1. Introduction

Microscope objective lens is the key component in any optical microscope. It is responsible for determining the resolution of the optical image the microscope can produce. Modern objective lenses made up of many glass lens elements and have attained a high quality and an excellence in performance based on the great extent of correction employed for primary optical aberrations. The manufacturers provide specification of the objective lens based on the parameters, such as numerical aperture (NA), magnification (M), working distance (WD), degree of aberration correction and certain other important characteristics, that are normally imprinted on the barrel of the objective lens. In many optical applications, these parameters are used to describe the focusing of light through the objective lens based on either the geometrical ray optics or the wave optics [1,2].

Geometric ray tracing is a simple way to describe the propagation of light rays through an optical system, allowing the image-forming for any optical system. Due to the simplicity, it is regarded as highly intuitive, particularly useful in teaching [3,4] and specifically suitable for optical design applications [5] without introducing the complex formulation of wave optics. Besides, in many laser based optical applications utilizing objective lenses, geometrical ray tracing is widely used to describe light propagation in the system, or to design an optimized optical system. Fallman et al. [6] and Mio et al. [7] used thin lens equation to describe the movement of a tight laser focus achieved by an objective lens, when the incident beam is controlled by a scanning mirror or a moving lens. Previously, we [8] have already used matrix based ray tracing method to find an optimized optical system that gives maximum volume for the laser focus. However, previous approaches [6–8] are effective only for describing the focus movement, but not accurate enough to describe the light focusing by the objective lens. This is because these approaches treat the whole objective lens as a single paraxial thin lens without considering the Abbe’s sine condition [9]. For this reason, focused light cone by the objective lens based on these approaches may show severe differences with exact light propagation through the objective lens. Gu et al. [10] presented the method of describing the focused light cone by the objective lens based on the Abbe’s sine condition for calculating the optical trapping force. This approach is useful for describing the cone shape at the exact focal point but, doesn’t remain valid for an asymmetrically focused beam.

In fact, to get exact light propagation through an objective lens, it is required to perform exact ray tracing based on the specifications of real lens elements used in an objective lens such as refracting surface radius, thickness, and refractive index. However, getting the specifications of real lens elements is difficult since the manufacturers only provide objective lens parameters but not the details of the each lens element. Thus exact ray tracing of objective lens is often restricted. For this reason, it is greatly needed to describe the light focusing of objective lens precisely, based on the given parameters of the objective lens. In this paper, we propose a simplified mathematical ray-optics model for oil immersion objective lens with consideration of Abbe’s sine condition using a paraxial thin lens formulation.

The rest of this paper is organized as follows. In section 2, we briefly introduce the basics of an objective lens and simplify the oil immersion objective lens using its design principle. Section 3 describes derivation of mathematical model for oil immersion objective lens using vectorial paraxial thin lens model. In section 4, we show the ray tracing results of the proposed model for a commercially available oil immersion objective lens and compare it with the exact ray tracing method based on the specification of real objective lens. Section 5, concludes the paper by summarizing our results and suggesting the future works.

2. Simplification of oil immersion objective lens

Modern objective lenses are mostly designed for infinite conjugate ratio i.e. the object of observation is placed in the front focal plane and its image is formed at infinity. This is aimed for easy incorporation of the accessories such as dichroic mirrors, polarizers and prisms between the objective lens and the eyepiece. For diffraction-limited resolution, it is very important to obtain a high level of correction for optical aberrations to design an objective lens. One of the basic methods for designing aberration-free objective lens is to obey the Abbe’s sine condition using an aplanatic front system [11]. When the objective lens satisfies Abbe’s sine condition, it can be treated as one thick lens with the principal planes shaping into spherical surfaces having centers at object and image points [12]. This can be considered as true due to the fact that the concept of principal planes is only valid when the angle between the rays and the optical axis becomes small enough fulfilling the condition sinα ≈ tanα ≈ α. Figure 1 shows a simplified schematic diagram of an infinity-corrected oil immersion objective lens. The object located at front focal plane FP₁ is protected with a thin cover glass with thickness t_g. Immersion oil having refractive index n is placed between the cover glass and the front face of the objective lens. Normally, the refractive index of immersion oil is 1.515, which comparably matches with the refractive index of the cover glass and the first lens. Due to the index matching, any ray originating from the front focal point F₁, at an angle α to the optical axis, keeps the same angle before being refracted by the first spherical surface SS centered at F₁. The spherical surface SS has a finite radius of nf due to the immersion oil. For comparison, first principal plane PP₁ is drawn at point O, having intersection of SS and the optical axis. Then, the ray passes though the second principle plane PP₂ at the same height, parallel to the optical axis. Note that PP₂ is the same as second spherical surface having its center at the image point. This is because of the fact that for an infinity-corrected objective lens, the image point is located at an infinite position from PP₂. Finally, the rays emerging from the back aperture propagate parallelly in the same direction.

 

Fig. 1. Simplified ray diagram of an infinity-corrected oil immersion objective lens. The rays leaving the front focal point F₁, refract at the spherical refracting surface SS instead of classical principal plane PP₁ in the paraxial regime.

Download Full Size | PDF

To collect more light and produce brighter images, the cones of light rays from the object point intercepted by SS should be as large as possible. Here, a measure of light gathering capability by the objective lens can be described by NA, which is a product of refractive index of the immersion medium and half-angle of the light cone, defined as nsinα [4]. By observing the geometry of light cone, the diameter of rays passing through the back aperture, which is also referred to as the diameter of back aperture, can be related to its numerical aperture as,

3. Simplified mathematical model for oil immersion objective lens

3.1 Derivation of vectorial ray tracing model for paraxial thin lens

In many optical experiments incorporating objective lens, the laser beam (or light) entering into the objective lens propagates in the direction opposite to that explained in sec. 2, and is focused at the front focal plane. To describe this, we need to derive a ray tracing model for the objective lens. For simplicity, first we will derive the ray tracing model for a paraxial thin lens. Then, the ray tracing model for objective lens will be derived. Figure 2 illustrates the ray propagation in a paraxial thin lens with focal length “f”. The dashed line in the Fig. 2 denotes ray propagation when the rays are directed towards the lens without rotation. The dotted line, whereas, denotes ray propagation when the incident rays are rotated at a certain angle with respect to the z-axis. Symbol “^” denotes to the unit vector and symbol “→” denotes to a general vector with a non-unit magnitude.

 

Fig. 2. Ray refraction by a paraxial thin lens

Download Full Size | PDF

This notation is preserved in the all figures and the following text. î is the unit direction vector of the input ray and ϕ is the rotation angle of the incident ray with respect to the z-axis. ρ⃗ denotes the vector from the lens center O to the ray intersection point A on the lens plane. r⃗ gives the refraction vector corresponding to the segment AF. r⃗′ denotes the vector introduced to derive the refraction vector (r⃗). In the paraxial approximation, focus is located at the same focal plane regardless of the ray rotation. By observing the geometric relations in Fig. 2, segment $\overline{\mathrm{OF}}$ can be represented as f/cosϕ, and vector $\overrightarrow{\mathrm{BC}}$ becomes ρ⃗cosϕ/f because the triangle ΔAOF and the triangle ΔCBA can be considered as similar triangles. Therefore, r⃗′ can be represented as,

The refraction vector r⃗ is now obtained by the multiplication of f/cosφ and r⃗′. Here, |r⃗| and r̃ denote the magnitude and direction of vector r⃗, respectively.

This is the vectorial refraction equation for a paraxial thin lens that can describe the ray refraction even when the incident beam rotates and the convergence/divergence angle of the incident beam changes.

3.2 Derivation of the ray tracing model for oil immersion objective lens

As explained in sec. 2, the light between spherical surface SS and second principal plane PP2 can be assumed to propagate as parallel, if the objective lens satisfies Abbe’s sine condition. Therefore, the ray propagation between the first principal plane PP₁ and second principal plane PP₂ in Fig. 1 can be neglected for simplicity. Note that the ray propagation between PP₁ and PP₂ is replaced with a plane located at O where z-axis and SS intersect each other. Due to this, if we locate a paraxial thin lens at point O, the ray propagation in an infinity-corrected oil immersion objective lens can be illustrated as shown in Fig. 3, showing the rotation of incident rays around the center of back aperture. In Fig. 3, ĩ denote the unit direction vector of the input ray entering into the objective lens. Detailed description of the generation of incident rays will be described in sec. 3.3. r⃗ denotes refraction vector for a thin lens and r⃗o denotes the refraction vector for an objective lens directing from the refracting point on the spherical surface SS of radius nf, to the front focus F ₁. r⃗u is the vector introduced for calculating the refraction vector (r⃗_o) of the objective lens. Subscript “u” is used to denote “uncorrected vector”. s⃗ is the vector connecting the ray intersection point on the thin lens to SS in ĩ direction for correction of r⃗u. As shown in Fig. 3, incident rays enter into the thin lens parallelly, then, the rays propagate parallelly before being refracted at SS and focused at front focus F ₁.

 

Fig. 3. Ray trace modeling for infinity corrected oil immersion objective lens considering sine condition using paraxial thin lens

Download Full Size | PDF

Here, our final concern of this derivation is to find r⃗o. However, the Snell’s law cannot be applied for calculating r⃗_o. This is because of the fact that r⃗_o will always be normal to SS regardless of the height of incident rays from the optical axis at the thin lens plane. For this reason, we introduce r⃗_u and s⃗ to calculate r⃗_o. First, r⃗_u can be derived using the refraction by the thin lens located at O. Since, the focus displacement of thin lens in lateral direction, when the incident beam rotates, is defined as f tanϕ so, the thin lens should have a focal length f equivalent to the effective focal length of objective lens regardless of the immersion medium of the objective lens. So, the refracted rays by the thin lens will be focused at F ₁*, which is displaced from F ₁ by a value of f(n-1). By introducing the focus difference vector “a⃗” between F ₁ and F ₁ *, r⃗_u can be represented as,

Now, correction vector s⃗ can be represented using ρ⃗ and the radius of SS as,

where s⃗ has the same direction as ĩ. Hence we can conclude that the refraction vector of an infinity-corrected oil immersion objective lens (r⃗_o) can be expressed using Eq. (4) and Eq. (5) as,

3.3 Generation of incident beam

To utilize the objective lens model represented in Eq. (6), for simulation of the objective lens, incident beam towards the objective lens needs to be defined. This can be achieved in parametric form using starting positions and unit direction vectors of individual rays of the incident beam. If the incident beam rotates, starting positions and unit direction vectors of the individual rays need to be recalculated to the transformed incident beam. For example, let’s consider an incident beam having diameter D, propagating in parallel without rotation towards the objective lens as shown in Fig. 4. In Fig. 4, O is marked as origin and P ₁ denotes the starting position of a particular ray in the incident beam. r gives the radial distance to the P ₁ from the axis of the beam, θ is the clockwise angle along the z-axis, from the x-axis to the P ₁, and l shows the distance along z-axis from O to the starting position of the center ray.

 

Fig. 4. Schematic view of the incident beam propagating in parallel towards the objective lens along the positive z-axis

Download Full Size | PDF

Incident ray with starting position P ₁ and direction ĩ can be represented in the parametric form as,

where ĩ₁ is [0, 0, 1].

Here, starting position of a ray P ₁ can be defined by,

where 0≤r≤D/2 and 0≤θ≤2π.

If the incident beam rotates around point C displaced at a distance d from the origin O, on the z-axis, with rotation angle ϕ_x and ϕ_y around x- and y-axis, then P ₁ and î₁ should be also transformed accordingly. This can be done in three steps. Firstly, P ₁ is translated by a value of d to match rotation center C with O. Secondly, the translated *P ₁ and î₁ are rotated around x- and y-axis by ϕ_x and ϕ_y to form P ₂ and î₁. And lastly, P ₂ is moved back by a value of d in negative z direction to form P. These steps are represented as,

• Translation of P ₁ by a value of d in positive z direction: P*₁=P ₁-C

where C is (0, 0, d) and P*₁ gives the new position after the translation along z-axis

• Rotation of P*₁ and î₁ to give P ₂ and î1:

P ₂=RP*₁

î=Rî₁

where R is the 3×3 rotation matrix resulting from the successive rotations around x-and y-axis as:

$R=R_y{R}_x=\left[\begin{array}{ccc}\cos {\varphi }_y& 0& -\sin {\varphi }_y\\ 0& 1& 0\\ \sin {\varphi }_y& 0& \cos {\varphi }_y\end{array}\right]\left[\begin{array}{ccc}\cos {\varphi }_x& \sin {\varphi }_x& 0\\ -\sin {\varphi }_x& \cos {\varphi }_x& 0\\ 0& 0& 1\end{array}\right].$

• Translation of P ₂ by a value of d in negative z direction: P = P ₂+C

The intersection point on the SS in Fig. 3 with the ray defined by P and î together with Eq. (6) completely determines the location of the focal point as well as the trajectory of an incident ray of a beam.

4. Results and discussions

4.1 Ray tracing results of the proposed objective lens model

Ray tracing is performed by assuming that the beam (light), composed of the particle or wave, can be modeled as a large number of very narrow rays. Figure 5 illustrates ray tracing results for commercially available infinity-corrected oil immersion objective lens (Olympus, #UPLSAPO 100XO, 100x, f: 1.8 mm, NA: 1.4, WD: 0.13 µm, t_g: 0.17 mm) with the proposed model. For the simulation of light focusing by the objective lens, the diameter of the incident beam for objective lens is calculated using Eq. (1) to be 5.04 mm. The incident beam for the objective lens is generated by using the method explained in sec. 3.3. As can be seen in the Fig. 5(a), incident beam normal to the lens is tightly focused on the top surface of cover glass through the refractive index matching using the immersion oil (n_g = n = 1.515). Also, light cone of the objective lens has a half angle of 67.5° satisfying the designed NA of the objective lens (α=sin^-1 (NA/n)).

 

Fig. 5. Results of the proposed ray tracing model, when implemented on the infinity-corrected oil immersion objective lens (Olympus, #UPLSAPO 100XO, 100x, f: 1.8 mm, NA: 1.4, WD: 0.13 µm, tg: 0.17 mm). (a) On-axis incident beam, (b) Rotation of on-axis incident beam around the center of the objective back aperture, (c) Off-axis incident beam (h: 0.5 mm).

Download Full Size | PDF

When the incident beam rotates around the center of the objective back aperture, resulting movements of focus on the focal plane can be easily described by Fig. 5(b). For off-axis case, when the center of incident beam is moved some distance from the optical axis in the direction normal to the optical axis, the beam focus is illustrated in Fig. 5(c) when the beam is shifted to a distance (h) of 0.5 mm towards right. It is evident from the illustration that some portion of the incident beam is blocked by the objective back aperture creating asymmetry in the light cone angles. However, the focus still remains in the same position as in Fig. 5(a). For centering experiments, generally the off-axis shift of beam is very small. However, for proving the validity of the proposed method, a huge distance of 0.5mm off-axis shift is considered.

4.2 Ray tracing results of the exact ray tracing

In order to verify that the proposed model for objective lens is valid and in proper agreement with reality, we performed exact ray tracing based on the real objective lens (100x, f: 1.8 mm, NA: 1.41, WD: 0.13 µm, oil, t_g: 0.17 mm) obtained from U.S. patent [13]. The objective lens has twenty one refracting surfaces by a combination of various lens elements such as singlet, doublet and triplet. The diameter of the incident beam is 5.076 mm, calculated using Eq. (1). The incident beam is generated by using the method explained in sec. 3.3. At each refracting surface, the refraction vector is calculated using vectorial refraction equation [14]. For the description of the process, the parallel beam advances towards the front lens and each ray is tested for intersection at every surface of the each lens element. This calculates the new ray direction once a collision is found. From each location, a new ray is sent out and the process is repeated until the generation of a complete path through the entire objective lens. Figure 6, shows the results of the exact ray tracing. Figures 6(a) and 6(b) denote the ray diagram for on-axis incident beam before and after rotation to about 1.8 degrees, respectively. Ray diagram for off-axis incident beam is illustrated in Fig. 6(c) when a shift distance of 0.5 mm is applied to the incident beam towards the right.

 

Fig. 6. Result of ray tracing based on real objective lens specifications (Olympus, 100x, f: 1.8 mm, NA: 1.41, WD: 0.13 µm, oil, tg: 0.17 mm), (a) On-axis incident beam, (b) Rotation of onaxis incident beam around the center of the objective back aperture to about 1.8 degrees, (c) Off-axis incident beam with an axis of 0.5 mm.

Download Full Size | PDF

As shown in Fig. 6, incident beam is focused on the top surface of the cover glass after passing through the entire lens elements. To understand this in detail, front focal length (f₁), back focal length (f₂) and the distance to back focal plane from the first refracting surface (p) are calculated with ABCD matrix of the optical system [4]. From the calculation, we could get that f ₁ and f ₂ are nf and f according to the proposed model. By exact ray tracing, we can find that the back focal plane is located at 23.4mm from the front lens. The front part of the objective lens in Fig. 6(a), circled with dashed line, is enlarged in the right side. The contour of the traced rays by the proposed model is drawn on the enlarged front part to compare it with the exact ray tracing. As can be seen, both the proposed model and the exact ray tracing result in the same refracting angle while leaving the front lens, forming similar light cone with a half cone angle of 68.5°. When incident beam rotates, as shown in Fig. 6(b), the ray propagation inside the lens elements is changed accordingly, creating an asymmetric light cone. This results in the focus movement at the focal plane similar to Fig. 5(b). For off-axis case, as shown above, some portion of the incident beam is blocked by the objective back aperture. It makes light cone asymmetric similar to Fig. 5(c). For the inside lens elements, the ray propagation is relatively complex and varies based on the state of incident beam as shown in Fig. 6. However, the cone shape at the focus remains very similar with the proposed model, even though the beam propagation in other lens elements is different. Figure 7 shows the deviation of the incident angle of a fan of rays from the surface normal at each refracting surface for the incident beam illustrated in Fig. 6. Figures 7(a)–7(c) represent the results for angles of the on-axis incident beam, for the rotation of on-axis incident beam to about 1.8 degrees, and for the 0.5mm off-axis incident beam.

 

Fig. 7. Deviation of the incident angle of a fan of rays from the surface normal at each refracting surface (a) On-axis incident beam, (b) Rotation of on-axis incident beam around the center of the objective back aperture to about 1.8 degrees, (c) Off-axis incident beam with a shift of 0.5 mm

Download Full Size | PDF

Surface index is numbered from the bottom surface to the top. The sign of the incident angle is determined based on the intersection point between the incident ray and the refracting surface. When the intersection point is the left from the optical axis, the sign of the incident angle is negative, and positive, otherwise. Circle marker represents the incident angle of the center ray while maximum and minimum angle at each surface index denote the angle by boundary rays. As can be seen, the incident angle for on-axis incidence is symmetric with optical axis. However, as expected, the incident angles for rotated on-axis incident beam and off-axis incident beam are varying non-symmetrically within the whole lens elements. The proposed model also holds its validity for these cases because many optical applications need to move the focus for specific purposes. These purposes could be the variation of the shape of light cone due to the rotation of the incident beam, the extent of vignetting due to the improper location of rotation center of the incident beam and the focus displacement when the incident beam rotates. As an example, the objective lens specifications, obtained from previous U.S. patent [13], are used for comparing the proposed model to the exact ray tracing in terms of the said criteria for on-axis and off-axis incident beams.

4.3 Comparison of the proposed model and the exact ray tracing

It is important to compare the change in the shape of light cone due to the rotation of incident beam. This is of importance because of the fact that it changes the maximum convergence angle at the boundary rays. Figure 8 illustrates the comparisons of half cone angle errors, when the on-axis incident beam rotates in clockwise direction. This includes the comparison between the proposed model and the exact ray tracing, and between the thin lens model and exact ray tracing. The results of the thin lens model are calculated using Eq. (3) for the incident beam rotating around the center of the lens plane as shown in Fig. 2. These correspond to the results obtained using previous approaches [6–8]. For proposed model and exact ray tracing, incident beam is rotated around the center of the objective back aperture to alleviate the loss of light energy resulting from the blockage of rays by the objective back aperture. As shown in Fig. 6, the back aperture of the objective lens is generally located inside the lens and the first lens element is placed at the bottom most position. This lens element has finite dimensions and due to this, some portion of the incident rays cannot be transmitted inside the objective lens, when the rotation angle is larger than the maximum allowable angle. The maximum angle is calculated from the exact ray tracing and found to be 2.75°.

 

Fig. 8. Comparison of half cone angle errors between the proposed model and the exact ray tracing, and between the thin lens model and exact ray tracing when the incident beam rotates, using real objective lens specifications (100x, f: 1.8 mm, NA: 1.41, WD: 0.13 µm, oil, t_g: 0.17 mm)

Download Full Size | PDF

For ease of comparison, the light cone is described by left-half cone angle (α_l) and right half cone angle (α_r) as shown in the inset of Fig. 8. Half cone angle errors are calculated by subtraction of the half cone angles calculated by the proposed model and the thin lens model from the half cone angles calculated by the exact ray tracing. In Fig. 8, solid and dotted lines denote the angle error of right-half cone (Δα_r) and the angle error of left-half cone (Δα_l), respectively. Circle marker distinguishes the results for the proposed model and the thin lens model. When the incident beam rotates, α_l and α_r become different, resulting in the asymmetry of the light cone, as shown in the inset. It is evident from Fig. 8 that the half angle errors for the thin lens model are very large, making the thin lens model improper to describe the light focusing by the objective lens. However, the proposed model closely follows the exact ray tracing for α_l. At the maximum angle, Δα_l reaches about 0.5°. For α_r, angle error is much larger than that for α_l. However, the extent of the angle error remains at about 2.8° even at the maximum rotation angle.

Figure 9 illustrates the variation of half cone angle errors between the proposed model and the exact ray tracing for the off-axis incident beam. An off-axis shift is applied to the right as described earlier and illustrated in Fig. 5(c) and Fig. 6(c). Figure 9(a) shows the half cone angle error with respect to the off-axis distance. Circle and square markers denote angle error of right-half cone and left-half cone, respectively.

 

Fig. 9. Comparison of half cone angle errors between the proposed model and the exact ray tracing with respect to off-axis distance (100x, f: 1.8 mm, NA: 1.41, WD: 0.13 µm, oil, tg: 0.17 mm). (a) Variation of half cone angle errors with off-axis distance, (b) Variation of left-half cone angle error for different off-axis distances when incident beam rotates, (b) Variation of right-half cone angle error for different off-axis distances when incident beam rotates

Download Full Size | PDF

Upon an increase in the off-axis shift, Δα_r linearly increases, while Δα_l forms a plateau after an abrupt rise. However, the extent of half cone angle error is very small, less than 0.35 degrees over the 0.75 mm off-axis shift. To see the influence of the rotation of off-axis incident beam, the variation of half cone angle errors for different off-axis distances are evaluated by implementing a clockwise rotation of the incident beam. This is shown in Fig. 9(b) for Δαr and Fig. 9(c) for Δα_l. Circle, square, and triangle markers denote the result of 0.25 mm, 0.5 mm, and 0.75 mm off-axis distance, respectively. As explained earlier, even a shift of 0.5mm is large enough for the practical implementation of the proposed method. However, the graphs are plotted for even a higher shift of 0.75 mm, to show the validity of the proposed method. It is evident that the half cone angle errors increase over all with the increase in the rotation angle of incident beam. For α_l, angle error increase like parabolic curve with rotation angle and larger off-axis distance shows relatively smaller angle error. For α_r, angle error fluctuates in a zig-zag pattern with rotation angle showing an overall increase. However, considering the extent of off-axis distance from the lens center in real experiments, more than 0.5mm off-axis is easily observable by naked eye. At a shift of 0.5mm, maximum half cone angle errors are relatively comparable to the results of on-axis incident beam. In scanning applications, half cone angle error becomes much smaller because the rotation angle of the incident beam is generally meaningful within the field of view (FOV) of the optical imaging system. For example, the objective lens having a magnification of 100x and a focal length of 1.8 mm shows 76 µm of focus displacement for a ±1.2° rotation of the incident beam by f tanϕ. However, the largest dimension in FOV is about 64 µm for a 1/2 inch CCD with a sensor area of 6.4 mm×4.8 mm [15]. Within 1.2° of rotation angle, the maximum angle errors for on-axis and off-axis are less than 1.0° and 1.4°, respectively. This enables the proposed model to be effective to alternate the exact ray tracing for the scanning applications.

Furthermore, mechanical vignetting happens when the rotation center of incident beam does not exist at the back focal plane due to its incorrect positioning. This is because the incident rays are partially blocked by the objective back aperture. Figure 10 illustrates the number of transmitted rays through the objective back aperture when the rotation center of incident beam is displaced from the back focal plane.

 

Fig. 10. Comparison of the number of rays passing through objective back aperture for proposed model and exact ray tracing (100x, f: 1.8 mm, NA: 1.41, WD: 0.13 µm, oil, tg: 0.17 mm) when rotation center of incident beam is displaced from the back focal plane. (a) On-axis incident beam, (b) Off-axis incident beam (h: 0.25 mm), (c) Off-axis incident beam (h: 0.5 mm).

Download Full Size | PDF

Figure 10(a) illustrates the results for on-axis incident beam whereas, Fig. 10(b) and Fig. 10(c) illustrate the results for incident beam with an off-axis shift of 0.25 mm and 0.5 mm, respectively. For symmetric and uniform distribution of rays, polar form of the incident beam, composed of 5000 number of rays, is used, as shown in the inset. In Fig. 10, solid, dashed and dotted lines denote the results of 5 mm, 10 mm, and 15 mm offset rotation from the back focal plane, respectively. Circle marker distinguishes the results of the proposed model from the exact ray tracing. For off-axis incidence, percentage transmitted rays are initially dropped due to the blockage by objective back aperture and this become larger for large off-axis distance as shown in Fig. 10(b) and Fig. 10(c). When the rotation angle increases, percentage of transmitted rays decreases while difference of the percentage of transmitted rays between the proposed model and the exact ray tracing increases. This becomes severe for large offset rotation from the back focal plane and for large off-axis shifts. However, the difference of percentage of transmitted rays is quite small, i.e. about 0.4% for Fig.10(a) and 0.6% for Fig. 10(b), at an angle smaller than 1.2° for an off-axis shift of less than 0.25 mm. For 0.5mm off-axis incidence, the difference of percentage of transmitted rays is significantly increased for 15 mm offset rotation. This is one limitation of the proposed model, but the difference of percentage of transmitted rays is relatively small, about 0.8% within 1.2° rotation angle for 10 mm offset rotation. Therefore, we can say that the proposed model nicely represents the beam propagation in case of the offset rotation from the center of objective back aperture, if the offset rotation is not much high.

Figure 11 illustrates focus displacement error at the focal plane, for the proposed model and the exact ray tracing when the incident beam rotates around the center of objective back aperture. Focus displacement error is evaluated by subtracting two focus displacements obtained from the proposed model and the exact ray tracing. The focus displacements are determined by finding the intersection point at the front focal plane between the center ray and the one of the boundary rays. In Fig. 11, circle marker denotes the result for on-axis incidence and square and triangle markers denote the results for 0.25 mm and 0.5 mm off-axis shift, respectively. Focus displacement errors increase with the rotation angle of incident beam. However, the extent of the errors is very small, about 0.13 µm, 0.1 µm, and 0.09 µm, even at the maximum rotation angle. The maximum rotation angle error is found to be less than 0.15 % (i.e. 0.13 µm/86.4 µm). This result shows that the proposed model nicely deals with the description of the focus displacement of a real objective lens when the incident beam rotates.

 

Fig. 11. Focus displacement error between proposed model and exact ray tracing (100x, f: 1.8 mm, NA: 1.41, WD: 0.13 µm, oil, t_g: 0.17 mm) when the incident beam rotates around the center of the objective back aperture.

Download Full Size | PDF

5. Conclusions

In this paper, we proposed a model for a commercially available oil immersion objective lens by considering the Abbe’s sine condition based on a paraxial thin lens model. To check the validity of the proposed model, ray tracing was performed for a commercially available oil immersion objective lens and graphical ray diagrams were presented. In addition, detailed comparison between the proposed model and the exact ray tracing of the objective lens was presented in terms of several important criteria including variation of the shape of light cone, extent of vignetting, and focus displacement for the case when the incident beam rotates. As demonstrated by the exemplary simulations, the proposed model has shown good agreements with the exact ray tracing in terms of the extent of vignetting as well as the variation of the shape of light cone due to the rotation of the incident beam. Also, the proposed model precisely describes the focus displacement of a real objective lens when the incident beam rotates around the center of objective back aperture. We anticipate that the proposed model is really effective to describe the light propagation in complex optical systems with oil immersion objective lens, that need to have the geometrical inspection of the optical system, e.g. analysis of the propagation of focused beam in microscopic chamber [16]. For the future, we plan to use this model for the calculation of optical trapping force exerted on a microscopic particle [17] when the beam is asymmetrically focused to move the trapped particle. We foresee that calculation of the optical force for asymmetrically focused beam will minimize the discrepancy of the optical trapping forces between computation and experimental work according to the trapped positions in the defined FOV.

For a high NA objective lens, there are several reports that dealt with the polarization effects on the focus field based on the wave optics approach. It is also well known that the point spread function (PSF) of linearly polarized beam focused by a high NA objective lens doesn’t remain rotationally symmetric [18] since the focused spot is broaden by the polarization effects. On the contrary, the PSF for circularly and radially polarized beam [19] shows a rotational symmetry. Currently those polarization effects cannot be quantified by the proposed model. However, by introducing the electric field vector in the incident beam and taking into account the change of the polarization and ray direction vector of each ray after the refraction by SS1, electric field in the focus can be evaluated to quantify the polarization effects [20]. In near future, we aim to study this problem and analyze the effects of polarization in the oil immersion objective lens.