Dynamic regulating of lasing mode in a whispering-gallery ... - lasing mode

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Thus the relationship between gaussian filter FWHM in K-space to the FWHM in Image-space can be determined by taking the Fourier Transform (FT) of the K-space gaussian. But a gaussian with in the numerator is just another gaussian with in the denominator. By equating the exponents and replacing , the Image-space can be determined. From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

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From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

Actual magnifying power will vary slightly, depending upon working distance, eye relief distance and the characteristics of the observer's eye.

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Cementing three lenses together produces a triplet lens. Triplets produce a better quality image, are corrected for three colors, and give little or no image distortion. They are best used for jobs that require a great deal of precision at high magnifying levels.

Multiplication in K-space is equivalent to convolution in Image-space. Thus the relationship between gaussian filter FWHM in K-space to the FWHM in Image-space can be determined by taking the Fourier Transform (FT) of the K-space gaussian. But a gaussian with in the numerator is just another gaussian with in the denominator. By equating the exponents and replacing , the Image-space can be determined. From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

The distance from the magnifier to the object viewed is the working distance. This distance is an important consideration with regard to the type of work that must be done under the magnifier. If your work requires the use of tools, a magnifier with a long working distance will allow enough space to both use the tools and comfortably view the object. Small working distance magnifiers with higher powers are preferred for close-up inspection work.

The distance between the closest and furthest points at which a magnifier in a fixed position stays in focus. The depth of field decreases as power increases.

By equating the exponents and replacing , the Image-space can be determined. From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

The for the HWHM radius, h, is given in Equation 2. Multiplication in K-space is equivalent to convolution in Image-space. Thus the relationship between gaussian filter FWHM in K-space to the FWHM in Image-space can be determined by taking the Fourier Transform (FT) of the K-space gaussian. But a gaussian with in the numerator is just another gaussian with in the denominator. By equating the exponents and replacing , the Image-space can be determined. From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

Due to physical laws, the outer part of the image formed by a simple lens may appear out of focus. This is caused by the curvature of the lens. The greater the magnification - and the greater curvature of the lens - the greater the problem. This can be easily overcome by designing a magnifier that has more than one lens. A triplet has a "flat field" which means the entire area of view is in focus and undistorted.

The field of view is the area seen through the magnifier. As power increases, lens diameter and field of view decrease. At 5 power (5X), field of view is about 1.5". At 10 power (10X), it is about 0.5". Usually, it is best to use low power for scanning larger surfaces and high power for small areas.

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To choose the correct magnifier for the job, first determine what tools are to be used on the job; then determine the size and the character of the subject; and finally, analyze the object's surface character. Then review the following aspects of magnifiers:

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The K-space gaussian filter has a HWHM (Half Width - Half Maximum) equal to the radius specified in Radius field. The FWHM (Full Width - Half Maximum) is simply equal to twice the radius. The values, g(r), of the gaussian filter are given for one dimension in Equation 1 for a radius = h and an image width of N pixels. The for the HWHM radius, h, is given in Equation 2. Multiplication in K-space is equivalent to convolution in Image-space. Thus the relationship between gaussian filter FWHM in K-space to the FWHM in Image-space can be determined by taking the Fourier Transform (FT) of the K-space gaussian. But a gaussian with in the numerator is just another gaussian with in the denominator. By equating the exponents and replacing , the Image-space can be determined. From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

Knowledge Center/ Application Notes/ Microscopy Application Notes/ Magnifying Lenses: How to Choose a Magnifier

Lens surfaces coated with special anti-reflection coatings will reduce light loss and are particularly useful for low-level light applications.

The doublet lens is two simple lenses used in conjunction with each other but not cemented together. The doublet produces an image of better quality because it corrects some of the outer image color distortion.

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An achromat is a positive simple lens cemented to a negative simple lens. The primary advantage is that it is corrected for two colors and works well at high powers. Most high quality magnifiers use achromats to eliminate color fringing at the edge of objects.

A perfect magnifier would be lightweight, have a large diameter, provide a wide viewing area, and offer high, distortion-free magnification. However, incorporating all of these features into one unit is optically impossible. The magnifying power of a lens depends on its focal length (fl). The focal length, in turn, depends on the lens curvature; the greater the curvature, the shorter the focal length and the greater the power. In the design of a simple, inexpensive magnifier, the lens diameter will typically decrease as the curvature increases to provide higher power. Conversely, as the curvature is decreased to lower the power, the diameter generally increases with a resulting increase in viewing area. In addition, distortion generally increases with an increase in curvature. Thus, a magnifier with a large diameter typically offers more viewing area and less power. So, both wide field of view and high magnifying power cannot be incorporated into a single design without elaborate, weighty, high-cost lenses.

The maximum distance the eye can be from the magnifier and still provide a full field of view. Longer eye reliefs generally provide more comfortable viewing.

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Because of physical laws, the lens may produce a prism effect giving the image false color fringes known as chromatic aberration. Simple lenses focus various colors at different points. Achromats with two simple lenses cemented together correct this by causing many colors to focus at the same point.

The simple lens is a single positive lens. Simple lenses are satisfactory for work that requires only low power magnifiers, such as 2X or 3X reading magnifiers. Simple lens magnifiers distort color on the outer fringes of the image and thereby lose sharpness.

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But a gaussian with in the numerator is just another gaussian with in the denominator. By equating the exponents and replacing , the Image-space can be determined. From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

10" is assumed to be the closest distance the human eye can focus for comfortable vision. An object only 1" from your eye would be 10 times larger, but out of focus. A magnifier's function is to assist your eye in focusing closer. Since a 1" focal length lens brings clear vision down to 1" from the eye, an object at this distance is clearly seen and appears to be 10 times closer than it does when viewed from 10" away. Such a magnifier is commonly called a 10X or 10 power. Using this definition, the magnifying power of a lens can be approximated as follows: MP = 10/FL if the focal length is specified in inches. If the focal length is specified in mm, the formula will be MP=250/FL.

How much you spend on a magnifier should be determined by the application for which it is being used. Buying the least expensive magnifier could lead to unsatisfactory and frustrating results. Any magnifier you buy should be a tool to fit the rigors of the environment in which it will be used. Additionally, it may be unwise to expect the same magnifier to satisfy the requirements of several functions. Below are several factors affecting the quality of the magnifier, as well as the function for which it is best suited.

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If it is desired to reduce high frequency 2D spatial noise, a LPF (Low Passs Filter) can be used by selecting a LPF choice. Then prior to the FFT, the fid image is multiplied by the specified 2D filter. The K-space gaussian filter has a HWHM (Half Width - Half Maximum) equal to the radius specified in Radius field. The FWHM (Full Width - Half Maximum) is simply equal to twice the radius. The values, g(r), of the gaussian filter are given for one dimension in Equation 1 for a radius = h and an image width of N pixels. The for the HWHM radius, h, is given in Equation 2. Multiplication in K-space is equivalent to convolution in Image-space. Thus the relationship between gaussian filter FWHM in K-space to the FWHM in Image-space can be determined by taking the Fourier Transform (FT) of the K-space gaussian. But a gaussian with in the numerator is just another gaussian with in the denominator. By equating the exponents and replacing , the Image-space can be determined. From equation 2 the Image-space HWHM, , is Substituting for and noting that FWHM = 2h John Paul Strupp Wed Jan 29 11:44:13 CST 1997

A single lens is satisfactory for low powers. Higher power magnifiers require two or more lens elements for improved resolution and correction of chromatic or other aberrations.