Plane Waves and Math Review - Homework 01

This is the first homework assignment for Lasers and Optomechanics at Syracuse University.

It is due 5pm on Friday, January 23, 2026

You will need to complete the questions in this jupyter notebook and submit it via your git repo

%matplotlib widget
from ipywidgets import *
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt

plt.style.use('dark_background')

fontsize = 14
mpl.rcParams.update(
    {
        "text.usetex": True,
        "figure.figsize": (9, 6),
        "figure.autolayout": True,
        "font.family": "serif",
        "font.serif": "georgia",
        # 'mathtext.fontset': 'cm',
        "lines.linewidth": 1.5,
        "font.size": fontsize,
        "xtick.labelsize": fontsize,
        "ytick.labelsize": fontsize,
        "legend.fancybox": True,
        "legend.fontsize": fontsize,
        "legend.framealpha": 0.7,
        "legend.handletextpad": 0.5,
        "legend.labelspacing": 0.2,
        "legend.loc": "best",
        "axes.edgecolor": "#b0b0b0",
        "grid.color": "#707070",  # grid color"
        "xtick.color": "#b0b0b0",
        "ytick.color": "#b0b0b0",
        "savefig.dpi": 80,
        "pdf.compression": 9,
    }
)

Approximations Review

In this course, you will need to remember and use some basic approximations.
These approximations all come from taking the Taylor Expansion of a function $f(x)$ about some point $x = a$ :

f(x)\Bigr|_{x \rightarrow a} \approx f(a) + f'(a) (x-a) + \dfrac{1}{2!} f''(a) (x-a)^2 = \displaystyle \sum_{n=0}^\infty \dfrac{f^{(n)}(a)}{n!}(x - a)^n

(1)

1Question 1: Binomial Approximation¶

The binomial approximation to first order in $x$ is as follows:

\begin{align} (1 + x)^{n} \approx 1 + n x. \end{align}

(2)

1.1Question 1A¶

Derive the binomial approximation using the Taylor Expansion to first order about $x = 0$

1.2Question 1B¶

Find the second and third order terms of the binomial approximation

1.3Question 1C¶

Plot the binomial function on $x \in [-1, 1]$ for $n = \dfrac{1}{2}$ .
Compare to plots of the first, second, and third order binomial approximation.
At what $x > 0$ does each approximation fail, becoming greater than 5% error?

1.4Question 1A Solution: (This example filled out for you)¶

Let

f(x) = (1 + x)^n,

(3)

then at $x = 0$ ,

f(0) = 1.

(4)

Then the first derivative $f'(x)$ is

f'(x) = n (1 + x)^{n-1}

(5)

and the derivative evaluated at $x = 0$ is

f'(0) = n

(6)

The Taylor Expansion to first order then becomes

\begin{align} f(x)\Bigr|_{x \rightarrow 0} &\approx f(0) + f'(0) x\\~\\ &= 1 + n x \end{align}

(7)

1.5Question 1B Solution:¶

Taking the second and third derivatives, and evaluating at 0 yields

\begin{align} f''(x) &= n (n-1) (1 + x)^{n-2} \\ f'''(x) &= n (n-1) (n-2) (1 + x)^{n-3} \\~\\ f''(0) &= n (n-1)\\ f''(0) &= n (n-1) (n-2) \end{align}

(8)

The second order expansion is

\begin{align} f(x)\Bigr|_{x \rightarrow 0} &\approx f(0) + f'(0) x + \dfrac{1}{2!} f''(0) x^2\\~\\ &= 1 + n x + \dfrac{1}{2} n (n-1) x^2 \end{align}

(9)

The third order expansion is

\begin{align} f(x)\Bigr|_{x \rightarrow 0} &\approx f(0) + f'(0) x + \dfrac{1}{2!} f''(0) x^2 + \dfrac{1}{3!} f'''(0) x^3\\~\\ &= 1 + n x + \dfrac{1}{2} n (n-1) x^2 + \dfrac{1}{6} n (n-1) (n-2) x^3 \end{align}

(10)

def binom(xx:float, nn:float):
    """Binomial function (1 + xx)^nn
    
    Inputs:
    -------
    xx: float or array of floats
        binomial variable
    nn: float
        binomial exponent

    Output:
    -------
    binom: float or array of floats
        binomial expansion
    """
    return (1 + xx)**nn

# Parameter definitions.  Protip: never make single-letter variable names
nn = 0.5
xx = np.linspace(-1, 2, 100)

taylor0 = 1
taylor1 = taylor0 + nn * xx
taylor2 = taylor1 + 0.5 * nn * (nn - 1) * xx**2
taylor3 = taylor2 + (1/6) * nn * (nn - 1) * (nn - 2) * xx**3

# At which x does the error become greater than 10%?
# First, we divide the approximation by the real function,
# Second, we subtract 1 from that ratio
# Third, we take the absolute value of the subtraction
# Fourth, we look for the first location where the final result is greater than 0.1
# Fifth, we find where x > 0
# Sixth, we take the intersection of the indices found
# Seventh, we find the first index where the error is large for plotting
error = 0.05
model = binom(xx, nn)

xx_errors = np.array([])
for taylor in [taylor1, taylor2, taylor3]:
    abs_errors = np.abs(taylor/model - 1) # final result
    indices_error = np.argwhere(abs_errors > error)
    indices_x = np.argwhere(xx > 0)
    
    indices_final = np.intersect1d(indices_error, indices_x)
    index = indices_final[0]
    
    xx_errors = np.append(xx_errors, xx[index])
print(xx_errors)

[0.87878788 1.3030303  1.42424242]

/var/folders/t1/fq8mx5mj0bx4kn1hlgb4hh840000gn/T/ipykernel_60905/3003632814.py:14: RuntimeWarning: divide by zero encountered in divide
  abs_errors = np.abs(taylor/model - 1) # final result

fig, s1 = plt.subplots(1)

s1.plot(xx, binom(xx, nn), label="Binomial Function")
s1.plot(xx, taylor1, ls="--", label="Taylor 1")
s1.plot(xx, taylor2, ls="--", label="Taylor 2")
s1.plot(xx, taylor3, ls="--", label="Taylor 3")

s1.axvline(x=0, label=f"$x = 0$")

for ii, xx_error in enumerate(xx_errors):
    s1.axvline(x=xx_error, color=f"C{ii+1}",ls=":", label=f"Taylor {ii+1} Error")

s1.set_title("Binomial Approximations about $x = 0$ for $n = " + f"{nn}" + "$")
s1.set_xlabel("$x$")
s1.set_ylabel("$f(x)$")
s1.legend()
s1.grid()
plt.show()

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2Question 2: Sine and Cosine Approximations¶

2.1Question 2A: Sine¶

Repeat the Taylor Expansion approximations for sine to second order about $x = 0$ .
Make the plots, but you don’t need to calculate the 5% error point.

2.2Question 2B: Cosine¶

Repeat the Taylor Expansion approximations for cosine to second order about $x = 0$ .
Make the plots, but you don’t need to calculate the 5% error point.

3Question 3: Complex Number Review¶

3.1Question 3A:¶

Plot the following complex function on a domain of $\phi \in [0, 2 \pi]$ :

\begin{align} z_1(\phi) &= 2 + e^{i \phi}\\ z_2(\phi) &= \dfrac{3}{2 - e^{i \phi}}\\ z_3(\phi) &= e^{(\sigma + i \omega) t} \end{align}

(11)

where for $z_4$ , $\sigma = -0.5$ , and $\omega = 1$ .

3.2Question 3B:¶

Calculate the magnitude $r(\phi)$ and argument $\theta(\phi)$ for each $z_i$ .

3.3Question 3C:¶

Calculate the velocity of the phasors with respect to $\phi$ , and draw them for each $z_i$ evaluated at $\phi = \left\{0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2} \right\}$

3.4Question 3D:¶

What is the primary difference between $z_1$ and $z_2$ ?

3.5Question 3E:¶

For $z_3$ , substitute $t$ for $\phi$ , and calculate the normalized time derivatives : $\dfrac{\dot{z_3}}{z_3}$ , $\dfrac{\ddot{z_3}}{z_3}$

and find expressions for the normalized real polar coordinates $\dfrac{\dot{r}}{r}, \dfrac{\ddot{r}}{r}, \dot{\theta}, \ddot{\theta}$ .

Discuss how the expressions you found for the polar coordinates relate to the path you plotted for $z_3$ in part A.

What happens if $\sigma = +0.5$ ?

4Question 4: Electric field propogating in 2D¶

In class, we assumed that an plane wave was propogating in the $\hat{k} = \hat{z}$ direction, with the electric field oscillating in the $\hat{x}$ direction.
Suppose now that the is oscillating in the $\dfrac{1}{\sqrt{2}} (\hat{x} + \hat{y})$ direction:

\begin{align} \boldsymbol{E} = E_0 \cos(\vec{k} \cdot \vec{r} - \omega t) \dfrac{1}{\sqrt{2}} (\hat{x} + \hat{y}) \end{align}

(12)

4.1Question 4A:¶

What direction of propogation $\hat{k}$ and magnetic field vector $\boldsymbol{B}$ are now possible?
Draw a diagram of the electric field vector and the plane of propogation.

4.2Question 4B:¶

What are the expressions for $\hat{k}$ and $\boldsymbol{B}$ if we constrain the direction of propogating to be (partially) in the positive $\hat{x}$ direction?

5Question 5: Spherical Plane Wave Intensity and Radiation Pressure¶

Suppose you have a sinusoidal spherical plane wave source a distance $d$ away along the $\hat{z}$ axis from a cylindrical mirror with radius $a$ .
Use the center of the spherical wave as the origin, and the distance from that center as the variable $r$ .
Assume that the cylinder is in the $xy$ plane.
Also assume that the spherical wave is emitting total power $P_\mathrm{total}$ in all directions.

5.1Question 5A:¶

What is the Poynting vector $\boldsymbol{S}$ for the spherical waves?
Hint: Equation 9.49 of Griffith’s E&M may be helpful here

5.2Question 5B:¶

What is the Poynting vector $\boldsymbol{S}$ incident on the mirror center?
What about the mirror edge?
Write an expression for the Poynting vector incident anywhere on the mirror’s surface.

5.3Question 5C:¶

Using your result from Question 5B, find the intensity $I$ incident on the mirror.

5.4Question 5D:¶

Find the total power $P$ incident on the mirror.
Compare to the total power emitted by the spherical plane wave.

5.5Question 5E:¶

Calculate the radiation pressure $p_\mathrm{rad}$ incident on the mirror.
Also find the radiation pressure force $F_\mathrm{rad}$ .
Assume the mirror is a perfect reflector.
If the mirror has a mass $m$ , what is its acceleration?