CALCULATION OF MODULATION TRANSFER FUNCTION OF AN OPTICAL...

CALCULATION OF MODULATION TRANSFER FUNCTION OF AN OPTICALSYSTEM BY USING SKEW RAY TRACING

Kuo-Hwa Tseng1, Chieh Kung2, Te-Tan Liao1, Hao-Peng Chang3

1Department of Mechanical Engineering, Far East University

2Department of Computer Application Engineering, Far East University

3Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C.

E-mail: [email protected]

Received July 2009, Accepted September 2009

No. 09-CSME-36, E.I.C. Accession 3122

ABSTRACT

The resolution and performance of an optical system can be characterized by a quantityknown as the modulation transfer function (MTF), which is a measurement of an opticalsystem’s ability to transfer contrast from the specimen to the intermediate image plane at aspecific resolution. Accordingly, this study employs skew ray tracing based on a 464homogeneous coordinate transformation matrix and Snell’s law to develop a detailedmethodology for determining the spot diagram on the image plane when light rays passthrough the optical system. And the authors present calculations of the MTF of an opticalsystem by using the spot diagram on the image plane. The numerical results of the proposedmethodology are demonstrated using a symmetrical optical system.

Keywords: modulation transfer function; skew ray tracing; spot diagram.

CALCUL DE LA FONCTION DE TRANSFERT DE MODULATION D’UN SYSTEMEOPTIQUE UTILISANT LE LANCER DE RAYONS OBLIQUES

RESUME

La resolution et la performance d’un systeme optique peuvent etre decrites par une quantiteconnue sous le nom de fonction de transfert de modulation (FTM). Cette derniere est unemesure de la capacite d’un systeme optique d’operer le transfert de contraste d’un specimen aune image plane intermediaire ayant une resolution specifique. Par consequent, dans cette etudeon emploie le lancer de rayons obliques sur une matrice de transformation homogene de 464 etla loi de Snell pour developper une methodologie detaillee pour determiner le diagramme depointe sur l’image plane au moment ou les rayons de lumiere passent a travers le systemeoptique. Les auteurs presentent le calcul du FTM d’un systeme optique en utilisant lediagramme de pointe sur l’image plane. Les resultats numeriques de la methode proposee sontdemontres en utilisant un systeme optique symetrique.

Mots-cles : fonction de transfert de modulation; lancer de rayons obliques; diagramme depointe.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3, 2009 429

1. INTRODUCTION

The determination of the modulation transfer function (MTF) enables the image quality of asystem to be evaluated and characterizes the spatial-frequency response of an imaging system[1]. The MTF of an optical system is an accepted way of describing its optical properties [2]. Itsconvenience results from the simple conversion of the convolution operation in the image spaceto multiplication in the spatial frequency domain. Describing a system by its MTF is mostuseful when the optics are complicated, in which case it gives a good measure of the spatialresolution attainable, and deviations of the measured MTF from a desired MTF can easily bequantified.

In most lenses there are many small variations of the design from with nearly equivalentperformance or modulation transfer function (MTF). Note that the merit function and theMTF are very rarely well correlated (if at all). The calculation of MTF can be done byautocorrelating the aberrated wavefront against itself, shifted laterally by a function of theaperture, the shift corresponding to the spatial frequency [3].

It is difficult to establish an exact measure of image quality since quality is an entity whichusually finds itself subject to human judgement. Nevertheless, certain criteria have been setforth in an attempt to quantify image quality, one of these being the cut-off spatial frequency.This, however, gives no information about the shape of the spatial frequency curve. The MTF(or OTF), which is about to be described, is a better criterion since it applies to the shape of theentire spatial frequency range.

The modulation transfer function (MTF), identical to the absolute value of the opticaltransfer function, is a better criterion for establishing an exact measure of the quality ofimaging, and it applies to features over the entire spatial-frequency range. The MTF measuresthe frequency response of an optical system, and over the years a variety of techniques havebeen developed [4]. Other applications include the work by Koopipat et al. [5] in which a newmeasurement method was proposed to separately measure the effect of ink spread and opticaldot gain on the MTF of printed images from an ink jet printer. Madanipour and TaghiTavassoly [6] introduced a simple and reliable technique for evaluating the MTFs of printers.They used a pair of similar gratings which provided the MTF for several frequencies. The mostcommon techniques can be divided into four categories: the sine-wave method, the bar-targetmethod, the edge-gradient method and a series-expansion method. In the sine-wave method, anobject target with a sine-wave variation in intensity is imaged, and the MTF is the ratio ofimage-to-object modulation depth [7]. This is the most direct method for measuring the MTF,but it requires a different target and a different set of measurements for each of the spatialfrequencies sampled. In the bar-target method, the object has square-wave variation in intensity[8,9]. In the edge-gradient method, the object is a knife edge, and the image data are the edge-spread function [10]. The MTF is obtained as the Fourier transform of the derivative of theedge-spread function. In the series-expansion method [11], a bar target is used but the methodof analysis more closely resembles that of the edge-gradient technique.

When designing optical systems, an accurate knowledge of the light path is required in orderto predict the performance of the system [12]. As a light ray passes though an optical system, itis reflected and refracted at the boundaries between different media. In practice, the light pathcan be determined by applying a ray tracing technique, in which the optical laws of reflection orrefraction are applied at each boundary encountered by the light ray [13,14]. The light rays in anoptical system may be axial, meridional or skew rays. Axial rays and meridional rays can betraced using relatively simple trigonometric formulae, or even graphically if a very low level of


precision is adequate. A crude knowledge of the paths followed by meridional rays in theparaxial regions of an optical system can also be obtained via successive applications of matrixproduction [15]. However, skew rays, which represent the most general type of ray, are far moredifficult to trace. Nonetheless, without tracing their paths, it is impossible to accurately modeloptical systems or to evaluate their performance. Lin [16] reformulated the traditional laws ofreflection and refraction in terms of revolution geometry, and then performed skew ray tracingand a sensitivity analysis to determine the changes in a light ray path as it crossed the boundarybetween different media.

The homogeneous coordinate representation of a vector in 3-space is a 4-space entityspecified such that a particular perspective projection recreates the 3-space. The transformationof a vector in 3-space to a homogeneous coordinate representation involves the addition of anextra coordinate (i.e. a scale factor) to each vector such that the vector retains the same meaningif each of its components, including the scale factor, is multiplied by a constant [17]. In theanalysis presented in this paper, the position vector Pixi+Piyj+Pizk in 3-space is written in theform of a column matrix jPi~ Pix Piy Piz 1½ �T, in which the pre-superscript ‘‘j’’ of theleading symbol, jPi, indicates that the vector is referred with respect to the coordinate frame(xyz)j. Given a point jPi, its transformation kPi is represented by the matrix product kPi 5 kAj

jPi,where kAj is a 4 6 4 matrix defining the position and orientation (referred to hereafter as theconfiguration) of a frame (xyz)j with respect to another frame (xyz)k. The same notation rulesare also applied to the unit directional vector j‘i~ ‘ix ‘iy ‘iz 0½ �T. Note that for vectorsreferred to the world frame (xyz)0, the pre-superscript ‘‘0’’ is omitted for reasons ofconvenience.

2. SKEW RAY TRACING AT OPTICAL BOUNDARY SURFACES

A fundamental feature of many optical elements is that their boundaries are surfaces ofrevolution. Therefore, when performing a geometrical analysis of the optical performance of asystem, it is first necessary to define the various boundary surfaces within the system in terms oftheir respective revolution geometries. A ray tracing technique can then be used to determinethe paths followed by the skew rays as they undergo successive reflection and refractionoperations at the various optical surfaces they encounter as they travel through the system.

Fig. 1. Skew ray tracing at medium boundary surface iri.


Figure 1 illustrates the skew ray paths at a typical boundary surface in an optical system. Theboundary surface iri can be defined by rotating its generating curve in the xizi plane, i.e.ili~ RiCbi 0 RiSbi 1½ �T, about the optical zi axis, i.e.

iri~Rot(zi,ai)ili~

Cai {Sai 0 0

Sai Cai 0 0

0 0 1 0

0 0 0 1

26664

37775

RiCbi

0

RiSbi

1

26664

37775

~ RiCbiCai RiCbiSai RiSbi 1½ �T,

(1)

where Rot(zi,ai) is the rotation transformation matrix about the zi axis, and C and S indicatecosine and sine functions, respectively.

The unit normal ini to this boundary surface is given by ini~si(Liri

Lbi

|Liri

Lai

)=Liri

Lbi

|Liri

Lai

��, where

the value of si is specified as either +1 or –1 such that the cosine of the incident angle has apositive value, i.e.Chi . 0. In other words, the unit normal at each boundary surface iri can beexpressed as

ini~{si CbiCai CbiSai Sbi 0½ �T: (2)

Note that iri and ini are both expressed with respect to the boundary coordinate frame (xyz)i.The configuration of the world frame (xyz)0 with respect to the boundary coordinate frame isgiven by

iA0~Ai0~

Iix Jix Kix tix

Iiu Jiy Kiy tiy

Iiz Jiz Kiz tiz

0 0 0 1

2664

3775: (3)

Where vectors Iix Iiy Iiz 0½ �T, Jix Jiy Jiz 0½ �T and Kix Kiy Kiz 0½ �T describe theorientation of the three unit vectors of frame (xyz)0 with respect to frame (xyz)i. Vectortix tiy tiz 1½ �T is the position vector of the origin of frame (xyz)0 with respect to frame (xyz)i.

The unit normal with respect to the world frame, i.e. ni, can be obtained as

ni~ nix niy niz 0½ �T~A0iini~A{1

i0ini

~{si

IixCbiCaizIiyCbiSaizIizSbi

JixCbiCaizJiyCbiSaizJizSbi

KixCbiCaizKiyCbiSaizKizSbi

0

26664

37775:

(4)

Figure 1 shows the general case where a light ray originating at point Pi{1~ Pi{1x Pi{1y½Pi{1z1�T and directed along a unit directional vector ‘i{1~ ‘i{1x ‘i{1y ‘i{1z 0½ �T is reflected /refracted at an optical medium boundary surface, iri. The incident point Pi, refracted ray ‘i, andreflected ray ‘i are given by the following formulae [18]:

Pi~ Pi{1xz‘i{1xli Pi{1yz‘i{1yli Pi{1zz‘i{1zli 1½ �T, (5)


‘i~

‘ix

‘iy

‘iz

0

2664

3775~

{nix

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1{N2

i z(NiChi)2

qzNi(‘i{1xznixChi)

{niy


i z(NiChi)2

qzNi(‘i{1yzniyChi)

{niz


i z(NiChi)2

qzNi(‘i{1zznizChi)

0

2666664

3777775

, (6)

‘i~

‘ix

‘iy

‘iz

0

2664

3775~

‘i{1xz2nixChi

‘i{1yz2niyChi

‘i{1zz2nizChi

0

2664

3775, (7)

where li is the magnitude of vector PiQi, Ni~jmedium,i{1

�jmedium,i is the relative refractive index

of medium i-1 with respect to medium i, and hi is the incident angle. Following reflection /refraction at the boundary surface, the light ray proceeds along its new path with point Pi as itsnew point of origin and ‘i(‘i) as its new unit directional vector(s).

Figure 2 provides a schematic illustration of a generic optical system comprising variousspherical and flat boundary surfaces. Under the assumption that the light source located at P0

radiates light uniformly over an imaginary hemispherical surface area, the unit directionalvector ‘0~ ‘0x ‘0y ‘0z 0½ �T of any ray originating from this light source can be expressed interms of the coordinates of spherical angles W and Y as ‘0~ SW CWCY CWSY 0½ �T. As aresult, any intermediate point along an arbitrary ray emitted from P0 has the form

Q1~P0z‘0l~ P0xzlSW P0yzlCWCY P0zzlCWSY 1½ �T: (8)

Without loss of generality, the following analysis traces a light ray originating at Pi{1~Pi{1x Pi{1y Pi{1z 1½ �T and having a unit directional vector of ‘i{1~ ‘i{1x ‘i{1y½‘i{1z0�T (i 5 1,2,3,…., n21) which is refracted / reflected at an optical boundary surface iri.To trace the path of the light ray as it travels through the optical system, it is first necessary tolabel the individual boundary surfaces within the system sequentially from 1 to n, where n is thetotal number of boundary surfaces in the system. Any ray can then be traced via the applicationof a conventional skew ray tracing algorithm at each boundary surface i 5 1, i 5 2, i 5 3,…. n-1.The point of input ray path to the system’s sensor plane rn is given by

Qn~ Pn{1xz‘n{1xl Pn{1yz‘n{1yl Pn{1zz‘n{1zl 1½ �T: (9)

In Eq. (9), Qn is referred to the world frame (xyz)0. Its expression with respect to the sensorframe (xyz)n is given by nQn~An0Qn, in which

An0~

1 0 0 tnx

0 1 0 tny

0 0 1 tnz

0 0 0 1

2664

3775 (10)

is the configuration of the world frame (xyz)0 with respect to the sensor frame (xyz)n. Theparameter l 5 ln can then be obtained as


ln~{(Pn{1zztnz)

‘n{1z: (11)

The sensor readings nPnxnPny½ �T of the system are given by

nPnxnPny

� �~

Pn{1xz‘n{1xlnztnx

Pn{1yz‘n{1ylnztny

� �: (12)

Equation (12) provides the basis for determining the sensor readings nPnxnPny½ �T using a ray

tracing approach. And the spot diagram of the object on the image plane can be obtained.

3. CALCULATION OF MTF BY USING RAY TRACING

MTF is one of the measures of the performance of an optical system. Consider a linearsystem, in which there is signal of sinusoidal wave with a frequency denoted as n, its luminousintensity distribution is,

I(x)~I0zIacos2pnx: (13)

The modulation of the light wave is

Fig. 2. Schematic illustration of a generic optical system comprising various spherical and flatboundary surfaces.


M~(Imzx-Imin)

(ImaxzImin)~

Ia

I0, (14)

where Imax is the maximal luminance, Imin the minimal luminance, Ia the amplitude of thesinusoidal light wave, and I0 the averaged amplitude of the wave. For an optical system, theMTF is defined as

MTF~Mi

M0, (15)

in which Mi and M0 are the modulations of an image and of an object, respectively. Taking thefrequency n or half diagonal length of a film, i.e. from the center to one of the vertices, as theabscissa and MTF as the ordinate, one may connect the various points to form a MTF curve.The domain of MTF is [0, 1]. In general, the value of MTF decreases with the frequency, n. Acut-off frequency is where MTF equals zero. Those signals whose frequencies are higher thanthe cut-off frequency can not be passed by the system. The cut-off frequency acts like a low-passfilter.

A spheroidal wave emitting from a point light source will enter into the entrance pupil andcome out from the exit pupil. The spheroidal wave produces diffraction and focuses on theimage coordinates to form a minimal diffraction diagram. If there is no image differencepresent, the system is called a diffraction-limited system.

Under the diffraction-limited situation, the MTF is related to the size of the diffractiondiagram and is a function of numerical aperture and wave length of the light. The MTF of adiffraction-limited system is written as

MTF(n)~2

p½cos-1k-k(1-k2)

12�, (16)

where K~ln

2NA, l is the wave length in mm, n is the spatial frequency in cycles/mm, NA is the

numerical aperture, and the cut-off frequency is

n0~2NA

l~

1

l(f=#): (17)

An optical system is similar to a low-pass filter, thus all signals whose frequency is greaterthan the cut-off frequency cannot be resolved.

Presented in Figure 3 is the symmetrical optical system for simulation study in this paper.The parameters of this system are as follows. The numerical aperture (NA) is -0.045, f-numberis 11.1, effective focal length (EFL) is 99.99mm, back focal length (BFL) is 179.375mm andobject distance (OD) is 179.38mm. The constructional parameters of lenses are listed inTable 1.

In this study three conditions of light sources position and aperture radius are considered forray tracing simulation. In Condition 1, the light source is on the optical axis and the radius ofthe aperture is 4.5mm (light source 1 in Fig. 3). In Condition 2, the light source is on the opticalaxis and the radius of the aperture is 1.5mm (light source 1 in Fig. 3). In Condition 3, thedistance between the light source and optical axis is 50mm and the radius of the aperture is1.5mm (light source 2 in Fig. 3). The OD is kept 179.38mm for all three conditions. To beginthe tracing simulation, the aperture is first divided into a 40640 grid array. Light rays are


emitted from the light source with different incident angle to the aperture. Light rays that passthrough the array grids are considered to simulate ray tracing. In this study, there are 1576 lightrays traced. Subsequently, the spot diagram of the light rays on the focal plane of the opticalsystem is obtained. The calculation of the MTF of the optical system is performed thereafter.

The spot diagram in the image plane is obtained by means of the skew ray tracing. Thedistribution of the luminance of light in the image plane is then estimated from the spot diagramby a scheme described as following. The spot diagram is divided into eighteen rectangularzones. These zones are equal in width and are symmetrical with the ordinate (Y axis); that is,nine striped zones are in positive X-axis half plane and nine in negative X-axis half plane. Thenumber of the spots in each zone is normalized by the total number of the light spots in themiddle two striped zones. As such, a line spread function (LSF) can be drawn to further obtainthe MTF curve of the optical system. Fig. 4 shows the MTF curve of the symmetrical opticalsystem under diffraction-limited condition. From Fig. 4, it is seen that the end spatial-frequency is 153 cycles/mm.

Fig. 3. The optical system for the calculation of MTF.

Table 1. Data of lens in the symmetrical optical system (unit: mm).

Boundary (i) Radius (mm) Thickness (mm) Material Index

1 21.907 9.020 SSK5 1.6582 19.003 1.950 air 13 42.090 1.096 LF2 1.5894 13.928 0.180 air 15 14.402 4.097 SK8 1.6116 74.290 3.812 air 17 3.812 air 18 -74.290 4.097 SK8 1.6119 -14.402 0.180 air 1

10 -13.928 1.096 LF2 1.58911 -42.090 1.950 air 112 -19.003 9.020 SSK5 1.65813 -21.907 179.375 air 1


The results of Condition 1 tracing simulation are presented in Fig. 5. Fig. 5(a) shows thespot diagram and Fig. 5(b) the MTF curve. Note that for Condition 1 the object distance (OD)is 179.38mm, the light source is on the optical axis and radius of the aperture is 4.5mm. FromFig. 5(a), the spot diagram is perfectly symmetrical and the spots are clearly focused. Thedistance between the origin and the rightmost spot on the abscissa is calculated to be 0.0452mm.The radius of the area where the light spots populate most densely is estimated to be 0.00662mmand the number of the light spots count in this area is 302. Note that the distance as well as theradius is obtained via a computer code which is developed during the course of the study.Because of symmetry, the number of light spots in each zone in the negative half plane is presentin Table 2. Based on the data in Table 2, a line spread function is drawn to obtain the MTFcurve as shown in Fig. 5(b). From Fig. 5(b), one can observe that the spatial-frequency is 127cycles/mm.

In Fig. 6, the spot diagram and the curve of the MTF value are presented for Condition 2simulation where the object distance (OD) is 179.38mm, the light source is on the optical axisand the radius of the aperture is 1.5 mm. The spot diagram shown in Fig. 6(a) is also perfectlysymmetrical. The radius of the spot diagram is 0.00131 mm. The radius of the area where thelight spots populate most densely is estimated about 0.000348mm and the number of the lightspots count in this area is 1042. The number of light spots in each zone in the negative half planeis present in Table 3. Comparing the MTF’s shown in Fig. 5(b) and Fig. 6(b), it is suggestedthat for a particular spatial frequency, the MTF values increases with the radius of the aperture,and the MTF curve becomes smoother with a larger aperture.

Fig. 7 presents the spot diagram and the MTF curve for Condition 3 simulation condition ofthe optical system. The object distance (OD) is 179.38mm, the distance between the light sourceand optical axis is 50 mm and the radius of the aperture is 1.5 mm. From Fig. 7(a), one canobtain that the spot diagram is 0.002286mm and 0.0196mm in X and Y direction, respectively.It is interesting to see that the spots are distributed quite uniformly in the diagram. There is noarea of dense spot as compared with those seen for Condition 1 and Condition 2. The numberof light spots in each zone in the negative half plane is present in Table 4. The MTF curve is

Fig. 4. The MTF of the optical system under Diffraction-limited.


obtained accordingly and is shown in Fig. 7(b). The MTF curve decreases rapidly below thespatial frequency 60 lp/mm and fluctuates moderately thereafter. There is no obvious spatial-frequency.

Table 2. Distribution of the spot number for simulation Condition 1 wherein the light source is inthe optical axis and the radius of the aperture is 4.5 mm.

Zone numberRange of each zone in

x-coordinateNumber of the spots

in the zone

1 -0.0452 , -0.0405 52 -0.0405 , -0.0355 53 -0.0355 , -0.0304 44 -0.0304 , -0.0254 45 -0.0254 , -0.0203 66 -0.0203 , -0.0152 117 -0.0152 , -0.0101 128 -0.0101 , -0.0051 259 -0.0051 , origin 141

Fig. 5. The MTF of the optical system under simulation Condition 1 wherein the OD is 179.38mm,the light source is in the optical axis and the radius of the aperture is 4.5mm.


Fig. 6. The MTF of the optical system under simulation Condition 2 wherein the OD is 179.38mm,the light source is in the optical axis and the radius of the aperture is 1.5 mm.

Table 3. Distribution of the spot number for simulation Condition 2 wherein the light source is inthe optical axis and the radius of the aperture is 1.5 mm.

Zone number Range of each zone in x-coordinate Number of the spots in the zone

1 -0.0013 , -0.0012 42 -0.0012 , -0.0010 103 -0.0010 , -0.00087 104 -0.00087 , -0.00072 105 -0.00072 , -0.00058 166 -0.00058 , -0.00043 187 -0.00043 , -0.00029 728 -0.00029 , -0.00014 769 -0.00014 , origin 288


Fig. 7. The MTF of the optical system under simulation Condition 3 wherein the OD is 179.38mm,the distance between the light source and optical axis is 50 mm and the radius of the aperture is1.5 mm.

Table 4. Distribution of the spot number for simulation Condition 3 wherein the distance betweenthe light source and optical axis is 50 mm and the radius of the aperture is 1.5 mm.

Zone number Range of each zone in x-coordinate Number of the spots in the zone

1 -0.0114 , -0.0102 272 -0.0102 , -0.0089 293 -0.0089 , -0.0076 444 -0.0076 , -0.0063 305 -0.0063 , -0.0051 306 -0.0051 , -0.0038 467 -0.0038 , -0.0025 488 -0.0025 , -0.0013 329 -0.0013 , origin 51


4. CONCLUSION

In this paper the spot diagram on the focal plane of an optical system is obtained by usingskew ray tracing. The MTF value of an optical system based on the luminous intensitydistribution which is computed from a spot diagram is calculated. Three important topics areaddressed in this study. First, the spot diagram obtained by using ray tracing of point lightsource is larger than that obtained from diffraction limit. Thus, The MTF curve is lower andprovide comparatively lower analytical capability. Second, the MTF curve calculated for off-axis light source decreases rapidly below a certain spatial frequency range and then fluctuatesbeyond that frequency. In addition, there is no obvious spatial-frequency to be identified for theoff-axis light source condition. Third, with a smaller aperture, one may obtain smoother MTFcurve with grater values.

ACKNOWLEDGMENTS

The financial support provided to this study by the National Science Council of Taiwanunder grant NSC98-2221-E-269-015 is gratefully acknowledged.

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