams whose inclination angles and / or azimuth angles are different from each other. n ↑, Amplitude A determined for every n n and phase exp(iα n ), and a constant ΔS, the modulation value Δ′S(r↑) of the area shift amount at each lattice point is 2. The two-dimensional photonic crystal laser according to claim 1, characterized in that it is expressed by:

4. The two-dimensional photonic crystal laser according to claim 1, wherein the two-dimensional lattice is a first square lattice, wherein the multiple modified refractive index areas are arranged without deviation from first lattice points that are lattice points of the first square lattice, and each have an area determined by the area deviation amount, and further comprising a multiple second modified refractive index areas having a refractive index different from that of the base material, wherein the multiple second modified refractive index areas are arranged without deviation from second lattice points that are lattice points of a second square lattice that is a square lattice having the same periodic length as the first square lattice and arranged at a position different from the first square lattice, and the area of ​​the second modified refractive index area at each second lattice point is a value deviated from a predetermined second reference area by a second area deviation amount which is the modulation value in modified refractive index areas where the modulation value is a value between a predetermined second upper limit value and a predetermined second lower limit value at the second lattice point, the second upper limit value in modified refractive index areas where the modulation value exceeds the second upper limit value, and the second lower limit value in modified refractive index areas where the modulation value is less than the second lower limit value.

5. The two-dimensional photonic crystal laser according to claim 1, characterized in that the two-dimensional lattice is a first square lattice, and the multiple modified refractive index areas are arranged shifted by the positional shift amount from first lattice points that are lattice points of the first square lattice, and each have an area determined by the area shift amount, and further comprises a multiple second modified refractive index areas having a refractive index different from that of the base material, and the multiple second modified refractive index areas are arranged without shifting from second lattice points that are lattice points of a second square lattice that is a square lattice having the same periodic length as the first square lattice and arranged at a different position from the first square lattice, and each have the same area.

6. A two-dimensional photonic crystal laser as described in claim 4 or 5, characterized in that the second square lattice is arranged in the same direction as one of the two primitive translation vectors of the first square lattice, shifted by a period within the range of 0.4 to 0.6 periods.

7. Electric field distribution E in wave number space corresponding to the far-field pattern to be formed by the laser beam emitted by the two-dimensional photonic crystal laser far_iFFT (K↑) is an inverse Fourier transform of the electric field distribution in real space in a cross section of the two-dimensional photonic crystal layer parallel to the layer, using the position vector r↑ and the wave number vector K↑ of each lattice point. The target radiation electric field distribution E rad_iFFT Using (r↑) With the modulation phase Ψ(r↑) expressed by the following formula, Δd(r↑) which is the positional shift amount of each of the multiple modified refractive index areas and / or ΔS(r↑) which is the area shift amount of each of the multiple modified refractive index areas is expressed by the following formula, using Δd which is the upper limit value of the positional shift amount and / or ΔS which is the upper limit value of the area shift amount: Δd(r↑)=B・Δd・sin(Ψ(r↑)) (when |B・sin(Ψ(r↑))|≦1, B is a constant) Δd (when B・sin(Ψ(r↑)>1) -Δd (when B・sin(Ψ(r↑)<-1) or / and ΔS(r↑)=C・ΔS・sin(Ψ(r↑)) (when |C・sin(Ψ(r↑))|≦1) ΔS (when C・sin(Ψ(r↑)>1) -ΔS 2. The two-dimensional photonic crystal laser according to claim 1, which is expressed by the following formula: (when C·sin(Ψ(r↑)<-1).

8. A two-dimensional photonic crystal laser as described in claim 1, characterized in that the positional deviation amount and / or area deviation amount of each of the multiple modified refractive index areas is expressed as a function that uses the position of the lattice point as a variable and saturates at the upper limit value and the lower limit value, respectively.

9. A vector k indicating a combination of the position vector r↑ of each lattice point, the inclination angle and / or the azimuth angle of each of n (n is an integer of 2 or more) laser beams having different inclination angles and / or azimuth angles. n ↑, and the phase exp(iα n ) a function having a variable, Δd(σ) being the positional deviation amount of each of the plurality of modified refractive index areas and / or ΔS(σ) being the area deviation amount of each of the plurality of modified refractive index areas, using Δd being the upper limit value of the positional deviation amount and / or ΔS being the upper limit value of the area deviation amount, and a constant t 9. The two-dimensional photonic crystal laser according to claim 8, wherein the two-dimensional photonic crystal laser is represented by the formula:

10. Electric field distribution E in wave number space corresponding to the far-field pattern to be formed by the laser beam emitted by the two-dimensional photonic crystal laser far_iFFT (K↑) is an inverse Fourier transform of the electric field distribution in real space in a cross section of the two-dimensional photonic crystal layer parallel to the layer, using the position vector r↑ and the wave number vector K↑ of each lattice point. The target radiation electric field distribution E rad_iFFT Using (r↑) The modulation phase Ψ(r↑) expressed by d or σ S σ d = B Δd sin(Ψ(r↑)) σ S =C·ΔS·sin(Ψ(r↑)), and Δd(σ d ) or / and the area shift amount ΔS(σ S ) is calculated by using Δd, which is the upper limit of the positional deviation amount, and / or ΔS, which is the upper limit of the area deviation amount, 9. The two-dimensional photonic crystal laser according to claim 8, wherein the two-dimensional photonic crystal laser is represented by the formula:

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