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Fundamental Optics Material Properties Optical Specifications Gaussian Beam Optics Optical Coatings The Reflection of Light REFLECTIONS AT UNCOATED SURFACES Whenever light is incident on the boundary between two media, some light is reflected and some is transmitted (undergoing refraction) into the second medium. Several physical laws govern the direction, phase, and relative amplitude of the reflected light. For our purposes, it is necessary to consider only polished optical surfaces. Diffuse reflections from rough surfaces are not considered here. The law of reflection states that the angle of incidence equals the angle of reflection. This is illustrated in figure 5.1 which shows reflection of a light ray at a simple air/glass interface. The incident and reflected rays make an equal angle with the axis perpendicular to the interface between the two media. INTENSITY At a simple interface between two dielectric materials, the amplitude of reflected light is a function of the ratio of the refractive index of the two materials, polarization of the incident light, and the angle of incidence. When a beam of light is incident on a plane surface at normal incidence, the relative amplitude of the reflected light, as a proportion of the incident light, is given by (14 p) (1 + p) where p is the ratio of the refractive indices of the two materials (n 1 /n 2 ). Intensity is the square of this expression. air n = 1.00 glass n = 1.52 Figure 5.1 interface incident ray reflected ray v i v r v i = v r v t refracted ray sinv t n = air sinv i n glass Reflection and refraction at a simple air/glass (5.1) The amount of reflected light is therefore larger when the disparity between the two refractive indices is greater. For an air/glass interface with the glass having a refractive index of 1.5, the intensity of the reflected light will be 4% of the incident light. For an optical system containing ten such surfaces, this shows that the transmitted beam will be attenuated to 66% of the incident beam from reflection losses alone. INCIDENCE ANGLE The intensity of reflected and transmitted beams is also a function of the angle of incidence. Because of refraction effects, it is necessary to consider internal and external reflection separately at this point. External reflection is defined as reflection at an interface where the incident beam originates in the material of lower refractive index (i.e., air in the case of an air/glass or air/water interface). Internal reflection refers to the opposite case. EXTERNAL REFLECTION AT A DIELECTRIC BOUNDARY Fresnel’s laws of reflection precisely describe amplitude and phase relationships between reflected and incident light at a boundary between two dielectric media. It is convenient to think of incident radiation as the superposition of two plane-polarized beams, one with its electric field parallel to the plane of incidence (p-polarized) and the other with its electric field perpendicular to the plane of incidence (s-polarized). Fresnel’s laws can be summarized in the following two equations which give the reflectance of the s- and p-polarized components: ( ) ( ) ⎡sin v14v ⎤ 2 r s = ⎢ ⎥ ⎣⎢ sin v1 + v2 ⎦⎥ 2 ⎡ v14v2) ⎤ r p = ⎢ ⎥ . ⎣⎢ tan( v1 + v2) ⎦⎥ ⎛ r = n 4 1 ⎞ ⎜ ⎝ n + 1 ⎟ ⎠ . 2 2 (5.2) (5.3) In the limit of normal incidence in air, Fresnel’s laws reduce to the following simple equation: (5.4) It can easily be seen that, for a refractive index of 1.52 (crown glass), this gives a reflectance of 4%. This important result shows that about 4% of all illumination incident normal to an air-glass surface will be reflected. In a multielement lens systems, reflection losses would be very high if antireflection coatings were not used. The variation of reflectance with angle of incidence for both the s- and p-polarized components can be seen in figure 5.2. It can be seen that the reflectance remains close to 4% over about 30 degrees incidence, and that it rises rapidly to 100% at grazing incidence. In addition, note that the p-component vanishes at 56º 39′. This angle, called Brewster’s angle, is the angle at which the reflected light is 5.4 1 Visit Us OnLine! www.mellesgriot.com
PERCENT REFLECTANCE 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 Figure 5.2 External reflection at a glass surface (n = 1.52) showing s- and p-polarized components completely polarized (see figure 5.3). This situation occurs when the reflected and refracted rays are perpendicular to each other (v 1 + v 2 = 90º ). This leads to the expression for Brewster’s angle, v B: v 1 = v B = arctan (n 2 /n 1 ). s-plane p-plane ANGLE OF INCIDENCE IN DEGREES Under these conditions, electric dipole oscillations of the p- component will be along the direction of propagation and therefore cannot contribute to the reflected ray. At Brewster’s angle, reflectance of the s-component is about 15%. INTERNAL REFLECTION AT A DIELECTRIC BOUNDARY For light incident from a higher to a lower refractive index medium, we can apply the results of Fresnel’s laws in exactly the same way. The angle in the high-index material at which polarization occurs is smaller by the ratio of the refractive indices in accordance with Snell’s law. The internal polarizing angle is 33º2l′ for a refractive index of 1.52, corresponding to the Brewster angle (56º 39′) in the external medium as shown in figure 5.4. The angle at which the emerging refracted ray is at grazing incidence is called the critical angle (see figure 5.5). For an external medium of air or vacuum (n = 1), the critical angle is given by vc ( l ) = arc sin ⎛ 1 ⎞ ⎜ ⎟ ⎝ n( l) ⎠ and depends on the refractive index n(l), which is a function of wavelength. For all angles of incidence higher than the critical angle, total internal reflection occurs. v p (5.5) air or vacuum index n 1 p-polarized incident ray isotropic dielectric solid index n 2 dipole axis direction e 1 normal v 2 v 1 absent p-polarized reflected ray refracted ray dipole radiation pattern: sin 2 v p-polarized refracted ray Figure 5.3 Brewster’s angle (at this angle, the p-polarized component is completely absent in the reflected ray) n air n glass v c = critical angle d c b a vc Figure 5.4 Internal reflection at a glass surface (n = 1.52) showing s- and p-polarized components PERCENT REFLECTANCE 100 90 80 70 60 50 40 30 20 10 Brewster total reflection angle 33° PRODUCT 21' NUMBER A B 07 PHT 501/07 PHF 501 10 3 07 PHT 503/07 PHF 503 15 5 07 PHT 505/07 PHF 505 20 5 07 PHT 507/07 PHF 507 30 5 07 PHT 509/07 PHF critical 509 angle 40 5 r 07 PHT s 511/07 PHF 41° 5118' 50 5 r p 0 10 20 30 40 50 60 70 80 90 ANGLE OF INCIDENCE IN DEGREES Figure 5.5 Critical angle (at this angle, the emerging ray is at grazing incidence) a a b c b d c Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings Visit Us Online! www.mellesgriot.com 1 5.5
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Fundamental Optics 1 Introduction 1
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Paraxial Formulas SIGN CONVENTIONS
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object F 2 Figure 1.4 image Example
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Figure 1.6 Figure 1.7 f f 2 object
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Figure 1.8 Figure 1.9 INDIVIDUAL EL
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Performance Factors After paraxial
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third-order, monochromatic, spheric
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Positive lens elements usually have
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Lens Shape Aberrations described in
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ay f-numbers 2.7 3.3 4.4 6.7 13.3 S
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AIRY DISC DIAMETER = 2.44 l f/# Fig
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Lens Selection Having discussed the
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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 and 52: Centration The mechanical axis and
Page 53 and 54: PERFECT RECTANGULAR LENS The MDMTF
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: OEM and Special Coatings Melles Gri
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
Page 103 and 104: Dielectric High-Reflection Coatings
Page 105 and 106: PERCENT REFLECTANCE 100 80 60 40 20
Page 107 and 108: 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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