Optical Coatings

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Fundamental Optics Material Properties Optical Specifications Gaussian Beam Optics Optical Coatings Modulation Transfer Function The modulation transfer function (MTF), a quantitative measure of image quality, is far superior to any classic resolution criteria. MTF describes the ability of a lens or system to transfer object contrast to the image. Curves can be associated with the subsystems that make up a complete electro-optical or photographic system. MTF data can be used to determine the feasibility of overall system expectations. Bar-chart resolution testing of lens systems is deceptive because almost 20% of the energy arriving at a lens system from a bar chart is modulated at the third harmonic and higher frequencies. Consider instead a sine-wave chart in the form of a positive transparency in which transmittance varies in one dimension. Assume that the transparency is viewed against a uniformly illuminated background. The maximum and minimum transmittances are T max and T min , respectively. A lens system under test forms a real image of the sine-wave chart, and the spatial frequency (u) of the image is measured in cycles per millimeter. Corresponding to the transmittances T max and T min are the image irradiances I max and I min . By analogy with Michelson’s definition of visibility of interference fringes, the contrast or modulation of the chart and image are defined, respectively, as and M = c M = i T T max max 4 T + T Imax 4 I I + I max min min min min where M c is the modulation of the chart and M i is the modulation of the image. The modulation transfer function of the optical system at spatial frequency u is then defined to be MTF = MTF(u) = M /M i c . (3.1) (3.2) (3.3) The graph of MTF versus u is a modulation transfer function curve and is defined only for lenses or systems with positive focal length that form real images. It is often convenient to plot the magnitude of MTF (u) versus u. Changes in MTF curves are easily seen by graphical comparison. For example, for lenses, the MTF curves change with field angle positions and conjugate ratios. In a system with astigmatism or coma, different MTF curves are obtained that correspond to various azimuths in the image plane through a single image point. For cylindrical lenses, only one azimuth is meaningful. MTF curves can be either polychromatic or monochromatic. Polychromatic curves show the effect of any chromatic aberration that may be present. For a well-corrected achromatic system, polychromatic MTF can be computed by weighted averaging of monochromatic MTFs at a single image surface. MTF can also be measured by a variety of commercially available instruments. Most instruments measure polychromatic MTF directly. PERFECT CIRCULAR LENS The monochromatic, diffraction-limited MTF (or MDMTF) of a circular aperture (perfect aberration-free spherical lens) at an arbitrary conjugate ratio is given by the formula MDMTF(x) = 2 ⎡ ⎢arc cos (x) 4 x 14 x π ⎣ where the arc cosine function is in radians and x is the normalized spatial frequency defined by x = u u ic where u is the absolute spatial frequency and u ic is the incoherent diffraction cutoff spatial frequency. There are several formulas for u ic including u = ic = 1.22 r d ⎛ 1.22l ⎞ n ′′ D 14⎜ ⎝ n ′′ D ⎟ ⎠ ls′′ 2 ⎛ 1.22l ⎞ 2n ′′ sin(u ′′) 14⎜ n D ⎟ ⎝ ′′ ⎠ = l 2n ′′ sin(u ′′) = l = n ′′ D (3.6) ls′′ where r d is the linear spot radius in the case of pure diffraction (Airy disc radius), D is the diameter of the lens clear aperture (or of a stop in near-contact), l is the wavelength, s″ is the secondary conjugate distance, u″ is the largest angle between any ray and the optical axis at the secondary conjugate point, the product n″ sin(u″) is by definition the image space numerical aperture, and n″ is the image space refractive index. It is essential that D, l, and s″ have consistent units (usually millimeters, in which case u and u ic will be in cycles per millimeter). The relationship sin(u ′′) = D 2s′′ implies that the secondary principal surface is a sphere centered upon the secondary conjugate point. This means that the lens is completely free of spherical aberration and coma, and, in the special case of infinite conjugate ratio (s″ = f″), u = n D ic ′′ . l f 2 2 ⎤ ⎥ ⎦ (3.4) (3.5) (3.7) (3.8) 3.4 1 Visit Us OnLine! www.mellesgriot.com
PERFECT RECTANGULAR LENS The MDMTF of a rectangular aperture (perfect aberrationfree cylindrical lens) at arbitrary conjugate ratio is given by the formula MDMTF(x) = (14 x) where x is again the normalized spatial frequency u/u ic , where, in the present cylindrical case, u = ic 1 r d and r d is one-half the full width of the central stripe of the diffraction pattern measured from first maximum to first minimum. This formula differs by a factor of 1.22 from the corresponding formula in the circular aperture case. The following applies to both circular and rectangular apertures: u = ic 2n ′′ sin(u ′′) . l The remaining three expressions for u ic in the circular aperture case can be applied to the present rectangular aperture case provided that two substitutions are made. Everywhere the constant 1.22 formerly appeared, it must be replaced by 1.00. Also, the aperture diameter D must now be replaced by the aperture width w. The relationship sin(u″) = w/2s″ means that the secondary principal surface is a circular cylinder centered upon the secondary conjugate line. In the special case of infinite conjugate ratio, the incoherent cutoff frequency for cylindrical lenses is u = n w ic ′′ . l f (3.9) (3.10) (3.11) (3.12) IDEAL PERFORMANCE AND REAL LENSES In an ideal lens, the x-intercept and the MDMTF-intercept are at unity (1.0). MDMTF(x) for the rectangular case is a straight line between these intercepts. For the circular case, MDMTF(x) is a curve that dips slightly below the straight line. These curves are shown in figure 3.2. Maximum contrast (unity) is apparent when spatial frequencies are low (i.e., for large features). Poor contrast is apparent when spatial frequencies are high (i.e., small features). All examples are limited at high frequencies by diffraction effects. A normalized spatial frequency of unity corresponds to the diffraction limit. All real cylindrical, monochromatic MTF curves fall on or below the straight MDMTF(x) line. Similarly, all real spherical and monochromatic MTF curves fall on or below the circular MDMTF(x) curve. Thus the two ideal MDMTF(x) curves represent the perfect (ideal) optical performance. Optical element or system quality is measured by how closely the real MTF curve approaches the corresponding ideal MDMTF(x) curve (see figure 3.3). MTF is an extremely sensitive measure of image degradation. To illustrate this, consider a lens having a quarter wavelength of spherical aberration. This aberration, barely discernible by eye, would reduce the MTF by as much as 0.2 at the midpoint of the spatial frequency range. MDMTF 1.0 .8 .6 .4 .2 circular aperture rectangular aperture 0 .2 .4 .6 .8 1.0 NORMALIZED SPATIAL FREQUENCY, X Figure 3.2 MDMTF(x) vs x, as a function of normalized spatial frequency, x MTF 1.0 .8 .6 .4 .2 lens with 1 / 4 wavelength aberration MDMTF 0 .2 .4 .6 .8 1.0 NORMALIZED SPATIAL FREQUENCY, X Figure 3.3 MTF as a function of normalized spatial frequency, x Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings Visit Us Online! www.mellesgriot.com 1 3.5
Page 1 and 2: Fundamental Optics 1 Introduction 1
Page 3 and 4: Paraxial Formulas SIGN CONVENTIONS
Page 5 and 6: object F 2 Figure 1.4 image Example
Page 7 and 8: Figure 1.6 Figure 1.7 f f 2 object
Page 9 and 10: Figure 1.8 Figure 1.9 INDIVIDUAL EL
Page 11 and 12: Performance Factors After paraxial
Page 13 and 14: third-order, monochromatic, spheric
Page 15 and 16: Positive lens elements usually have
Page 17 and 18: Lens Shape Aberrations described in
Page 19 and 20: ay f-numbers 2.7 3.3 4.4 6.7 13.3 S
Page 21 and 22: AIRY DISC DIAMETER = 2.44 l f/# Fig
Page 23 and 24: Lens Selection Having discussed the
Page 25 and 26: Example 4: Diffraction-Limited Perf
Page 27 and 28: Aberration Balancing To improve sys
Page 29 and 30: Definition of Terms FOCAL LENGTH (f
Page 31 and 32: infinity (which is a comfortable vi
Page 33 and 34: For symmetric lenses (r 2 = 4r 1 ),
Page 35 and 36: Separation of Nodal Point from Corr
Page 37 and 38: Gaussian Beam Optics 2 Introduction
Page 39 and 40: Even if a Gaussian TEM 00 laser-bea
Page 41 and 42: eam expander INCORPORATING M 2 INTO
Page 43 and 44: M 2 AND THE LENS EQUATION For real-
Page 45 and 46: employed when laser power must be c
Page 47 and 48: Figure 2.13 Keplerian beam expander
Page 49 and 50: Optical Specifications 3 Wavefront
Page 51: Centration The mechanical axis and
Page 55 and 56: The permissible number of maximum-s
Page 57 and 58: Material Properties 4 Material Prop
Page 59 and 60: Introduction Glass manufacturers pr
Page 61 and 62: maximum value for homogeneity withi
Page 63 and 64: Melles Griot Lens Materials Melles
Page 65 and 66: Refractive Index of Five Schott Gla
Page 67 and 68: Synthetic Fused Silica Fused silica
Page 69 and 70: PERCENT INTERNAL TRANSMITTANCE 99.9
Page 71 and 72: Low-Expansion Borosilicate Glass Th
Page 73 and 74: ZERODUR ® Many optical application
Page 75 and 76: Optical Coatings 5 Optical Coatings
Page 77 and 78: OEM and Special Coatings Melles Gri
Page 79 and 80: PERCENT REFLECTANCE 100 90 80 70 60
Page 81 and 82: will depend on the ratio of optical
Page 83 and 84: v = angle of incidence PERCENT REFL
Page 85 and 86: PERCENT REFLECTANCE PER SURFACE 2.0
Page 87 and 88: Two-Layer Coatings of Arbitrary Thi
Page 89 and 90: ION-ASSISTED BOMBARDMENT Ion-assist
Page 91 and 92: Single-Layer MgF 2 Antireflection C
Page 93 and 94: PERCENT REFLECTANCE 5 4 3 2 1 norma
Page 95 and 96: EXTENDED-RANGE HEBBAR ANTIREFLECTIO
Page 97 and 98: V-Coatings V-coatings are multilaye
Page 99 and 100: Metallic High-Reflection Coatings W
Page 101 and 102: INTERNAL SILVER (/036) $ For intern
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Dielectric High-Reflection Coatings
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PERCENT REFLECTANCE 100 80 60 40 20
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MAXBRIte Coatings MAXBRIte (multi
Page 109 and 110:
Laser-Line MAX-R Coatings These mu
Page 111 and 112:
Ultrafast Coating Melles Griot has
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