1. Introduction

Structured illumination is an important optical technique whose basic idea emerged in the 1960’s based on holograms [1] such as detour phase [2] or kinoform [3]. It is widely used in super-resolution microscopy [4], three-dimensional (3D) metrology [5–7], and microscopic fringe projection profilometry [7]. Recently, high-speed 3D metrology has attracted much attention due to a growing demand for light detection and ranging (LiDAR) [8], facial recognition in smartphones [9], motion capture in gaming [10], and even for healthcare imaging [11]. In 3D metrology, structured illumination offers magnificent merits: it acquires a wide angular range instantly and does not need a scanning system. Illumination patterns are roughly categorized into multi-dot and fringe patterns. For example, the dot-projector in iPhone X, which is embedded as a part of a 3D sensor for facial recognition, can project 30,000 large-scale multi-dot patterns [5,9,12]. The dot projector is created by combining vertical-cavity surface-emitting lasers (VCSELs) and diffractive optical elements (DOEs) [13,14]. On the other hand, fringe patterns are used in Grey code patterns [5,6] or fringe projection profilometry including Fourier transform profilometry [7], and the phase shifting profilometry [5–7]. Particularly, the phase shifting profilometry in which shifting fringe patterns are alternatively projected to acquire phase information of the fringes is important in high-resolution 3D shape measurement since the resolution is drastically higher than period of the fringes.

To obtain structured illuminations, a combination of laser sources with well-tuned DOEs can provide most desired far field patterns (FFPs). However, it requires delicate optical coupling, which hinders compactness and robustness. Moreover, conventional DOEs are also accompanied by a zero-order beam due to imperfect phase modulation that disturbs the structured illumination [15]. Another practical approach for obtaining structured illuminations is to use a projector [6,7], but it takes up a rather large space. Thus, compactness is required to improve portability.

In this context, we proposed a novel light source called the “integrable spatial-Phase Modulating Surface-Emitting Lasers” (iPMSELs) [16], which are suitable as an ultra-compact light source since an arbitrary 2D beam pattern is projected from a needle-tip sized device. The basic idea of iPMSELs is to integrate holography into a compact coherent light source that enables projection of arbitrary beam patterns. For this purpose, we utilize photonic-crystal surface-emitting lasers (PCSELs) [17,18] as an ideal coherent light source. In iPMSELs, the photonic band edge mode of 2D photonic crystals (PCs) acts as an in-plane resonator, while spatial phase modulation for the desired beam patterns is introduced by the local shift of holes from a periodic position in a holographic manner. So far, we have clarified the origin of the beam patterns [19] and have demonstrated the removal of the zero-order beam in the surface normal that disturbs the beam patterns [20], visible wavelength [21], and electrical switching of beam patterns by using arrayed iPMSELs where eight devices are integrated on a TO-8 base [22,23].

If the multi-dot patterns and fringe patterns are obtained from iPMSELs, it offers an ultra-compact light source for 3D metrology that not only downsizes the conventional light source but also contributes to 3D inspections in narrow spaces such as dental and endoscope examinations. Therefore, this work aims to examine these beam patterns by using iPMSELs. To this end, we have demonstrated large-scale multi-dot patterns and fringe patterns both without a zero-order beam, which are important in 3D metrology.

In this work, we first describe the device structure and principle. Second, we present the design including the method to define the beam pattern project on flat screen and the capability of the phase shifting methods in iPMSELs even though the conjugate ±1^st order beams are superimposed. Third, the experimental results including FFP and projection images on a flat screen are shown and discussed.

2. Structure and principle of iPMSELs

In this work, we adopt M-point band edge based iPMSELs [20] that does not have a zero-order beam in the surface normal that disturbs the target beam pattern. A schematic of the device structure is shown in Fig. 1(a). First, the n-AlGaAs cladding layer, InGaAs/AlGaAs active layer, and i-GaAs layer are grown on n-GaAs substrates by using metal-organic chemical-vapor deposition (MOCVD). Then the octagonal holes are formed on top of the GaAs layer in the following manner: (1) the illumination pattern is defined in the wavenumber space, (2) the complex field distribution F(x,y) of a near field pattern (NFP) is calculated via an inverse Fraunhofer diffraction process of the illumination patterns, (3) the phase distribution ϕ(x,y) is acquired from an argument of the complex field distribution F(x,y), and (4) the center of gravity of the holes is rotated by angle ϕ(x,y) counterclockwise along the circle whose radius is r and center corresponds to the lattice point of a 2D PC [see Figs. 1(b) and 1(c)]. We named this positional shift method the circular shift method [19] and use it in this work. Here, the lasing wavelength is 940 nm, the lattice constant is a=202 nm, and the radius is r = 0.08a. The holes are fabricated by using electron beam (EB) lithography and a dry etching technique. The square emitter area is 200×200 µm² in an 800×800 µm² square chip. Note that since the emitter has approximately 1 mega scattering points, this structure enables 1-mega-pixel images in the wavenumber space, which is comparable to the images of a typical projector emitting several-mega-pixels image from several tens of centimeters. This layer modulates the spatial phase of in-plane lightwaves, so we call it the phase modulating (PM) layer hereafter. After the holes form in the PM layer, the p-AlGaAs cladding layer and p-GaAs contacting layer are successively grown by regrowth of MOCVD. Note that air-holes are formed during the regrowth because of the difference in the growth rate between the top and bottom of the holes. Then, square p-electrode and n-electrode that have an opening window with a SiN anti-reflection layer are formed by using photolithography.

 

Fig. 1. Schematics of (a) device structure of the iPMSELs. (b) Local positional shift of the holes from lattice point O of the 2D PC in accordance with the circular shift method. (c) Basic structure of the 2D PC at M-point band edge where the red arrows indicate basic in-plane lightwaves. Note that λ indicates the wavelength inside the cavity. (d) Intuitive image of spatial phase modulation of the wavefront of the output beam due to the positional shift of the holes. Note that the fluctuation of the wavefront is magnified for visibility.

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Next, the operating principle is followed. First, it is reasonable to assume that the in-plane resonance of the iPMSELs is similar to that of the PCSELs since the positional hole shift is small. Due to the diffraction of the 2D PC at the M-point band edge, four basic in-plane lightwaves are formed in the diagonal directions as shown in Fig. 1(c). These basic lightwaves are coupled and form a 2D standing wave that acts as an in-plane resonance cavity. Note that the surface normal diffraction is prohibited in the M-point band edge. This is why the zero-order beam in the surface normal is removed. On the other hand, the scattered basic lightwave at each of the holes is retarded/proceeds due to the local positional shift of the holes, so the wavefronts of the scattered lightwaves are locally modulated in accordance with the angle ϕ(x,y) as shown in Fig. 1(d). One question is how to output the target beam pattern out of plane even though the surface normal diffraction is prohibited. The answer is to add the original phase distribution ϕ(x,y) to the additional phase distribution ϕ_a(x,y). The additional phase distribution is expressed as

 

Fig. 2. Schematics of the zero and ±1^st order beam (a) without the additional phase distribution and (b) with the additional phase shift that tilts the target beam pattern (1^st order beam) from the direction along band edge mode (the basic lightwaves) to the surface normal.

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3. Design of the beam patterns for 3D metrology

3.1 Relationship of beam pattern between the wavenumber space and flat screen

In the 3D metrology using structured illumination, the position of the object is basically acquired by the principle of triangulation that requires precise angles of the beam pattern from the light source as well as the angle of the corresponding beam pattern from the image capturing device. In the case of a multi-dot pattern, a random dot pattern tends to be used to reduce misidentification of a specific dot by associating it with the neighboring dot patterns [6]. On the other hand, in the fringe pattern, the period of the fringe is usually constant on a flat projecting screen to acquire the phase by using the phase shifting profilometry [5–7]. In either case, the illumination patterns are a set of dots, and the relationship should be clarified between a point on a flat projecting screen and the corresponding point in the wavenumber space that is directly used in the inverse-Fourier transformation process as mentioned above. Here the intuitive relationship is briefly described. For a detailed description of the relationship, see Appendix A.

For intuitive understanding, we consider three typical examples of periodic grid patterns in various coordinates. First, a periodic grid pattern on a flat screen as shown in Fig. 3(a) is converted into the grid pattern in the angular space by using Eq. (5) as shown in Fig. 3(b) where the period of the grid becomes narrower the further it is from the origin (center). The conversion patterns in the wavenumber space are obtained by using Eq. (7) as shown in Fig. 3(c), which is a round square grid pattern where the outer grid lines approaching a circle.

 

Fig. 3. Relationship of periodic grid pattern among flat screen, angular space, and wavenumber space. Note that wavenumber is normalized by Eq. (2). (a-c) Periodic grid pattern on flat screen is shown on flat screen, in angular space, and in wavenumber space, respectively. (d-f) Periodic gird pattern in wavenumber space is shown on flat screen, in angular space, and in wavenumber space, respectively. (g-h) Periodic grid pattern in angular space is shown in angular space and wavenumber space, respectively.

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Second, the opposite case is a periodic grid pattern in the wavenumber space as shown in Fig. 3(f). The conversion pattern in angular space is obtained by using Eqs. (4) and (6) as shown in Fig. 4(e), while that on a flat screen is obtained by using Eq. (5) as shown in Fig. 3(f). Interestingly, grid lines in both patterns approaching hyperbolas as they move further from the origin. A similar tendency is seen in the dot-patterns of iPhone X [12], which indicates that the position of the dot-patterns is defined in the wavenumber space.

 

Fig. 4. Current-output characteristics and linear scale lasing spectrum of the iPMSELs under pulse room temperature conditions with respect to (a-b) the multi-dot patterns having 60×60 points and (c-d) the fringe patterns.

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Finally, in the case of periodic grid patterns in the angular space as shown in Fig. 3(g), the conversion pattern in the wavenumber space is obtained by using Eq. (7) as shown in Fig. 3(h), which has a complete circle border that corresponds to the light line. The grid interval become closer as it moves further from the origin. Note that all patterns in Fig. 3 are processed using Gaussian blur treatment in ImageJ for visibility.

In this work, we design the periodic multi-dot pattern and periodic fringe pattern on a flat screen for simplicity. However, the beam patterns in the other required coordinates can be designed easily.

3.2 Phase shifting profilometry of iPMSELs

The resolution of the 3D metrology by using structured illumination obviously depends on the resolution of structured illumination. However, high resolution of the illumination pattern requires a larger emission area. An approach to improve the resolution without changing the resolution of the illumination pattern is the phase shifting profilometry [5–7]. When a periodic fringe pattern is projected on the object, the fringes on the surface are distorted in accordance with the height of the surface in each pixel of the image capturing device. In the phase shifting profilometry, the distortion of the fringe is quantified as the phase. Specifically, slightly shifted periodic fringe patterns are projected alternatively where the shift is usually set to 1/N (N: integer) of the interval of the fringe, capturing each image by the image capturing device, and acquiring the phase in each pixel by simple calculation. The phase can be converted into depth, thus 3D shape information is acquired. Here, the fringe patterns need to be able to be shifted arbitrarily. Meanwhile, the conjugate ±1^st order beam patterns are projected from the iPMSELs, which disturb the target 1^st order beam. An approach to circumvent the competition of the ±1^st order beam is to separate the regions of both beam patterns, though this approach is limited to symmetric beam patterns. Therefore, what happens when the conjugate ±1^st order fringe patterns are superposed in the same area should be clarified.

The simple derivation of the intensity of the fringe patterns from the iPMSELs is described in Appendix B. The important results are summarized in Table 1, where K (=2π/Λ) is the wavenumber of the fringe, Λ is interval of the fringe, and θ is the phase shift of the fringe.

Table 1. Feature of the fringe pattern from iPMSELs.

View Table

At the beginning of the design, the amplitude in the wavenumber space is defined as the target beam patterns of the 1^st order beam, then the phase distribution is calculated by the inverse Fourier transformation process, and finally the output pattern, which is the superposition of the intensity of the ±1^st order beam patterns, is projected. The spatial frequency and phase shift of the designed fringe pattern with respect to amplitude need to be half of those with respect to the intensity. Conversely, the amplitude must have half the cycles and half the phase shift of the desired fringe patterns. Note that though absolute value of the amplitude pattern obtained by the Fourier transformation of the phase distribution shows similar tendency to that of the intensity pattern, both of them are not same.

4. Preparation of the iPMSELs

In this work, the iPMSELs are designed to project multi-dot patterns and fringe patterns. In the multi-dot patterns, the 1^st order beam patterns are designed as half of the dot patterns that have 20×40, 30×60, 40×80, and 60×120 points in half of the wavenumber space. Consequently, due to superposition of the ±1^st order beam patterns, the dot patterns become 40×40, 60×60, 80×80, and 120×120 points, respectively. The multi-dot is defined within a square region on the flat projection screen that angularly ranges approximately ±27.2° along the Xs-Ys axis on the screen where the dots are aligned uniformly.

In the fringe patterns, the 1^st order beam patterns are designed with respect to the amplitude as sinusoidal waves within the square region on a flat screen that have 18 cycles and range approximately ±57.5° along the Xs-Ys axis on the flat screen. Because the 1^st order beam pattern is designed on the wavenumber plane with respect to the amplitude, the resulting intensity patterns have double the dense fringes (36 cycles) and double the phase shift as discussed in previous section. For clarifying the capability of the phase shifting profilometry in the iPMSELs, two iPMSELs are prepared that have a different initial phase that differs π/2 with respect to amplitude of the 1^st order beam patterns.

5. Results and discussion

5.1 Lasing characteristics

The lasing characteristics of the iPMSELs including light output and spectrum are discussed. The measurement setup is the same as that in the previous work [20]. The light-output and spectrum of the iPMSELs are shown in Fig. 4. In the multi-dot patterns, all devices from 40×40 to 120×120 points show a similar tendency that we adopt the multi-dot patterns having 60×60 points, where the light-output is shown in Fig. 4(a). The threshold current and slope efficiency (SE) are 0.38 A and 0.27 W/A, under the room-temperature pulse condition. The pulse width, duty, and temperature are set at 50 ns, 1%, and 25 °C. Here the output power exceeds 1W, which is three times higher than the output power of the M-point iPMSELs in the previous work [20], while the maximum output power is limited by the limitation of the LD driver (ILX lightwave, LDP-3830). The lasing spectrum is shown in Fig. 4(b) where the peak wavelength is 938.37 nm. On the other hand, lower output power is observed in the fringe pattern as shown in Fig. 4(c) where the threshold current and SE are 0.53 A and 0.082 W/A. The lasing spectrum shown in Fig. 4(d) has a peak wavelength of 938.44 nm, which is almost the same as that of the multi-dot patterns. The lower output power of the fringe patterns might be caused by a fabrication error, because the multi-dot patterns in the same process as the fringe patterns also show lower output power that is slightly higher than the output of the fringe patterns. However, the output power can be increased by optimization.

5.2 Multi-dot patterns

The measurement setup of FFP is almost the same as in the previous work [20]. The setup has the FFP optics (Hamamatsu, A3267-12) for a wide range, while the magnified FFP is acquired by the high-resolution FFP optics (Hamamatsu, A3267-15). The FFP of the multi-dot patterns is shown in Fig. 5 under the room temperature pulse conditions. Here the operation current is 0.5 A, and the other operating conditions are the same as in Fig. 4. It is reasonable that the interval of the dots slightly shrinks in the periphery of the projecting area, which is peculiar to the conversion pattern of the periodic grid pattern from the flat screen to the angular space as shown in Fig. 3(b). Meanwhile the projection patterns are slightly round, which is peculiar to the conversion patterns not to angular space [Fig. 3(b)] but to the wavenumber space as shown in Fig. 3(c). This might be caused by a slight distortion of the FFP optics in the large angular region. Figure 6 shows magnified FFP under the same operating conditions. The divergence angle of an individual spot roughly ranges from 0.4° to 0.6° in 40×40 dots where the full width of half maximum (FWHM) is used as the measure since it is difficult to measure the 1/e2 value of beam divergence without the effect of the neighboring dots in such a dense dot array. Since the diffraction limit of the 200-µm² emitter of the iPMSELs is estimated to be 0.5° (see Appendix C), the experimental results are the same order as or slightly broader than the diffraction limit, but this divergence angle is sufficient for 3D metrology. If a narrower divergence angle is required, the emitter area must be made broader to reduce the diffraction limit.

 

Fig. 5. FFP of the iPMSELs projecting multi-dot patterns under pulse room temperature conditions. The operating current is 0.5 A, and the other conditions are the same as those in Fig. 4.

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Fig. 6. Detailed FFP of the iPMSELs projecting multi-dot patterns under pulse room temperature conditions. The operating conditions are the same as in Fig. 5. Note that the central bright areas are stray light.

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Almost periodic alignment is observed in the 40×40 dots. In the 60×60 dots, slight shifts are observed that are attributable to the conversion process from the flat screen to angular space. Meanwhile, the coalescence of the dots is observed in 80×80 dots and progresses in 120×120 dots. The coalescence can be suppressed by broadening the angular range, but we accept the coalescent in this work because the angular range is difficult to broaden with our setup. Note that because the dot pattern is recognized from clues of the neighboring dots, the coalescence of the dots does not lead to severe degradation of 3D metrology. In fact, such a coalescent dot pattern is used in the 3D measurement system [6]. Note that brightness and contrast of all FFPs in Figs. 5–7 are tuned automatically in ImageJ for visibility. In order to assess the quality of dots, we evaluated the line plot of a horizontal line of the 40×40 dots patterns in Fig. 6 which reveals the standard deviation against the averaged peak intensity is around 10%, while keeping the positional accuracy.

 

Fig. 7. FFP of the iPMSELs projecting fringe patterns under pulse room temperature conditions. The reference fringe is seen in (a), while the fringe having phase shift π/2 with respect to the amplitude is seen in (b). The operating current is 1 A, and the other conditions are the same as in Fig. 4. Note that the central bright areas in (a) and (b) are stray light.

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5.3 Fringe patterns

The FFPs of the fringe patterns are measured under the room temperature pulse conditions. Here the operation current is 1 A since a larger threshold current is required in the fringe pattern than in the multi-dot pattern, while the other operating conditions are the same as in Fig. 4. First, the fringes in the magnified FFP acquired by using the high-resolution FFP optics are shown in Fig. 7(a) as a reference. The magnified FFP shows roughly periodic fringes in the central angular range. Although a slight grained texture is observed that is intrinsically attributed to the hologram to obtain the phase distribution, the fluctuation of the intensity is clearly recognized. Note that the horizontal line plot at center of both of the fringe patterns in Fig. 7 reveals that root mean squared error against the amplitude are approximately 30% which should be improved for precise 3D measurement in future.

To verify the phase shifting profilometry by using iPMSELs, the shifting fringe is prepared that has a phase shift of π/2 with respect to amplitude. The FFP of the shifting fringe is shown in Fig. 7(b) as the magnified range. As discussed previously, the fringes are successfully shifted half a period as shown in Fig. 7(b), which corresponds to the phase shift of π in intensity that can be consistent with the relationship shown in Table 1. Thus, the phase shifting profilometry can be successfully used in iPMSELs. On the other hand, the periodic grids on the flat screen are used in the phase shifting profilometry, so we have also demonstrated the projection images on the flat screen, which are discussed in the next section.

5.4 Projection images on the flat screen

The see-through projection images on the flat screen were captured by a camera (Watec, WAT-902B) with the lens (Ricoh, TV lens) from the backside of the screen where the patterns were projected on the front side from the iPMSELs. The results are shown in Fig. 8, where the operating current is 1 A and the other conditions are the same as Fig. 4. In the multi-dot pattern, the projection region is almost square as designed, while the uniform interval is observed in the fringe pattern that is essential to apply the phase shifting profilometry. The slight distortion is caused by the tilt from the optical axis of the iPMSELs.

 

Fig. 8. See-through projection images of the iPMSELs under pulse room temperature conditions. (a) The multi-dot pattern (60×60). (b) The reference fringe pattern. The operating current is 1 A, and the other conditions are the same as in Fig. 4.

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6. Conclusion

We have demonstrated large-scale multi-dot patterns and fringe patterns both without the zero-order beam by using the M-point integrable spatial-phase-modulating surface-emitting lasers (iPMSELs) for 3D metrology. In the former case, we have demonstrated large-scale multi-dot patterns of more than 10,000 points from a 200×200-µm² emitter, which is the same order of magnitude as points of the dot-projector in commercialized face recognition systems. The output power exceeds 1 W in the multi-dot patterns, which is three times higher than that in previous work. In the latter case, we have developed a method to design the fringe patterns and successfully demonstrated the shift of the fringe even though the conjugate ±1^st order beams are superimposed in the iPMSELs. This result is essential for a phase shifting profilometry that enables high-resolution 3D shape measurement. We believe that the iPMSELs have potential to be an ultra-compact light source for 3D metrology that not only downsizes the conventional light source but also contributes to 3D inspection in narrow spaces such as dental and endoscope examinations.

Appendix A: relationship of the beam pattern in various coordinate systems

A point in the illumination pattern is related to a wavevector in the wavenumber space. Figure 9(a) indicates the wavevector in spherical coordinates of the wavenumber space. Here, the radial component of the wavevector, azimuthal angle, and polar angle are k, θ_rot, and θ_tilt, respectively. Since the band edge mode is utilized, the wavevector can be normalized by dividing the reciprocal vector G=2π/a as

(2)$$k = {a / \lambda }. $$

Note that the normalized wavevector k is 0.215 [2π/a] in this work and the radius of the light line is the k in the wavenumber space. Since, the in-plane components of the wavevector are geometrically expressed as

(3)$${\vec{k}_{/{/}}} = \left( {\begin{array}{c} {{k_x}}\\ {{k_y}} \end{array}} \right) = \left( {\begin{array}{c} {k\sin {\theta_{tilt}}\cos {\theta_{rot}}}\\ {k\sin {\theta_{tilt}}\sin {\theta_{rot}}} \end{array}} \right), $$

the azimuthal angle and polar angle are related to the wavevector by

(4)$$\left( {\begin{array}{c} {{\theta_{rot}}}\\ {{\theta_{tilt}}} \end{array}} \right) = \left( {\begin{array}{c} {{{\tan }^{ - 1}}({{{{k_y}} / {{k_x}}}} )}\\ {{{\sin }^{ - 1}}\left( {{{\sqrt {k_x^2 + k_y^2} } / k}} \right)} \end{array}} \right). $$

On the other hand, the screen coordinate in Fig. 9(b) is geometrically expressed as

(5)$$\left( {\begin{array}{c} {{X_S}}\\ {{Y_S}} \end{array}} \right) = \left( {\begin{array}{c} {D\tan {\theta_{tilt}}\cos {\theta_{rot}}}\\ {D\tan {\theta_{tilt}}\sin {\theta_{rot}}} \end{array}} \right) = \left( {\begin{array}{c} {D\tan {\theta_X}}\\ {D\tan {\theta_Y}} \end{array}} \right). $$

 

Fig. 9. (a) Wavevector in spherical coordinate of wavenumber space. (b) Projection of wavevector on flat screen in XYZ coordinate.

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Equation (5) immediately leads to the following relationship in angular space as

(6)$$\left( {\begin{array}{c} {{\theta_x}}\\ {{\theta_y}} \end{array}} \right) = \left( {\begin{array}{c} {{{\tan }^{ - 1}}({\tan {\theta_{tilt}}\cos {\theta_{rot}}} )}\\ {{{\tan }^{ - 1}}({\tan {\theta_{tilt}}\sin {\theta_{rot}}} )} \end{array}} \right). $$

Thus, the angles in angular space is related to angles in spherical coordinate as

(7)$$\left( {\begin{array}{c} {{\theta_{rot}}}\\ {{\theta_{tilt}}} \end{array}} \right) = \left( {\begin{array}{c} {{{\tan }^{ - 1}}({{{\tan {\theta_y}} / {\tan {\theta_x}}}} )}\\ {{{\tan }^{ - 1}}\left( {\sqrt {{{\tan }^2}{\theta_x} + {{\tan }^2}{\theta_y}} } \right)} \end{array}} \right). $$

According to equations (2-7), the coordinate system can be converted among the flat screen, angular space, and wavenumber space.

Appendix B: intensity of the fringe pattern in iPMSELs

Superposition of the conjugate ±1^st order fringe pattern in the same region is considered. The fringe pattern is assumed to be a sinusoidal wave in the phase shifting profilometry, so we simply assume the amplitude of the 1^st order target fringe beam patterns to be a combination of the sinusoidal wave and the basic lightwaves. It is written in

(8)$${A_1} = {a_1}\cos ({Kx + \theta } )\exp \{{j({\omega t - {k_x}x - {k_y}y - {k_z}z} )} \}, $$

where a₁ is the amplitude of the 1^st order beam attributed to the difference between the ideal and actual phase distributions corresponding to the position of the holes, K (K=2π/Λ) is the wavenumber of the fringe, Λ is the period of the fringe, θ is phase shift of the fringe, ω (ω=2π/λ) is the angular frequency of light, λ is the wavelength of light in vacuum, k=(k_x, k_y, k_z) is the wavevector of the lightwaves of the 1^st order beam, and (x, y, z) is the position vector. Similarly, the amplitude of the -1^st order beam pattern is written as

(9)$${A_{ - 1}} = {a_{ - 1}}\cos ({Kx + \theta } )\exp \{{j({\omega t + {k_x}x + {k_y}y - {k_z}z} )} \}, $$

and the resulting total amplitude is in addition to Eqs. (8) and (9) followed by

(10)$$A = \cos ({Kx + \theta } )\{{{a_1}\exp [{j\{{\omega t - {k_z}z - ({{k_x}x + {k_y}y} )} \}} ]+ {a_{ - 1}}\exp [{j\{{\omega t - {k_z}z + ({{k_x}x + {k_y}y} )} \}} ]} \}$$

The intensity is the squared magnitude of the amplitude as

(11)$$I = A\bar{A} = {\cos ^2}({Kx + \theta } )\{{a_1^2 + a_{ - 1}^2 + 2{a_1}{a_{ - 1}}\cos ({k_x}x + {k_y}y)} \}. $$

Since the wavenumber of the lightwaves k is sufficiently larger than that of the fringe K (k>>K), the intensity is practically written by averaging the term of the cosine in Eq. (11) as

(12)$$I \cong ({a_1^2 + a_{ - 1}^2} )\frac{{\cos \{{2({Kx + \theta } )} \}+ 1}}{2}. $$

As the result, the intensity of the fringe pattern has spatial frequency 2/Λ double that of the designed spatial frequency of the amplitude 1/Λ. Therefore, if we set half the spatial frequency and half the phase shift of the required fringe pattern, the fringe pattern with arbitrary spatial frequency and phase shift can be obtained in iPMSELs.

Appendix C: diffraction limit

The diffraction limit from the square emitter is considered since the emitter of the iPMSELs is square in this work. Due to the Fraunhofer diffraction [1], the width of the main lobe W of the projection beam from the square aperture is expressed as

(13)$$W = \frac{{\lambda z}}{{nw}}, $$

where λ is the wavelength in the air, z is the distance along propagation axis, n is the refractive index of the propagation space, and w is the half width of the square aperture. Then the diffraction limit can be expressed by converting the width W into the divergence angle θ_DL followed by

(14)$${\theta _{DL}} = 2{\tan ^{ - 1}}\left( {\frac{\lambda }{{2nw}}} \right). $$

In this work, the diffraction limit θ_DL is estimated as 0.54°.

Funding

Japan Science and Technology Agency (JST); Center of Innovation Program (COI) (JPMJCE1311).

Acknowledgments

The authors are grateful to A. Hiruma (President), T. Hara (Director), M. Yamanishi (Research Fellow), M. Niigaki, Y. Yamashita, K. Nozaki, H. Toyoda, T. Hirohata, T. Edamura, and A. Watanabe of HPK for their encouragement throughout this work and A. Higuchi and M. Hitaka for their assistance with epitaxial growth.

Disclosures

The authors declare no conflicts of interest.

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