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Mobile App Chapter 13: Problem 26
A symmetric dielectric slab waveguide has a slab thickness \(d=10 \mu \mathrm{m}\), with \(n_{1}=1.48\) and \(n_{2}=1.45\). If the operating wavelength is \(\lambda=1.3 \mu \mathrm{m}\), what modes will propagate?
Short Answer
Expert verified
Answer: The propagating modes in this waveguide are the TE0 and TM0 modes.
Step by step solution
01
Calculate the V number of the waveguide
The V number is given by the following formula:
\[V = \frac{2\pi d}{\lambda}\sqrt{n_1^2 - n_2^2}\]
where d is the slab thickness, λ is the operating wavelength, and n1 and n2 are the refractive indices of the core and cladding, respectively.
Now, let's plug in the values provided:
\[V = \frac{2 \pi (10 \times 10^{-6} \mathrm{m})}{(1.3 \times 10^{-6} \mathrm{m})}\sqrt{(1.48)^2 - (1.45)^2}.\]
02
Compute the V number
Now let's evaluate the expression for the V number:
\[V \approx \frac{2 \pi (10 \times 10^{-6} \mathrm{m})}{(1.3 \times 10^{-6} \mathrm{m})}\sqrt{(1.48)^2 - (1.45)^2} \approx 3.59064.\]
03
Determine the maximum number of modes
The maximum number of modes that can propagate in a slab waveguide can be determined using the V number. In this case, since we have a symmetric waveguide, the maximum number of even and odd modes can be calculated using the following formulas:
For even modes (m_e): \(m_e \leq \frac{V}{\pi}\)
For odd modes (m_o): \(m_o \leq \frac{V}{\pi}\)
Let's calculate the maximum number of even and odd modes:
\(m_e \leq \frac{3.59064}{\pi} \approx 1.14279.\)
\(m_o \leq \frac{3.59064}{\pi} \approx 1.14279.\)
04
Determine the propagating modes
Based on the results from step 3, we can now determine which modes will propagate through the waveguide:
Since m_e ≈ 1.14279 and m_o ≈ 1.14279, there is only one integer value less than these numbers (for both even and odd), which is 0. This means that there is only one mode for both even and odd, and it is the fundamental mode (TE0 or TM0).
The propagating modes in this waveguide are the TE0 and TM0 modes.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
V Number Calculation
In the study of optical waveguides, the V number (also known as the normalized frequency parameter) is critical for understanding the modes that a waveguide can support. The V number is calculated using the formula:
\[V = \frac{2\pi d}{\lambda}\sqrt{n_1^2 - n_2^2}\]
where:\
\[V = \frac{2\pi d}{\lambda}\sqrt{n_1^2 - n_2^2}\]
where:\
- \(d\) is the waveguide's thickness.
- \(\lambda\) is the wavelength of the light used.
- \(n_1\) and \(n_2\) are the refractive indices of the core and cladding, respectively.
Refractive Index
The refractive index is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the material:
\[n = \frac{c}{v}\]
where \(n\) is the refractive index, \(c\) is the speed of light in vacuum, and \(v\) is the speed of light in the material. In our exercise, the waveguide has core and cladding refractive indices \(n_1\) and \(n_2\) respectively. The difference in these refractive indices is what confines the light within the core due to total internal reflection, allowing the waveguide to function effectively.
\[n = \frac{c}{v}\]
where \(n\) is the refractive index, \(c\) is the speed of light in vacuum, and \(v\) is the speed of light in the material. In our exercise, the waveguide has core and cladding refractive indices \(n_1\) and \(n_2\) respectively. The difference in these refractive indices is what confines the light within the core due to total internal reflection, allowing the waveguide to function effectively.
Propagation of Electromagnetic Waves in Waveguides
Waveguides are structures that direct the propagation of electromagnetic waves from one point to another. The dielectric slab waveguide confines the light in one dimension and allows it to propagate in another. The light is confined by the higher refractive index of the core \(n_1\) compared to the cladding \(n_2\), leading to a phenomenon known as total internal reflection.
When light hits the boundary between two materials with different refractive indices at a steep angle, it's reflected back into the original material without any loss. This phenomenon keeps the light trapped within the core, allowing it to travel along the waveguide. The modes of propagation refer to the patterns of the electromagnetic field that can exist within the waveguide, which are reliant on the waveguide's geometry and the operating wavelength.
When light hits the boundary between two materials with different refractive indices at a steep angle, it's reflected back into the original material without any loss. This phenomenon keeps the light trapped within the core, allowing it to travel along the waveguide. The modes of propagation refer to the patterns of the electromagnetic field that can exist within the waveguide, which are reliant on the waveguide's geometry and the operating wavelength.
TE and TM Modes
TE (Transverse Electric) and TM (Transverse Magnetic) modes are types of electromagnetic field distributions, or modes, within a waveguide. TE modes have electric fields that are entirely transverse to the direction of propagation, meaning they have no electric field component in the direction of propagation. Meanwhile, TM modes have magnetic fields that are entirely transverse to the direction of propagation, without a magnetic field component in the direction of propagation.
Each mode represents a unique pattern of electric and magnetic field distribution within the waveguide. The TE0 or TM0 modes, which are also known as fundamental modes, have the simplest field distribution and are typically the first to propagate because they require the lowest cut-off frequency. Higher-order modes, TE1, TM1, etc., have more complex field distributions and can only propagate if the V number is high enough to surpass specific cut-off values.
Each mode represents a unique pattern of electric and magnetic field distribution within the waveguide. The TE0 or TM0 modes, which are also known as fundamental modes, have the simplest field distribution and are typically the first to propagate because they require the lowest cut-off frequency. Higher-order modes, TE1, TM1, etc., have more complex field distributions and can only propagate if the V number is high enough to surpass specific cut-off values.
