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ioLight has invented a portable microscope, with a resolution of better than 1μm, which produces beautiful pictures of animal and plant cells and displays them directly onto your tablet or mobile phone.

Phase contrast microscopes were invented to combat the problem of live cell study with a bright field microscope. Phase contrast microscopy is an optical microscopy technique in which phase shift is converted into change in amplitude/intensity of light. The phase shifts when light travels through dense medium and its velocity decreases, concurrently there is a shift in the phase. When the two waves meet at a certain point it will result in a destructive interference, decreasing amplitude and thereby density. Phase contrast microscopy is useful for looking at specimens that are both colourless and transparent.

The light microscope, or optical microscope, is a microscope that uses visible light and a system of lenses to magnify images. These days there are many complex designs of them which have been developed with the aim of improving resolution and sample contrast.

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This type of microscope was developed in response to drawbacks with fluorescence microscopes (principally that they use high intensity UV light which means the samples are continuously exposed to it, causing photo bleaching and blurring in some samples). Two major modifications were made to address this downside: use of laser light instead of mercury arch lamp and images taken using a digital camera with a pin hole. The pin hole functions to allow light of only one focal plane to be focused on the digital camera. A laser beam focused and scanned over the sample produces 3D and 2D images therewith.

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To first order approximation coherence length is inversly to the bandwidth of the light, but divergence of the bundle plays a role too. Coherence length is important in interference, eg why Newtons rings are only a few around the center of that lens for white sunlight, but hundreds when illuminated with a He/Ne laser.

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A polarising microscope is an optical microscope composed of a detector, lenses and polarising filters. Its process includes illumination of the sample with polarised light and is useful for better visualisation and understanding of birefringent materials (materials that have two different refractive indices). This microscope is operated through the use of a polarized filter can be turned and fixed in the light path beneath the specimen, usually below the stage. This particular device is known for its anti-reflective properties which is deemed essential for deep analysis of an isotropic particles that requires high integrity of light transmission.

Suppose i have two waves emanating from a point source. The waves start out completely in phase. Is the coherence length consistently defined as the length at which these two waves achieve a phase difference of 1 radian? That seems arbitrary to me, you could easily choose a phase difference of $ \frac {\pi}{10}$

Bright field vs dark field vsphase contrast

I'm kind of lost on this whole topic of wave coherence to be honest... I'd appreciate someone giving an overview for the example of two standard plane waves (with full mathematical description).

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Fluorescence microscopy is done with an optical microscope that uses a mercury arch lamp as a source of UV light. The microscope will also comprise excitation filter, dichromatic mirror and an emission filter. Fluorescence, used to observe the specimen, begins where a molecule absorbs light of high frequency and emits light of lower frequency. Fluorescence microscopy uses reflected light. In a fluorescence microscope the light source travels in a different trajectory than in the basic light microscope. An advantage of fluourescence microscopy is that it can be used to detect and visualise multiple fluorescent molecules e.g. cells glowing as they are doing their work. iOLight sell a microscope for mobile digital fluorescence microscopy, which is also great for field microscopy uses.

Now, I'm not sure there is a very quantitative definition of the correlation time. You could define it as the delay where the autocorrelation function drops below some arbitrary threshold. This is equivalent to setting a threshold on the visibility of the interference pattern. The relationship with the shape of the spectral line should also be apparent: the squared modulus of the Fourier transform of $A(t)$ is the shape of the line (the spectrum of the wave shifted by $-\omega$). It is also the Fourier transform of the autocorrelation function of $A(t)$. Thus, when the line is wide, the autocorrelation function is narrow,and the coherence time is short.

Coherence length is the maximal difference in way traveled by the to rays without losing the phase relation which allows interference. This lenghts can be some µmeters (eg for white light from a glowing body) or several meters (eg for very narrow lines from discharge lamps) or even more for mode selected/stabilized lasers.

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DIC creates contrast in a specimen by creating a high-resolution image of a thin optical section. With differential interference contrast microscopy, two closely spaced parallel rays are generated and made to interfere after passing through an unstained sample. The background is made dark and the interference pattern is particularly sharp at boundaries. Specimens will appear really bright in contrast to the dark background.

Fluorescence microscopy is done with an optical microscope that uses a mercury arch lamp as a source of UV light. The microscope will also comprise excitation filter, dichromatic mirror and an emission filter. Fluorescence, used to observe the specimen, begins where a molecule absorbs light of high frequency and emits light of lower frequency. Fluorescence microscopy uses reflected light. In a fluorescence microscope the light source travels in a different trajectory than in the basic light microscope. An advantage of fluourescence microscopy is that it can be used to detect and visualise multiple fluorescent molecules e.g. cells glowing as they are doing their work. iOLight sell a microscope for mobile digital fluorescence microscopy, which is also great for field microscopy uses.

To understand the coherence time, say you have a wave described, in complex notation, by $$ E(t) = A(t) e^{i \omega t} $$ where $A(t)$ is a slowly varying complex amplitude. You make this wave interfere with a delayed version of itself and collect the intensity $$ |E(t) + E(t-\tau)|^2 = |E(t)|^2 + |E(t-\tau)|^2 + 2\Re\big(E(t)E^*(t-\tau)\big). $$ where $\Re$ means real part and $^*$ means complex conjugate. The interference term is $$ 2\Re\big(E(t)E^*(t-\tau)\big) = 2\Re\big(A(t)A^*(t-\tau)e^{i \omega \tau}\big) $$ If $A(t)$ is constant, or roughly constant within a time interval $\tau$, then this becomes $$ 2|A(t)|^2 \cos(\omega \tau) $$ which is the interference pattern. On the other hand, if $A(t)$ fluctuates sufficiently fast, and $\tau$ is larger than its correlation time, then $A(t)A^*(t-\tau)$ averages to zero and there is no interference. Thus, the coherence time can be simply seen as the correlation time of the complex amplitude $A(t)$.

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Dark field vs bright field microscopy: Bright field microscopy uses the most basic and the common type of optical microscope. Bright field microscopes usually have many components and the light sources used are either a halogen lamp or LED. This type of microscope tends to have low contrast owning to the biological samples transmitting most of the light. Staining if often required to combat this problem, which comes with the disadvantage that live imaging is difficult due to staining killing the cells. Dark field microscopy is generally preferred therefore over light field. With a dark field microscope a special aperture is used to focus incident light meaning the background stays dark. The light does not pass directly through the sample being studied. Instead light is reflected off the specimen, making it appear to be emitting light. Brightfield microscopy shows clear magnification while the dark field image shows minute details.

Is the coherence length consistently defined as the length at which these two waves achieve a phase difference of 1 radian?

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