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Numerical ray tracing method for an eccentric radial gradient-index rod lens Shuma Horiuchi,* Shuhei Yoshida, and Manabu Yamamoto Department of Applied Electronics, Tokyo University of Science, 6-3-1 Niijuku, Katsushika-ku, Tokyo 125-8585, Japan *Corresponding author:
[email protected] Received June 2, 2014; revised August 6, 2014; accepted August 11, 2014; posted August 13, 2014 (Doc. ID 213299); published September 4, 2014 We propose a ray tracing method for a radial gradient-index (GRIN) rod lens with an eccentric refractive index distribution. Radial GRIN rod lenses are typically treated as being rotationally symmetric around the optical axis. However, there are several eccentricities of the refractive index distribution in the transverse section, and an eccentricity point is the position of the highest refractive index with respect to the rod axis. Some manufacturing techniques can introduce these eccentricities in the refractive index distribution, and the effect of eccentricity on the lens performance cannot be neglected in some cases. Ray tracing in an eccentric refractive index distribution is possible by extending the conventional method. This allows analysis of the imaging performance of a radial GRIN rod lens with an eccentric refractive index distribution. Since the proposed method builds on the conventional formalism for a rotationally symmetric refractive index distribution, it is simple and easy to implement. © 2014 Optical Society of America OCIS codes: (080.1753) Computation methods; (080.2740) Geometric optical design; (110.2760) Gradientindex lenses. http://dx.doi.org/10.1364/JOSAA.31.002131
1. INTRODUCTION
2. PRINCIPLE OF RADIAL GRIN LENSES
A radial gradient-index (GRIN) lens bends light through a continuous change in the refractive index within the lens and is used as an imaging lens or for fiber couplings. The SELFOC lens from Nippon sheet glass, manufactured using ion exchange, is a popular radial GRIN lens [1–3]. GRIN lenses can also be prepared using other methods [4–8]. However, some highly productive and relatively low-cost methods may result in GRIN lenses with an eccentric refractive index distribution instead of the ideal rotationally symmetric distribution around the optical axis. We considered manufacturing methods similar to those reported by Oda et al. [9] and developed a ray tracing method for GRIN lenses with an eccentric refractive index distribution to investigate their optical properties. Several methods for ray tracing in GRIN media have been described in the literature [10–16]. These previous works on radial GRIN rod lenses assume a rotationally symmetric system around the optical axis. Although a ray tracing method for the case of a tilted or decentered gradient index was proposed in Moore [17]. In this work we considered a GRIN rod lens with an eccentric refractive index distribution. We have extended the ray tracing method used for a rotationally symmetric GRIN lens to include the case of an eccentric refractive index distribution within the lens. Therefore, the power series expansion formula often used to describe a refractive index distribution can be used without modification, and simulations based on the conventional procedure, which involves numerical solution of differential equations, can be used without much change simply by specifying the eccentricity point. Figure 1 shows the refractive index distribution of the eccentric GRIN rod lens that is assumed in this paper.
In ray tracing, a frequently used ray equation is
1084-7529/14/102131-04$15.00/0
d dr n ∇n; ds ds
(1)
where r is a variable-position vector on the ray, n is the refractive index, and ds is a geometric increment along the ray path. Cartesian coordinates are used in Eq. (1), and the optical axis is chosen along the z direction. Denoting the components of an optical ray vector as ξ, η, and ζ, the relation ξ2 η2 ζ 2 n2
(2)
holds. Since ∂n∕∂z 0 for a radial GRIN lens, n
dz ζ 0 const: ds
(3)
When the GRIN lens is rotationally symmetric, the refractive index n is described by the radial distance r from the optical axis. Thus, the following two equations may be derived from Eq. (1): d2 x nr ∂nr ; 2 dz2 ζ 0 ∂x
d2 y nr ∂nr ; 2 dz2 ζ 0 ∂y
(4)
dy : dz
(5)
where ξ ζ0
dx ; dz
© 2014 Optical Society of America
η ζ0
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Horiuchi et al.
y
z
y x
x
E
n n0
n2 ~r n20 1 − g~r 2 h4 g~r 4 h6 g~r 6 ;
E x
Refractive index distribution
where h4 and h6 are higher-order refractive index coefficients and g is the gradient coefficient (1/length). Although n0 usually denotes the refractive index at the optical axis, here it denotes the refractive index at the eccentric point. We treat the eccentric distribution that satisfies the ratio r~ ∕R s∕l, where s is the distance between points E and Q, and l is the distance between points E and K. Note that this is the feature that makes the model specific and is not applicable to any other distribution. r~ is determined from the distances s and l, using the following equation: s ex − x2 y2 1∕2 r~ R R : l ex − kx 2 k2y
GRIN rod lens
Fig. 1. Refractive index distribution of an eccentric GRIN rod lens. The eccentricity point along the x axis, which is the position of the highest refractive index n0 , is E.
A. Determination of the Refractive Index and Slopes We now describe a method for determining the refractive index and its slopes at a coordinate point along a ray in the GRIN lens. Figure 2 shows the variation of refractive index distribution in an eccentric GRIN lens. The eccentricity point along the x axis is Eex ; 0, where 0 ≤ ex < R and Qx; y is a coordinate along the ray. The intersection of the line EQ and the outer rim of the GRIN lens is denoted by point Kkx ; ky . The refractive index ring at point Q is a true circle, and its ~ ex ; 0. Point E~ is dynamically determined using center is E~ point Q and the condition 0 ≤ e~ x ≤ ex . Denoting the distance between points E~ and Q by r~ , the rotationally symmetric refractive index distribution can be replaced by a function of r~ . The refractive index distribution can be described by the power series expansion [18], y K
Q mQE mQE~ R
φ
s ex − x2 y2 1∕2 ; ex 1 − e~ x ex 1 − l ex − kx 2 k2y
s ~ r
~
(8)
where kx and ky are the coordinates of the point of intersection between the line EQ and the outer circle of the GRIN lens and are given by
kx
8 2 2 2 2 2 1∕2 < ex τ −τ R −e2x R 1τ
x − ex < 0
:
ex τ2 τ2 R2 −e2x R2 1∕2 1τ2
x − ex > 0
;
ky τkx − ex : (9)
Here, τ
y : x − ex
(10)
When x ex , kx ex ;
ky
−R2 − e2x 1∕2
y < 0
R2 − e2x 1∕2
y > 0
:
(11)
The right-hand side of Eq. (4) equals 0 when x ex and y 0. Next, we provide an equation to determine the slope of the refractive index at point Q along the ray to obtain the partial derivatives in Eq. (4). The slope of the refractive index is denoted as m. The slope along the QE direction, mQE , is given by
l
θ E
(7)
e~ x is also given by
By numerically solving the differential equations in Eq. (4), the ray path in a radial GRIN lens can be determined.
3. ECCENTRIC RADIAL GRIN ROD LENSES
(6)
E
x
E′
Fig. 2. Schematic of an eccentric radial GRIN rod lens. The contours indicate rings with the same refractive index. Q is a coordinate of the ray, E is the eccentricity point, E~ is the central point of the circle at the point Q, E0 is an arbitrary eccentricity point, and K is the intersection point of the line EQ and the edge of the GRIN lens. s is the distance between points E and Q, and l is the distance between points E and K. mQE and mQE~ are the slopes of the refractive index along the QE and QE~ directions, respectively. R is the radius of the GRIN rod lens.
mQE
∂n~r ∂~r ∂n~r R ∂n~r : ∂s ∂s ∂~r l ∂~r
(12)
The slope of the refractive index at point Q lies along the QE~ direction. Denoting the angle between lines QE and QE~ as ϕ, we have cos ϕ
ex − x~ex − x y2 : r~ s
(13)
~ m ~ , is given by Therefore, the slope along direction QE, QE mQE~
mQE : cos ϕ
(14)
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Substituting Eqs. (7), (8), and (13) into Eq. (14) produces Eq. (15): mQE~
r~ 2 ∂n~r ex − x~ex − x y2 ∂~r h i −x2 y2 ∂n~r R2 eex−k 2 2 ∂~r x x ky : h i1∕2 2 2 −x y 2 − x y ex − x ex 1 − eex−k 2 2 k x
x
(15)
Decomposing the slope into x-axis and y-axis components and denoting these as mQE−x and mQE−y ~ ~ , respectively, we have
d2 x n~r 2 mQE−x ~ dz2 ζ0 n20 r~ x − e~ x 2 ζ 0 ex − x~ex − x y2 × −g2 r~ 2h4 g4 r~ 3 3h6 g6 r~ 5 ; dz2
n~r mQE−y ~ ζ 20
n20 r~ y ζ 20 ex − x~ex − x y2
y
u ux
uy
uz
0 T :
(21)
u0 M −1 u:
(22)
The rotation matrix M is multiplied after emission from the GRIN lens to convert from local coordinates to global coordinates [19]. In conclusion, use of the rotation matrix allows the use of the formalism of Eq. (17) for ray tracing, even when the eccentricity point in the GRIN lens is arbitrary.
4. EXAMPLE OF A RAY TRACING SIMULATION FOR AN ECCENTRIC GRIN ROD LENS Applicability of the ray tracing method detailed in the previous sections was demonstrated by investigating the properties of a GRIN rod lens with an eccentric refractive index distribution, using ray tracing simulation and analysis. Table 1 lists the parameters of the GRIN rod lens used in this simulation. Figure 3 shows spot diagrams for the varying eccentricity of the refractive index distribution in the GRIN lens described by the parameter values in Table 1. It is evident from Fig. 3 that the effect of off-axis aberrations increased with increasing eccentricity. Figure 4 shows the ray trajectories for eccentric GRIN rod lenses. Focusing on the lens area where the light
e0y < 0 e0y ≥ 0
:
(18)
The x coordinate of the eccentricity point E on the x axis after rotation, ex , is 02 1∕2 : ex e02 x ey
The rotation matrix around the z axis is given by
Lens length Working distance g n0 R
(17)
B. Expression for an Arbitrary Eccentricity Point Using the rotation matrix, we can write equations similar to those described in Subsection 3.A for an arbitrary position of the eccentricity point. Denoting the arbitrary eccentricity point in the GRIN lens as E0 e0x ; e0y , the angle of rotation θ is given by
x
0 1
Table 1. Parameters of the GRIN Lens
× −g2 r~ 2h4 g4 r~ 3 3h6 g6 r~ 5 :
8 h i e0x > < −cos−1 e02 e 02 1∕2 h x 0y i θ > : cos−1 02 ex02 1∕2 e e
0
(20)
(19)
4.30 2.80 0.849 1.617 0.130
[mm] [mm] [1/mm] [mm]
10 µm
5 µm
5 µm
d2 y
0
C 0 0C C: C 1 0A
After rotation the vector q is transformed to q0 and the direction cosine vector u is transformed to u0 :
Image height along the x-axis
z 1 T ;
q0 M −1 q;
The differential equation is applicable when the refractive index is eccentric can be obtained by substituting Eq. (16) into Eq. (4). If the refractive index is given by Eq. (6), Eq. (4) can be written as
1
Denoting the coordinate vector of the ray as q and the direction cosine vector as u,
mQE−x ~
(16)
0 0
cos θ
0
y
x − e~ x mQE~ r~ ∂n~r r~ x − e~ x ; ex − x~ex − x y2 ∂~r y mQE~ r~ r~ y ∂n~r : ex − x~ex − x y2 ∂~r
− sin θ
B B sin θ M B B @ 0
q x y
mQE−y ~
cos θ
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0 µm 0 µm
5 µm
10 µm
15 µm
ex Fig. 3. Spot diagrams for eccentric GRIN rod lenses. The distance ex is positioned as far as 15 μm along the x axis, and the image height increases up to 10 μm along the x axis in the simulations.
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J. Opt. Soc. Am. A / Vol. 31, No. 10 / October 2014 Focal plane 2.8 mm
0.26 mm
4.3 mm
GRIN lens
Exit pupil
2.8 mm
x
(a)
(b)
(c)
refractive index distributions and providing the eccentricity point for the eccentric GRIN rod lens. The proposed simulation method will be useful in the evaluation of GRIN lens design margins and the analysis of optical properties when the refractive index is eccentric.
REFERENCES Entrance pupil
z
Horiuchi et al.
(d)
Fig. 4. Ray trajectories in eccentric GRIN rod lenses. (a) No eccentricity, (b), (c), and (d) the distance ex is located at 5, 10, and 15 μm, respectively. The aspect ratio of the z axis to the x axis is 1∶10.
Fig. 5. Rendering images produced when a lattice pattern was imaged. (a) No eccentricity, (b), (c), and (d) the distance ex is located at 5, 10, and 15 μm, respectively. The number of pixels is 256 × 256.
is concentrated, we found that there was torsion when the GRIN rod lens was eccentric, and this resulted in degradation in the imaging performance. Figure 5 shows the results of the simulations when a lattice pattern was imaged using eccentric GRIN rod lenses. Rendering [20] is used in these simulations. The degradation in imaging performance due to the eccentricity of the refractive index distribution is also evident in Fig. 5.
5. CONCLUSION We have proposed a ray tracing method that can be used when the refractive index distribution is eccentric. This method allows the simulation of eccentric GRIN rod lenses by using the conventional ray tracing method for rotationally symmetric
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