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Multi-distance surface-emitting beam profile calculation method based on the FDTD method and the diffraction theory

Hiroshi Ohno, Rei Hashimoto, Kei Kaneko, Tsutomu Kakuno, and Shinji Saito

Open Access Open Access

Abstract

A hybrid method to calculate a multi-distance beam profile emitted perpendicular from a surface of a photonic crystal (PhC) is proposed here based on the finite-domain time-difference (FDTD) method and the diffraction theory. Although the FDTD method is available to calculate a near-field emitted from the PhC, it needs too many voxels to calculate mid- and far-fields. Thus, the diffraction theory is additionally applied to obtain the mid- and far-fields using the near-field calculated by the FDTD method. A surface-emitting quantum cascade laser (QCL) that consists of a PhC and an edge-emitting laser source is fabricated to demonstrate the validity of the hybrid method. A measured beam profile of the QCL agrees with that calculated using the hybrid method, which validates applicability of the method to a surface-emitting device.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

A photonic crystal (PhC) has a periodic optical structure that can control the motion of light, which, in principle, is applicable wherever the light must be manipulated [1–9]. Indeed, the PhC can provide a powerful solution for surface-emitting beam devices by vertical beam coupling of an incident beam to an emitting beam [10]. Besides the PhC, gradient-index materials are also available to control motion of the light [11–15]. There are several applications that need the vertical beam coupling for light communication devices [16], integrated optical circuits, and surface-emitting lasers [17,18]. The surface-emitting laser that emits light perpendicular from its top surface, unlike conventional edge-emitting lasers, is often favorable for two-dimensional-integration optical devices by virtue of its stable power extraction and insensitivity to cleaving conditions. Furthermore, large-area surface-emitting lasers are advantageous for realizing high output power with a narrow laser beam profile.

One type of surface-emitting laser, the quantum cascade laser (QCL) incorporating a PhC, has undergone rapid development [19–22]. QCLs are semiconductor lasers that emit in the mid- to far-infrared portion of the wavelength spectrum. Unlike typical inter-band semiconductor lasers, the QCL laser emission is achieved through the use of inter-sub-band transitions in a repeated stack of semiconductor multiple quantum well heterostructures. Because the wavelength of the QCL is much longer than that of visible light, fabrication and experimental handling are considered to be relatively easy.

In optical design of a surface-emitting beam device incorporating a PhC, beam profiles for several distances from the top surface of the PhC are one of the most important features. To calculate the beam profiles emitted from the PhC, one candidate is the finite-domain time-difference (FDTD) method [23–27]. Although near-field of the beam can be calculated by the FDTD method, mid- and far-fields need too many voxels in the case of the FDTD method. A hybrid method to calculate the mid- and far-fields for surface-emitting devices is therefore proposed here based on the diffraction theory using a near-field calculated by the FDTD method. The remainder of this paper is organized as follows. In section 2, the basic equations necessary to calculate mid- and far-field are derived based on the diffraction theory using a near-field as input data. Section 3 describes the structure of a QCL that consists of a PhC and an edge-emitting laser source where the PhC can couple an incident beam vertically to an emitting beam from the top surface. In section 4, a near-field of a beam emitted from the above-mentioned QCL is calculated based on the FDTD method. In-plane beam profiles in mid- and far-field are also shown to be calculated by the hybrid method using the near-field calculated by the FDTD method. The surface-emitting QCL is fabricated, and its beam profile is experimentally measured. The measured beam profile of the fabricated QCL is compared with the theoretically calculated results. Section 5 is devoted to discussions. Conclusions are described in section 6.

2. Surface-emitting beam profile calculation in mid- and far-field

Light can be described as an electromagnetic wave based on Maxwell’s equations. An electromagnetic field ϕ should therefore satisfy the following Helmholtz equation in a free space as

$$\left( {\Delta - \frac{{{\partial^2}}}{{{c^2}\partial {t^2}}}} \right)\phi = 0, $$
where c and t denote the light velocity and time, respectively. When the field is a monochromatic plane wave with a wavelength of λ0 and an angular frequency of ω, a wavenumber vector k of the monochromatic plane wave satisfies the following equations derived from Eq. (1) as
$${{\textbf k}^2} - {k_0}^2 = {k_x}^2 + {k_y}^2 + {k_z}^2 - {k_0}^2 = 0, $$
$${k_0} = \frac{\omega }{c} = \frac{{2\pi }}{{{\lambda _0}}}, $$
where kx, ky, and kz denote x-, y-, and z-components of a wavenumber vector of k in a Cartesian coordinate of (x, y, z).

Because Maxwell’s equations are linear, any superposition of plane waves within homogeneous linear media is also a solution of Maxwell’s equations. The electromagnetic field ϕ can therefore be described in the Fourier transformation form as

$$\phi ({x,y,z} )= \mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5pt {}} {d{k_x}} d{k_y}d{k_z}\delta ({{{\textbf k}^2} - {k_0}^2} )\Phi ({\textbf k} )\exp [{i({{\textbf k} \cdot {\textbf r}} )} ], $$
where δ and r denote the Dirac delta function and a position vector of (x, y, z), respectively. Note that the δ function imposes the condition represented by Eq. (2) derived from the Helmholtz equation. Equation (4) can be further transformed as
$$\phi ({x,y,z} )= \int\!\!\!\int\limits_{} {d{k_x}d{k_y}\Phi ({\textbf k} )\exp [{i({{k_x}x + {k_y}y + {k_z}z} )} ]}, $$
where the kz is a function of kx and ky, and can be written as
$${k_z} = \sqrt {k_0^2 - k_x^2 - k_y^2}. $$

Figure 1 shows a schematic perspective view of near-, mid-, and far-fields emitted from a top surface of a PhC that is layered on a substrate. A beam is incident on an edge of the PhC, and is vertically coupled to an emitting beam that propagates along z-axis. A plane of a near-field is set to z = 0 plane in this work. The Fourier transformed field Φ in the integrand of Eq. (5) can be obtained with the in-plane near-field ϕ (x, y, 0) using the inverse Fourier transformation as

$$\Phi ({\textbf k} )= \Phi ({{k_x},{k_y}} )= \frac{1}{{{{({2\pi } )}^2}}}\int\!\!\!\int\limits_{} {dxdy\phi ({x,y,0} )\exp [{ - i({{k_x}x + {k_y}y} )} ]}. $$

 figure: Fig. 1.

Fig. 1. Schematic perspective view of near-, mid-, and far-fields emitted from a top surface of a PhC that is layered on a substrate. A beam is incident on an edge of the PhC, and is vertically coupled to an emitting beam that propagates along z -axis.

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The function Φ is sometimes called the angular power spectrum [28]. Inserting Eq. (7) into Eq. (5), the field ϕ can be derived as

$$\phi ({x,y,z} )= \int\!\!\!\int\limits_{} {\frac{{d{k_x}d{k_y}}}{{{{({2\pi } )}^2}}}\int\!\!\!\int\limits_{} {dx^{\prime}dy^{\prime}\phi ({x^{\prime},y^{\prime},0} )\exp [{ - i({{k_x}x^{\prime} + {k_y}y^{\prime}} )} ]} \exp [{i({{k_x}x + {k_y}y + {k_z}z} )} ]}. $$

Equation (8) is strictly derived and represents the surface-emitting beam in multi-distance. The mid- and far-fields can therefore be treated in a unified manner. Note that the kz in Eq. (8) is a function of the kx and ky as represented by Eq. (6). In-plane mid- and far-fields can therefore be calculated using Eq. (8) under the condition represented by Eq. (6) when the in-plane near field ϕ (x, y, 0) is given. The near-field is calculable using the FDTD method.

As the value of z (i.e., distance from the near-field) increases, Eq. (8) becomes difficult to compute, because the rapid oscillations of the exponential phase factor require dense sampling of the functions. In the far-field, however, the stationary-phase approximation becomes applicable [29]. Based on the stationary-phase approximation, the wavenumber components of kx and ky in the integrand of Eq. (8) can be respectively written with small values of Δkx and Δky around respective stationary values as

$$k_x^{} = {k_0}\frac{x}{r} + \Delta k_x^{}, $$
$$k_y^{} = {k_0}\frac{y}{r} + \Delta k_y^{}, $$
where the r denotes a radius with respect to the coordinate origin as
$$r = \sqrt {{x^2} + {y^2} + {z^2}}. $$

Using Eqs. (9)–(11), the z-component of the wavenumber kz represented by Eq. (6) can be approximated with the Taylor expansion as

$${k_z} \simeq {k_0}\frac{z}{r} + \frac{{\partial k_z^{}}}{{\partial k_x^{}}}\Delta k_x^{} + \frac{{\partial k_z^{}}}{{\partial k_y^{}}}\Delta k_y^{} + \frac{{{\partial ^2}k_z^{}}}{{2{\partial ^2}k_x^{}}}\Delta k_x^2 + \frac{{{\partial ^2}k_z^{}}}{{2{\partial ^2}k_y^{}}}\Delta k_y^2 + \frac{{{\partial ^2}k_z^{}}}{{\partial k_x^{}\partial k_y^{}}}\Delta k_x^{}\Delta k_y^{}. $$

Inserting Eq. (6) into the right-hand side of Eq. (12), the following equation can be derived as

$${k_z} = {k_0}\frac{z}{r} - \frac{x}{z}\Delta k_x^{} - \frac{y}{z}\Delta k_y^{} + \frac{r}{{{k_0}z}}\left( { - \frac{1}{2}\left( {1 + \frac{{{x^2}}}{{{z^2}}}} \right)\Delta k_x^2 - \frac{1}{2}\left( {1 + \frac{{{y^2}}}{{{z^2}}}} \right)\Delta k_y^2 - \frac{{xy}}{{{z^2}}}\Delta k_x^{}\Delta k_y^{}} \right). $$

Using Eqs. (9), (10), and (13), the phase of the exponential phase factor in Eq. (8) can be written as

$${k_x}x + {k_y}y + {k_z}z = {k_0}r\left( {1 - \frac{1}{2}\left( {1 + \frac{{{x^2}}}{{{z^2}}}} \right)\frac{{\Delta k_x^2}}{{k_0^2}} - \frac{1}{2}\left( {1 + \frac{{{y^2}}}{{{z^2}}}} \right)\frac{{\Delta k_y^2}}{{k_0^2}} - \frac{{xy}}{{{z^2}}}\frac{{\Delta k_x^{}\Delta k_y^{}}}{{k_0^2}}} \right). $$

Inserting Eq. (14) into Eq. (8), Eq. (8) can be analytically integrated over the Δkx and Δky in the far-field limit as

$$\phi ({x,y,z} )={-} \frac{i}{r}\exp [{2\pi ir} ]\Phi \left( {{k_0}\frac{x}{r},{k_0}\frac{y}{r}} \right)\cos \theta, $$
where the θ denotes a zenith angle defined as
$$\cos \theta = \frac{z}{r}. $$

Equation (15) represents an in-plane beam profile at the far-field limit whereas Eq. (8) represents the mid- and far-field. These equations need the Fourier transformed near-field represented by Φ that is calculable using the FDTD method.

3. Surface-emitting QCL consisting of PhC and edge-emitting laser source

Figure 2 shows (a) a perspective view of a surface-emitting QCL consisting of a PhC and a ridge waveguide laser source (i.e., edge-emitting laser source), and (b) a cross-sectional view of the PhC in x – z plane. The ridge waveguide laser source that is monolithically connected with the PhC has a multiple quantum well (MQW) and a n-InP substrate, which can emit a laser beam into an edge of the PhC. The MQW is composed of about 600 stacked layers of AlInAs and InGaAs. The PhC is composed of stacked layers including the MQW in which the incident beam propagates, and has a grating of InGaAs with a periodical pitch Λ on its top surface. An electrode contact is attached only to the ridge waveguide laser source. The laser will not be absorbed in the PhC because the absorption wavelength of the PhC that is not electrically pumped is shifted from that of the electrically pumped laser source.

 figure: Fig. 2.

Fig. 2. (a) Perspective view of surface-emitting QCL consisting of a PhC and a ridge waveguide laser source, and (b) cross-sectional view of the PhC in x - z plane. The ridge waveguide laser source has a multiple quantum well (MQW) and n-InP substrate, which can emit a laser beam into an edge of the PhC. The PhC is composed of stacked layers including the MQW where the incident beam propagates, and has a grating with a periodical pitch Λ on the top surface. The width of the ridge waveguide is W0. The length of the PhC in the x-direction is L0. The depth of the QCL without the substrate is D0.

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The width of the ridge waveguide W0 is set to 15 μm. The length of the PhC in the x–direction, L0, is set to 1000 μm. The depth of the QCL without the substrate, D0, is set to 8.2 μm.

The PhC couples the incident beam vertically to a beam emitting from its top surface. In this work, a diffraction caused by the grating is used for the vertical coupling. The grating pitch thus satisfies the following condition as

$$\Lambda = \frac{{{\lambda _0}}}{{{n_\textrm{e}}}}, $$
where ne represents an effective refractive index of the PhC. In this work, ne is calculated to be 3.23 on the basis of the FDTD simulation. A wavelength of the ridge waveguide laser source is set to 4.39 μm. The periodical pitch of the grating therefore can be calculated using Eq. (17) to be 1.36.

4. Results of the surface-emitting beam profile

4.1 Near-field calculation based on the FDTD method

A near-field of the surface-emitting QCL mentioned in the previous chapter can be calculated using the FDTD method [30]. Although the three-dimensional FDTD simulation could be performed in principle, the computational time and the required memory would be inordinately large due to the long grating length L0 of 1000 μm. The two-dimensional FDTD simulation is therefore applied to the x - z cross-sectional plane of the QCL. A one-dimensional near-field along the x-axis can thus be obtained by the two-dimensional FDTD simulation. The parameters used for the FDTD simulation are listed in Table 1.

Tables Icon

Table 1. Parameters used for the FDTD simulation.

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A two-dimensional near-field in the x - y plane (i.e., z = 0 plane) is assumed to be rectangular where the one-dimensional field distribution along the x-axis is obtainable by the FDTD simulation. The two-dimensional near-field can thus be obtained with a length of L0 and a width of W0 corresponding to the width of the incident beam on the edge of the PhC. The divergence of the incident beam from the edge, however, should be considered for more accurate calculation, which is neglected in this work.

Figure 3 shows the in-plane (i.e., x - y plane) near-field calculated using the FDTD simulation. An incident beam is set to a TM (Transverse Magnetic) slab mode of the ridge waveguide. Field ϕ is set to y-component magnetic field. An intensity of the field is normalized by the maximum value and contoured in gray scale. A wavelength is set to 4.39 μm. The grating pitch of the PhC is set to 1.36, which satisfies Eq. (17) with the effective refractive index ne of 3.23.

 figure: Fig. 3.

Fig. 3. In-plane near-field calculated using the FDTD method in the x - y plane. An incident beam is set to TM (Transverse Magnetic) slab mode of ridge waveguide. Field ϕ is set to y-component magnetic field. An intensity of the field is normalized by the maximum value and contoured in gray scale. A wavelength is set to 4.39 μm. The grating pitch of the PhC is set to 1.36.

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4.2 Mid-field calculation based on the diffraction theory

 figure: Fig. 4.

Fig. 4. In-plane mid-fields for several distances of z = 0, 25, 50, 100, and 200 μm. In each plot, field ϕ is set to a magnetic field. A log-scale intensity of each field is normalized by the maximum value of the field at z = 0 and contoured in gray scale. These fields are calculated using Eq. (8) under the condition represented by Eq. (6) with the near-field calculated by the FDTD method

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Mid-fields at several distances can be calculated using Eq. (8) under the condition represented by Eq. (6) with the near-field previously calculated by the FDTD method. Figure 4 shows mid-fields in the x – y planes for several distances of z = 0, 25, 50, 100, and 200 μm. In each plot, field ϕ is set to a magnetic field. A log-scale intensity of each field is normalized by a maximum value of the field at z = 0 and contoured in gray scale.

It can be found from these mid-fields for various distances that the beam rapidly spread in the y-direction along the z-axis whereas it remains relatively collimated in the x-direction along the z-axis.

4.3 Far-field calculation based on the diffraction theory

 figure: Fig. 5.

Fig. 5. (a) In-plane (i.e., θx - θy plane) angle-resolved far-field intensity calculated using Eqs. (18)–(20) is contoured in gray scale on a logarithmic scale. (b) Perspective view of the angle-resolved far-field intensity is plotted with wire-surface. The intensity is normalized by the maximum value.

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In far-distance from the top surface of the PhC, Eq. (8) becomes difficult to compute because the rapid oscillations of the exponential phase factor require dense sampling of the functions. Therefore, the stationary-phase approximation is applied to Eq. (8). A far-field can thus be calculated using Eq. (15) with the near-field calculated previously by the FDTD method.

An angle-resolved intensity distribution, I, of the far-field can be written using Eq. (15) as

$$I({{\theta_x},{\theta_y}} )= {|{\phi ({x,y,z} )} |^2} = \frac{1}{{{r^2}}}{\left|{\Phi \left( {{k_0}\frac{x}{r},{k_0}\frac{y}{r}} \right)} \right|^2}{\cos ^2}\theta, $$
where angles of θx and θy as arguments of the intensity distribution function of I are respectively defined as
$${\theta _x} \equiv \arcsin \left( {\frac{x}{r}} \right), $$
$${\theta _y} \equiv \arcsin \left( {\frac{y}{r}} \right). $$

As shown in the left-hand side of Fig. 5, an in-plane (i.e., θx - θy plane) angle-resolved far-field intensity calculated using Eqs. (18)–(20) is contoured in gray scale on a logarithmic scale. In the right-hand side of Fig. 5, a perspective view of the angle-resolved far-field intensity is plotted with wire-surface. The intensity is normalized by the maximum value.

4.4 Experimental results

The surface-emitting QCL consisting of the PhC and ridge waveguide laser source as mentioned above is fabricated. Figure 6 shows (a) a top view of the fabricated surface-emitting QCL, (b) an intensity spectrum of the laser source measured by FTIR (Fourier Transform Infrared Spectroscopy), and (c) a perspective view of SEM (Scanning Electron Microscope) image of the PhC. The peak wavelength of the laser is close to 4.39 μm. A grating of the PhC consists of multiple triangular pillars on the top surface. The triangular pillar is employed to enhance the beam extraction efficiency due to the asymmetry effect.

 figure: Fig. 6.

Fig. 6. (a) Top view of fabricated surface-emitting QCL consisting of PhC and ridge waveguide laser source, (b) intensity spectrum of the laser source measured by FTIR (Fourier Transform Infrared Spectroscopy), and (c) perspective view of SEM (Scanning Electron Microscope) image of the PhC. A grating of the PhC consists of multiple triangular pillars on the top surface.

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Figure 7 shows measured beam profiles with respect to (a) x-direction angle θx and (b) y-direction angle θy. In each plot, the vertical axis indicates normalized intensity and the horizontal axis indicates angle in degrees. The simulation results previously calculated using Eqs. (18)–(20) are also shown with solid line in each plot. The experimental results are denoted with circle mark. The experiment is performed using a rotating stage with a bolometer (NEC, IRV-T0831). The experimental results agree with the simulation results.

 figure: Fig. 7.

Fig. 7. Measured beam profiles with respect to (a) x-direction angle and (b) y-direction angle. In each plot, the vertical axis indicates normalized intensity and the horizontal axis indicates angle in degrees. The simulation results are also shown with solid line in each plot. The experimental results are denoted with circle mark.

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5. Discussions

Although the experimental results almost agree with the simulation results, the experimental beam profile with respect to the x-direction angle of θx of FWHM (full width at half maximum) of about 2 degrees is larger than that of the simulation of FWHM of about 0.4 degrees. This might be due to the multi-wavelength generation of the emitting beam around the peak wavelength in the experiment whereas only a single wavelength is considered in the simulation.

In the calculation of the rectangular near-field in the x - y plane, the divergence of the incident beam on the edge of the PhC is assumed to be negligible. In practice, however, the divergence of the incident beam should occur to some extent, which might affect the beam profile.

6. Conclusions

A hybrid method to calculate a multi-distance beam profile emitted perpendicular from a surface of a PhC is proposed here based on the FDTD method and the diffraction theory. Although the FDTD method is available to calculate a near-field emitted from the PhC, it needs too many voxels to calculate mid- and far-fields. Thus, the diffraction theory is additionally applied to obtain the mid- and far-fields using the near-field calculated by the FDTD method. A surface-emitting QCL that consists of a PhC and an edge-emitting laser is designed, and shows that its surface-emitting beam calculated by the hybrid method spreads rapidly in the y-direction whereas it remains relatively collimated in the x-direction. An experimentally measured beam profile of the fabricated QCL in far-field agrees with that calculated using the hybrid method, which validates applicability of the method to a surface-emitting device.

Funding

Innovative Science and Technology Initiative for Security by Acquisition, Technology & Logistics Agency, Japan (JPJ004596).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Schematic perspective view of near-, mid-, and far-fields emitted from a top surface of a PhC that is layered on a substrate. A beam is incident on an edge of the PhC, and is vertically coupled to an emitting beam that propagates along z -axis.

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Fig. 2.
Fig. 2. (a) Perspective view of surface-emitting QCL consisting of a PhC and a ridge waveguide laser source, and (b) cross-sectional view of the PhC in x - z plane. The ridge waveguide laser source has a multiple quantum well (MQW) and n-InP substrate, which can emit a laser beam into an edge of the PhC. The PhC is composed of stacked layers including the MQW where the incident beam propagates, and has a grating with a periodical pitch Λ on the top surface. The width of the ridge waveguide is W0. The length of the PhC in the x-direction is L0. The depth of the QCL without the substrate is D0.

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Fig. 3.
Fig. 3. In-plane near-field calculated using the FDTD method in the x - y plane. An incident beam is set to TM (Transverse Magnetic) slab mode of ridge waveguide. Field ϕ is set to y-component magnetic field. An intensity of the field is normalized by the maximum value and contoured in gray scale. A wavelength is set to 4.39 μm. The grating pitch of the PhC is set to 1.36.

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Fig. 4.
Fig. 4. In-plane mid-fields for several distances of z = 0, 25, 50, 100, and 200 μm. In each plot, field ϕ is set to a magnetic field. A log-scale intensity of each field is normalized by the maximum value of the field at z = 0 and contoured in gray scale. These fields are calculated using Eq. (8) under the condition represented by Eq. (6) with the near-field calculated by the FDTD method

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Fig. 5.
Fig. 5. (a) In-plane (i.e., θx - θy plane) angle-resolved far-field intensity calculated using Eqs. (18)–(20) is contoured in gray scale on a logarithmic scale. (b) Perspective view of the angle-resolved far-field intensity is plotted with wire-surface. The intensity is normalized by the maximum value.

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Fig. 6.
Fig. 6. (a) Top view of fabricated surface-emitting QCL consisting of PhC and ridge waveguide laser source, (b) intensity spectrum of the laser source measured by FTIR (Fourier Transform Infrared Spectroscopy), and (c) perspective view of SEM (Scanning Electron Microscope) image of the PhC. A grating of the PhC consists of multiple triangular pillars on the top surface.

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Fig. 7.
Fig. 7. Measured beam profiles with respect to (a) x-direction angle and (b) y-direction angle. In each plot, the vertical axis indicates normalized intensity and the horizontal axis indicates angle in degrees. The simulation results are also shown with solid line in each plot. The experimental results are denoted with circle mark.

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Tables (1)

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Table 1. Parameters used for the FDTD simulation.

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Equations (20)

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( Δ − ∂ 2 c 2 ∂ t 2 ) ϕ = 0 ,
k 2 − k 0 2 = k x 2 + k y 2 + k z 2 − k 0 2 = 0 ,
k 0 = ω c = 2 π λ 0 ,
ϕ ( x , y , z ) = ∫ ∫ ∫ ⁡ d k x d k y d k z δ ( k 2 − k 0 2 ) Φ ( k ) exp ⁡ [ i ( k ⋅ r ) ] ,
ϕ ( x , y , z ) = ∫ ∫ d k x d k y Φ ( k ) exp ⁡ [ i ( k x x + k y y + k z z ) ] ,
k z = k 0 2 − k x 2 − k y 2 .
Φ ( k ) = Φ ( k x , k y ) = 1 ( 2 π ) 2 ∫ ∫ d x d y ϕ ( x , y , 0 ) exp ⁡ [ − i ( k x x + k y y ) ] .
ϕ ( x , y , z ) = ∫ ∫ d k x d k y ( 2 π ) 2 ∫ ∫ d x ′ d y ′ ϕ ( x ′ , y ′ , 0 ) exp ⁡ [ − i ( k x x ′ + k y y ′ ) ] exp ⁡ [ i ( k x x + k y y + k z z ) ] .
k x = k 0 x r + Δ k x ,
k y = k 0 y r + Δ k y ,
r = x 2 + y 2 + z 2 .
k z ≃ k 0 z r + ∂ k z ∂ k x Δ k x + ∂ k z ∂ k y Δ k y + ∂ 2 k z 2 ∂ 2 k x Δ k x 2 + ∂ 2 k z 2 ∂ 2 k y Δ k y 2 + ∂ 2 k z ∂ k x ∂ k y Δ k x Δ k y .
k z = k 0 z r − x z Δ k x − y z Δ k y + r k 0 z ( − 1 2 ( 1 + x 2 z 2 ) Δ k x 2 − 1 2 ( 1 + y 2 z 2 ) Δ k y 2 − x y z 2 Δ k x Δ k y ) .
k x x + k y y + k z z = k 0 r ( 1 − 1 2 ( 1 + x 2 z 2 ) Δ k x 2 k 0 2 − 1 2 ( 1 + y 2 z 2 ) Δ k y 2 k 0 2 − x y z 2 Δ k x Δ k y k 0 2 ) .
ϕ ( x , y , z ) = − i r exp ⁡ [ 2 π i r ] Φ ( k 0 x r , k 0 y r ) cos ⁡ θ ,
cos ⁡ θ = z r .
Λ = λ 0 n e ,
I ( θ x , θ y ) = | ϕ ( x , y , z ) | 2 = 1 r 2 | Φ ( k 0 x r , k 0 y r ) | 2 cos 2 θ ,
θ x ≡ arcsin ⁡ ( x r ) ,
θ y ≡ arcsin ⁡ ( y r ) .
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