• Understanding the Michelson Interferometer

    Tony Francis • 03/09/2025 at 18:16 • 0 comments

    A Michelson interferometer splits a beam of light into two paths using a beamsplitter.  These paths are then reflected back and recombined, creating an interference pattern.  The intensity of this pattern depends on the path difference between the two beams

    Superposition of Two Sine Waves: Resultant Intensity at the Michelson Interferometer Detector

    Example: Two Sine Waves

    Let's consider two sine waves representing the two beams:

    • Wave 1: y1 = A * sin(kx)
    • Wave 2: y2 = A * sin(kx + φ)

    Where:

    • A is the amplitude.
    • k is the wave number.
    • x is the position.
    • φ is the phase difference, which is related to the path difference (Δ).

    The path difference (Δ) and phase difference (φ) are related by: φ = (2π/λ) * Δ, where λ is the wavelength.

    Specific Cases:

    1. Path Difference Δ = 0 (φ = 0): Constructive Interference
      • y2 = A * sin(kx)
      • y_total = y1 + y2 = 2A * sin(kx)
      • The waves are in phase, and the resulting wave has double the amplitude (constructive interference). The intensity is maximum.
    2. Path Difference Δ = λ/2 (φ = π): Destructive Interference
      • y2 = A * sin(kx + π) = -A * sin(kx)
      • y_total = y1 + y2 = A * sin(kx) - A * sin(kx) = 0
      • The waves are completely out of phase, and they cancel each other out (destructive interference). The intensity is zero.
    3. Path Difference Δ = λ (φ = 2π): Constructive Interference
      • y2 = A * sin(kx + 2π) = A * sin(kx)
      • y_total = y1 + y2 = 2A * sin(kx)
      • The waves are in phase, and the resulting wave has double the amplitude (constructive interference). The intensity is maximum.

    The detector output, representing the interference of the two waves, is visualized as an intensity variation. When you digitize the interference pattern of a monochromatic laser in a Michelson interferometer with a mirror moving at a constant velocity, here's what you can infer:

    Key Relationship:

    • The crucial relationship is between the frequency of the detected sinusoidal waveform, the velocity of the moving mirror, and the wavelength of the incident light. It's expressed by the equation:
      • f = 2v / λ
      • Where:
        • f is the frequency of the detector's output signal.
        • v is the velocity of the moving mirror.
        • λ is the wavelength of the incident light.

    Detector Output:

    • A sinusoidal waveform: The detector will capture a signal that oscillates like a sine wave. This is because the movement of the mirror at a constant velocity causes a linear change in the path difference between the two beams of light. This linear change in path difference results in a periodic variation of the interference pattern's intensity.
    • Frequency proportional to velocity: The frequency of this sinusoidal waveform is directly proportional to the velocity of the moving mirror. A faster mirror movement will result in a higher frequency sine wave, and a slower movement will result in a lower frequency.

    Inference of the Sine Wave (recorded at the detector):

    • Measurement of small displacements: The ability to detect changes in the interference pattern allows for very precise measurements of small displacements. This principle is used in various applications, including laser Doppler velocimetry and precision metrology.
    • Characterization of light source: In some cases, analyzing the sine wave can also provide information about the characteristics of the laser light source, such as its stability and coherence.
    • Determination of Wavelength: Conversely, if the velocity of the mirror is accurately known, the wavelength of the incident light can be determined. By rearranging the formula (λ = 2v/f), and measuring the frequency of the sinewave, the wavelength can be calculated. 

    So far, we've focused on one frequency of light. In the next section, we'll see how the interferometer reacts to light with a mix of frequencies

  • Unravelling Light: The Michelson Interferometer and Fourier Transforms

    Tony Francis • 03/07/2025 at 18:05 • 0 comments

    The Michelson Interferometer, a clever optical device, allows us to split a beam of light and then recombine it, creating interference patterns that reveal a wealth of information about the light itself. It's the heart of many scientific instruments, and it's especially crucial in Fourier Transform Spectroscopy (FTS).

    How a Michelson Interferometer Works

    Imagine a beam of light hitting a "beam splitter"—a special mirror that transmits half the light and reflects the other half. This creates two separate beams.

    • Beam 1: Travels to a fixed mirror, reflects back to the beam splitter.
    • Beam 2: Travels to a movable mirror, reflects back to the beam splitter.

    The two beams then recombine at the beam splitter and are directed towards a detector. The path length difference between the two beams determines whether they interfere constructively (bright) or destructively (dark).

    The Interferogram: A Dance of Light and Dark

    • Monochromatic Light (Single Color):
      • If you use a laser, which emits a single wavelength (monochromatic light), you'll see a simple sinusoidal (wave-like) pattern as you move the movable mirror. This pattern, called an "interferogram," represents the intensity of the combined light as a function of the path length difference.
      • When the path lengths are identical, the waves align perfectly (constructive interference), creating a bright spot. As you move the mirror, the path difference changes, and the waves start to cancel each other out (destructive interference), creating a dark spot. This oscillation between bright and dark creates the sine wave like interferogram.
    Interferogram from Michelson Interferometer: Red Laser
    • Broadband Light (Multiple Colors):
      • Now, let's use white light, which contains a spectrum of many wavelengths (broadband light). Each wavelength will produce its own sinusoidal interferogram.
      • When the movable mirror is at zero path difference (both paths are equal), all wavelengths interfere constructively, resulting in a strong central peak in the interferogram.
      • As you move the mirror, the different wavelengths start to interfere differently. Shorter wavelengths oscillate faster than longer wavelengths. This causes the interferogram to become a complex pattern, where the central peak rapidly decreases in amplitude as the mirror moves.
      • The interferogram for broadband light looks like a central peak that rapidly decreases in intensity with distance from the zero path difference.
      • The broadband Inteferogram image from Wikipedia 

    From Interferogram to Spectrum: Fourier Transform Magic

    The key to Fourier Transform Spectroscopy lies in the fact that the interferogram contains all the information about the wavelengths present in the light. To extract this information, we use a mathematical tool called the Fourier Transform.

    • The Fourier Transform takes the interferogram (intensity versus path length difference) and converts it into a spectrum (intensity versus wavelength or frequency).
    • Essentially, it breaks down the complex interferogram into its individual sinusoidal components, revealing the intensity of each wavelength present in the light.
    • This is the same math that is used to decompose a sound wave into its constituent frequencies.
    • Because the Fourier Transform is a mathematical process, it can be done very quickly with computers. This is one of the reasons that Fourier Transform Spectrometers are so powerful.

    Why Use a Fourier Transform Interferometer?

    • High Throughput: FTS instruments collect all wavelengths simultaneously, making them much faster than traditional spectrometers that scan through wavelengths one by one.
    • High Accuracy: The precise measurement of path length difference in the interferometer allows for very accurate wavelength determination.
    • High Signal-to-Noise Ratio: FTS instruments can achieve high signal-to-noise ratios, allowing for the detection of weak signals.

    Next Steps

    The Michelson Interferometer, when combined with the power of the Fourier Transform, provides a powerful tool for analyzing...

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