When light incident at 45° is reflected off of a dichroic beamsplitter with a slight bend, then third- and higher-order aberrations (such as astigmatism) can degrade the quality of a focused spot size (for a collimated beam) after a focusing lens or of an image after an imaging lens. These effects can be accurately modeled using ray optics with standard optical modeling software. As an example, the graph shows the spot size obtained at an image plane that results from a perfect point source object after reflecting off of a dichroic beamsplitter with various radii of curvature.

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Allen wrenches come in a variety of sizes. Each size is designed to fit a specific size of hexagonal screw or bolt. They are measured in millimeters or inches, with the size of the wrench often printed or stamped on the tool itself.

Allen wrenches, also known as hex keys, Allen keys or hex wrenches, are small, hand tools that are designed to fit into hexagonal screws and bolts. Special security hex keys have a recess that allows it to be fitted to the pin in a security screw. Constructed out of steel and other resilient materials, these wrenches often have a layer of chrome or a similar protective coating to guard against oxidation and deterioration.

Another benefit of using Allen keys is their precision. Unlike other tools, which can easily slip or strip screws and bolts, Allen wrenches provide a secure grip that allows for precise and accurate torque. This is especially important in industries where precise measurements are necessary to ensure the proper functioning of equipment and machinery.

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Allen wrenches are also known for their durability. Because they are made of high-quality steel or other durable materials, they can withstand a lot of wear and tear without breaking or losing their shape. This makes them a reliable tool that can be used repeatedly without needing frequent replacement.

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When light is reflected off of a spherically curved surface with radius R, the light behaves as if it were transmitted through a lens of focal length R/2. If the light is subsequently focused by a lens of focal length f, the curvature causes a focal shift (Δf) compared to reflection off of a flat surface.  Based on a first-order optics approximation, the relative focal shift is Δf/f ~ 2f/R. In order to maintain a relative focal shift Δf/f below a certain required value, the radius of curvature of the bent filter surface needs to exceed

where D is the diameter of the beam at the focusing lens and λ is the wavelength of light. Note that the required radius of curvature is independent of the focal length of the lens. As an example, for a 1 mm diameter laser beam the radius of curvature of the dichroic should be greater than about 3 meters. Semrock's laser-quality dichroic beamsplitters are generally specified with a maximum beam diameter, D, for which the focal shift will be less than one Rayleigh Range at the edge wavelength of the beamsplitter. For instance, the Di02-R488-25x36 is a BrightLine® Laser Dichroic which will cause less than one Rayleigh Range of focal shift for a 488 nm laser beam as large as 2.5 mm in diameter. For a 1 mm diameter beam, the focal shift will be less than 1/6th of the Rayleigh Range.  For advanced microscopy applications our  Super-resolution / TIRF grade of Flatness dichroic beamsplitters provide optimal solutions for much larger beam diameters, by minimizing reflected wavefront distortions.

Suppose a laser beam is reflected off of a dichroic beamsplitter before being focused to a spot at the sample plane by a 40X, 0.75 NA microscope objective, which has a focal length of f = 5 mm, and suppose we want to keep the focal shift below 5 µm, or the relative focal shift below 0.1%. Then the radius of curvature of the dichroic must be greater than about 10 meters.

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Optical filters are generally comprised of multi-layered thin-film coatings on plane, parallel glass substrates. All Semrock filters use a single substrate with coatings on one or both sides to maximize transmission and reliability.  The glass substrate is not always perfectly flat before coating and the intrinsic stress of hard coatings can cause slight bending of the substrate.  Fortunately, this bending has no noticeable effect on light transmitted through an optical filter at or near normal incidence.   For light incident at high angles of incidence, like for a 45° dichroic beamsplitter, the bent substrate creates a slight divergence of the transmitted beam axis similar to the effect of a small wedge in the substrate. Unless the radius of curvature is very small (much less than one meter) the quality of the transmitted beam remains largely unaffected.

Allen wrenches are a common hand tool used in various industries including automotive, manufacturing, construction, and home improvement. These small but mighty tools are essential for tightening and loosening hexagonal screws and bolts.

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Allen wrenches are an essential tool in many industries because they provide a quick and easy way to tighten and loosen hexagonal screws and bolts. These screws and bolts are commonly used in machinery and equipment, making Allen keys a crucial part of any maintenance or repair job.

One of the main benefits of using Allen wrenches is their ability to fit into tight spaces. Because of their small size and L-shaped design, they can easily reach into narrow spaces and corners where other tools cannot. This makes them especially useful in automotive and machinery maintenance, where screws and bolts are often located in hard-to-reach areas.

Reflected image quality can be worse than the ideal diffraction-limited response for dichroics that are not perfectly flat, though it should be noted that the true spot size at the image plane can be appreciably larger than the diffraction-limited spot size in an actual system. Nevertheless, care should be taken to select properly optimized, flatter dichroic beamsplitters when working with reflected light. Dichroics designed to reflect laser light ("Laser" or "Super-resolution / TIRF" [7] Flatness dichroics) ensure negligible focal shift for laser beams up to several mm in diameter even for most demanding applications such as Super-resolution and TIRF microscopy utilizing large diameter beams. Dichroics designed to reflect imaging beams ("Image-splitting" grade of flatness dichroics) have the most extreme flatness requirements, since they must effectively eliminate the effects of astigmatism for beams as large as 1 cm or more. For complete listing of Image-splitting dichroics review our product page.

As an image-splitting example, let's assume a 40X, 0.75 NA microscope objective, for which the beam diameter is about 7.5 mm. Thus the radius of the dichroic must exceed about 160 meters for yellow-green light, in order to observe no noticeable focal shift. For such advanced microscopy applications our Image-splitting grade of Flatness dichroic beamsplitters provide optimal solutions for much larger beam diameters for image-splitting applications.

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In most cases, a small shift of the focal plane is not problematic because a lens or camera adjustment can be made to compensate for the shift, but there are certain applications in which a focal shift cannot be tolerated. For example, when a laser beam is reflected off of a dichroic beamsplitter in a Total Internal Reflection Fluorescence (TIRF) microscope arranged in the epifluorescence configuration, the laser beam must be focused at the back focal plane of the objective and too much focal shift caused by a bent dichroic can make it impossible to achieve this condition.

Choosing the right Allen wrench for the job is essential to ensure that the job is done correctly and safely. When selecting an Allen wrench, consider the following factors:

This plot is based on a typical epifluorescence microscope configuration, assuming a perfect point source at the sample location, imaged onto the image plane (e.g., CCD surface) by an ideal 40X, 0.75 NA objective and a tube lens with a 180 mm typical focal length (industry standard tube length focal lengths range between 160 and 200 mm). The resulting beam diameter is 6.75 mm. The reflection off of the dichroic is assumed to occur mid-way between the objective and the tube lens. The field of view of the system is assumed to be limited by a 20 mm diameter field size at the camera plane. The light is assumed to have a wavelength of 510 nm (peak of GFP emission). For comparison, the diffraction-limited spot size that would result from a perfect objective and tube lens and a perfectly flat dichroic is 16.6 μm.

There are several different types of Allen wrenches, each with its unique features and benefits. The most common types include:

The most common shape of an Allen key is the "L" shape, but some styles may feature a different shape, such as a straight or T-shaped design. The "L" shape is the most popular because it provides the user with the leverage needed to apply the necessary force to turn the screw or bolt.

When working with gaussian beams (often an excellent model for a laser beam), the depth of focus is quantified by the "Rayleigh Range," which is the distance from the beam waist (smallest focused spot) to the point where the waist has increased by a factor of √2. Practically, the Rayleigh Range defines the depth of focus because a beam (or image point) focused to a location within one Rayleigh Range of the waist appears to be in focus. If we use the criterion that the focal shift should be less than one Rayleigh Range, then the radius of curvature of a reflecting filter in front of a focusing lens should exceed

Reflected light, however, is significantly impacted by a bent filter substrate. This situation typically arises when an excitation beam is reflected off of a dichroic beamsplitter before reaching a sample, or when an imaging beam is split into two colors using a dichroic beamsplitter. Two main effects may occur: (1) the position of the focal plane shifts and (2) the size (or shape) of the focused spot changes or the quality of the image is compromised.