Question

How many lines per centimeter must a grating have if there is to be no second- order spectrum for any visible wavelength \((400-750 \mathrm{nm})\) ?

Short answer

Answer: The grating must have at least 66666 lines per centimeter.

Step-by-step solution

Write down the grating equation

The grating equation is given by: \(d \cdot \sin{\theta} = m \cdot \lambda\) where \(d\) is the distance between the grating lines (in meters), \(\theta\) is the angle of diffraction, \(m\) is the order of the spectrum, and \(\lambda\) is the wavelength of light (in meters).

Set the longest visible wavelength and the condition to avoid the 2nd-order spectrum

The longest visible wavelength \(\lambda\) is 750 nm, or \(750 \times 10^{-9}\) m. To avoid the second-order spectrum, the angle of diffraction \(\theta\) should be equal to or greater than 90° when \(m = 2\): \(d \cdot \sin{90^\circ} \geq 2 \cdot (750 \times 10^{-9}\mathrm{m})\)

Find the distance between the grating lines and the number of lines per centimeter

We can now solve for \(d\): \(d \geq 2 \cdot (750 \times 10^{-9}\mathrm{m})\) The number of lines per centimeter, \(N\), is the reciprocal of the distance between lines, \(d\), converted to centimeters: \(N = \frac{1}{d (in\:cm)}\) Substitute the value we found for \(d\) in the previous step: \(N = \frac{1}{2 \cdot (750 \times 10^{-7}\mathrm{cm})}\)

Calculate the number of lines per centimeter

Now we just need to calculate the value of \(N\): \(N = \frac{1}{1.5 \times 10^{-5}\mathrm{cm}}\) \(N = 66666.67\) Since there can't be a fraction of a line, we round down to the nearest whole number, which gives us 66666 lines per centimeter. So, the grating must have at least 66666 lines per centimeter to avoid a second-order spectrum for any visible wavelength.

Key concepts

Grating Equation

One of the most important tools in understanding how a diffraction grating works is the grating equation. This equation helps determine how light spreads out into different colors as it passes through a grating. Here's the equation:\[ d \cdot \sin{\theta} = m \cdot \lambda \]Let's break it down:

• **\( d \):** This is the distance between each line on the grating. It tells us how tightly packed the lines are.
• **\( \theta \):** This is the angle at which light is diffracted or bent.
• **\( m \):** This represents the order of the spectrum, or how many times the original pattern repeats itself.
• **\( \lambda \):** This stands for the wavelength of the light, which changes for different colors of light.

To use the equation, you plug in the values for the order \( m \) and the wavelength \( \lambda \), then calculate the angle \( \theta \) or the distance \( d \), such as we need when arranging the lines in a diffraction grating based on certain requirements.

Visible Spectrum

Light we see every day consists of various colors, forming what is called the 'Visible Spectrum.' The wavelengths in the visible spectrum range from about 400 to 750 nanometers (nm).

• **400 nm:** This is the blue end of the spectrum, where light has a shorter wavelength and more energy.
• **750 nm:** At this point, we reach the red end of the spectrum, which has a longer wavelength and less energy.

Understanding the visible spectrum is crucial in this problem because we need to adjust the grating so that no second-order spectrum (\( m = 2 \)) falls within this visible range. By calculating for the longest wavelength in the visible range, or 750 nm, we ensure that only the first order is visible and at its fullest extent.

Order of Spectrum

In diffraction grating, the term 'order of spectrum' refers to how many repetitions, or orders, the spectrum appears. It is represented by \( m \) in the grating equation.

• **First Order (\( m = 1 \)):** This is the most prominent and typically the clearest set of colors seen when light passes through the grating.
• **Second Order (\( m = 2 \)):** Double the wavelength causes this set to form. It can cause overlaps with other spectra, which may complicate observations.

In the exercise, avoiding a second-order spectrum (\( m = 2 \)) within the visible range is essential. This means designing the grating such that any light diffracted in a higher order shifts out of the visible range, or the second-order light will overlap with sections we do not want in regular observation.