Since the recent dramatic growth slowdown in the mobile phone market, studies of wearable devices are actively being carried out to find a new market for wearable devices. Among the many types of wearable devices, the head mounted display (HMD) could be one of the most approachable products in the optical field [1]. According to Tractica, a foreign professional research center, virtual reality-related markets for HMD products are expected to grow explosively, to up to 21 trillion KRW by 2020 [2]. The HMD is a kind of viewfinder applied optical system where the optical part is detached from the camera to be worn on the face [3]. There are two types of HMDs [4]: a see-through type which allows the users to view the projected images, as well as to realize the augmented reality; and a see-close type where the user can only view the virtual reality on a display screen.

The HMD proposed in this paper must have the smallest size and weight possible to be worn on a face like glasses. Therefore, we used plastic materials for less weight and included the least number of lenses for smaller size. In Korea, people aged 65 or older reached 13.2% of the total population in 2015, and we are rapidly entering into an aging society [5]. For effective use by elderly people with presbyopia, whose crystalline lens has lost its elasticity and flexibility due to aging, a focus-adjustable product design was adopted in this paper. After considering the proper size of the product, the proposed system was designed to realize the same sensation as watching a 56" screen from a distance of 2 m, which corresponds to a field of view (FOV) of approximately 40°. In addition, the product must be focus-adjustable because each person has unique eyesight. Therefore, the system to be designed in this paper must have a lens that can be shifted to adjust its focus while the overall length of the product remains the same.

Measurement differences between designed parts and fabricated parts are called errors, and the error factors may be divided into symmetric and asymmetric ones. The symmetric error factors are the errors with respect to rotational symmetry about an optical axis, such as the number of Newton fringes, thickness, refractive index errors, etc., and the asymmetric error factors are the ones with no symmetry with respect to an optical axis, which are caused by lens tilt and apparatus fabrication errors, such as decentering, tilt, etc. [6]. Here, the error factors affecting the back focal length (BFL) include only symmetric error factors such as curvature of each side, thickness, and index. [7]. Only the rotational symmetric tolerance was calculated in this paper, and the focus-adjusting lens was shifted in order to adjust the image surface variations resulting from the symmetric error factors.

In order to evaluate mass production in terms of pricing, a tolerance analysis for afocal optical systems, such as HMDs, was performed to determine the fabricating errors of the acceptable parts, and the corresponding yield was calculated [8]. The design technique and tolerance analysis method for the afocal optical system proposed here may be applied to other fields, and this method can be widely used in camera viewfinders, not to mention HMDs, and afocal system products [9]. Besides HMDs, some afocal optical systems may also be applied to Head Up Displays (HUDs), which reflect the enlarged screen of a small image element onto a windshield [10]. Moreover, the basic design method of HMD optical systems can be used as is because the optical system used for the Fundus camera, which is not an afocal system but a retinoscope, is similar to an HMD optical system [11]. Therefore, this paper will specifically describe how to develop an HMD optical system, and related applications that are highly marketable, for tolerance analysis.

The rays from an object go through an eye-piece system, enter the human eye, and then focus on the retina, and here the eye works as a video receiver [12]. To see the object and focus the images on the retina, the rays must be parallel to the eyes. An optical system which allows the parallel light to be incident on the last surface of the optical system (the eye) is called an afocal optical system [13].

In Fig. 1, the eye is located where the stop is, and the image element is placed where the image surface is. A virtual image is seen as the image hits the retina after the image ray passes through the optical systems and reaches the stop, almost parallel to the eye. The larger the diameter of the stop, the wider the range of eye movement, which makes for easier viewing of the image; yet, this increases the design difficulty. In general, the diameter of the person’s pupil under an interior reading light is about 2 mm. However, considering the movement of the eye, we propose the stop size of the Type A design to be 8 mm and that of Type B to be 6 mm, as will be shown in Chapter 3.

In the afocal optical system, the lateral aberration is expressed in the angular difference between the chief ray and an arbitrary ray for aberration calculation at the stop surface, where the aberration is expressed in angular units [14]. Thus, the reciprocal of the distance between the stop surface and the convergence point of the chief and the arbitrary rays needs to be utilized. In general, the longitudinal aberration is expressed in diopters, the reciprocal of the meter unit, in the afocal system.

The Gaussian bracket is used to calculate the exact lens shift, but the longitudinal sensitivity may be used for an approximate calculation when the lens shift is minimal. Because the lens shifts very little while correcting the variation of the image surface in this system, we used the longitudinal sensitivity to calculate the approximate lens shift. Eq. (1) is used for calculating the longitudinal sensitivity, where m_i refers to a transverse sensitivity of the i^th group, N refers to the number of groups, and p_i refers to the longitudinal sensitivity of the i^th group. Eq. (2) is also used for calculating the approximate changes of the image surface when the i^th group is shifted, where Δf refers to the changes in the image surface, and Δa_i refers to the amount shifted for the i^th lens. Additionally, Δf in Eq. (3) refers to the changes in the image surface, which can be derived from the Gaussian Imagery Equation. Here, l refers to the first principal point and f refers to the effective field length (EFL).

In a product fabrication with design data determined by the optical design, the actual production of the product is not in general the same as originally designed due to the limitations of the fabricating and assembly precision of the parts. If an error occurs during the fabricating stage of the product, the resolution, such as the modulation transfer function (MTF), usually becomes lower than the designed value. However, a lower resolution than designed does not necessarily mean it is defective. Despite the somewhat lower resolution performance, the product can be regarded to be of fair quality if it meets the initial target performance. The optical system is designed to have a higher performance than the initial target during the design stage in order to compensate for the performance degradation. The maximum limit of acceptable errors to satisfy the target optical performance is called the tolerance.

Only the symmetric tolerance was calculated in this paper and the symmetric tolerance factors include the curvature, thickness, and refractive index, which are the major design variables of the optical system. Because the aberration changes by the tolerance are minimal in this paper, having very little effect on the yield, we only considered the yield with respect to the BFL. In addition, the performance changes of the systems when a certain tolerance is applied to the design variables are called the sensitivity, and the yield of the product can be calculated using the system sensitivity with respect to each tolerance.

The system performance can be determined by the sensitivity analysis with respect to the tolerance. Here, we used the longitudinal aberration rather than the MTF to obtain the system performance because it is much faster to calculate the aberration than the MTF, and the performance changes from the errors at each surface can be linearly summed.

Eq. (4) is an expression for the BFL in terms of the curvature, thickness, and refractive index. Eq. (4) can also be expanded using a Taylor expansion of the design variables. Here, the Taylor expansion must be expanded using the given initial design values, where, R, d, and n in Eq. (4) refer to the radius of curvature, thickness, and refractive index of each product, respectively; and R₀ is the initial design value of the radius of curvature, d₀ is the initial design value of the thickness, and n₀ is the initial design value of the refractive index. Lastly, f_BFL is a function of the BFL in terms of R, d, and n; and f_BFL0 refers to the initial BFL.

The number of terms in Eq. (4) will increase depending on the lenses in the optical systems. In order to find out the performance changes, however, the variation in the image surface (ΔBFL) due to variations in the design variables must be obtained. To solve for ΔBFL, Eq. (5) is used, expressed in terms of the difference of each term in Eq. (4) and the design variables. In Eq. (5), ΔR = R − R₀, Δd = d − d₀, and Δn = n − n₀ are the variations in the radius of curvature, thickness, and refractive index, respectively, which represent the errors in the initial design values. Each design variable’s differential for BFL indicates the sensitivity of the corresponding tolerance item. This sensitivity becomes the proportional coefficient to find the ΔBFL of the optical system when a certain tolerance is applied to the design values [16]. Because there is almost no interaction and the effect of higher-order terms caused by the combination of the variations in each design value, the linear approximation would work relatively well. Therefore, ΔBFL can be expressed approximately as the right side of Eq. (5) [16].

There are spherical aberrations, which degrade the on-axis resolution, and astigmatism, which compromises the off-axis resolution. Besides, if the optical system, like HMD, has a large distortion, the screen will look as if it rotates depend on the variation of the pupil position. This is called the swing phenomenon, which causes dizziness when wearing an HMD optical system. This phenomenon is similar to the keystone that occurs in the projection system [17]. Therefore, distortion should also be included as one of the evaluation factors in an HMD optical system even though it does not lower the resolution. Along with the BFL, spherical aberration, astigmatism, and distortion should be added to the performance evaluation factors. In Eq. (5), spherical aberration, astigmatism, and distortion should be considered rather than the BFL for the same calculation, and the sensitivity for each design variable must be calculated. However, due to the nature of the afocal optical system, the eye focuses on each image surface, and almost no performance degradation from the image surface curvature is detected, so it can be omitted.

Even if each lens and device part is fabricated or assembled within the tolerance in general, assembling those parts without additional adjustments would cause performance degradation and lead to a low yield. As a workaround, and to increase the yield, the value of an arbitrary design variable is adjusted to correct the image surface shift and aberration changes. Among the many design variables of the optical systems, the curvature and refractive index cannot be used as adjustment variables in this paper because they are fixed in the product fabrication stage. The HMD optical system to be designed in this paper has a movable lens for focus adjustment. By using this, the image surface shift and aberration changes due to the tolerance will be adjusted. Here, the space before and after the movable lens must satisfy the values for the focus adjustment as well as for the performance adjustment. If the space is not enough, the tolerance range must be reduced. If the tolerance range cannot be reduced due to the fabrication technique limitations or cost, the optical system must be redesigned. The greater the tolerance range, the easier it is for parts fabrication and assembly, which enables the lower unit price; however, the optical system performance will be compromised. Therefore, the appropriate tolerance level should be assigned by considering the cost of production as well as fabrication and assembly techniques.

The product error, on the other hand, would have a statistical distribution for the quantity and quality management of the products, where ΔR, Δd, and Δn become the probability samples with a particular statistical distribution [18]. Thus, the aberration function expressed as a linear combination of the design variables has a statistical distribution proportional to the error. The function describing the statistical distribution is called a probability density function, and the accumulated probability density function is a distribution function [19, 20]. Among the many types of probability density functions, the most typical one is the normal distribution [18]. In case the design variables are not well managed and randomized fabrication and assembly errors exist, they would have the same probability distribution for the error range within the tolerance. Such a statistical distribution follows a uniform distribution, which is the worst-case scenario of totally unmanaged fabrication and assembly errors. However, if the design variables are distributed around the design values thanks to a well-managed fabrication and assembly process, they follow the normal distribution. The statistical distributions of the errors in this paper are assumed to follow the worst-case scenario of a uniform distribution. The standard deviation of the normal distribution is generally that of the standard deviation of the uniform distribution [21]

Even if the design variables, such as the fabrication and assembly errors of the parts, follow the uniform distribution, the performance distribution for the changes of the overall design variables will follow the normal distribution by the central limit theorem [22]. The top quality management level in the mass production stage would be achieved when the standard deviation of the statistics following such a normal distribution becomes 1/6 that of the range of acceptable performance; that is, when the range of acceptable performance is six times the standard deviation of the performance distribution [23]. Considering the unknown causes of fabrication errors at this point, the average of the performance distribution may be slightly different from the design values. In general, an average shift resulting from the unknown causes of fabrication errors of as much as 1.5 times the standard deviation (σ) is acceptable in the quality management [24]. Taking into account the average shift in reality, the highest level of quality management would be achieved when the acceptable performance range become 4.5 times the standard deviation of the performance distribution [24]. This corresponds to 6.7 defective parts per million (DPPM); that is, a theoretical minimum defective percentage [24]. If enough lens shift distance can be obtained to adjust the ΔBFL distribution of 4.5σ, which occurs during the fabrication and assembly, the maximum yield may therefore be achieved.

Two types of optical systems were designed in this paper: Type A, which is relatively large, and Type B, which is relatively small, as will be shown in Chapter 3. The yields of Type A and Type B can be calculated with Eq. (6), where σ refers to the standard deviation, DOF refers to the depth of focus, x refers to the units of each product, and μ refers to the mean value of each production unit with respect to the design variables.

The optical system in this paper was designed to compensate the ΔBFL by adjusting the focus-adjustable lens in order to increase the yield, and Eq. (2) can be used to determine the lens shift when the image surface varies by as much as 4.5σ. If there is not enough space for the lens to be moved forwards and backwards, the optical system must be redesigned. Otherwise, the system will meet the theoretical maximum yield.

Based on the data for the initial optical system in Tables 1 and 2, the Type A optical system was determined [26]. The Type A is an optical system with a diagonal HFOV (Half Field of View) 20 degrees and its specifications include an entrance pupil diameter of 8 mm, and an HD (1280 × 720) level of pixels. The distance between the human eye and the first lens from the eye is called the eye relief [27]. The eye relief was set to 15 mm to allow for glasses wearers to use it as well. For the optical layout in Fig. 2, an image from an image surface is focused on the inner part of a beam splitter by a relay system, then the rays are collimated at the stop after passing through the optical parts with the reflector.

Figure 3(a) is the layout of the optical system with an adjusted ratio of image size: the EFL to meet the system specifications before designing the Type A. Moreover, in order to design the smallest possible optical system, the refracting power of the cemented and above convex lenses (14^th to 18^th surfaces) was set to be identical to be that of the cemented lens and the above single convex lens (17^th to 18^th surfaces); they were replaced with a single lens. Figure 3(b) is the optical layout of the system where the refracting power of 14^th to 18^th surfaces is replaced with a single convex lens (17^th to 18^th surfaces).

An aspherical lens was used instead of a spherical lens to correct the distortion, and the 4^th order aspherical coefficient was calculated to make the Seidel aberration for the distortion zero. Distortion (DST) of the present optical system is 0.049438 mm (0.783%). In the present lens data, the 12^th and 13^th surfaces are spherical, but the 12^th surface was considered to be aspherical to calculate the 4^th order aspherical coefficient.

In Eq. (7), Seidel aberration coefficient S^*_Vi is used for correction of the distortion by the aspheric effect of the i^th surface; K_i is a conic coefficient of the i^th surface; c_i is a curvature of the i^th surface; and A_4i is the 4^th order aspherical coefficient of the i^th surface. Eq. (8) is derived from Eq. (7) to find the 4^th order aspherical coefficient to remove the 3^rd order distortion for the entire surface. Here, DST is the 3^rd order distortion for the entire surface. h_i refers to the paraxial image height of the axial ray at the i^th surface, hi refers to the paraxial image height of the chief ray at the i^th surface, u_i refers to the paraxial angle of the axial ray at the i^th surface, ui refers to the paraxial angle of the chief ray at the i^th surface, n_i refers to the refractive index of the i^th surface, r_i refers to the radius of curvature of the i^th surface, n refers to the refractive index of the image surface, and u refers to the paraxial angle of the axial ray at the image surface. Eqs. (7) and (8) are used to make the 3^rd order distortion for the surfaces of the system zero. Note here that the calculated result of A_4i in Eq. (8) is to offset the resultant aberration; therefore, the opposite sign to the existing aberration must be used.

The calculated 4^th order aspherical coefficient of the 12^th surface is −0.147850, and the 3^rd order distortion after applying this value is zero. By applying the lens data in Table 3 and the 4^th order aspherical coefficient of the 12^th surface, the initial design data of the Type A were obtained. The design data in Tables 4 and 5 were also obtained through optimization, and the optical layout of the final Type A is shown in Fig. 4. The final system was designed to be smaller than the initial system in Fig. 2 (overall length of 101.27 mm, beam splitter length of 26 mm). Type A was designed within −4 D to +1 D to allow focus adjustment, and Fig. 5 shows the layout of the longitudinal aberration of Type A in the case of −1 D.

The flow chart in Fig. 6 describes the design process for Type B. The Type B system is designed based on an existing viewfinder patent (JP2012-093478 EX1). The optical layout as shown in Fig. 7 can be obtained when the patent information is applied. The specifications of the Type B are the same as that of the Type A, i.e., EFL of 17.5 mm, eye relief of 15 mm, and HD. However, because of decreased freedom in the design for using fewer optical surfaces, the Type B was designed to have a smaller diameter of the entrance pupil (EP) of 6 mm, compared to the Type A. First, the image size and EFL ratio were adjusted to meet the target specifications of the system. Here, it is important to adjust the ratio of the surface next to the stop to avoid a possible decrease in eye relief. Because the initial data for Type B mainly consists of the radius of curvature, thickness, and refractive index, the surface type of every surface is changed to a spherical one.

The thickness of the biconvex lens (4^th-5^th surfaces) was increased to 10 mm discretely to ensure the space for a beam splitter. It will change the EFL of the biconvex lens; to keep the EFL the same as before the thickness change, the curvature of the front or back surface of the biconvex lens must be changed. To do so, the EFL equation for a thick lens is used to find the accurate curvature of the front or back surface of the biconvex lens. From the two choices of curvature between the front (4^th) and back (5^th) surfaces of the biconvex lens, the curvature of the surface with the smaller ray refraction angle should be chosen to ensure little change in the performance. Since the front (4^th) surface of the biconvex lens has a relatively smaller ray refraction angle, the curvature of the front surface can be changed to realize the existing EFL before the thickness of the beam splitter is increased.

Eq. (9) is an equation of the EFL for thick lenses, and this can be used to obtain Eq. (10). In Eqs. (9) and (10), c₁ and k₁ refer to the curvature and the refracting power of the 1^st surface of the biconvex lens, c₂ and k₂ refer to the curvature and the refracting power of the 2^nd surface of the biconvex lens, K refers to the overall refracting power of the biconvex lens, n₁ refers to the refractive index of the biconvex lens, and d₁ refers to the thickness of the biconvex lens.

Using Eq. (10), the radius of curvature of the front surface of the biconvex lens was found to be 8.651 mm so that the EFL would have the same value as before changing the thickness of the beam splitter. However, because the curvature and thickness of the biconvex lens were changed, the distance of principal points between the biconvex lens and the concave lens would also be different from that of the initial system. To maintain the characteristics of the initial system, the distance between the two lenses must be adjusted to correct the position of each lens with the interval of the principal points of the initial system. Table 6 shows the principal points of the biconvex lens before and after the change, as well as the principal points of the concave lens.

The distance between the secondary principal point of the biconvex lens before changing its properties and the primary principal point of the concave lens is 2.99768 mm; and the distance between the secondary principal point of the biconvex lens after changing its properties and the primary principal point of the concave lens is 5.75573 mm. Because of the increased distance of 2.75805 mm between the principal points, the thickness at the 5^th surface, the distance between the biconvex lens and concave lens, was reduced by as much as 2.75805 mm to make it the same as the distance between the principal points of the initial system.

Although the overall EFL of the current system was set at the initial design target of 17.5 mm, the thickness of the 5^th surface between the biconvex and concave lenses turned out to be a negative value of −2.04162 mm. In this case, physical fabrication of the product would be impossible. Therefore, the thickness should be adjusted to an arbitrary positive number. To do so, the distance between the biconvex lens and the concave lens needs to be adjusted to an arbitrary value and the Gaussian bracket is used to keep the overall EFL unchanged. Then, the thickness between the concave lens and the meniscus lens (8^th-9^th surfaces) can be found.

Here, in Eq. (11), k₁ refers to the refracting power of the biconvex lens, z₁ refers to the distance between the secondary principal point of the biconvex lens and the primary principal point of the concave lens, Δz₁ refers to an arbitrary value added to make the distance between the biconvex lens and the concave lens a positive value, k₂ refers to the refracting power of the concave lens, z₂ refers to the distance between the secondary principal point of the concave lens and the primary principal point of the meniscus lens, k₃ refers to the refracting power of the meniscus lens, and K refers to the overall EFL of the target specifications. d₂ in Eq. (12) refers to the distance between the concave lens and the meniscus lens, H₂ refers to the secondary principal point of the concave lens, and H₁ refers to the primary principal point of the meniscus lens.

Here, 3.1 mm is added to the thickness of the 5^th surface, the gap between the biconvex lens and the concave lens, to ensure an approximately 1 mm space for focus adjustment with the concave lens. In Eq. (12), d₂ refers to the thickness of the 7^th surface, and the calculated value of 9.80294 mm was obtained when considering the secondary principal point of the concave lens and the primary principal point of the meniscus lens (0.91865 mm). Because the image surface has a negative value and exists within the lens, however, the thickness of the 7^th surface was reduced by as much as 8.4 mm at our discretion to obtain a positive position for the image sensor. Then, the rate was adjusted as per the overall EFL to get the design data in Table 7.

In order to have a system with minimized aberration, the aspheric 4^th order coefficient was used to adjust the 3^rd order aberration. The curvature of the field is not important due to the characteristics of the afocal system, thus other aberrations except the curvature of the field were adjusted. However, if the spherical aberration, coma aberration, astigmatism, and distortion are all adjusted, it would not be easy to fabricate the shape of the system; thus, only the 3^rd order aberrations for the three types of aberration except the spherical aberration, which is relatively small, were adjusted. The aberrations to be adjusted in the present optical system include the tangential coma (TCO) of −0.533186 mm, tangential astigmatism (TAS) of −1.301066 mm, sagittal astigmatism (SAS) of −0.528742 mm, and DST of −1.110859 mm. The TCO and DST need to be adjusted to zero, and the SAS and TAS to have the same value.

In Eq. (13), S^*_Iii refers to the Seidel aberration coefficient of the coma aberration due to the aspherical effect of the i^th surface, K_i refers to the conic aberration of the i^th surface, c_i refers to the curvature of the i^th surface, and A_4i refers to the 4^th order asphericity of the i^th surface. Eq. (14) was derived from Eq. (13). It is used to find the 4^th order asphericity to eliminate the coma aberration of the zero direction for the entire surface, where the TCO is the 3^rd order coma aberration of the entire surface. In Eq. (15), S^*_IIIi refers to the Seidel aberration coefficient of the astigmatism due to the aspherical effect of the i^th surface, and Eq. (15) can be derived from Eq. (16). Eq. (16) is to find the 4^th order asphericity to eliminate the astigmatism for the entire surface. n_i refers to the refractive index of the i^th surface, n refers to the refractive index of the image surface, u refers to the paraxial angle of an axial ray at the image surface, h_i refers to the paraxial image height of an axial ray on the i^th surface, and hi refers to the paraxial image height of the chief ray on the i^th surface. For the 3^rd order distortion, Eqs. (7) and (8) are used.

To adjust the Seidel aberration using the aspherical coefficient, just the A₄ instead of K may be used. If just the A₄ were used, three different equations would be needed in total to adjust the coma aberration, astigmatism, and distortion. If the 4^th order aspherical coefficients of the 4^th, 5^th, and 8^th surfaces were used to adjust the astigmatism, coma aberration, and distortion, respectively, Eq. (17) would be used to find the 4^th order aspherical coefficient solutions for those three equations for all zero aberrations at a time.

As shown in Fig. 8, the aspherical coefficients where the TCO and DST are zero and the difference between TAS and SAS is zero are A₄ = −8.992560e^-5 at the 4^th surface, i.e., the 1^st surface of the biconvex lens; A₄ = 1.112215e^-4 at the 5^th surface, i.e., the front of the concave lens; and A₄ = −1.221848e^-4 at the 8^th surface, i.e., the front of the meniscus lens. Figure 9 is the optical layout of the adjusted aberrations data. Figure 8 shows the 3^rd order aberrations after adjusting the Seidel aberration for the 4^th, 5^th, and 8^th surfaces.

The lens data in Table 7 and the 4^th order asphericities at the 4^th, 5^th, and 8^th surfaces were used for the initial design data of the Type B system. Afterwards, these data were optimized to design the Type B. Table 8, Table 9, and Fig. 10 show the design data and the layout of the final Type B system. The Type B is also a focus-adjustable design in the range of −4 D to +1 D, and Fig. 11 is the longitudinal aberration layout of the Type B in the case of −1 D. Table 10 shows the comparison between the Type A and Type B.

Using Eqs. (1)-(3), the longitudinal sensitivity, image surface shift at 1 D, and lens shift of the Type A and Type B can be calculated. The Type A and Type B systems in this paper were designed to be focus-adjustable within −4 D to +1 D, where the image surfaces of each system were shifted by as much as 0.3142 mm and 0.35949 mm, respectively, as the image surface (image sensor in this case) changed by up to 1 D. Table 11 shows the image surface shift and lens shift due to the change of 1 D, which is calculated using the longitudinal sensitivity and DOF.

Figures 12 to 17 show the aberration changes when a certain tolerance is assigned to the designed optical systems. When the tolerances of the five-fringe Newton Ring for the radius of curvature, 0.05 mm for the thickness, and 0.001 for the refractive index were assigned, the changes of each aberration can be expressed in terms of the sensitivity for each tolerance. Here, the material of the cover glass is BSC7 by the Hoya Corporation, which allows a tolerance of 0.0003. Because the off-axis does not carry more weight than the on-axis in general, the sensitivity data was calculated at the area where its FOV is 0.8 times that of the object side. Table 12 lists the standard deviations of the Type A and Type B, which can be used to calculate the yield.

Since the variations of longitudinal spherical aberration and astigmatism are all less than 1 D and maximum changes in the distortion (−0.270%, +0.084%) are very little for both Type A and Type B, they have little effect on the yield. However, the standard deviations of the image surface shift were greater than the depth of focus (Type A: 0.0186 mm and Type B: 0.0255 mm) affecting the performance changes, which might cause defective products. Therefore, only the image surface shift was considered to get the yield. Eq. (6) was used for the yield of Type A and Type B, and it can be calculated using the Norm.dist function in Excel. This function returns the normal distribution for the given average and specific standard deviation. As a result, Type A and Type B for the current image surface shift had yields of 5.05% and 17.36%, respectively. To increase the yield, the system was designed to adjust its focus-adjustable lens to correct the image surface shift in this paper, and Eq. (2) was used to obtain the lens shift when the image surface was changed by as much as 4.5σ as in Table 13. The focus-adjustable lens in Type A and Type B must be shifted by as much as 0.65804 mm and 0.35923 mm, respectively. Unless there is enough space ahead of and behind the focus-adjustable lens, the system must be redesigned. Fortunately, this optical system had all the necessary spaces and satisfied the maximum theoretical yield.

A see-through type HMD optical system was designed to realize the augmented reality. Affordable plastic was used to lighten the system, and Type A system followed by Type B was designed in a smaller and different shape. In addition, a piece of lens was configured to be shifted from −4 D to +1 D for focus adjustment in order to compensate the individual eyesight.

Although fabrication errors are unavoidable during mass production, the product can be regarded to be of fair quality as long as the target performance is satisfied. The system yield was calculated after analyzing its sensitivity in relation to the tolerance, which is the maximum margin of error satisfying the system performance. Only the symmetric tolerance was dealt with in this paper, and a Newton ring of five fringes, thickness of 0.05 mm, and refractive index of 0.001 were assigned as the tolerances to analyze the sensitivity for each case. The system performance could be checked via the sensitivity analysis in relation to the tolerance, and a longitudinal aberration was used instead of the MTF. This was because the aberration can be calculated much faster than with the MTF, and the performance changes for the error at each surface can be linearly summed. The sensitivity analysis was performed on the longitudinal aberration and the variation of the image surface for the on-axis performance, and on the astigmatism and distortion for the off-axis performance. The yield was calculated only for the image surface shift with greater variation because the longitudinal spherical aberration, astigmatism, and distortion of Types A and B are relatively small. Here, a focus-adjustable lens was used to adjust the image surface shift for a greater yield. Because the shift in axis for the standard deviation was assumed to be 1.5σ with respect to the fabrication tolerance, the image surface correction was assigned as 4.5 times the standard deviation of the image surface shift to reach the maximum theoretical yield.