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Another specification can be “Plan Fluor” for fluorite and “APO” for apochromatic. Next we have the magnification, numerical aperture, and the immersion medium. As mentioned before, dry objective lenses usually have a NA no larger than 0.95, but that number can be considerably higher in immerse objectives. We next have an infinity symbol, meaning that the lens is infinity corrected.

There are three design variables that can help us calculate the microscope objective resolution: the system wavelength, the light cone captured by the objective (also known as numerical aperture), and the refractive index between the first lens of the objective and the sample. This can be expressed by the following formula:

Most off the shelf microscope objectives have several body markings to better identify them. Typical markings can be seen in Figure 2.

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Fig. 3. The Gaussian distribution for various . The standard deviation determines the width of the distribution. The shape of the Gaussian is shown in Fig. 3 which illustrates this distribution for various . The significance of as a measure of the distribution width is clearly seen. As can be calculated from (19), the standard deviation corresponds to the half width of the peak at about 60% of the full height. In some applications, however, the full width at half maximum (FWHM) is often used instead. This is somewhat larger than and can easily be shown to be (20) This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

In the previous calculation, I assumed an angle of acceptance of 72-degrees with a reasonable upper limit when working with air (that angle gives us a NA of 0.95). However, by immersing the sample and microscope in oil or another liquid, it is possible to have a larger NA. This affects not only the resolution of our image but also its brightness (the brightness is calculated as the square of its NA).

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2.3 The Gaussian or Normal Distribution The Gaussian or normal distribution plays a central role in all of statistics and is the most ubiquitous distribution in all the sciences. Measurement errors, and in particular, instrumental errors are generally described by this probability distribution. Moreover, even in cases where its application is not strictly correct, the Gaussian often provides a good approximation to the true governing distribution. The Gaussian is a continuous, symmetric distribution whose density is given by (19) The two parameters µ and 2 can be shown to correspond to the mean and variance of the distribution by applying (8) and (9). Fig. 3. The Gaussian distribution for various . The standard deviation determines the width of the distribution. The shape of the Gaussian is shown in Fig. 3 which illustrates this distribution for various . The significance of as a measure of the distribution width is clearly seen. As can be calculated from (19), the standard deviation corresponds to the half width of the peak at about 60% of the full height. In some applications, however, the full width at half maximum (FWHM) is often used instead. This is somewhat larger than and can easily be shown to be (20) This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

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The Gaussian is a continuous, symmetric distribution whose density is given by (19) The two parameters µ and 2 can be shown to correspond to the mean and variance of the distribution by applying (8) and (9). Fig. 3. The Gaussian distribution for various . The standard deviation determines the width of the distribution. The shape of the Gaussian is shown in Fig. 3 which illustrates this distribution for various . The significance of as a measure of the distribution width is clearly seen. As can be calculated from (19), the standard deviation corresponds to the half width of the peak at about 60% of the full height. In some applications, however, the full width at half maximum (FWHM) is often used instead. This is somewhat larger than and can easily be shown to be (20) This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

In conclusion, microscope objective lenses are an essential part of a microscope and are used to magnify the specimen being observed. They consist of several components that work together to produce a clear image, and their magnification can vary depending on the intended use of the microscope.

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(21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

Where R is the resolution, ? is the light wavelength, n is the refractive index, and θ is the half angle of the acceptance light cone (NA is the numerical angle defined as sin(θ)). For example, a microscope objective that works with visible light, with air surrounding the sample, and an acceptance half-angle cone of 72-degrees, will have a minimum resolution of 256 nm. If we surround the sample in a liquid with a refractive index of 1.5, our resolution will improve to 171 nm.

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(20) This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

In previous entries, we have talked about the design of scanning microscopes, infinity corrected microscopes, confocal microscope design, and Koehler illumination systems-a common illumination system in microscopes. The most essential microscope element in a borescope design is the objective lens.

Objective lenses can have just a couple of lens elements, (an achromat and simple lens, for example) or multiple groups of elements. Even two microscope objectives with the same magnification can have a completely different design, as shown in Figure 1.

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An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

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The magnification of the objective lens can vary, depending on the intended use of the microscope. For example, objective lenses for biological applications typically range from 4x to 100x, while those used for metallurgical applications can range up to 200x or more [1].

The shape of the Gaussian is shown in Fig. 3 which illustrates this distribution for various . The significance of as a measure of the distribution width is clearly seen. As can be calculated from (19), the standard deviation corresponds to the half width of the peak at about 60% of the full height. In some applications, however, the full width at half maximum (FWHM) is often used instead. This is somewhat larger than and can easily be shown to be (20) This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

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where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

The simplest designs are usually called ‘achromat objectives’ and contain only a front lens and a couple of achromatic doublets to correct for aberrations. On the other hand, we have Apochromat microscope objectives in which several apochromatic doublets are used, in addition to some achromats for a better image quality. For a better explanation of the difference between achromatic and apochromatic lenses, please read the linked articles.

The Gaussian or normal distribution plays a central role in all of statistics and is the most ubiquitous distribution in all the sciences. Measurement errors, and in particular, instrumental errors are generally described by this probability distribution. Moreover, even in cases where its application is not strictly correct, the Gaussian often provides a good approximation to the true governing distribution. The Gaussian is a continuous, symmetric distribution whose density is given by (19) The two parameters µ and 2 can be shown to correspond to the mean and variance of the distribution by applying (8) and (9). Fig. 3. The Gaussian distribution for various . The standard deviation determines the width of the distribution. The shape of the Gaussian is shown in Fig. 3 which illustrates this distribution for various . The significance of as a measure of the distribution width is clearly seen. As can be calculated from (19), the standard deviation corresponds to the half width of the peak at about 60% of the full height. In some applications, however, the full width at half maximum (FWHM) is often used instead. This is somewhat larger than and can easily be shown to be (20) This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

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This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

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(19) The two parameters µ and 2 can be shown to correspond to the mean and variance of the distribution by applying (8) and (9). Fig. 3. The Gaussian distribution for various . The standard deviation determines the width of the distribution. The shape of the Gaussian is shown in Fig. 3 which illustrates this distribution for various . The significance of as a measure of the distribution width is clearly seen. As can be calculated from (19), the standard deviation corresponds to the half width of the peak at about 60% of the full height. In some applications, however, the full width at half maximum (FWHM) is often used instead. This is somewhat larger than and can easily be shown to be (20) This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

Objective lenses for microscopes typically have several components, including the front lens, the rear lens, the aperture, the lens barrel, and the thread. Each component plays an important role in determining the objective’s performance. For example, the aperture determines the resolution and depth of field of the objective lens, while the thread allows the objective to be attached to the microscope.

Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

Microscope objective lenses are a crucial part of a microscope, responsible for magnifying the specimen being observed. They are used to gather light from the object being observed and focus the light rays to produce a real image. The objective lens is one of the most important parts of a microscope, as it determines the microscope’s basic performance and function [3].

In cases where the objective is not meant to be used in infinity corrected microscopes, there will be a number, usually 160) referring to the length of the microscope tube. Some microscope objectives will show the letters “DIN” which stands for “Deutsche Industrial Normen.” that sets a length of 160 mm.

Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

The two parameters µ and 2 can be shown to correspond to the mean and variance of the distribution by applying (8) and (9). Fig. 3. The Gaussian distribution for various . The standard deviation determines the width of the distribution. The shape of the Gaussian is shown in Fig. 3 which illustrates this distribution for various . The significance of as a measure of the distribution width is clearly seen. As can be calculated from (19), the standard deviation corresponds to the half width of the peak at about 60% of the full height. In some applications, however, the full width at half maximum (FWHM) is often used instead. This is somewhat larger than and can easily be shown to be (20) This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM). Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and 2 = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation (21) where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian. Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.

The microscope objective will show the manufacturer (not shown in the figure), followed by the type of aberration correction; in our image, we have a “Plan Achromat” which produces a flat surface at the image plane and achromat for the type of chromatic aberration.

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With over 15 years of experience and 500+ unique optical systems designed, Optics for Hire specializes in advanced optical engineering. If it uses light, we've worked on it.