Question

When performing optical spectroscopy (for example, photoluminescence or Raman spectroscopy), a laser beam is focused on the sample to be investigated by means of a lens having a focal distance \(f\). Assume that the laser beam exits a pupil \(D_{o}\) in diameter that is located at a distance \(d_{\mathrm{o}}\) from the focusing lens. For the case when the image of the exit pupil forms on the sample, calculate a) at what distance \(d_{\mathrm{i}}\) from the lens is the sample located and b) the diameter \(D_{i}\) of the laser spot (image of the exit pupil) on the sample. c) What are the numerical results for: \(f=10.0 \mathrm{~cm},\) \(D_{o}=2.00 \mathrm{~mm}, d_{\mathrm{o}}=1.50 \mathrm{~m} ?\)

Short answer

Question: Calculate the distance from the lens to the sample and the diameter of the laser spot on the sample for an optical spectroscopy setup with a focal distance of 10.0 cm, object diameter of 2.00 mm, and object distance of 1.50 m. Answer: The distance from the lens to the sample is 11.1 cm, and the diameter of the laser spot on the sample is 0.148 mm.

Step-by-step solution

Lens Formula

The lens formula is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) We need to find \(d_i\) in terms of \(f\) and \(d_o\).

Rearrange the Lens Formula to Solve for \(d_i\)

By rearranging the lens formula, we can solve for \(d_i\): \( d_i = \frac{fd_o}{d_o - f} \)

Find the Magnification Formula

The magnification formula relates the distances and sizes of the object and the image: \( M = \frac{D_i}{D_o} = \frac{d_i}{d_o} \)

Rearrange Magnification Formula to Solve for \(D_i\)

By rearranging the magnification formula, we can solve for \(D_i\): \( D_i = D_o \cdot \frac{d_i}{d_o} \)

Substituting Values

Now we will substitute the given values \(f=10.0 \mathrm{~cm},\) \(D_{o}=2.00 \mathrm{~mm}, d_{\mathrm{o}}=1.50 \mathrm{~m}\) into the derived formulas to find the values of \(d_i\) and \(D_i\): a) \(d_i = \frac{(10.0 \mathrm{~cm})(1.50 \mathrm{~m})}{1.50 \mathrm{~m} - 10.0 \mathrm{~cm}}\) b) \(D_i = (2.00 \mathrm{~mm}) \cdot \frac{d_i}{1.50 \mathrm{~m}}\)

Calculate Results

Calculate the numerical results for \(d_i\) and \(D_i\): a) \(d_i =\frac{(10.0\,\mathrm{cm})(150.0\,\mathrm{cm})}{150.0\,\mathrm{cm} - 10.0\,\mathrm{cm}} = 11.1\,\mathrm{cm}\) b) \(D_i = (2.00\,\mathrm{mm}) \cdot \frac{11.1\,\mathrm{~cm}}{150.0\,\mathrm{cm}} = 0.148\,\mathrm{mm}\) So, the distance from the lens to the sample (\(d_i\)) is 11.1 cm, and the diameter of the laser spot on the sample (\(D_i\)) is 0.148 mm.

Key concepts

Lens Formula

The lens formula is a fundamental principle used in optics to relate the distances and focal length of a lens. It is expressed as \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \(f\) is the focal distance of the lens, \(d_o\) is the distance from the lens to the object, and \(d_i\) is the distance from the lens to the image. By knowing two of these distances, you can calculate the third. This is especially useful in experiments involving optical spectroscopy, such as focusing a laser beam on a sample.

For example, in the provided exercise, understanding this formula allows the calculation of the distance from the lens to the sample when the image of the laser exit pupil (object) is formed on the sample. It is essential to apply the formula correctly and ensure that all units are consistent to obtain accurate results.

Focal Distance

Focal distance or focal length is the distance from the center of a lens to its focus, where rays of light converge or diverge. It is a critical property that determines how the lens will bend light and the size of the image relative to the object. In the context of laser beam focusing, as utilized in spectroscopic techniques, the focal distance determines how tightly the beam can be focused. Shorter focal lengths result in smaller, more concentrated spots on the sample, potentially increasing the intensity and precision of the measurements.

The example in the exercise given the focal length \(f=10.0 \mathrm{~cm}\), illuminates how a known focal distance allows for the determination of the position and size of a focused laser spot. The shorter focal length would focus the beam to a smaller spot size, enhancing the spectroscopic analysis.

Magnification Formula

In optical systems, the magnification formula \(M = \frac{D_i}{D_o} = \frac{d_i}{d_o}\) is used to determine the size relationship between an image and an object. Here, \(D_i\) is the diameter of the image, \(D_o\) is the diameter of the object, \(d_i\) is the distance from the lens to the image, and \(d_o\) is the distance from the lens to the object. This formula is particularly useful when we need to calculate how large an image will appear compared to the original object size.

In our example, we see that understanding the magnification allows for calculating the diameter of the laser spot on the sample (image of the exit pupil). A high magnification might imply a more intense and narrower laser beam, which is crucial in techniques like Raman spectroscopy, where the detail and precision of the illuminated spot significantly impact the quality of the spectral data obtained.

Laser Beam Focusing

Laser beam focusing is the act of concentrating a laser beam to a small point or area. Achieving a sharp focus is essential for applications requiring precise spatial resolution, such as optical spectroscopy or laser cutting. In typical setups, lenses are used to focus the beam by bending the light paths towards a focal point. The diameter of the laser spot (image of the exit pupil) on the sample is a critical parameter as it affects the intensity and resolution of the measurements.

By applying the lens and magnification formulas, such as those in our exercise, we can accurately predict the size and position of the focused spot. For spectroscopic applications, a tight focus increases the interaction between the light and the material, which may lead to enhanced signal strength in the spectra acquired.

Raman Spectroscopy

Raman spectroscopy is a technique used to observe vibrational, rotational, and other low-frequency modes in a system. It is based on the inelastic scattering of monochromatic light, typically from a laser. When laser light interacts with molecular vibrations, it results in the energy of the laser photons being shifted up or down—this is known as the Raman effect.

The precision of the laser focus has a significant impact on the Raman signal quality. A well-focused laser beam leads to higher intensity and better resolution of the Raman spectra. This makes understanding the concepts of lens formula, focal distance, and magnification very relevant to scientists and engineers working with Raman spectroscopy. In the exercise context, accurately calculating the spot size and distance from the lens to the sample ensures that the Raman spectroscopy experiment is optimally configured for maximum data quality.