Question

Assume a semiconductor laser has a length of $800\mu \mathrm{m}$. Laser emission can occur when the cavity length is equal to an integer number of half wavelengths. What wavelengths in the range $650\mathrm{nm}<\lambda <$ $652\mathrm{nm}$ can this laser emit, and in each case, list the cavity length in wavelengths.

Short answer

The laser does not emit at any wavelengths between 650 nm and 652 nm.

Step-by-step solution

Understand the Problem

The laser emits light at specific wavelengths, determined by the length of the cavity being an integer multiple of half-wavelengths. We need to find which wavelengths between 650 nm and 652 nm can fit this condition in an 800 μm long cavity.

Convert Cavity Length to Consistent Units

Convert the laser cavity length from micrometers to nanometers for consistency with the wavelengths given:$$800\text{ μm}=800\times {10}^3\text{ nm}=800000\text{ nm}$$

Determine the Condition for Emission

The condition for emission is that the cavity length must be an integer number of half-wavelengths. Therefore, if $m$ is the integer, then:$$800000=m\cdot \frac{\lambda }{2}$$Rearranging gives:$$m=\frac{2\cdot 800000}{\lambda }$$

Calculate Wavelengths and Integer Values

Substitute the wavelengths between 650 nm and 652 nm to find for which $\lambda$, $m$ is an integer:For $\lambda =650\text{ nm}$:$$m=\frac{2\cdot 800000}{650}=2461.54$$Not an integer, so 650 nm is not a valid wavelength.For $\lambda =650.5\text{ nm}$:$$m=\frac{2\cdot 800000}{650.5}\approx 2460.99$$Not an integer.For $\lambda =651\text{ nm}$:$$m=\frac{2\cdot 800000}{651}\approx 2460.06$$Not an integer.For $\lambda =651.5\text{ nm}$:$$m=\frac{2\cdot 800000}{651.5}=2459.27$$Not an integer.For $\lambda =652\text{ nm}$:$$m=\frac{2\cdot 800000}{652}\approx 2458.47$$Not an integer.

Conclude with Valid Wavelengths

Since none of the tested wavelengths between 650 nm and 652 nm result in an integer number of half-wavelengths, this particular semiconductor laser does not emit at any of these wavelengths under the given conditions.

Key concepts

Wavelength Calculation for Semiconductor Lasers

To determine the range of wavelengths that a semiconductor laser can emit, you need to understand how to calculate wavelengths based on certain conditions. The wavelength in a laser system is often controlled by the cavity length, which is the distance between the two mirrors at either end of the laser. Here, the problem involves a laser with a cavity length of 800 μm.
The wavelengths emitted are those that can form standing waves in the cavity, specifically, the cavity length must be an integer multiple of half the wavelength. This condition ensures that peaks and troughs of the light waves reinforce each other to form a coherent beam. When calculating possible emission wavelengths:

• Convert all measurements to the same unit to avoid errors. For this problem, measurements are converted into nanometers because the possible wavelengths are given in this unit.
• Use the relationship between the cavity length and wavelength to find allowable wavelengths: $$m=\frac{2\cdot L}{\lambda }$$
• Make sure to check only within the specified ranges. In this problem, 650 nm to 652 nm is the range we're investigating.

Understanding the Half-Wavelength Condition

The half-wavelength condition is crucial for laser emission, particularly in semiconductor lasers. It essentially dictates the formation of standing waves within the laser cavity. More formally, the cavity length ($L$) must be an integer multiple of half the wavelength ($\lambda /2$).
Here's how you interpret it:

• The integer ($m$) represents the number of half-wavelengths fitting into the cavity. This integer must be counted in whole numbers, as partial standing waves will not sustain laser action.
• The formula $$800000=m\cdot \frac{\lambda }{2}$$ indicates the relationship, which can be rearranged to solve for $m$: $$m=\frac{2\cdot 800000}{\lambda }$$
• A workable wavelength is one for which $m$ calculated is a whole integer. Non-integer values mean the wavelength cannot be emitted by the laser within the given parameters.

How Cavity Length Influences Laser Operation

The cavity length in a semiconductor laser is vital because it defines the physical boundaries where light waves must fit as standing waves. A key factor is that only certain wavelengths will satisfy the standing wave condition. These are the wavelengths that ensure internal consistency of the wave pattern:

• The length of the laser cavity ($L$) acts as a restriction, allowing only certain harmonics or resonant frequencies to exist within its space.
• If the cavity is too short or too long for a particular wavelength, coherent light emission breaks down since the wave cannot seamlessly reflect back and forth within the cavity.
• This exercise explores the impact of an 800 μm cavity on specific wavelengths and shows how crucial precise measurement and calculation are in determining laser emission viability.

Overall, understanding how cavity length impacts laser emissions helps in designing lasers for specific applications, ensuring they can produce the desired coherent light effectively.