X和Y两方向独立求解
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(a) (1)扩散角：
(2)焦深：
(3)曲率半径：
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(b)当W0x=2W0y时: (1)z=0
(2)z=M>>z0
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3.2-1 Beam Focusing. An argon-ion laser produces a Gaussian beam of wavelength =488 nm and waist radius W0=0.5 mm. Design a single-lens optical system for focusing the light to a spot of diameter 100 um. What is the shortest focal-length lens that may be used?
光电子学作业 参考答案
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Fundamentals of Photonics （括号内为中文教材上题目 的位置）：
• 1.4-1 • 1.4-4 • 3.1-2 (P41-2) • 3.1-3 (P41-3) • 3.2-1 (P41-4)
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1.4-1 Ray-Transfer Matrix of a Lens System.
借助公式：
可求解，计算较麻烦
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• 由方程组可得
• 由(2)得:• 解得： • 所以或者（舍去）作业1
3.1-3 The Elliptic Gaussian Beam. The paraxial Helmholtz equation admits a Gaussian beam with intensity in the z=0 plane, with beam waist radii W0x and W0y in the x and y-directions respectively. The contours of constant intensity are therefore ellipses instead of circles. Write expressions for the beam depth of focus, angular divergence, and radii of curvature in the x and y directions, as functions of W0x W0y, and the wavelength . If W0x=2W0y sketch the shape of the beam spot in the z=0 plane and in the far field (z much greater than the depths of focus in both transverse directions).
• Determine the ray-transfer matrix for an optical system made of a thin convex lens of focal length f and a thin concave lens of focal length -f separated by a distance f. Discuss the imaging properties of this composite lens.
可列式求解：
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(a)过长度为d的自由空间，光线角度不变，位置改变
(近轴光线
)
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(b)薄透镜厚度d 0, y-z面上为焦距为f的薄透镜
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3.1-2 Beam Identification by Two Widths. A Gaussian beam of wavelength =10.6 pm (emitted by a CO2 laser) has widths W1=1.699mm and W2=3.38mm at two points separated by a distance d=10 cm. Determine the location of the waist and the waist radius.
f
-f
f
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注意矩阵相乘的顺序
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1.4-4 4X4 Ray-Transfer Matrix for Skewed Rays. Matrix methods may be generalized to describe skewed paraxial rays in circularly symmetric systems, and to astigmatic (noncircularly symmetric) systems. A ray crossing the plane z=0 is generally characterized by four variables-the coordinates (x, y) of its position in the plane, and the angles ( ) that its projections in the x-z and y-z planes make with the z axis. The emerging ray is also characterized by four variables linearly related to the initial four variables. The optical system may then be characterized completely, within the paraxial approximation, by a 4X4 matrix. (a)Determine the 4X4 ray-transfer matrix of a distance d in free space. (b)Determine the 4X4 ray-transfer matrix of a thin cylindrical lens with focal length f oriented in the y direction (Fig. P1.4-4). The cylindrical lens has focal length f for rays in the y-z plane, and no focusing power for rays in the x-z plane.