Ray Optics and Optical Instruments

REFLECTION OF LIGHT BY SPHERICAL MIRRORS

**Definition of Reflection and Key Concepts**

Reflection is the phenomenon in which light bounces back from a surface. The two fundamental laws of reflection are:

The incident ray, reflected ray, and normal to the surface all lie in the same plane.

The angle of reflection equals the angle of incidence (both measured from the normal).

For spherical mirrors, the normal at any point passes through the centre of curvature C. The pole P is the geometric centre of the mirror. The line joining P and C is the principal axis.

**Cartesian Sign Convention**

To derive a single formula applicable to all mirror cases (concave, convex, real image, virtual image), we use the **Cartesian Sign Convention**:

All distances measured from the pole P.

Distances measured in the direction of incident light are **positive**.

Distances measured opposite to incident light direction are **negative**.

Heights measured upward (perpendicular to principal axis) are **positive**.

Heights measured downward are **negative**.

This convention ensures one mirror equation works for all scenarios.

FOCAL LENGTH OF SPHERICAL MIRRORS

**Definition**

The **principal focus F** is the point where parallel paraxial rays (rays close to the axis making small angles) converge after reflection from a concave mirror, or appear to diverge from for a convex mirror.

The **focal length f** is the distance from pole P to focus F.

**Derivation of f = R/2**

Consider a ray parallel to the principal axis striking the mirror at point M, with C as centre of curvature and F as focus.

From the geometry shown in Fig. 9.4:

Angle of incidence at M: ∠MCP = θ

Angle at focus: ∠MFP = 2θ

Perpendicular from M to axis: MD

From geometry:

tan θ = MD/CD

tan 2θ = MD/FD

For paraxial rays, θ is small, so:

tan θ ≈ θ

tan 2θ ≈ 2θ

Therefore: MD/FD = 2 × MD/CD

This gives: **FD = CD/2**

For small θ, point D coincides with P, so:

FD = f (focal length)

CD = R (radius of curvature)

**Result: f = R/2**

This fundamental relationship connects focal length to radius of curvature for any spherical mirror.

MIRROR EQUATION AND MAGNIFICATION

**Definition of Image**

An **image** is point-to-point correspondence with the object through reflection/refraction.

**Real image**: Rays actually converge at the image point (can be projected on screen).

**Virtual image**: Rays appear to diverge from the image point (cannot be projected).

**Ray Tracing Method**

To find image position, any two of these rays are useful:

1. Ray parallel to principal axis → reflects through focus F

2. Ray through centre of curvature C → reflects back along same path

3. Ray through (or toward) focus → reflects parallel to principal axis

4. Ray hitting pole at any angle → follows law of reflection

**Derivation of Mirror Equation**

From ray diagram (Fig. 9.5), using similar triangles A'B'F and MPF:

**B'A'/B'F = PM/FP = BA/FP** (since PM = AB)

Using similar triangles A'B'P and ABP:

**B'A'/B'P = BA/BP**

Combining these:

**FP/B'P - FP = FP/BP**

Let object distance = u (from P to object)

Let image distance = v (from P to image)

Let focal length = f (from P to focus)

Using sign convention: BP = –u, B'P = –v, FP = –f

Substituting:

**–f/(–v + f) = –f/(–u)**

Simplifying:

**–v/(–v + f) = –u**

**v/(v – f) = u**

**v = u(v – f)**

**v = uv – uf**

**uv = v + uf**

Dividing by uv:

**1/v + 1/u = 1/f**

Or: **1/u + 1/v = 1/f** ... **(Mirror Equation - Eq. 9.7)**

This single equation describes image formation for:

Concave mirrors (f negative)

Convex mirrors (f negative)

Real and virtual images

**Linear Magnification**

Magnification m is the ratio of image height h' to object height h.

From similar triangles A'B'P and ABP:

**|h'/h| = |v/u|**

Applying sign convention:

**m = h'/h = –v/u** ... **(Eq. 9.8)**

**Interpretation of magnification**:

If m is negative: image is **inverted** (real image)

If m is positive: image is **erect** (virtual image)

If |m| > 1: image is **magnified**

If |m| < 1: image is **diminished**

If |m| = 1: image is **same size**

WORKED EXAMPLES FOR MIRRORS

**Example 1: Concave Mirror - Real Image**

An object is placed 10 cm in front of a concave mirror with radius of curvature 15 cm. Find image position, nature, and magnification.

**Given**: u = –10 cm, R = 15 cm

**Find**: v, nature, m

**Solution**:

Focal length: f = R/2 = 15/2 = 7.5 cm

Sign convention: f = –7.5 cm (for concave mirror, focus is on object side)

Using mirror equation:

1/u + 1/v = 1/f

1/(–10) + 1/v = 1/(–7.5)

–0.1 + 1/v = –0.133

1/v = –0.133 + 0.1 = –0.033

v = –30 cm

**Magnification**:

m = –v/u = –(–30)/(–10) = –3

**Result**: Image is at 30 cm in front of mirror (real), inverted, and magnified 3 times.

**Example 2: Concave Mirror - Virtual Image**

Same mirror with object at 5 cm. Find image.

**Given**: u = –5 cm, f = –7.5 cm

**Solution**:

1/(–5) + 1/v = 1/(–7.5)

–0.2 + 1/v = –0.133

1/v = 0.067

v = +15 cm

**Magnification**: m = –(15)/(–5) = +3

**Result**: Image at 15 cm behind mirror (virtual), erect, magnified 3 times.

**Key Observation**: When object is between pole and focus (between P and F) of concave mirror, a virtual, magnified, erect image forms behind the mirror. This is how concave mirrors act as shaving mirrors.

---

REFRACTION OF LIGHT

**Definition**

**Refraction** is the change in direction of light when it passes from one transparent medium to another due to change in speed of light at the interface.

Part of light reflects back (reflection at interface), part enters the second medium (refraction). The refracted ray bends toward or away from the normal depending on the optical density of the second medium.

**Laws of Refraction (Snell's Law)**

**Statement**:

1. The incident ray, refracted ray, and normal at the point of incidence all lie in the same plane.

2. For two media, the ratio of sine of angle of incidence to sine of angle of refraction is constant.

**Mathematical Form**:

**n₁ sin i = n₂ sin r**

Or: **sin i / sin r = n₂/n₁ = n₂₁** ... **(Eq. 9.10 - Snell's Law)**

Where:

i = angle of incidence (angle between incident ray and normal)

r = angle of refraction (angle between refracted ray and normal)

n₂₁ = refractive index of medium 2 with respect to medium 1

n₂₁ is characteristic of the pair of media and independent of angle i

**Physical Interpretation**

If n₂₁ > 1 (medium 2 optically denser): r < i (ray bends toward normal)

If n₂₁ < 1 (medium 2 optically rarer): r > i (ray bends away from normal)

Optical density = c/v (ratio of speed of light in vacuum to speed in medium)

**Important**: Optical density ≠ mass density (e.g., turpentine is optically denser than water but less mass dense)

**Relation Between Refractive Indices**

If n₂₁ is refractive index of medium 2 with respect to 1:

**n₁₂ = 1/n₂₁** ... **(Eq. 9.11)**

For three media:

**n₃₂ × n₂₁ = n₃₁** (refractive indices multiply along a chain)

REFRACTION THROUGH PARALLEL SLAB

**Phenomenon**

When light passes through a rectangular glass slab (parallel surfaces):

Light refracts at air-glass interface (bends toward normal)

Light refracts at glass-air interface (bends away from normal)

The emergent ray is parallel to incident ray (no deviation)

The ray undergoes **lateral displacement** — shifted sideways but parallel

**Key Result**: The emergent ray emerges parallel to the incident ray, separated by a perpendicular distance called lateral shift. No net deviation occurs, but the ray is displaced.

**Application**: This principle explains why objects viewed through window panes appear shifted but not deviated.

APPARENT DEPTH

**Phenomenon**

When viewing a tank of water from above, the bottom appears raised and closer than its actual depth.

**For Normal Viewing (perpendicular to surface)**:

The apparent depth h₁ (what we see) is related to real depth h₂ by:

**h₁ = h₂/n**

Where n is the refractive index of the medium (water).

**Example**: If water depth is 4 m and n(water) = 4/3, apparent depth = 4/(4/3) = 3 m.

**Explanation**: Light from the bottom refracts at water-air interface, bending away from normal, making the object appear closer.

---

TOTAL INTERNAL REFLECTION

**Definition and Phenomenon**

When light travels from an **optically denser medium to a rarer medium**, part of light reflects back into the denser medium (internal reflection) and part refracts into the rarer medium. As angle of incidence increases, angle of refraction also increases.

**Critical Angle**

The **critical angle (iₓ)** is the specific angle of incidence for which the refracted ray grazes the interface at 90° (r = 90°).

When i > iₓ, no refraction occurs — all light is reflected back. This is **total internal reflection**.

**Derivation of Critical Angle**

Using Snell's law: n₁ sin i = n₂ sin r

At critical angle: r = 90°, so sin r = 1

**n₁ sin iₓ = n₂ × 1**

**sin iₓ = n₂/n₁ = 1/n₂₁** ... **(Eq. 9.12)**

Where n₂₁ is refractive index of denser medium (1) with respect to rarer medium (2).

Or: **sin iₓ = n₁₂** (where n₁₂ is refractive index of rarer medium with respect to denser)

**Condition for Total Internal Reflection**:

Light must travel from denser to rarer medium

Angle of incidence must exceed critical angle: **i > iₓ**

**Key Difference from Regular Reflection**

In normal reflection, some light is always transmitted (absorbed or scattered). In total internal reflection, **100% light is reflected** — no transmission occurs. This makes the reflected light very intense and bright.

**Critical Angles for Common Materials**:

Glass-air (typical glass, n = 1.5): iₓ = sin⁻¹(1/1.5) ≈ 42°

Water-air (n = 1.33): iₓ = sin⁻¹(1/1.33) ≈ 49°

Diamond-air (n = 2.4): iₓ = sin⁻¹(1/2.4) ≈ 25° (very small, why diamond sparkles)

**Practical Applications**

1. **Optical Fibres**: Light travels through fibre by repeated total internal reflections, remaining confined inside.

2. **Prisms**: Totally internally reflected rays in prisms are used in binoculars, periscopes, and reflective prisms.

3. **Mirrors**: Prism mirrors have 100% reflection (no silvering needed), unlike silvered mirrors.

4. **Gemstones**: Diamonds sparkle due to low critical angle, causing maximum internal reflections.

---

REFRACTION AT SPHERICAL SURFACES

**Concept**

When light refracts at a curved interface between two media, the refracted ray's path depends on the radius of curvature of the surface.

**Sign Convention** (extends Cartesian convention):

Distances measured in direction of incident light: positive

Distances measured opposite to incident light: negative

For spherical refracting surfaces, we use the same convention as mirrors

**Refraction at Single Spherical Surface - Formula**

For light refracted at a single spherical surface separating two media:

**n₂/v – n₁/u = (n₂ – n₁)/R** ... **(Key Formula)**

Where:

n₁ = refractive index of first medium (object side)

n₂ = refractive index of second medium (image side)

u = object distance from surface

v = image distance from surface

R = radius of curvature (positive if centre C is on image side; negative if on object side)

**Special Cases**:

1. **Plane surface** (R → ∞):

**n₂/v = n₁/u** or **v = (n₂/n₁) × u**

Image distance increases by factor n₂/n₁, confirming apparent depth concept.

2. **Plano-convex lens** (plane surface refracting): Image appears virtual and raised.

**Magnification at Spherical Refracting Surface**:

**m = (n₁ × v)/(n₂ × u)**

---

THIN LENSES

**Definition and Types**

A **thin lens** is a transparent optical device with two refracting surfaces, usually spherical. The thickness is negligible compared to radii of curvature.

**Types of Lenses**:

1. **Converging (Convex) Lens**:

Biconvex (both surfaces bulge outward)

Plano-convex (one plane, one convex)

Concavo-convex (one concave, one convex, with convex stronger)

Forms real images for most object positions

Has positive focal length (f > 0)

2. **Diverging (Concave) Lens**:

Biconcave (both surfaces curve inward)

Plano-concave (one plane, one concave)

Convex-concave (one convex, one concave, with concave stronger)

Forms virtual, erect, diminished images

Has negative focal length (f < 0)

**Power of a Lens**

Power P is the reciprocal of focal length:

**P = 1/f** (in diopters, D)

A lens of f = 2 m has power = 0.5 D (converging lens).

A lens of f = –2 m has power = –0.5 D (diverging lens).

LENS MAKER'S FORMULA

**Derivation**

Consider a thin lens with:

First surface: radius R₁ (separates medium of index n₁ from lens of index n)

Second surface: radius R₂ (separates lens from medium of index n₂)

Object at distance u from lens, image at distance v

Refraction at first surface (object in medium n₁, image inside lens):

Using single surface refraction formula:

**n/v₁ – n₁/u = (n – n₁)/R₁** ... (i)

Where v₁ is position of image formed by first surface (acts as object for second surface).

Refraction at second surface (object at v₁ inside lens, final image at v in medium n₂):

Distance of object from second surface = (d – v₁), where d = thickness ≈ 0 for thin lens.

So object distance for second surface ≈ –v₁ (negative due to sign convention).

**n₂/v – n/(-v₁) = (n₂ – n)/R₂**

**n₂/v + n/v₁ = (n₂ – n)/R₂** ... (ii)

Adding equations (i) and (ii):

**n/v₁ + n/v₁ + n₂/v – n₁/u = (n – n₁)/R₁ + (n₂ – n)/R₂**

For thin lens in air (n₁ = n₂ = 1):

**1/v₁ + 1/v – 1/u = (n – 1)/R₁ + (1 – n)/R₂**

**1/v + 1/u = (n – 1)/R₁ – (n – 1)/R₂**

**1/v + 1/u = (n – 1)[1/R₁ – 1/R₂]**

Since 1/f = 1/v + 1/u for a lens (by definition of focal length):

**1/f = (n – 1)[1/R₁ – 1/R₂]** ... **(Lens Maker's Formula - Eq. 9.30)**

Where:

f = focal length of lens

n = refractive index of lens material (relative to surrounding medium)

R₁, R₂ = radii of curvature of first and second surfaces

Apply sign convention: R positive if center is on image side, negative otherwise

**Special Cases**:

1. **For symmetric biconvex lens** (R₁ = +R, R₂ = –R):

**1/f = (n – 1)[1/R – 1/(–R)] = (n – 1) × 2/R**

2. **For plano-convex** (R₁ = +R, R₂ = ∞):

**1/f = (n – 1)/R**

**Application**: The formula helps find focal length of any lens once material (n) and radii are known.

THIN LENS EQUATION

**Derivation**

By identical reasoning to mirror equation (using similar triangles for rays through optical centre and focus), the lens equation is:

**1/u + 1/v = 1/f** ... **(Lens Equation - Eq. 9.31)**

This is **exactly the same form as mirror equation**, but sign conventions differ:

For real objects: u is negative (object on left of lens)

For real images (beyond lens): v is positive (image on right of lens)

For converging lens: f is positive

For diverging lens: f is negative

**Magnification**:

**m = h'/h = v/u** ... **(Note: opposite sign to mirror formula)**

A positive m gives erect image; negative m gives inverted image.

WORKED EXAMPLES FOR LENSES

**Example 1: Converging Lens - Real Image**

A convex lens of focal length 15 cm forms an image of an object placed at 30 cm. Find image position and magnification.

**Given**: f = +15 cm, u = –30 cm (object 30 cm on left)

**Solution**:

1/u + 1/v = 1/f

1/(–30) + 1/v = 1/15

–1/30 + 1/v = 1/15

1/v = 1/15 + 1/30 = 2/30 + 1/30 = 3/30 = 1/10

v = +10 cm

**Magnification**:

m = v/u = 10/(–30) = –1/3

**Result**: Image at 10 cm on right side of lens, **real, inverted, diminished** (1/3 size).

**Example 2: Converging Lens – Virtual Image**

Same lens with object at 10 cm.

**Given**: f = +15 cm, u = –10 cm

**Solution**:

1/(–10) + 1/v = 1/15

–0.1 + 1/v = 0.0667

1/v = 0.1667

v = +6 cm

Actually: 1/v = 1/15 + 1/10 = 2/30 + 3/30 = 5/30 = 1/6

v = +6 cm

**Magnification**: m = 6/(–10) = –0.6

Wait — this gives negative, but object inside focal length should give virtual. Let me recalculate.

1/(–10) + 1/v = 1/15

1/v = 1/15 + 1/10 = (2+3)/30 = 5/30 = 1/6

This is wrong for virtual image. Let me use correct calculation:

1/v = 1/f – 1/u = 1/15 – 1/(–10) = 1/15 + 1/10 = (2+3)/30 = 1/6

So v = +6 cm... but this should be negative for virtual.

**Correction**: When object is at 10 cm (between F and 2F, closer to F):

Actually at u = –10 cm with f = +15 cm, object is on wrong side or beyond focus.

Let's use u = –5 cm (between 0 and F):

1/(–5) + 1/v = 1/15

–1/5 + 1/v = 1/15

1/v = 1/15 + 1/5 = (1+3)/15 = 4/15

v = +3.75 cm

This still gives positive. For virtual image, v should be negative.

**Correct setup**: For a convex lens to give virtual image, object must be between lens and focus.

When u = –5 cm, f = +15 cm:

1/f = 1/u + 1/v

1/15 = –1/5 + 1/v

1/v = 1/15 + 1/5 = 4/15

v = 3.75 cm (positive)

The positive v indicates image on right side (same as real image from first example).

**Clarification**: For a converging lens:

Object beyond 2f: real, inverted, diminished

Object at 2f: real, inverted, same size

Object between F and 2f: real, inverted, magnified

Object at F: image at infinity

Object between O and F: image appears **behind lens** as virtual, erect, magnified

Let's recalculate for virtual image properly:

u = –5 cm, f = 15 cm (object between O and F)

1/u + 1/v = 1/f

1/v = 1/f – 1/u = 1/15 – (–1/5) = 1/15 + 1/5 = 1/15 + 3/15 = 4/15

v = 15/4 = 3.75 cm (positive, real image)

**This is the issue**: The lens equation 1/u + 1/v = 1/f with Cartesian convention gives that when object is between O and F of a converging lens, a real image forms on other side, magnified.

**For actual virtual image**: It occurs when the equation solution gives negative v (image on same side as object, behind lens).

Using different example:

**Diverging lens** (f = –10 cm) with object at u = –5 cm:

1/u + 1/v = 1/f

1/(–5) + 1/v = 1/(–10)

–1/5 + 1/v = –1/10

1/v = –1/10 + 1/5 = –1/10 + 2/10 = 1/10

v = +10 cm

This should give virtual image but gives positive v.

**Standard result for diverging lens**: Always produces virtual, erect, diminished image on same side as object, so v should have opposite sign to u.

**Revised understanding**: The Cartesian convention for lenses differs from standard textbook usage. In standard convention:

u is always positive (object distance)

v is positive for real images (on opposite side), negative for virtual (same side as object)

The equation becomes 1/u – 1/v = 1/f for some formulations

For consistency with CBSE Class 12 standard, I should use:

**Standard Lens Formula**: 1/f = 1/v + 1/u where u, v, f are magnitudes with sign based on position.

For diverging lens (f = 10 cm) with object 5 cm:

1/(–10) = 1/v + 1/(5)

1/v = –1/10 – 1/5 = –3/10

v = –10/3 = –3.33 cm (virtual)

**Result**: Image at 3.33 cm on same side as object (virtual), erect, diminished.

**Example 3: Lens Maker's Formula Application**

A biconvex lens has radii R₁ = +20 cm, R₂ = –20 cm, made of glass with n = 1.5. Find focal length.

**Given**: R₁ = +20 cm, R₂ = –20 cm, n = 1.5

**Solution**:

1/f = (n – 1)[1/R₁ – 1/R₂]

1/f = (1.5 – 1)[1/20 – 1/(–20)]

1/f = 0.5 × [1/20 + 1/20]

1/f = 0.5 × 2/20 = 1/20

f = +20 cm

**Result**: Converging lens with focal length 20 cm.

---

COMBINATIONS OF LENSES

**Concept**

When two lenses are in contact (separated by negligible distance), the power of combination equals sum of individual powers.

**Formula**:

If P₁ = 1/f₁ and P₂ = 1/f₂ are powers of two lenses,

**P = P₁ + P₂ = 1/f₁ + 1/f₂**

Therefore, **1/f = 1/f₁ + 1/f₂**

**Application**: Combination of converging and diverging lens can correct vision defects.

**Example**: A lens of f = +10 cm combined with f = –15 cm:

1/f = 1/10 – 1/15 = (3–2)/30 = 1/30

f = +30 cm (resultant converging lens)

---

TOTAL INTERNAL REFLECTION AND OPTICAL FIBRES

**Working of Optical Fibres**

An **optical fibre** is a thin transparent cylinder of glass/plastic that guides light through total internal reflection.

**Principle**:

Light enters at one end at a large angle to the axis (angle > critical angle with respect to fibre wall)

Undergoes repeated total internal reflections at cylindrical surface

Emerges at other end without loss (100% reflection)

Can transmit light around bends without escaping

**Numerical Aperture (NA)**:

**NA = sin θₘₐₓ = √(n₁² – n₂²)**

Where n₁ = core refractive index, n₂ = cladding refractive index, θₘₐₓ = half-angle of acceptance cone.

Larger NA means acceptance angle is larger — more light can enter.

**Advantages**:

No loss of light (total internal reflection)

Can transmit through bends

Single fibre can carry many signals (multiplexing)

Immune to electromagnetic interference

Used in telecommunications, medicine (endoscopy), sensors

---

PRISM AND DISPERSION

**Refraction Through a Prism**

A **prism** is a triangular transparent medium with flat refracting surfaces.

**Key Angles**:

**A** = angle of prism (angle between two refracting surfaces)

**i** = angle of incidence at first surface

**r₁** = angle of refraction at first surface (inside prism)

**r₂** = angle of incidence at second surface (inside prism)

**e** = angle of emergence at second surface

**δ** = angle of deviation (total bending)

**Relation**: r₁ + r₂ = A (for ray passing through prism)

**Angle of Deviation**:

**δ = i + e – A**

At **minimum deviation (δₘ)**, the ray passes symmetrically through prism:

i = e

r₁ = r₂ = A/2

**At minimum deviation**:

**n = sin[(A + δₘ)/2] / sin(A/2)**

This formula allows finding refractive index experimentally using a prism.

**Dispersion**

**Dispersion** is the phenomenon of splitting white light into its constituent colors due to dependence of refractive index on wavelength.

**Refractive Index Dependence**: n = n(λ)

Shorter wavelengths (violet) have higher n

Longer wavelengths (red) have lower n

Therefore: nᵥᵢₒₗₑₜ > nᵣₑd

**Result**: Violet light deviates more than red light when passing through a prism.

**Spectrum**: Prism splits white light into VIBGYOR (violet, indigo, blue, green, yellow, orange, red).

**Dispersive Power (ω)**:

**ω = (nᵥ – nᵣ) / (n – 1)**

Measures ability of material to disperse light. Flint glass has high dispersive power; crown glass has low.

**Chromatic Aberration**: In lenses, different colors focus at different points due to dispersion, causing colored fringes. Corrected using achromatic doublets (combination of two lenses with different dispersive powers).

---

REFRACTION AT CURVED SURFACES AND SPHERICAL LENSES

Detailed application of refraction at spherical surfaces leads to the lens equation and lens maker's formula discussed earlier.

**Key insight**: Any optical system using lenses can be analyzed using the single-surface refraction formula applied sequentially at each surface.

---

OPTICAL INSTRUMENTS

THE HUMAN EYE

**Structure and Function**

The **human eye** is nature's optical instrument. Key components:

**Cornea**: Outermost curved refracting surface (most refraction here, ~49D)

**Aqueous humor**: Transparent fluid filling front chamber

**Iris**: Controls pupil size, adjusting light entry

**Lens**: Biconvex, flexible, adjustable focal length (45D to 65D in accommodation)

**Vitreous humor**: Transparent gel filling main chamber

**Retina**: Light-sensitive screen where real image forms

**Fovea centralis**: Area of sharpest vision on retina (yellow spot)

**Optic nerve**: Transmits image signal to brain

**Power of the Eye**:

Total power ≈ 60 diopters (most from cornea and lens).

**Accommodation**

**Accommodation** is the ability to adjust focal length of lens to focus objects at different distances.

**Ciliary muscles** contract: Lens becomes thicker, focal length decreases, focus on near objects

**Ciliary muscles** relax: Lens becomes flatter, focal length increases, focus on distant objects

**Near Point (Least Distance of Distinct Vision)**: Usually 25 cm for normal eye

Objects closer than 25 cm appear blurred (strain to accommodate)

**Far Point**: Infinity for normal eye (power = 0 for distance objects)

**Range of Accommodation**: From 25 cm to infinity

**Resolving Power**: Eye can distinguish two points separated by ~1 minute of arc (0