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Q1. A 1 mW HeNe laser has a wavelength, \lambdaλ = 632.8 nm, and a beam waist, w = 0.5 mm. What is the far field divergence angle, \thetaθ, of this laser? Express your answer in radians.

Answer: θ = λ / (π * w) Where: λ = 632.8 nm = 632.8 x 10^-9 m (wavelength) w = 0.5 mm = 0.5 x 10^-3 m (beam waist) θ = (632.8 x 10^-9 m) / (π * 0.5 x 10^-3 m) = 0.402 rad

Q2. For the previous situation, what is the Rayleigh range of the laser, z0? Express your answer in meters.

Answer: z0 = (π * w^2) / λ Where: λ = 632.8 nm = 632.8 x 10^-9 m (wavelength) w = 0.5 mm = 0.5 x 10^-3 m (beam waist) z0 = (π * (0.5 x 10^-3 m)^2) / (632.8 x 10^-9 m) ≈ 0.198 m

Q3. For the previous question, what is the beam radius at 2z0? Express your answer in millimeters.

Answer: 2z0 = 2 * 0.198 m = 0.396 m

Q4. For the previous question, what is the radius of curvature of the wave front at z0? Express your answer in meters.

Answer: Radius of curvature = z0 / 2 = 0.198 m / 2 = 0.099 m

Q5. For the previous situation, what is the peak intensity of the laser at the waist? Carefully integrate the intensity profile of the Gaussian beam to relate the peak intensity to the power. Express your answer in W/m^2.

Answer: I(r) = P / (π * w^2) * e^(-2 * r^2 / w^2) Where: P = 1 mW = 1 x 10^-3 W (power) w = 0.5 x 10^-3 m (beam waist) r is the radial distance from the beam axis. To find the peak intensity, you integrate this expression over the entire beam profile. The peak intensity is the maximum value of this profile. I_peak = P / (π * w^2) I_peak = (1 x 10^-3 W) / (π * (0.5 x 10^-3 m)^2) I_peak ≈ 127323 W/m^2 So, the peak intensity of the laser at the waist is approximately 127,323 W/m^2.

Q1. A Gaussian beam from a frequency-doubled Nd:YAG laser, with wavelength 532 nm, is measured to have a beam radius of 1.6 mm and 3.0 mm at two points to the right of the laser separated by a distance of 1.0 m.

What is the location of the beam waist relative to the 1.6 mm measurement point? Express your answer as a positive distance in units of meters with two significant figures.

Answer: The location of the beam waist relative to the 1.6 mm measurement point is approximately 2.67 meters.

Q2. For the previous question, what is the size of the beam waist, w_0w

0

? Express your answer in units of micrometers with two significant figures. For your own benefit, calculate the Rayleigh range, z_0z

0

. How does this impact the solution method you can use for this problem?

**Answer: The size of the beam waist (w0) is approximately 1.07 micrometers. The Rayleigh range (z0) would be approximately 0.536 meters. The Rayleigh range impacts the solution method as it’s used to calculate the beam waist size.**

Q3. A Gaussian beam, with wavelength, \lambdaλ, and waist size, w_0w

0

, is incident normally on a solid block with an index of refraction, n, and thickness, L, as shown below.

What is the far-field diffraction angle, theta, of the output beam, that is, after exiting the block ? Express your answer in terms of the given variables (lambda, w_0, L, n).

Preview will appear here…

Answer: The far-field diffraction angle (theta) of the output beam after exiting the block can be expressed as:

Q4. Assume that the block is moved sufficiently far to the left such that the waist is in air on the right. The waist location will shift in z due to this operation. How far does the waist location shift due to the block? Give your answer for the location of the new waist relative to z = 0. Express your answer in terms of the given variables (lambda, w_0, L, n).

Preview will appear here…

Answer: Moving the block to the left will shift the waist location by a distance approximately equal to -L. So, the new waist location relative to z = 0 is -L.

Q5. Set up Gaussian beam propagation using ABCD matrices or the two-ray method. Setting up a simple program in Matlab would be a good way to do this. Sketch out some simple problems like the fiber collimator example in class using this to get some practice. This will be useful for the next few problems and in general when doing design with Gaussian beams.

A Gaussian beam is relayed by a single lens as shown below. The waist and divergence rays have been sketched to the left of the lens, as well as the beam radius w(z). Sketch the beam radius w(z) to the right of the lens.

What is the distance from the lens to the beam waist to the right of the lens, measured in units of one square in the drawing? Express your answer with two significant figures.

Answer: The distance from the lens to the beam waist to the right of the lens, measured in units of one square in the drawing, is 4.67 squares.

Q6. For the previous problem, what is the waist size, w_0w

0

, to the right of the lens, in units of one square in the drawing? Express your answer with two significant figures.

Answer: The waist size (w0) to the right of the lens, in units of one square in the drawing, is 0.88 squares.

Q7. A HeNe laser (\lambdaλ = 632.8 nm) consists of a linear laser cavity that is 0.50 m long, with two mirrors, both with radii of curvature of 5.0 m, and both concave inward (toward the center of the laser cavity).

What is the location of the beam waist in the laser cavity? Express your answer in meters, measured from the mirror on the left, with two significant figures.

Answer: The location of the beam waist in the laser cavity is approximately 0.12 meters, measured from the mirror on the left.

Q8. For the situation described in question 7, what is the beam waist size, w_0w

0

? Express your answer in millimeters (mm) with two significant figures.

Answer: The beam waist size (w0) is approximately 0.12 millimeters (120 micrometers).

Q9. For the situation described in question 7, what is the radius of the beam spot size on the right cavity mirror? Express your answer in millimeters (mm) with two significant figures.

Answer: The radius of the beam spot size on the right cavity mirror is approximately 0.12 millimeters (120 micrometers).

Q10. For the situation described in question 7, what is the beam divergence

angle for this laser? Express your answer in milliradians (mrad) with two significant figures.

Answer: The beam divergence angle for this laser is approximately 1.05 milliradians (mrad).

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