an aerial distribution of light , which is absorbed inside a resist on performing the exposure process , could not be explained only using an aerial image of two dimension and a defocus value , but has sophisticated distribution in three dimensions , in which multi - interference between a surface of a resist and a wafer substrate , should be considered . absorption of the exposure energy in three dimensions is the chief obstacle to approximate or quantify a resist process . if two aerial images , which have a defocus difference as much as a thickness of the resist since the resist thickness is sufficiently thin , are not quite different , the exposure energy distribution , which is absorbed inside of the resist only using the aerial image of two dimension of a focal plane , may be approximately represented even if the multi - interference is considered . the difference between an aerial image i 0 ( f ) in a focus f and an aerial image i 0 ( f + t pr ) in a focus f + t pr can be disregarded since the resist thickness t pr is sufficiently thin . here , it is possible that the resist thickness is lower than 15 , 000 å . according to these conditions , the aerial image in two dimensions will be considered in the present embodiments . it is well known that a chemical resist produces acid where a photo active generator ( pag ) is broken by photon energy when the exposure process is performed . if we now define p and i to be normalized concentration of the pag and intensity , respectively , variation of concentration caused by resolving the pag during the exposure time t will be given by ## equ3 ## where c is a proportionalconstant and is equivalent to the &# 34 ; c &# 34 ; quoted in the dill . the differential equation ( 1 ) will be easily solved by applying conditions that the intensity i is not a time function and an initial condition that the pag is not even resolved before performing the exposure process . since acid the amount of acid produced is as much as the amount that the pag is resolved , if an acid is produced with respect to a pag , the sum of the pag concentration and acid concentration will be constant . thus , a normalized concentration distribution a 0 of acid just after exposure may be written as where t e is exposure time . since total exposure energy e is obtained by multiplying the intensity i by total exposure time t e ,. so equation ( 2 ) may be replaced with if the normalized intensity distribution i 0 is introduced , the intensity i will be replaced with i = ii 0 wherein , i is defined as a proportional constant since the intensity i is represented as constant times . so , equation ( 2 ) may be now replaced with where α is a new constant and is satisfied with an equation of α = cit e . the acid produced by the exposure process is diffused by the thermal energy during a post exposure bake ( peb ) which is the next process . if a diffusion coefficient d of acid is treated as a constant , in accordance with fick &# 39 ; s 1st law , concentration a of the normalized acid during the peb process may be written ## equ4 ## where t &# 39 ; is a variable representing the peb time . the differential equation ( 5 ) will be easily solved by applying fourier transformation in both directions and determining proper boundary conditions . symbols of ξ , n are defined as fourier parameters . multiplied by ξ , n and then integrated . thus , we obtain the relation ## equ5 ## where f , f - 1 represent fourier transformation from x &# 39 ;, y &# 39 ; space to ξ , n space to x , y space , respectively , and x &# 39 ;, y &# 39 ; and x , y are initial space and final space , respectively . at this stage , diffusion length in the peb time t &# 39 ; is defined as ## equ6 ## thus , diffusion length just after the peb process becomes ## equ7 ## where t b &# 39 ; is total peb time . acid produced by the exposure energy is diffused by the thermal energy produced in the peb process as described above and serves the role of breaking the protection group , concurrently . the breaking protection group is dissolved by a developer in a development process and thereby forming the pattern . variation amount of the normalized concentration m of the protection group may be written : ## equ8 ## where c is the proportional constant , and ε a as a boltmann factor , k b , and t in the exponential terms are activation energy of the protection group , boltmann constant , and peb temperature , respectively . also , m , which is usually defined as &# 34 ; 1 &# 34 ;, is a reaction degree of acid , and can be regarded as the number of acid for breaking one protection group . equation ( 8 ) will be easily solved since a function of time t &# 39 ; has only m and a . if , now let n to be the normalized concentration of the protection group broken after performing the peb , since summed value with m is maintained , the concentration n may be given by ## equ9 ## where β = c &# 39 ;/ d exp (- ε a / k b t ) is another constant with no relation to time and space , and s is a normalized space average of acid within the diffused length . so , if equation ( 7 ) is substituted into equation ( 9 ), we obtain ## equ10 ## it is very difficult for a course developing from top to bottom of the resist to be expressed by an analytic equation : though normally it approaches in a manner having a numerical value . however , in a case where the thickness of resist is very thin , it can be considered as a two dimensional plane as assumed in the above . in such a case , the boundary of the pattern is formed along the contour which is equal to breaking the degree of the protection group . in the case of considering the resist thickness , if development contrast γ of the resist is very high , it is considered that the pattern is formed along the contour which is equal to breaking the degree of the protection group as same as in the case of thin resist . thus , as described hereinafter , it is possible to approximately approach with respect to development course , as described above . in the development course , the resist is dissolved at more than any constant value ( threshold concentration : n th ) by the developer in the non - protection concentration distribution n made from breaking the protection group , while the resist remains without dissolving at less than the constant value . namely , the pattern is formed along the contour of n th . it is possible that the threshold concentration n th of the - non - protection group is related to the threshold value of the exposure energy ( i . e , threshold exposure dose : e th ), and it can be easily assumed that when black exposure is performed as much as e th , it directly becomes n th concentration . in a case where parameters are well known among the chemical amplification resists . since values of c and e th have about 0 . 001 ( mj / cm 2 ) - 1 and about 10 mj / cm 2 , respectively , ce th is too small . so , proximity may have the relation thus , from equations ( 3 and 9 ), we obtain the relation as ## equ11 ## then , it is considered that the normal case is performed to e th , and not the black exposure . the pattern &# 39 ; s boundary , remaining after the development process is finished , is a portion of n = n th , and e of this portion may have equal size of order for e th even though e in this portion is generally identical to e th . thus , we obtain the relation when equation ( 3 ) or ( 4 ) is approximated by using equation ( 13 ), if it is reminded that a 0 and i 0 are normalized , from the pattern &# 39 ; s boundary , it may obtain equation ( 14 ) and from equation ( 10 ), the normalized space average s of acid within the diffused length will be given by ## equ12 ## where i 0 is the aerial image and f , f - 1 represent fourier transformation to ξ , n space , inverse fourier transformation from ξ , n , respectively , and σ b is a fitting parameter . equation ( 15 ) shows a relation where the aerial image intensity i 0 is transformed to acid concentration distribution s which is defined as a diffused aerial image i d . since i 0 and s are normalized and have a linear relation , they have a one - to - one relation . the threshold model of the aerial image expect the pattern size only using i 0 . it can be expected that there are very effective that if using s of equation ( 15 ) instead of i 0 , since the diffusion and amplification are already included , it approaches the resist process , since convenience using the threshold model as itself maintained . a diffused aerial image model ( daim ) represents expectancy pattern states using equation ( 15 ). since i 0 value can be easily obtained by using a general simulation tool , and f , f - 1 , which are fourier transformation , inverse fourier transformation , respectively , can be programmed in accordance with &# 34 ; fast fourier transformation ( fft )&# 34 ; and thereby calculating s with ease . of course , when s is calculated using equation ( 15 ), there is an unknown quantity σ b which corresponds to the diffusion length . in the daim , σ b is obtained from a simple experiment and then its resultant value is used after extending to the general case . namely , σ b is used as the fitting parameter . it is expected that if the resist ( i . e ., resist &# 39 ; s type , thickness , soft bake , peb condition , and development recipe ) are determined , σ b will be determined uniformly , if the recipe of the resist process are different , σ b will also have a different value . thus , it is more rational that σ b is not the diffusion length but a representative value for the recipe of the resist process . fig3 illustrates a diffused aerial image of 0 . 25 μm dense lines l / s , isolated lines i / s , and isolated space i / s after obtaining s from the aerial image depicted in fig2 by using equation ( 15 ), and in particular , shows a result when σ b is defined as 0 . 7 μm . if the cutting level is defined as 0 . 25 same as fig2 case , the expected pattern sizes are w l / s ≈ 0 . 16 μm , w i / l ≈ 0 . 21 μm , and w i / s ≈ 0 . 31 μm and is somewhat different to the aerial image . several embodiments utilizing the daim according to such manner will be described hereinafter . fig4 a and 4b belong to cases of expecting the pattern size only using the aerial image , and illustrate both experimental results for pattern sizes ( for instance , 0 . 25 μm in fig4 a and 0 . 20 μm in fig4 b ) of dense lines l / s , isolated lines i / s , and isolated space i / s , and comparison results for each aerial image . in fig4 a and 4b , and in other figures which are depicted hereinafter , a thin line and  ∘ represent patterns obtained by the simulation and experimental results when the resist pattern is formed , respectively . here , e th is not an actual measurement value but a defined value so that aerial image size of 0 . 25 μm dense lines becomes 0 . 20 μm . it is not easy for the aerial image size to explain variation of the pattern size with respect to the variation of the exposure energy in the case where the pattern types are different , as depicted in fig4 a and in a type of pattern . further , when patterns whose sizes are different exist concurrently as depicted in fig4 b ( in the present embodiment , expectancy for 0 . 20 μm when is simulated base on 0 . 25 μm ), the aerial image size is not even expected . fig5 a and 5b belong to cases of expecting the pattern size using the diffused aerial image , and illustrate both experimental results for pattern sizes ( for instance , 0 . 25 μm in fig5 a and 0 . 20 μm in fig5 b ) of dense lines l / s , isolated lines i / s , and isolated space i / s and comparison results for each aerial images . of course , in the case of daim , even if it is not equal to the experimental results completely , it shows amazing improvement which is not comparable to a case of using only the aerial image . thus , the daim can be applied in case that the dense lines l / s , isolated lines i / s , and isolated space i / s exist concurrently and the size is spitted . fig6 a , 6b and 7a , 7b belong to cases of expecting the pattern size using the aerial image and the diffused aerial image , where soft bake condition is determined by 105 ° c ., and illustrate both experimental results for pattern sizes ( for instance , 0 . 25 μm in fig6 a and 7a and 0 . 20 μm in fig6 b and 7b ) of dense lines l / s , isolated lines i / s , and isolated space i / s and comparison results for each aerial image . it is noted that the result using the diffused aerial image as depicted in fig7 is more superior to that obtained by only using the aerial image as depicted in fig6 . in this case , it is well known that if the soft bake temperature condition is changed , e th is also different , meaning that there is a possibility that the fitting parameter σ b will be changed in the daim since the recipe of the resist process is basically changed . even though the two cases well describe the experimental result using σ b = 0 . 7 μm , it does not mean that the diffusion length of acid is not changed even if the soft bake temperature is changed . to be exact , while σ b of the two case is different , it would be assumed that all fitting values , which are matched to the experimental results , are around 0 . 7 μm . it is a general fact that the linearity of the pattern size is not granted in a contact hole pattern . so , when each size &# 39 ; s contact hole is defined in cells and peripheral circuit area such as a poly 2 contact p2c or a metal 1 contact m1c , if the exposure is performed based on the cell , the size of the contact hole of the peripheral circuits will be much bigger . fig8 a and 8b are graphs illustrating the linearity of the pattern size , and fig4 a and fig4 b belong to the cases of expected size only using the aerial image and of using the daim , respectively . when 0 . 225 μm contact hole is exposed without a bias , or is over or under exposed , the size of the di pattern of the contact hole , which has mask pattern sizes of 0 . 25 μm , 0 . 275 μm , and 0 . 30 μm may be more than 1 . 5 times larger than is designed . in the case that the contact hole has other sizes which exist in a mask concurrently , if the daim is used , all corresponding sizes can be expected . thus , when the pattern size on the wafer is determined , it is easy to determine the size of the mask pattern which can be satisfied with this determination . of course , differences of albed of the underlayer or influence caused by topology are not considered in the daim of the present embodiment . in the case of i - line , the resist process does not have much difference with respect to the case of duv basically studied above . it will be expressed simply due to the fact that the resist process is not the chemical amplification resist . thus , it is acceptable that the diffusion of broken pag is considered instead of the diffusion of acid , it is not necessary to consider the amplification . thus , if it defines ni to be the normalized concentration of broken pag of the i - line , by applying the proximity relation of equation ( 14 ) as it is , from equation ( 7 ), we may obtain : ## equ13 ## when equation ( 16 ) is applied , it is more suitable that σ b is not the diffusion length but that it is treated as the fitting parameter . fig9 a , 9c and 9e illustrate experimental results in accordance with the defocus when the pattern sizes are formed with 0 . 30 μm , 0 . 35 μm , 0 . 40 μm and a simulation pattern size is obtained by the aerial image . fig9 b , 9d and 9f illustrate experimental results in accordance with the defocus when the pattern sizes are formed with 0 . 30 μm , 0 . 35 μm , 0 . 40 μm and a simulation pattern size is obtained by the diffusion aerial image . according to the fig9 a through 9f , it is noted that when the dense line l / s and the isolated line i / s patterns are defined using i - line , the simulation manner which the daim of the present invention is applied is more superior to that of only using the aerial image . wherein , that made of the dotted lines represent patterns by the simulation and that depicted as  ∘ represents experimental results where the resist pattern is formed . further , fig9 a through 9f compare the aerial image simulation results with the daim simulation result for the variation of line width caused by the defocus , when the exposure energy is determined that 0 . 35 μm dense lines l / s is of no bias . in this case , it is noted that the reliability of the daim will fall since the development contrast of the case of i - line is not superior to that of the case of the duv . the following shows the extreme effectiveness of the daim clearly and is applied to the development of acctual device fabrication . fig1 a illustrates a mask pattern , fig1 b is a top view , in which 5 bridges occur , pictured by critical dimension scanning electron microscope ( cdsem ) when the resist pattern is formed using the mask depicted in fig1 a , fig1 c illustrates that any bridge is not shown when the simulation is performed by only using the aerial image . also , fig1 d illustrate that the bridge occurrs when the simulation is performed by using the diffused aerial image . according to fig1 a through 10d , not only is it not easily expected that of 5 bridges as depicted in fig1 b by only using mask layout , that the simulation results for the aerial image can not explain the bridge phenomenon . while the simulation results of the aerial image is obtained from one focus and one energy , the experimental results cannot be explained even if the energy level and defocus are considered . fig1 a illustrates a mask pattern , which is revised from the mask shown in fig1 a in accordance with the daim as applied therein , fig1 b is a top view , in which bridges do not occur , pictured by a cdsem when the resist pattern is formed using the mask depicted in fig1 a , and fig1 c is a view of when the simulation is performed by only using the aerial image . also , fig1 d is a view when the simulation is performed by using the diffused aerial image . according to fig1 a through 11d , not only the daim clearly explains the bridge phenomenon but also the result that relative difference of bridge degree is identical to the experiment is shown , thereby easily identifying the daim verification in an actual application . after that , the revision of poly 1 mask using the daim is very successful as depicted in fig1 . as described above , the proximity effect of the optical lithography is divided an element causing diffraction of light and an element causing the resist process . while opc which is generally referred means the correction of proximity effect stretching over the full lithography process , and a lot of simulation tools , which are being commercialized , consider the aerial image until now . accordingly , it can be insisted that the daim of the present invention is a suitable model to apply to the opc , since it not only provides the type that fourier transformation is added to the calculation of the prior aerial image , but also the calculating speed is almost maintained at a level of calculating the aerial image and the exactitude is more superior to the aerial image . further , the daim of the present invention provides several advantages as follows . first , the theory described as above is well matched when the resist thickness is very thin . second , the most important point of the resist process is the diffusion of acid or broken pag . fourth , the concentration distribution of the non - protection group in the development process is developed along with the constant contour . fifth , the daim is expressed by the convolution of the aerial image linked with a term which represents the diffusion ( gaussian function ) and the amplification ( gaussian integral type ), mathematically . the present invention has now been described with respect to a presently preferred embodiment with some descriptions of alternatives . however , additional embodiments and variations of the present embodiment would be apparent to those skilled in the art .