Principle of beam generation in on-chip 2D beam pattern projecting lasers

Yu Takiguchi; Kazuyoshi Hirose; Takahiro Sugiyama; Yoshiro Nomoto; Soh Uenoyama; Yoshitaka Kurosaka

doi:10.1364/OE.26.010787

1. Introduction

The concept of structured coherent light illumination with a combination of laser light sources [1] and digitally generated phase-only holograms arose in the 1960s [2]. The recent technical advances in highly accurate optical devices and fabrication methods boosted the capability of current high-end optical systems. If the target structure illumination pattern is well-defined, modulating the phase distribution of a collimated laser source with diffractive optical elements (DOEs) can manipulate the far-field light pattern (FFP) to almost any desired intensity profile [3]. These optics, however, require additional optical coupling elements, and several approaches are considered to generate patterned light directly from compact semiconductor lasers. These approaches include using the intra-cavity metasurface [4] and direct creation of DOE on the electrode of a semiconductor surface [5]. We recently experimentally demonstrated a direct emitting wavefront modulated semiconductor laser by implementing phase-only hologram information inside the resonator. These devices are called integrable phase-modulating surface-emitting lasers (iPMSELs) [6, 7]. When the desired FFP is determined in wavenumber space, the corresponding electromagnetic (EM) field on the surface plane of a device, i.e., the near field pattern (NFP), can be obtained by inverse Fraunhofer diffraction of its FFP [8]. Although the EM field is expressed as a complex number, the wavefront can be introduced as the phase distribution of the field based on assumption that only the wavefront of the NFP is required for the reconstruction of the FFP intensity distribution. The idea of intra-wavefront modulation was introduced by integrating a phase-modulating layer into semiconductor lasers [7]. For the purpose, we based on the lasing resonator of photonic-crystal surface-emitting lasers (PCSELs) [9, 10] which is an ideal light source to produce surface-emitting plane waves invented by Prof. Noda et al. [9]. In iPMSELs, the position of holes consisting of a two-dimensional photonic-crystal (2D-PC) in PCSELs are locally modulated in holographic manner to control the NFP. In the viewpoint of the device derived from PCSELs, the modulated PCSELs which emits a deflected beam is also proposed [11–13]. While the epitaxial structure of the resonator is similar to that of PCSELs, in the iPMSELs, the photonic-crystal resonator layer is replaced by a holographic phase-modulating layer where holes are allocated in the vicinity of the lattice point in a 2D-PC to obtain a two-dimensional beam pattern. By shifting the in-plane position of holes in PCSELs in the manner of holography, the basic (in-plane) standing waves comprising the band edge mode in a 2D-PC are modulated when surface emission occurs. This is due to the in-plane positional shift from the lattice point, which retards or advances the phase of diffracted lightwaves. The in-plane positions of holes are shifted according to the phase distribution in a fixed manner described in latter section. In this work, we clarify the mechanism of wavefront modulated beam generation from our theoretical derivation and demonstrated their responses.

2. Concept and structure of iPMSELs

We start from basic idea of the iPMSELs, which project static arbitrary two-dimensional (2D) beam patterns. In iPMSELs, a phase-modulating layer plays an important role in realizing concurrent formation of in-plane resonances similar to PCSELs and modulation of wavefront. This results in the emission of a modulated wavefront from a semiconductor surface, while uniform wavefronts are emitted in the case of PCSELs. The modulated wavefront is realized by shifting the in-plane position of holes in a fixed manner. Note that the position shift should be set within the limit so as not to disturb standing wave formation. At that time, the band edge mode is not severely disturbed, and an arbitrary modulated plane wave is emitted from the semiconductor surface. Here, we can determine the positional shift of holes for a desired beam pattern from the following procedure. The beam pattern is related to NFP via diffraction theory [8]. Fraunhofer diffraction theory gives the relationship between the FFP and NFP. Based on this theory, the desired FFP is related to the NFP via an inverse Fourier transformation:

N (x, y) = A \cdot I F T [F (k_{x}, k_{y})] .

A is a constant term, IFT denotes the inverse Fourier transform operation, N(x,y) is the EM field of the NFP, and F(k_x,k_y) is EM field of the FFP in wavenumber space as discussed in Appendix A. As aforementioned, it is enough to reconstruct the FFP from argument of the NFP. Thus, the phase distribution ϕ(x,y) is defined as

N (x, y) = I (x, y) \cdot \exp (i ϕ (x, y)),

where I(x,y) is the amplitude of the NFP. Then, the in-plane positions of holes are shifted according to the phase distribution ϕ(x,y) in a fixed manner as described later. As a result, one obtains the desired 2D beam patterns. The intuitive image of wavefront modulation via spatial phase modulation of the surface emission is shown in Fig. 1. In contrast to PCSELs, vertical plane waves are modulated due to the in-plane shift of holes in iPMSELs, because the in-plane positional shift from the lattice point retards or advances the phase of diffracted lightwaves when surface emission occurs. This is the concept of iPMSELs that we proposed in [6,7].

Fig. 1 Schematic drawing explaining wavefront modulation: (a) square lattice PCSEL, (b) iPMSEL.

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Next, the device structure is presented. An iPMSEL schematic is shown in Fig. 2. The basic structure of iPMSELs is similar to that of PCSELs, which utilize the band edge mode of a 2D PC as a resonant cavity and show large-area coherent oscillation [9, 10]. However, the holes in the 2D PC layer are no longer periodic since the holes are shifted from their lattice points according to the phase distribution. In this work, the manner of position shift of the holes are defined as circular and linear shift methods, as shown in Fig. 3. In the case of circular shift method, the center of gravity of the holes is rotated by a phase angle ϕ(x,y) on a circle, where the center corresponds to the lattice point of a 2D PC with radius r. In the case of the linear shift method, the center of gravity of the holes is shifted linearly by a phase angle ϕ(x,y) on a line segment, where the midpoint corresponds to the lattice point of a 2D PC with length 2L.

Fig. 2 Structure of iPMSELs.

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Fig. 3 Schematic of phase-modulating method based on (a) circular shift and (b) linear shift. In this study, the angle θ is 45°.

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The detailed device structure and fabrication processes are almost same as those presented in [6], except for the phase-modulating (PM) layer. In the PM layer, the hole shapes and lattice constant a are right-isosceles triangular and 282 nm, respectively. The filling factor of the holes (the area of holes against area of unit cells) in the 2D PC before the regrowth process is 30%. In this work, we adopted the linear shift method for spatial phase modulation, where the L is set to 0.08a.

We should note that we initially proposed a structure which has additional holes at the lattice point in the circular shift method [7], where the holes at the lattice point support the band edge resonance mode, while holes on the circle modulate the phase of vertical plane waves. We found later that the additional holes can be removed when the hole shifts are small [6]. In addition, we noticed that the additional holes on the lattice point are theoretically useful for concurrent mode stabilization and suppression of the 0th order beam in a circular shift method when the hole shifts become large. This will be described later. Here it should be stressed that the general concept of projecting arbitrary 2D beam patterns is employed in the initial models.

3. Theory: the origin of beam patterns in iPMSELs

We have demonstrated 2D arbitrary beam patterns based on circular shifts for the first time [6]. Thus far, the iPMSELs show other subsidiary beam patterns, which have not been discussed in detail, particularly regarding their physical origin. From the point of view of engineering the beam patterns, it is important to clarify the origin of these beam patterns in iPMSELs, as well as reveal their physical background. For this purpose, we focus on the actual phase distribution of the diffracted lightwaves as attributed to their geometric position. Generally, the ideal phase distributions of output lightwaves differ from the actual phase distribution of the diffracted lightwaves. There are several reasons this occurs. First, the position shift is not proportional to the ideal phase distribution in the case of the circular shift method. Second, a specific arrangement of holes provides different phase distributions between different directional lightwaves. Third, even in a single basic lightwave, it is difficult to generate a 2π phase shift because it requires that the holes have a large position shift. This disturbs stable lasing and also causes neighboring holes to coalesce. Moreover, an insufficient phase shift below 2π causes subsidiary diffraction, which will be described later.

In this work, we utilize the Γ₂ band edge of a square-lattice 2D PC [10,14] as the standard structure. At that time, four basic lightwaves of the band edge mode in the 2D PC should be taken into account. We denote the four basic basic lightwaves which propagating upward, downward, leftward, rightward as U, D, L, and R, as shown in Fig. 3. In this section, we discuss the theoretical beam patterns of iPMSELs.

3.1 Beam pattern of an ideal NFP

As discussed in the second section, the NFP phase distribution ϕ(x,y) is obtained by the method based on Fourier optics [8]. In the case of the Γ₂ band edge of a square lattice 2D PC [10,14], the in-plane wave vector component of the four basic lightwaves of the band edge mode is zero when surface emission occurs. Thereby, it is appropriate to define the phase difference for a vertical plane wave obtained by diffraction in the Γ₂ point Δϕ_Γ(x,y) as follows:

Δ ϕ_{Γ} (x, y) = \exp {i \cdot ϕ (x, y)} .

This corresponds to the phase distribution of the NFP. Therefore, the FFP are obtained as a Fourier transform of Eq. (3). As an example, we show the phase distribution of the beam pattern of characters “ABCD” in Fig. 4(a). The beam pattern is obtained via Fourier transform in wavenumber space, as shown in Fig. 4(b).

Fig. 4 (a) Phase angle distribution of the beam pattern “ABCD”. (b) Corresponding beam pattern of the phase angle distribution in wavenumber space normalized by 2π/a.

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For preparing the later discussion, we should define the general order phase distributions as follows:

Δ ϕ_{Γ n} (x, y) = \exp {i \cdot n ϕ (x, y)},

In Eq. (4), n is an integer (n = 0, ± 1,…) corresponding to the phase distribution of the NFP for each diffraction order. The calculated 0th order and −1st order beam of the character beam pattern “ABCD” are shown in Fig. 5. As shown here, the 0th order beam shows the corresponding point beam in the surface normal direction without any spatial phase modulation, while the −1st order beam is symmetric to the 1st order beam against the surface normal since the components of wave vector are conjugates.

Fig. 5 Beam patterns of Fig. 4 in wavenumber space normalized by 2π/a. (a) 0th order and (b) −1st order.

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3.2 Circular shift method

Here, we discuss the circular shift spatial phase modulation methods that were implemented in a previous paper [6]. When the phase distribution is given by ϕ(x,y), the phase difference for plane wave in the vertical direction obtained by diffraction of U, D, R, and L are given as,

Δ ϕ_{U} (x, y) = \exp {i \frac{2 π r}{a} \cdot \sin ϕ (x, y)},

Δ ϕ_{D} (x, y) = \exp {- i \frac{2 π r}{a} \cdot \sin ϕ (x, y)},

Δ ϕ_{R} (x, y) = \exp {i \frac{2 π r}{a} \cdot \cos ϕ (x, y)},

Δ ϕ_{L} (x, y) = \exp {- i \frac{2 π r}{a} \cdot \cos ϕ (x, y)},

where r is the radius of the circle in Fig. 3(a) and a is the lattice constant. It is clear that the phase distribution is converted by a trigonometric function due to circular position shift. Since a wavefront having phase difference of exp(iϕ(x,y)) projects the designed beam patterns, it is useful to rewrite the above equations by expansion of exp(iϕ(x,y)). The above equations are expanded mathematically as

Δ ϕ_{U} (x, y) = \sum_{n = - \infty}^{\infty} J_{n} (\frac{2 π r}{a}) \cdot \exp (i n ϕ (x, y)),

Δ ϕ_{D} (x, y) = \sum_{n = - \infty}^{\infty} J_{- n} (\frac{2 π r}{a}) \cdot \exp (i n ϕ (x, y)),

Δ ϕ_{R} (x, y) = \sum_{n = - \infty}^{\infty} e^{- j \frac{n π}{2}} J_{- n} (\frac{2 π r}{a}) \cdot \exp (i n ϕ (x, y)),

Δ ϕ_{L} (x, y) = \sum_{n = - \infty}^{\infty} e^{- j \frac{n π}{2}} J_{n} (\frac{2 π r}{a}) \cdot \exp (i n ϕ (x, y)),

where n is an integer (n = 0, ± 1, …) and J_n(x) is the nth order Bessel function. The derivation of Eqs. (6) is given in Appendix B. From Eq. (6), one finds that the amplitudes of the nth order wavefront U are presented as J_n(2πr/a). Here, the 1st order beam corresponds to the designed beam patterns, while the −1st order is symmetric to the 1st order beam pattern against the surface normal, and the 0th order beam is the surface normal beam described in previous section. Based on the relation J_-n(x) = (−1)ⁿJ_n(x), the 1st order beam of each lightwave has the same intensity as that of its symmetric −1st order beam. While the total output beam is expressed as a linear combination of Eq. (6), the intensity of the 1st and −1st order beams in each of the basic lightwaves are same. Therefore, a symmetric beam pattern is obtained in the iPMSELs based on the circular shift method.

For further understanding, the behavior of the 0th and ± 1st order beams are studied. By ignoring the constant term which accounts for the contribution of each of the basic lightwaves, the amplitude of the 0th order beam (the surface normal beam) is represented as J₀(2πr/a). Similar to the 0th order beam, the amplitude of the ± 1st order beam is represented as J _± ₁(2πr/a). Here, the amplitude and intensity dependence of the ± 1st and 0th order beams versus positional shift r in the circular shift method are shown in Fig. 6. Here, the maximum shift r is set within 0.5a in order to prevent coalescence of neighboring holes. Note that the intensity of the −1st order beam is same as 1st order. The intensity of 1st order beam increases until r is below 0.30a, then gradually decreases with increasing r. Meanwhile, the 0th order beam decreases with increasing r below 0.38a. When r is 0.38a, the intensity of the 0th order beam is successfully suppressed. When r is larger than 0.38a, the phase of the 0th order beam becomes inverted. In this specific region, the 0th order beam causes destructive interference with diffraction at the lattice points. It means that if we put an additional hole at the lattice point in this region, the 0th order beam causes destructive interference with the surface normal diffraction at the lattice point. Furthermore, the additional holes at lattice point might play an important role in stabilizing lasing, even with such a large position shift.

Fig. 6 Relationship between the ± 1st and 0th order beams for a single lightwave vs. positional shift r in the circular shift method. (a) Amplitude. (b) Intensity.

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3.3 Linear shift method

Although the symmetric beam pattern is obtained in the circular shift methods in principle, it is desirable to output an asymmetric beam pattern for more sophisticated beam control. As discussed in the previous section, the origin of a symmetric beam pattern is due to the same amount intensity of the ± 1st order beam, which is attributed to trigonometric spatial phase modulation. Thus, it is necessary to implement the other spatial phase modulating method to obtain an asymmetric beam pattern. To eliminate the trigonometric spatial phase modulation term in the circular shift methods, the simple solution requires shifting the holes on a line, as shown in Fig. 3(b). We named this type of spatial phase modulation as the linear shift method. Similar to the previous section, the beam patterns based on linear shift methods are discussed as follows.

When the phase distribution is given by ϕ(x,y), the phase differences for a plane wave in the vertical direction obtained by diffraction of U, D, R, and L are given as

Δ ϕ_{U} (x, y) = \exp [i \cdot \frac{2 L \sin θ}{a} {ϕ (x, y) - ϕ_{0}}],

Δ ϕ_{D} (x, y) = \exp [- i \cdot \frac{2 L \sin θ}{a} {ϕ (x, y) - ϕ_{0}}],

Δ ϕ_{R} (x, y) = \exp [i \cdot \frac{2 L \cos θ}{a} {ϕ (x, y) - ϕ_{0}}],

Δ ϕ_{L} (x, y) = \exp [- i \cdot \frac{2 L \cos θ}{a} {ϕ (x, y) - ϕ_{0}}],

where L is half the line length in Fig. 3(b), a is the lattice constant, θ is the angle shown in Fig. 3(b), and ϕ₀ is the initial phase angle at the lattice point, which is set as 0° in this work. To ensure homogeneous contributions from the horizontal (L, R) and vertical (U, D) lightwaves, we have taken θ = 45° in this work. This simplifies Eqs. (7) as follows:

Δ ϕ_{U} (x, y) = Δ ϕ_{R} (x, y) = \exp [i \cdot \frac{\sqrt{2} L ϕ (x, y)}{a}]

Δ ϕ_{D} (x, y) = Δ ϕ_{L} (x, y) = \exp [- i \cdot \frac{\sqrt{2} L ϕ (x, y)}{a}]

In contrast to the circular shift methods, it is obvious that the trigonometric terms are successfully removed in the linear shift method. It is also useful to expand Eqs. (8) by the complex exponential function of ϕ(x,y) as

Δ ϕ_{U} (x, y) = Δ ϕ_{R} (x, y) = \sum_{n = - \infty}^{\infty} A_{n} \cdot \exp {i n ϕ (x, y)},

Δ ϕ_{D} (x, y) = Δ ϕ_{L} (x, y) = \sum_{n = - \infty}^{\infty} - A_{n} \cdot \exp {i n ϕ (x, y)},

A_{n} = \exp {i π (\frac{\sqrt{2} L}{a} - n)} \cdot \sin c {π (\frac{\sqrt{2} L}{a} - n)},

where sinc(x) is defined as sin(x)/x, A_n represents the nth order diffraction component for each lightwave (the derivation of Eqs. (9) is given in Appendix C). From Eqs. (9), the amplitudes of the 0th and ± 1st order beams are

A_{0} = \sin c (\frac{\sqrt{2} π L}{a}),

A_{1} = \exp {i π (\frac{\sqrt{2} π L}{a} - 1)} \cdot \sin c {π (\frac{\sqrt{2} π L}{a} - 1)},

A_{- 1} = \exp {i π (\frac{\sqrt{2} π L}{a} + 1)} \cdot \sin c {π (\frac{\sqrt{2} π L}{a} + 1)},

Although the amplitudes of the 0th and ± 1st order components are expressed as linear combinations of the four lightwaves, the tendency can be simply evaluated by the magnitude of the components for a single lightwave. Moreover, since the magnitude of the complex exponential term in Eqs. (10b) and (10c) are unity, they can be evaluated by the sinc function:

A_{0}' = \sin c (\frac{\sqrt{2} π L}{a}),

A_{1}' = \sin c {π (\frac{\sqrt{2} L}{a} - 1)},

A_{- 1}' = \sin c {π (\frac{\sqrt{2} L}{a} + 1)},

The amplitude and intensity dependence of the ± 1st and 0th order beams against position shift L in the linear shift method are shown in Fig. 7. Here, the maximum position shift L is set within

L \leq a / \sqrt{2}

in order to prevent coalescence of neighboring holes. As a result, the 1st order component is different from the −1st order component, in contrast to circular shift method. Thus, the linear shift method is capable of generating an asymmetric beam pattern from a single basic lightwave in principle. With increasing positional shift L, the 0th order component decreases while the 1st order component increases. When L is greater than 0.35a, the intensity of the 1st order component becomes larger than that of the 0th order component. Meanwhile, intensity of −1st order component increases with increasing L less than 0.30a, then it decreases when L is greater than 0.30a. The intensity of the 1st order beam is larger than the intensity of the −1st order beam. The intensity of the 1st order beam becomes twice as large as that of −1st order beam at L~0.12a. The difference between them becomes larger as L becomes larger than 0.12a. Note that in the case of

L = a / \sqrt{2}

, the spatial phase modulation is given in the form of exp{iϕ(x,y)}, which corresponds to the ideal phase distribution. Therefore the 0th and −1st order beams vanish. It should be noted that it is necessary to have an additional mechanism for achieving an asymmetric beam pattern since the 1st order beam of counter propagating lightwaves overlaps with the −1st order beam. In addition, this requires either an intensity or diffraction efficiency difference between counter propagating lightwaves.

Fig. 7 Relationship of the ± 1st and 0th order beam of a single lightwave vs. maximum distance L in the linear shift method. (a) Amplitude. (b) Intensity.

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4. Measured FFP

In order to verify the theoretical results, we measured the FFP from the iPMSELs based on linear shift methods. We used a far-field optical system (Hamamatsu, A3267-12), a camera (Hamamatsu, ORCA), and a beam profiler (Hamamatsu, LEPAS-12) to measure the FFP. We used a pulsed current source (ILX Lightwave, LDP-3830), where the pulse width and cycle are 50 ns and 10 kHz, respectively. The temperature was set to 25 °C using a Peltier controller (Daitron, DPC-100). Figure 8(a) shows the FFP operated at 4 A under pulsed conditions, while Fig. 8(b) shows a line plot of the integrated intensity in the vertical axis of the FFP. It is clear that the FFP shows an asymmetric beam pattern. Note that the central 0th order beam is saturated in order to improve visibility of the ± 1st order beam patterns. Diffraction efficiency of the ± 1st order beam patterns at 4 A are 6.9% in total. This value was obtained by comparing the total output power and that of the ± 1st order beams which was approximately evaluated by subtracting the output power of the 0th order beam from total output power. We measured the total output power and that of the 0th order beam by using the thermal detector (Ophir, 3A-FS) at approximately 3 mm and 50 mm distance, respectively (see [6]). It should be noted that it increases by increasing the maximum shift distance L as shown in Figs. 7. In addition, recently we have experimentally confirmed that the central 0th order beam can be eliminated (This will be given elsewhere). In this experimental result, we have focused on only the ± 1st order beams. Here, the ± 1st order beam at the right-upper side beam pattern consist of the 1st order beam of the L and D lightwaves and the −1st order beam of the U and R lightwaves, and vice versa. The intensity of the right-upper side and left-lower side beam patterns are evaluated by comparing the integrated intensity inside the same areal yellow square regions in Fig. 8. As a result, the intensity of the right-upper side beam pattern is 40% higher than that of the left-lower side. According to Fig. 7, the intensity of the 1st order beam for a single lightwave is approximately 57.5% higher than that of the −1st order beam when L is 0.08a. This differs from the experimental result. The disagreement in both results could be attributed to the effects of linear combination of the four lightwaves. In this experiment, the right-upper side beam pattern consists of the 1st order beam of the D/L lightwaves, and the −1st order beam consists of the U/R lightwaves. On the other side, the left-lower side beam pattern consists of the −1st order beam of the D/L lightwaves and the 1st order beam of the U/R lightwaves. Thereby, one of the reasons might be the intensity difference and contribution among these lightwaves.

Fig. 8 FFP of the iPMSEL based on axial displacement methods. (a) Measured FFP. (b) Line plot of the FFP.

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It should also be taken into account that the vertical inclination of the air-hole structure in the phase-modulating layer causes the vertical diffraction difference between the counter propagating lightwaves. However, this is not discussed in this work because it is outside the scope of this paper. Here, it is important to note that the demonstration of an asymmetric beam pattern is based on a systematic theoretical consideration of the linear shift methods. We can also conclude from the theoretical study that it is suitable to obtain the symmetric beam pattern from the conventional iPMSELs based on the circular shift method since the intensity of 1st and −1st order beams are equal.

5. Conclusion

We clarified the mechanism of wavefront modulated beam generation from iPMSELs based on systematic theoretical derivation, and we experimentally demonstrated their responses. First, we presented the principle of iPMSELs and the beam pattern of an ideal NFP. Second, we presented the two fundamental types of spatial phase modulation methods, which utilize position shifts of holes from the lattice point in a square lattice 2D PC. Here, we categorize the beam patterns according to their order, and the output beam patterns are discussed theoretically. Our study reveals that conventional iPMSELs, based on the circular shift method, output a symmetric beam pattern, while novel iPMSELs, based on the linear shift method, output asymmetric beam patterns in principle. Finally, we demonstrated asymmetric beam patterns in iPMSELs based on linear shift methods. These results support our theoretical results. We believe that the theoretical insight obtained in this work leads to more sophisticated control of output beam patterns.

Appendix A relationship between NFP and FFP via Fourier optics

First, the wavefront along the z axis is defined as F(X,Y,Z) and is approximately related to the NFP N(x,y,0) via Fraunhofer diffraction [8]:

F (X, Y, Z) = \frac{e^{i k Z} e^{i \frac{k}{2 Z} (X^{2} + Y^{2})}}{i λ Z} \cdot \iint N (x, y, 0) \cdot \exp [- i \frac{k}{Z} (x X + y Y)] d x d y,

In Eq. (12), i is of the complex unit, k is the wavenumber defined as 2πn_e/λ, λ is the wavelength, and n_e is refractive index in the space where light propagates. When we consider the intensity of the FFP, the integer term in Eq. (12) is substantial since the term 1/iλZ indicates decay along the z axis. Thus, the intensity of the exponential term outside the integer equals unity. In addition, by considering spherical coordinates, Eq. (12) can be rewritten as

\begin{array}{l} F (k_{x}, k_{y}) = A \iint N (x, y, 0) \cdot \exp {- i (k_{x} x + k_{y} y)} d x d y, \\ k_{x} = k \frac{X}{Z} = k \sin θ_{t i l t} \cos θ_{r o t}, \\ k_{y} = k \frac{Y}{Z} = k \sin θ_{t i l t} \sin θ_{r o t}, \end{array}

where A is the coefficient in Eq. (12), k_x and k_y are the tangential wavenumber components along the x and y axis respectively, and θ_tilt and θ_rot are inclination and azimuth angles in spherical coordinates [6]. According to Eq. (13), one can readily see that the FFP and NFP are related via a Fourier transformation.

Note that in this paper, the Fraunhofer diffraction equations are adopted to approximate Kirchhoff's formula and calculate the far field electric fields in order to generate the 2D pattern. Although other approximations, e.g. Fresnel diffraction in the near field, is also a strong candidate for calculating wave propagation in a three-dimensional (3D) volume. In other words, iPMSEL has the possibility of generating multiple spots in 3D space, namely 3D control of beam, by extending the wavefront calculation method, which enhances their capability in a very wide range of optical systems.

Appendix B derivation of Eq. (6)

Equation (6) is derived as follows. The definition of the Bessel function [15] is expressed as

e^{\frac{z}{2} (t - \frac{1}{t})} = \sum_{n = - \infty}^{\infty} t^{n} J_{n} (z),

where z is a real number not equal to zero or infinity, t is a complex number, and n is an integer (n = 0, ± 1, …). By substituting t as t = e^jθ, we can obtain the following mathematical formula:

e^{i z \sin θ} = \sum_{n = - \infty}^{\infty} e^{i n θ} J_{n} (z) .

Since sinusoidal functions can be easily converted as sinθ = cos(π/2-θ), we can derive the other equations of Eqs. (6).

Appendix C derivation of Eq. (9)

Equation (9) is simply derived by Laurent expansion of Eq. (8) as follows. First, Eq. (8) is substituted with the function f(z) as

f (z) = z^{c} = \sum_{m = - \infty}^{\infty} B_{m} \cdot z^{m},

B_{m} = \frac{1}{2 π i} \oint_{C} \frac{f (t)}{t^{m + 1}} d t,

where z is set as

z = e^{i ϕ}

, c is

\sqrt{2} L / a

in Eq. (9), and C is the unit circle in the complex plane. Since

d z = i e^{i ϕ} d ϕ

in the complex plane, B_m is written as

B_{m} = \frac{1}{2 π} \int_{0}^{2 π} e^{i (c + 1 - m - 1) ϕ} d ϕ .

Calculating the integral in Eq. (18) yields Eq. (9).

Acknowledgments

The authors are grateful to A. Hiruma (President), T. Hara (Director), M. Yamanishi (Research Fellow), K. Takizawa, M. Niigaki, Y. Yamashita, K. Nozaki, T. Takemori, H. Toyoda, T. Hirohata and T. Edamura of HPK for their encouragement throughout this work; A. Higuchi and M. Hitaka for their assistance with epitaxial growth. This work was supported in part by the innovative Photonics Evolution Research Center (iPERC) in Hamamatsu, in scheme of COI STREAM of MEXT, Japan.

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Principle of beam generation in on-chip 2D beam pattern projecting lasers

Abstract

1. Introduction

2. Concept and structure of iPMSELs

3. Theory: the origin of beam patterns in iPMSELs

3.1 Beam pattern of an ideal NFP

3.2 Circular shift method

3.3 Linear shift method

4. Measured FFP

5. Conclusion

Appendix A relationship between NFP and FFP via Fourier optics

Appendix B derivation of Eq. (6)

Appendix C derivation of Eq. (9)

Acknowledgments

References and links

Cited By

Figures (8)

Equations (34)

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