Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
About All Issues Feature Issues

Discussion of temperature, TV distortion, and lateral color of a 4-megapixel DLP projector lens

Wen-Shing Sun, Cian-Ming Huang, and Jhe-Syuan Lin

Open Access Open Access

Abstract

This paper explores lens design through thermal analysis and correction of 4-megapixel DLP projector lenses at temperatures ranging from 10°C to 80°C, and analyzes and discusses the TV distortion and lateral color with a focal length of 24 mm, F/# of 1.71, projection distance of 7.2 m, and projection screen size of 200 inches. Appropriate lens materials are selected and changes are made to the lens barrel materials to achieve athermalization. At temperatures from 10°C to 80°C, the image quality of lens is as follows: MTF (93 lp/mm) is greater than 0.554, lateral color is less than 1.39 µm, optical distortion is less than 0.147%, TV distortion is less than 0.079%, and relative illumination less than 92.52%.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Studies regarding the effect of temperature on lenses can be traced back to 1948. Grey [1] in their introduction of object-image relationships, proposed that the focal planes would varies with the temperature and developed a formula for calculation of the variation of the refractive index of a lens with the temperature. Duggin [2] and Horman [3] proposed temperature analysis algorithms for dual wavelength band application. Jamieson [4] and Rogers [5] studied color corrections for temperature changes for wide wavebands, and Tamagawa and Tajime [6] analyzed the design of dual band optical systems. Rayces and Lebich [7] discussed the athermalization of double-mirror and three-mirror systems, as well as the design of a method to reduce chromatic aberration. Conventional optical designs are based on the normal temperature of 22°C. When the temperature changes, not only do the physical parameters of the optical components change because of differences in their thermal expansion coefficients, but the refractive indices of the optical components are also altered by the temperature. The structure of the optical system is thus changed, which leads to the displacement of the image plane and the occurrence of defocusing, and consequently, degradation of image quality. Hence, compensation methods have been developed to prevent dramatic changes in image quality with temperature changes. The method for maintaining image quality and overcoming the effects of temperature changes is called athermalization. The compensation mechanism are mainly divided into two types [8]: active compensation and passive compensation. In active compensation, electronic and mechanical applications are combined. The current temperature is received through a temperature sensor installed on the body, and the information received is converted to a set of programmed calculations which can be used to calculate the movement needed by the compensating lenses at this temperature. The other method, passive compensation, can be divided into mechanical compensation and optical compensation. Mechanical compensation involves selection of the appropriate materials for the mechanism components to overcome the effects of environmental changes, in particular materials with high thermal expansion coefficients. When the temperature rises, materials with high thermal expansion coefficients move backwards from the last lens of the camera, facilitating refocusing to achieve athermalization. Optical compensation for temperature changes [9] is related to selection of the materials for lenses. In this compensation method, the temperature coefficients of the refractive indices and thermal expansion coefficients of the lens materials must be considered.

2. Design theory

2.1 Relation between the refractive index and wavelength

The refractive index of a light source moving through a medium varies with the different wavelengths of the light, the characteristic causing dispersion. The relation between the refractive index of glass and the wavelength light can be obtained from Eq. (1) [10]

$${n^2}(\lambda ) - 1 = \frac{{{B_1}{\lambda ^2}}}{{({\lambda ^2} - {C_1})}} + \frac{{{B_2}{\lambda ^2}}}{{({\lambda ^2} - {C_2})}} + \frac{{{B_3}{\lambda ^2}}}{{({\lambda ^2} - {C_3})}},$$
where, n(λ) represents the refractive index of a wavelength; B1, B2, B3, C1, C2 and C3 are the constant values of the dispersion formula for glass; and λ is the wavelength in units of µm. Hence, rays with different wavelengths travel at different speeds through the same lens, and the refractive indices are different.

2.2 Effects of temperature changes on lens design

Temperature changes will affect the refractive index of a lens, causing thermal expansion and contraction of the lenses and lens barrels, and can even affect the diopters of the lenses.

2.2.1 Relation between the refractive index of glass and temperature

The refractive indices of the media also vary with temperature changes; therefore, the effects of temperature on the refractive indices must be considered. When the temperature changes, the relation between the refractive index and temperature changes can be obtained from the Sellmeier dispersion formula, as shown in Eq. (2) [11]:

$$\frac{{dn(\lambda ,T)}}{{dT}} = \frac{{{n^2}(\lambda ,{T_0}) - 1}}{{2n(\lambda ,{T_0})}} \times \left( {{D_0} + 2{D_1} \cdot \Delta T + 3{D_2} \cdot \Delta {T^2} + \frac{{{E_0} + 2{E_1} \cdot \Delta T}}{{{\lambda^2} - \lambda_{TK}^2}}} \right),$$
where, T0 is the reference temperature of 22°C; T is the temperature in °C; $\varDelta$T is the difference between the temperature and reference temperature (22°C); λ is the wavelength of the light source in µm; D0, D1, D2, E0, E1 and λTK are the temperature coefficients; n(λ, T) is the refractive index of a wavelength at temperature T; and n(λ, T0) is the refractive index of the same wavelength at reference temperature (22°C)T0.

The refractive indices at different temperatures are shown in Eq. (3) [11].

$$n(\lambda ,T) = n(\lambda ,{T_0}) + \frac{{dn(\lambda ,T)}}{{dT}}\Delta T,$$
where $\varDelta$T is the difference between temperature T and reference temperature To(22°C).

2.2.2 Effects of temperature changes and the thermal expansion coefficient on lens parameters

When the temperature changes, the thermal expansion and contraction between lens groups will cause structural changes in the overall design, which affect the radius of curvature (R), thickness (D), air thickness (L), length of lens barrel, and aspherical coefficient. The factor α that affects these changes, is called the thermal expansion coefficient. The changes in the parameters of the lens after temperature changes are listed in Table 1 [12].

Tables Icon

Table 1. Temperature changes of lens parameters and the thermal expansion coefficient

View Table | View all tables in this article

Here, α is the of the thermal expansion coefficient of the lens material; however, regarding the air space, the thermal expansion coefficient (β) of the lens barrel material is changed.

The equation of an aspherical surface is shown in Eq. (4)

$$z = \frac{{C{r^2}}}{{1 + \sqrt {1 - (1 + K){C^2}{r^2}} }} + {a_4}{r^4} + {a_6}{r^6} + {a_8}{r^8} + \ldots \ldots .\textrm{,}$$
where z is the sag of the aspherical surface; C represents the central curvature of the surface; r is the vertical height from any point on the surface to the optical axis; a4, a6, a8 and a10 are high-order coefficients of the aspherical surface; and K is the conic constant.

2.2.3 dn/dt relation of diopter changes

The lens power ϕ is the reciprocal of the effective focal length. In a single lens, the changes in the lens power resulting from temperature changes, are shown in Eq. (5),

$$\frac{{d\phi }}{{dT}} = \phi (\frac{1}{{n - 1}}\frac{{dn}}{{dT}} - \alpha ) = \gamma \phi ,$$
where α is the thermal expansion coefficient of the lens and γ is the thermal glass constant; as show in Eq. (6)
$$\gamma = \frac{1}{{n - 1}}\frac{{dn}}{{dT}} - \alpha .$$
In Eq. (5), the variation of the lens power with temperature is proportional to the thermal glass constant. This method also applies to lens assembly. Therefore, the two conditions in Eq. (7) and Eq. (8) must be met for athermalization of the optical systems.
$$\textrm{Total power}:\sum\limits_{i = 1}^j {{\phi _i} = \phi } ,$$
$$\textrm{Athermalization :}\sum\limits_{i = 1}^j {\frac{{d{\phi _i}}}{{dT}} + \beta \cdot \phi = 0} ,$$
where ϕi is the power of the i-th lens i in a lens; ϕ is the total power; and β is the linear thermal expansion coefficient of the lens barrel material.

2.3 DMD (digital micromirror device)

This section describes the DMD specifications and DMD offset in the vertical direction.

2.3.1 DMD specifications

Table 2 shows the DMD specifications used in this work; the model number is TI DLP660TE 0.66′′ [13] and the number of pixels is 2716×1528.

Tables Icon

Table 2. TI DLP 660TE 0.66′′ specifications

View Table | View all tables in this article

2.3.2 DMD offset in the vertical direction

Let A be the half horizontal dimension of the DMD (A = 7.333 mm) and B the half vertical dimension of the DMD (B = 4.126 mm). The DMD is at the object plane of a projection lens and the projection screen is at the image plane of the projection lens. Now let C be the offset of the center point of the DMD on the object plane in the lens projection system, and the offset percentage is calculated, as shown in Eq. (9),

$$\textrm{y offset} = \left( {\frac{C}{B}} \right) \times 100\%,$$
The maximum height h of the lens object is calculated, as shown in Eq. (10)
$$h = \sqrt {{A^2} + {{({B + C} )}^2}} .$$
In this paper, the effective area of the DMD is 14.666 mm × 8.251 mm, and the downward offsetting of the central point of the DMD in the vertical direction is C = B = 4.126 mm, namely, the y offset is 100%, thus, the maximum object height (h = 11.038 mm) for the design can be calculated from Eq. (10).

2.4 Optical distortion and TV distortion

2.4.1 Definition of optical distortion

The optical distortion is defined as Eq. (11).

$$\textrm{optical distortion} = \frac{{\textrm{real image height} - \textrm{ideal image height}}}{{\textrm{ideal image height}}} \times 100\%. $$

2.4.2 Definition of TV distortion

Let the screen size be 200 inches, and the scale be 16:9. Let the diagonal height be 1.0 field; then the horizontal height is 0.87 field, the vertical height is 0.49 field. The dashed line shows the ideal image height and the solid line shows the real image height in Fig. 1. TV distortion can be divided into horizontal TV distortion and vertical TV distortion, where horizontal TV distortion is the percentage of the horizontal line warping Δ1 from 0.49 field to 1.0 field to the ideal image height hVp of the maximum vertical field, as in Eq. (12).

$${TV}{{D}_{H}} = \frac{{{\Delta _1}}}{{{h_{Vp}}}}100\%.$$
As shown in Fig. 2 and Eq. (13), vertical TV distortion is the percentage of vertical line warping Δ2 from 0.87 field to 1.0 field to the ideal image height hHp of the maximum horizontal field.
$${TV}{{D}_{V}} = \frac{{{\Delta _2}}}{{{h_{Hp}}}}100\%.$$

 figure: Fig. 1.

Fig. 1. Schematic diagram of horizontal distortion

Download Full Size | PDF

 figure: Fig. 2.

Fig. 2. Schematic diagram of vertical distortion

Download Full Size | PDF

2.5 Setting of DMD object field

In a projection optical system, the location of the DMD is set as the object, and the location of the screen is set as the image. In the optical system designed for this paper, due to the effects of 100% offset of the DMD in the vertical direction, its maximum object height is 11.038 mm, as shown in Fig. 3. Let the maximum object height of 11.038 mm be 1.0 field, then the horizontal height of 7.333 mm is 0.66 field and the vertical height of 8.251 mm is 0.75 field.

 figure: Fig. 3.

Fig. 3. Image circle of projection lens

Download Full Size | PDF

2.6 Definition and calculation of horizontal line warping on screen

2.6.1 Definition of horizontal line warping

On the screen, different degrees of distortion will cause horizontal line warping. Horizontal line warping in the screen image is formulated based on the field of the real image height. Figure 4 shows the horizontal line warping of the screen image, where h0.75R, h0.8R, h0.9R and h1.0R are the real image heights of 0.75 field, 0.8 field, 0.9 field and 1.0 field, respectively, and h⊥0.75, h⊥0.8, h⊥0.9 and h⊥1.0 are the real vertical heights of images of 0.75 field, 0.8 field, 0.9 field and 1.0 field, and h0.75R= h⊥0.75 respectively. The horizontal line warping (Δ1) is the value obtained by subtracting the minimum vertical height (h⊥min) from the maximum vertical height (h⊥max) of the real image height in the range from 0.75 field to 1.0 field, as shown in Eq. (14).

$${\Delta _1} = |{{h_{ \bot MAX}} - {h_{ \bot MIN}}} |.$$

 figure: Fig. 4.

Fig. 4. Horizontal line warping in screen image

Download Full Size | PDF

2.6.2 Definition of the angle between the ideal image height and its vertical height

To calculate the horizontal warping, the angles between the ideal image heights of various fields in the horizontal direction and their vertical heights must be defined. In Fig. 5, the red dashed line shows the ideal image height, and if we let the distortion be positive, then the black solid line represents the pincushion distortion. The colors of the arrows and the symbol descriptions show the ideal image heights, that is, h0.75p, h0.8p, h0.9p and h1.0p are the ideal image heights of 0.75 field, 0.8 field, 0.9 field and 1.0 field, respectively, and θ2, θ3 and θ4 are the included angles between the ideal image heights and their vertical heights of 0.8 field, 0.9 field and 1.0 field, respectively, namely, ${\theta _\textrm{2}} = {cos ^{ - 1}}({{h_{\textrm{0}\textrm{.75}p}}/{h_{\textrm{0}\textrm{.8}p}}} )$, ${\theta _\textrm{3}} = {\cos ^{ - 1}}({{h_{\textrm{0}\textrm{.75}p}}/{h_{\textrm{0}\textrm{.9}p}}} )$ and ${\theta _\textrm{4}} = {cos ^{ - 1}}({{h_{\textrm{0}\textrm{.75}p}}/{h_{\textrm{1}\textrm{.0}p}}} )$.

 figure: Fig. 5.

Fig. 5. The angles between the ideal image height in the horizontal direction and vertical height

Download Full Size | PDF

2.6.3 Calculation of horizontal line warping

To calculate horizontal line warping Δ1, which is the value obtained by subtracting the minimum vertical height from the maximum vertical height of the real image height in the range from 0.75 field to 1.0 field, as shown in Fig. 6, and the real vertical heights of all fields in the screen image must be calculated. For instance, for h⊥0.75, h⊥0.8, h⊥0.9 and h⊥1.0, the real image height of 0.75 field is its vertical height, namely h⊥0.75= h0.75R. The relations between the real image height and the vertical height of a field are ${h_{ \bot \textrm{0}\textrm{.8}}} = {h_{\textrm{0}\textrm{.8}R}}cos {\theta _2}$, ${h_{ \bot \textrm{0}\textrm{.9}}} = {h_{\textrm{0}\textrm{.9R}}}cos {\theta _3}$ and ${h_{ \bot \textrm{1}\textrm{.0}}} = {h_{\textrm{1}\textrm{.0}R}}cos {\theta _4}$, which are used to calculate horizontal warping.

 figure: Fig. 6.

Fig. 6. Warping in the horizontal direction

Download Full Size | PDF

2.7 Definition and calculation of vertical line warping on the screen

2.7.1 Definition of vertical line warping

Figure 7 shows the vertical line warping of the screen image, where h0.66R, h0.7R, h0.8R, h0.9R and h1.0R are the real image heights of 0.66 field, 0.7 field, 0.8 field, 0.9 field and 1.0 field, respectively, and $h_{\Vert0.66}$, $h_{\Vert0.7}$, $h_{\Vert0.8}$, $h_{\Vert0.9}$, and $h_{\Vert1.0}$ are the real vertical heights of images of 0.66 field, 0.7 field, 0.8 field, 0.9 field and 1.0 field, respectively, and h0.66R= $h_{\Vert0.66}$. As lens imaging is symmetric, the right and the left vertical lines have the same warping. The vertical line warping Δ2 is the value obtained by subtracting the minimum horizontal height $h_{\Vert\textrm{min}}$ from the maximum horizontal height $h_{\Vert\textrm{max}}$ of the real image height in the range from 0.66 field to 1.0 field, as shown in Eq. (15).

$${\Delta _2} = \textrm{ }|{{h_{{\parallel} max}} - \textrm{ }{h_{{\parallel} min}}} |.$$

 figure: Fig. 7.

Fig. 7. Vertical line warping in the screen image

Download Full Size | PDF

2.7.2 Definition of the included angle between the ideal image height and its horizontal height

To calculate the vertical warping, the angles between the ideal image heights of various fields in the vertical direction and its horizontal heights must be defined. As shown in Fig. 8, the red dashed line shows the ideal image height, and the black solid line represents the pincushion distortion. The colored arrows show the ideal image heights, that is, h0.66p, h0.7p, h0.8p, h0.9p and h1.0p are the ideal image heights of 0.66 field, 0.7 field, 0.8 field, 0.9 field and 1.0 field, respectively, and θ2′, θ3′, θ4′ and θ5′ be the included angles between the ideal image heights and their horizontal heights, respectively, then ${\theta ^{\prime}_\textrm{2}} = {\cos ^{ - 1}}({{h_{\textrm{0}\textrm{.66}p}}/{h_{\textrm{0}\textrm{.7}p}}} )$, ${\theta ^{\prime}_\textrm{3}} = {\cos ^{ - 1}}({{h_{\textrm{0}\textrm{.66}p}}/{h_{\textrm{0}\textrm{.8}p}}} )$, ${\theta ^{\prime}_\textrm{4}} = {\cos ^{ - 1}}({{h_{\textrm{0}\textrm{.66}p}}/{h_{\textrm{0}\textrm{.9}p}}} )$, ${\theta ^{\prime}_\textrm{4}} = {\cos ^{ - 1}}({{h_{\textrm{0}\textrm{.66}p}}/{h_{\textrm{1}\textrm{.0}p}}} )$.

 figure: Fig. 8.

Fig. 8. The angles between the ideal image height in the vertical direction and horizontal height

Download Full Size | PDF

2.7.3 Calculation of vertical line warping

Vertical line warping Δ2 is the value obtained by subtracting the minimum horizontal height $h_{\Vert\textrm{min}}$ from the maximum horizontal height $h_{\Vert\textrm{max}}$ for the real image height in the range from 0.66 field to 1.0 field. Thus, the real horizontal heights of all fields in the screen image must be calculated, namely, $h_{\Vert0.66}$, $h_{\Vert0.7}$, $h_{\Vert0.8}$, $h_{\Vert0.9}$, and $h_{\Vert1.0}$. The real image height of 0.66 field is its horizontal height, namely, h0.66R= $h_{\Vert0.66}$, as shown in Fig. 9. The relations between the real image height and the horizontal height of a field are ${h_{||\textrm{0}\textrm{.7}}} = {h_{\textrm{0}\textrm{.7}R}}\cos {\theta _2}^{\prime}$, ${h_{||\textrm{0}\textrm{.8}}} = {h_{\textrm{0}\textrm{.8}R}}\cos {\theta _3}^{\prime}$, ${h_{||\textrm{0}\textrm{.9}}} = {h_{\textrm{0}\textrm{.9}R}}\cos {\theta _4}^{\prime}$, ${h_{||\textrm{1}\textrm{.0}}} = {h_{\textrm{1}\textrm{.0}R}}\cos {\theta _5}^{\prime}$, which are used to calculate vertical warping.

 figure: Fig. 9.

Fig. 9. Warping in the vertical direction

Download Full Size | PDF

2.8 Definition of the resolution of horizontal and vertical lines in the screen image

The warping of horizontal and vertical lines in the screen image is affected by optical distortion. The resolution of the human eye is at least 1′ (1 arc-minute), thus, the resolution of horizontal and vertical lines in the screen image is also at least 1′. If the resolution of horizontal and vertical lines in the screen image is less than 1′, the human eye cannot perceive the warping of horizontal and vertical lines, so all the person would see is a straight line. This study establishes the horizontal line resolution Θ‖ and the vertical line resolution Θ⊥, as shown in Eq. (16) and Eq. (17). Here, Δ1 and Δ2 are the horizontal line warping and the vertical line warping, respectively, and L can be set to 250 mm, which is the visual distance of the human eye, or 4000 mm, which is the distance between the audience in the first row of a large conference hall and the screen. As shown in Fig. 10, the locations of the DMD, the human eye, and the screen in the projection system are established (1′ equals to 1/60 degree).

$${\Theta _{\textrm{||}}} = 60\ {\tan ^{ - 1}}(\frac{{{\Delta _1}}}{{L}}),$$
$${\Theta _ \bot} = 60\ tan^{ - 1}(\frac{{{\Delta _{2}}}}{L}{),}$$
where Δ1 is the horizontal line warping; and Δ2 is the vertical line warping.

 figure: Fig. 10.

Fig. 10. Locations of the DMD, human eye and screen in the projection system

Download Full Size | PDF

2.9 Relationship between TV distortion and optical distortion

2.9.1 Relationship between horizontal TV distortion and horizontal line warping

The relationship between horizontal TV distortion and horizontal line warping is shown in Fig. 11, where h0.75R denotes that the real image height of 0.75 field and the vertical height; haR, hbR are the real image heights of a field and b field; h⊥a is the real maximum vertical height of a field from 0.75 field to 1.0 field; h⊥b is the real minimum vertical height of b field from 0.75 field to 1.0 field; θa is the included angle between the ideal image height hap of a field and its vertical height h0.75P, namely, θa=cos−1(h0.75P / haP); θb is the included angle between the ideal image height hbp of b field and its vertical height h0.75P, namely, θb=cos−1(h0.75P / hbP), as shown in Fig. 12.

 figure: Fig. 11.

Fig. 11. Relationship between horizontal TV distortion and horizontal warping

Download Full Size | PDF

 figure: Fig. 12.

Fig. 12. Definition of angles θa and θb in the horizontal direction

Download Full Size | PDF

h⊥a is the real maximum vertical image height of a field, as shown in Eq. (18),

$${h_{ \bot a}} = {h_{aR}}\cos {\theta _a} = {h_{aR}} \times \frac{{{h_{0.75p}}}}{{{h_{ap}}}}.$$
h⊥b is the real minimum vertical image height of b field, as shown in Eq. (19),
$${h_{ \bot b}} = {h_{bR}}\cos {\theta _b} = {h_{bR}} \times \frac{{{h_{0.75p}}}}{{{h_{bp}}}}.$$
Horizontal line warping Δ1 is the value obtained by subtracting the minimum vertical height h⊥b from the maximum vertical height(h⊥a of the real image height in the range from 0.75 field to 1.0 field, as shown in Eq. (20),
$${\Delta _1} = |{{h_{ \bot a}} - {h_{ \bot b}}} |= \left|{{h_{aR}} \times \frac{{{h_{0.75p}}}}{{{h_{ap}}}} - {h_{bR}} \times \frac{{{h_{0.75p}}}}{{{h_{bp}}}}} \right|.$$
The relationship between horizontal TV distortion and horizontal line warping is shown in Eq. (21),
$$\begin{aligned} {TV}{{D}_{H}} &= \frac{{{\Delta _1}}}{{{h_{0.75p}}}}100\%= \left|{\frac{{{h_{aR}}}}{{{h_{ap}}}} - \frac{{{h_{bR}}}}{{{h_{bp}}}}} \right|\times 100\%\\ &= \left|{\frac{{{h_{aR}} - {h_{ap}}}}{{{h_{ap}}}} \times 100\%- \frac{{{h_{bR}} - {h_{bp}}}}{{{h_{bp}}}} \times 100\%} \right|= |{O{D_a} - O{D_b}} |. \end{aligned}$$
ODa is the maximum optical distortion from 0.75 field to 1.0 field, and ODb is the minimum optical distortion from 0.75 field to 1.0 field.

2.9.2 Relationship between vertical TV distortion and vertical line warping

The relationship between vertical TV distortion and vertical line warping is shown in Fig. 13, where h0.66R represents that the real image height of 0.66 field equals the horizontal height; h‖a is the real maximum horizontal image height of a field from 0.66 field to 1.0 field; and h‖b is the real minimum horizontal image height of b field from 0.66 field to 1.0 field.

 figure: Fig. 13.

Fig. 13. Relationship between vertical TV distortion and vertical warping

Download Full Size | PDF

The vertical line warping Δ2 is obtained by subtracting the minimum horizontal height h‖b from the maximum horizontal height h‖a from 0.66 field to 1.0 field, as shown in Eq. (22),

$${\Delta _2} = |{{h_{{\parallel} a}} - {h_{{\parallel} b}}} |.$$
The relationship between vertical TV distortion and vertical line warping is shown in Eq. (23),
$${TV}{{D}_{V}} = \frac{{{\Delta _2}}}{{{h_{0.66p}}}}100\%= |{O{D_a} - O{D_b}} |.$$
ODa is the maximum optical distortion from 0.66 field to 1.0 field, and ODb is the minimum optical distortion from 0.66 field to 1.0 field.

2.10 Definition of lateral color and its resolution

2.10.1 Definition of lateral color

After lens imaging, due to different image heights arising from the different wavelengths of the light sources, the white light from the off-axis position on the object plane causes different lateral colors on the image plane. The absolute difference between the maximum image height and the minimum image height from the short-wavelength ray to the long-wavelength ray is the lateral color Δλ, as shown in Fig. 14.

 figure: Fig. 14.

Fig. 14. Schematic diagram of lateral color

Download Full Size | PDF

2.10.2 Definition of lateral color resolution (Θλ)

The screen images are meant to be observed so it is necessary for difference in color in the screen images to be determined by the human eye. As the resolution of the human eye is at least 1′, the lateral color resolution must also be at least 1′. Δλ is set as the lateral color, and L can be set to 250 mm, which is the visual distance of the human eye, or 4000 mm, which is the distance between the audience in the first row of a large conference hall and the screen. The definition of lateral color resolution is shown in Eq. (24).

$${\Theta _\lambda } = 60\ {\tan ^{ - 1}}\frac{{{\Delta _\lambda }}}{L}.$$

3. Projection lens design

Lens specifications, athermalization design, and lens image quality analysis for projection lens design are discussed.

3.1 Design specifications of projection lens

The dimensions of the projection lens are 4427 mm × 2490 mm, its diagonal size is 200 inches, the projection distance is 7.2m, the magnification of projection from the DMD position to the screen is −301.832, the lens focal length is 24.06 mm, the ratio of projection distance to screen width is 1.62, the lens F/#=1.71, DMD offset is 100%, the maximum imaging height at DMD position is 11.038 mm, and the temperature required by the lens is from 10°C to 80°C; the design specifications of the projection lens are shown in Table 3.

Tables Icon

Table 3. Design specifications of projection lens

View Table | View all tables in this article

3.2 Athermalization design of projector lens

From 2.2.3 section athermalization Eq. (8), The projector lens athermalization is related to the $\frac{{d{\varphi _i}}}{{dT}}$, total power φ and the linear thermal expansion coefficient β of the lens barrel material. Where $\frac{{d{\varphi _i}}}{{dT}}$ is related to each lens power φi, the lens refractive index nd, $\frac{{d{n_d}}}{{dT}}$ and the thermal expansion coefficient α of lens. The projection lens design consists of 12 spherical glass lenses and 2 pieces of plate glass. First, aluminum is used as the lens barrel material for the athermalization design, the athermalization value is calculated to be $\mathop \sum \nolimits_{i = 1}^j \frac{{d{\varphi _i}}}{{dT}} + \beta \cdot \varphi $ $ = 9.58 \times {10^{ - 6}}({m{m^{ - 1}}} )$. Then, plastic PMMA is used as the lens barrel material, the athermalization value is calculated to be $\mathop \sum \nolimits_{i = 1}^j \frac{{d{\varphi _i}}}{{dT}} + \beta \cdot \varphi = 3.29 \times {10^{ - 6}}({m{m^{ - 1}}} )$, a reduction in the athermalization value of 65% as shown in Table 4.

Tables Icon

Table 4. Parameters related to athermalization of the lens

View Table | View all tables in this article

3.3 Projector lens drawing and lens data

A drawing of the projection lens and the design information are shown in Fig. 15. The design data and aspheric coefficients of the lens are listed in Table 5 and Table 6.

 figure: Fig. 15.

Fig. 15. Drawing of projection lens

Download Full Size | PDF

Tables Icon

Table 5. Data for the projection lens

View Table | View all tables in this article
Tables Icon

Table 6. Aspherical coefficient data for the projection lens

View Table | View all tables in this article

3.4 TV distortion and lateral color

According to the Eq. (21) in Section 2.9.1, the horizontal TV distortion is the absolute value of maximum optical distortion value minus the minimum optical distortion value from 0.75 field to 1.0 field. According to the Eq. (23) in Section 2.9.2, vertical TV distortion is the absolute value of maximum optical distortion value minus the minimum optical distortion value from 0.66 field to 1.0 field. Therefore, we can constraint the amount of optical distortion changes from 0.66 field to 1.0 field to optimize TV distortion. The largest TV distortion is at 80 °C. The horizontal TV distortion is 0.0788%; The horizontal line warping Δ1 is 6.51µm; The horizontal line resolution is Θ‖, $\Theta_{\Vert}({\textrm{L} = 4000\ \textrm{mm}} )= 60\ {tan ^{ - 1}}(\frac{{0.00651}}{{4000}}) = 0.00559^{\prime}$, $\Theta_{\Vert}({\textrm{L} = 250\ \textrm{mm}} )= 0.08952{^{\prime}}$; The vertical line warping Δ2 is 5.14 µm; The vertical line resolution is $\Theta_{\bot}$, $\Theta_{\bot}({\textrm{L} = 4000\ \textrm{mm}} )= 60\ {tan ^{ - 1}}(\frac{{0.00514}}{{4000}}) = 0.00442^{\prime}$, $\Theta_{\bot}({\textrm{L} = 250\ \textrm{mm}} )= 0.07068^{\prime}$; The lateral color Δλ is 1.39 µm; The lateral color resolution is $\Theta_{\lambda}$, $\Theta_{\lambda}({\textrm{L} = 4000\ \textrm{mm}} )= 60\ {tan ^{ - 1}}(\frac{{0.00139}}{{4000}}) = 0.00119^{\prime}$, $\Theta_{\lambda}({\textrm{L} = 250\ \textrm{mm}} )= 0.01911^{\prime}$.

3.5 Lens image quality of athermalization at temperature from 10°C to 80°C

Table 7 shows the results of image quality analysis at temperatures from 10°C to 80°C, where the horizontal line warping changes from 2.89 µm to 6.51 µm at different temperatures, and the horizontal TV distortion changes from 0.0350% to 0.0788%. The horizontal line resolution changes from 0.00248′ to 0.00559′ at a distance of about 4000 mm, and from 0.03974′ to 0.08952′ at the visual distance of 250 mm. The vertical line warping changes from 2.57 µm to 5.14 µm, and the vertical TV distortion changes from 0.0350% to 0.0700%. The vertical line resolution changes from 0.00221′ to 0.00442′ at a distance of 4000 mm, and from 0.03534′ to 0.07068′ at a visual distance of 250 mm. The optical distortion changes from 0.1102% to 0.1467% with temperature, and the lateral color changes from 0.79 µm to 1.39 µm at different temperatures. The lateral color resolution changes from 0.00067′ to 0.00119′ at a distance of 4000 mm, and from 0.01086′ to 0.01911′ at a visual distance of 250 mm. The relative illumination changes from 92.52% to 93.95% with temperature, and the MTF changes from 0.554 to 0.613 at different temperatures.

Tables Icon

Table 7. Lens image quality at temperature from 10°C to 80°C

View Table | View all tables in this article

4. Conclusion

A projection lens was designed for used at temperatures from 10°C to 80°C. Temperature changes can cause variation in the shape of the lens surface, thus, the refractive index of the material can also be changed, which affects lens image quality. When the temperature rises, the shape of the lens surface changes due to thermal expansion causing variation in the focal length. However, different lens materials with different refractive indices can be selected to offset the changes in the lens focal length, the athermalization process. A projection lens projects images onto a screen to be viewed by the human eye. In order to match the resolution of the human eye, which is 1′ it is necessary for the horizontal and vertical line resolutions and the lateral color resolution to be less than 1′. When the projection lens projects images onto the screen at temperatures from 10°C to 80°C, the horizontal TV distortion must be less than 0.079%, the vertical TV distortion less than 0.070%, the optical distortion less than 0.1467%, the lateral color less than 1.39 µm, the relative illumination less than 92.52%, and the MTF(93 lp/mm) greater than 0.554.

Funding

Ministry of Science and Technology, Taiwan (MOST 108-2221-E-008-090).

Acknowledgements

This study was supported in part by the Ministry of Science and Technology, under project numbers MOST 108-2221-E-008-090.

Disclosures

The authors declare no conflicts of interest.

References

1. D. S. Grey, “Athermalization of Optical Systems,” J. Opt. Soc. Am. 38(6), 542–546 (1948). [CrossRef]  

2. M. J. Duggin, “Discrimination of targets from background of similar temperature, using two-channel data in the 3.5-4.1-µm and 11–12-µm regions,” Appl. Opt. 25(7), 1186–1195 (1986). [CrossRef]  

3. M. H. Horman, “Temperature analysis from multispectral infrared data,” Appl. Opt. 15(9), 2099–2104 (1976). [CrossRef]  

4. T. H. Jamieson, “Ultrawide waveband optics,” Opt. Eng. 23(2), 111–116 (1984). [CrossRef]  

5. M. Roberts and P. J. Rogers, “Wide waveband infrared optics,” Proc. SPIE 1013, 84–91 (1989). [CrossRef]  

6. Y. Tamagawa and T. Tajime, “Dual-band optical systems with a projective athermal chart: design,” Appl. Opt. 36(1), 297–301 (1997). [CrossRef]  

7. J. L. Rayces and L. Lebich, “Thermal compensation of infrared achromatic objectives with three optical materials,” Proc. SPIE 1354, 752–759 (1991). [CrossRef]  

8. P. J. Rogers, “Athermalized FLIR optics,” Proc. SPIE 1354, 742–751 (1991). [CrossRef]  

9. Y. Tamagawa, S. Wakabayashi, T. Tajime, and T. Hashimoto, “Multilens system design with an athermal chart,” Appl. Opt. 33(34), 8009–8013 (1994). [CrossRef]  

10. Schott, “TIE-29: Refractive index and dispersion,” in Proc. Schott Technical information (Schott Inc., Germany, 2015).

11. Schott, “TIE-19: Temperature coefficient of the refractive index,” in Proc. Schott Technical information (Schott Inc., Germany, July 2008).

12. Y. Bai, T. W. Xing, W. M. Lin, and W. M. Xie, “Athermalization of middle infrared optical system,” J. Appl. Opt. 33(1), 181–185 (2012).

13. TEXAS INSTRUMENT, “DLP 660TE 0.66 4 K UHD DMD,” http://www.ti.com/lit/ds/symlink/dlp660te.pdf

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)
Fig. 1.
Fig. 1. Schematic diagram of horizontal distortion

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Schematic diagram of vertical distortion

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Image circle of projection lens

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Horizontal line warping in screen image

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. The angles between the ideal image height in the horizontal direction and vertical height

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. Warping in the horizontal direction

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. Vertical line warping in the screen image

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. The angles between the ideal image height in the vertical direction and horizontal height

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. Warping in the vertical direction

View in Article | Download Full Size | PDF

Fig. 10.
Fig. 10. Locations of the DMD, human eye and screen in the projection system

View in Article | Download Full Size | PDF

Fig. 11.
Fig. 11. Relationship between horizontal TV distortion and horizontal warping

View in Article | Download Full Size | PDF

Fig. 12.
Fig. 12. Definition of angles θa and θb in the horizontal direction

View in Article | Download Full Size | PDF

Fig. 13.
Fig. 13. Relationship between vertical TV distortion and vertical warping

View in Article | Download Full Size | PDF

Fig. 14.
Fig. 14. Schematic diagram of lateral color

View in Article | Download Full Size | PDF

Fig. 15.
Fig. 15. Drawing of projection lens

View in Article | Download Full Size | PDF

Tables (7)

Tables Icon

Table 1. Temperature changes of lens parameters and the thermal expansion coefficient

View Table | View all tables in this article
Tables Icon

Table 2. TI DLP 660TE 0.66′′ specifications

View Table | View all tables in this article
Tables Icon

Table 3. Design specifications of projection lens

View Table | View all tables in this article
Tables Icon

Table 4. Parameters related to athermalization of the lens

View Table | View all tables in this article
Tables Icon

Table 5. Data for the projection lens

View Table | View all tables in this article
Tables Icon

Table 6. Aspherical coefficient data for the projection lens

View Table | View all tables in this article
Tables Icon

Table 7. Lens image quality at temperature from 10°C to 80°C

View Table | View all tables in this article

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

n 2 ( λ ) − 1 = B 1 λ 2 ( λ 2 − C 1 ) + B 2 λ 2 ( λ 2 − C 2 ) + B 3 λ 2 ( λ 2 − C 3 ) ,
d n ( λ , T ) d T = n 2 ( λ , T 0 ) − 1 2 n ( λ , T 0 ) × ( D 0 + 2 D 1 ⋅ Δ T + 3 D 2 ⋅ Δ T 2 + E 0 + 2 E 1 ⋅ Δ T λ 2 − λ T K 2 ) ,
n ( λ , T ) = n ( λ , T 0 ) + d n ( λ , T ) d T Δ T ,
z = C r 2 1 + 1 − ( 1 + K ) C 2 r 2 + a 4 r 4 + a 6 r 6 + a 8 r 8 + … … . ,
d ϕ d T = ϕ ( 1 n − 1 d n d T − α ) = γ ϕ ,
γ = 1 n − 1 d n d T − α .
Total power : ∑ i = 1 j ϕ i = ϕ ,
Athermalization : ∑ i = 1 j d ϕ i d T + β ⋅ ϕ = 0 ,
y offset = ( C B ) × 100 % ,
h = A 2 + ( B + C ) 2 .
optical distortion = real image height − ideal image height ideal image height × 100 % .
T V D H = Δ 1 h V p 100 % .
T V D V = Δ 2 h H p 100 % .
Δ 1 = | h ⊥ M A X − h ⊥ M I N | .
Δ 2 =   | h ∥ m a x −   h ∥ m i n | .
Θ || = 60   tan − 1 ( Δ 1 L ) ,
Θ ⊥ = 60   t a n − 1 ( Δ 2 L ) ,
h ⊥ a = h a R cos ⁡ θ a = h a R × h 0.75 p h a p .
h ⊥ b = h b R cos ⁡ θ b = h b R × h 0.75 p h b p .
Δ 1 = | h ⊥ a − h ⊥ b | = | h a R × h 0.75 p h a p − h b R × h 0.75 p h b p | .
T V D H = Δ 1 h 0.75 p 100 % = | h a R h a p − h b R h b p | × 100 % = | h a R − h a p h a p × 100 % − h b R − h b p h b p × 100 % | = | O D a − O D b | .
Δ 2 = | h ∥ a − h ∥ b | .
T V D V = Δ 2 h 0.66 p 100 % = | O D a − O D b | .
Θ λ = 60   tan − 1 Δ λ L .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use